Because education should be the accumulation of understanding, not just an accumulation of facts, I have tried to write a textbook that emphasizes the fundamental concepts of electromagnetics, wave propagation, network analysis, and design principles as applied to modem microwave engineering. Although I have avoided the handbook approach, in which a large amount of information is presented with little or no explanation or context, a considerable amount of material in this book is related to the design of specific microwave circuits and components, for both practical and motivational value. I have tried to present the analysis and logic behind these designs so that the reader can see and understand the process of applying fundamental concepts to arrive at useful results. The engineer who has a firm grasp of the basic concepts and principles of microwave engineering, and has seen how these can be applied toward a specific design objective, is the engineer who is most likely to be rewarded with a creative and productive career. Modern microwave engineering involves predominantly distributed circuit analysis and design, in contrast to the waveguide and field theory orientation of earlier generations. The majority of microwave engineers today design planar components and integrated circuits without direct recourse to electromagnetic analysis. Microwave computer-aided design (CAD) software and network analyzers are the essential tools of today's microwave engineer, and microwave engineering education must respond to this shift in emphasis to network analysis, planar circuits and components, and active circuit design. Microwave engineering will always involve electromagnetics {many of the more sophisticated microwave CAD packages implement rigorous field theory solutions), and students will still benefit from an exposure to subjects such as waveguide modes and coupling through apertures, but the change in emphasis to microwave circuit analysis and design is clear. Microwave and RF technology is more pervasive than ever. This is especially true in the commercial sector, where modern applications include cellular telephony, personal communications systems, wireless local area data networks, millimeter wave collision avoidance vehicle radars, direct broadcast satellites for radio and television, global positioning systems, radio frequency identification tagging, ultra wideband radio and radar systems, and microwave remote sensing systems for the environment. Defense systems continue to rely heavily on microwave technology for passive and active sensing, communications, and weapons control systems. This state of affairs suggests that there will be no shortage of challenging problems in RF and microwave engineering in the foreseeable future, and a clear need for engineers having both an understanding of the fundamentals of microwave engineering as well as the creativity to apply this knowledge to problems of practical interest. v vi Preface The success of the first two editions of Microwave Engineering has been gratifying. For this edition we solicited detailed feedback from teachers and readers for their thoughts about what topics should be deleted and added. There was almost no agreement on specific material to remove {it seemed that almost every topic in the book was being used by someone). There was, however, fairly uniform agreement in favor of more material on active circuit design and related topics. To this end we have increased the number of chapters from 12 to 13 and have added new material on noise, nonlinear effects, RF MEMs, diode and transistor device characteristics,. txansistorp^weTamplifiers, FET mixers, transistor oscillators, oscillator phase noise, and frequency multipliers. Section? on intermodulation products, dynamic range, mixers, antennas, and receiver design have been completely rewritten. Numerous new or revised examples and problems, have been added, with several of these related to practical design problemsinvolving pranarcircnits and components. Another new feature of this edition is a list of Answers to Selected Problems at the end of the book, Topics that have been cut for this edition include the uniqueness theorem, Fabry-Perot resonators, electronic warfare, and some examples related to waveguides. This text is written for a two-semester course in microwave engineering, for seniors or first-year graduate students. If students have a good background in undergraduate electromagnetics, the material in Chapters 1 and 2 can be reviewed fairly quickly. Students with less background should study this material in more detail. Chapters 3—13 can then be followed in sequence, but it is likely that the instructor will want to choose between a field theory emphasis (Chapters 3-9, 13), or more of a circuit design emphasis (Chapters 4-8, 10-12). Alternatively, it is possible to focus exclusively on microwave circuit design by selectively covering Chapters 2,4-8, and 10-13, avoiding the material on electromagnetic analysis. Two important items that should be included in a successful course on microwave engineering are the use of computer-aided design (CAD) simulation software and a microwave laboratory experience. Providing students with access to CAD software allows them to verify results of the design-oriented problems in the text, giving immediate feedback that builds confidence and makes the effort more rewarding. Because the drudgery of repetitive calculation is eliminated, students can easily try alternative approaches and explore problems in more detail. The effect of line losses, for example, is explored in several examples and problems—this would be effectively impossible without the use of modern CAD tools. In addition, classroom exposure to CAD tools provides useful experience upon graduation. Most of the commercially available microwave CAD tools are very expensive, but several manufacturers provide academic discounts or free "student versions" of their products. Ansoft Corporation, for example, has a student version of their popular SERENADE package available for free download at their Web site (www.ansoft.com). A hands-on microwave instructional laboratory is expensive to equip but provides the best way for students to develop an intuition and physical feeling for microwave phenomena. A laboratory with the first semester of the course might cover the measurement of microwave power, frequency, standing wave ratio, impedance, and 5-parameters, as well as the characterization of basic microwave components such as tuners, couplers, resonators, loads, circulators, and filters. Important practical knowledge about connectors, waveguides, and microwave test equipment will be acquired in this way. Alternatively, a more advanced laboratory session can consider topics such as noise figure, intermodulation distortion, and mixing. Naturally, the type of experiments that can be offered is heavily dependent on the test equipment that is available. With this edition we are able to make available several resources for students and instructors on the Wiley Web site. A sample instructional laboratory manual, along with SERENADE circuit files for many of the problems and examples in the text, can be found at www.wiley.com/college/pozar. An on-line solution manual for all problems in the Preface vii text is available to qualified instructors, who may apply for access through the Web site www.wiley.com/college/pozar and going to the Instructor's Companion Site. ACKNOWLEDGMENTS Many people deserve my thanks for their help in completing this book, especially the numerous students and teachers who have used the first two editions of Microwave Engineering. I would also like to thank my colleagues in the microwave engineering group at the University of Massachusetts at Amherst for their support and collegiality over the years. In particular, Keith Carver and Bob Jackson made several helpful suggestions and contributions. Juraj Bartolic (University of Zagreb) provided the simplified derivation of the ^-parameter stability criteria in Chapter 11.1 am grateful to the following people for providing photographs: Dr. Naresh Deo of Millitech Corp., Dr. John Bryant of the University of Michigan, Mr. Harry Syrigos of Alpha Industries, Professor Cal Swift, Professor Bob Jackson, and Mr. B. Hou of the University of Massachusetts, Mr. J. Wendler of M/A-COM, Dr. Mike Adlerstein and Mr. Mark Russell of Raytheon Company, Mr. Hugo Vifian of Hewlett-Packard, and Dr. M. Abouzahra of Lincoln Laboratory. Finally, I would like to thank Bill Zobrist and the staff of John Wiley & Sons for iheir invaluable help and professionalism during this project. David M. Pozar Amherst, MA Microwave Engineering Third Edition David M. Pozar University of Massachusetts at Amherst 2 2 0 6 2 2 ITRC- WILEY John Wiley & Sons, Inc. Executive Editor Senior Production Editor Senior Design Manager '> Illustration Editor Bill Zobrist Petřina Kulek 2_oo 5 Karin Kíncheloe Gene AieDo Cover design/illustration Carol C Grobe This book was set in Times Roman 1 (VI2 by TechBooks and printed and bound by Hamilton Printing. The cover was printed by Phoenix Color, This book is printed on acid Tree paper. © Copyright © 2005 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the iy?6 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01523, (97») 750-8400, fax (978) 750-4470. Requests to the Publisher for permission should he addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hobokeru NJ 07030, (201) 748-6011, fax (201) 748-6008, E-Mail: PERMREQ@WILEYCOM, To order books or for customer service please call 1-800-CALL WILEY(225-5945). library of Congress Caiahging-in-Publkation Data Pozar. David M, Microwave engineering/David M. Pozar,—3sd ed. p. cm. Includes index. ISBN (M71-44878-8 (doth) 1. Microwaves, 2, Microwave devices. 3. Microwave circuits, I. Title. TK7S76.P69 2005 621.381'3-^dc22 20O3O650OI Printed in the United States of America 10 937654321 Contents ELECTROMAGNETIC THEORY 1 1.1 Introduction to Microwave Engineering 1 Applications of Microwave Engineering 2 A Short History of Microwave Engineering 3 1.2 Maxwell's Equations 5 1.3 Fields in Media and Boundary Conditions 9 Fields at a General Material Interface 11 Fields at a Dielectric Interface Fields at the Interface with a Perfect Conductor (Electric Wall) 13 The Magnetic Wall Boundary Condition 14 The Radiation Condition 1.4 The Wave Equation and Basic Plane Wave Solutions 14 The Helmholtz Equation 14 Plane Waves in a Lossless Medium 15 Plane Waves in a General Lossy Medium 16 Plane Waves in a Good Conductor 18 1.5 General Plane Wave Solutions 19 Circularly Polarized Plane Waves 23 1.6 Energy and Power 24 Power Absorbed by a Good Conductor 26 1.7 Plane Wave Reflection from a Media Interface 27 General Medium 28 Lossless Medium 29 Good Conductor 30 Perfect Conductor 32 The Surface Impedance Concept 32 1.8 Oblique Incidence at a Dielectric Interface 34 Parallel Polarization 35 Perpendicular Polarization 36 Total Reflection and Surface Waves 38 1.9 Some Useful Theorems 40 The Reciprocity Theorem 40 Image Theory 42 x Contents 2 TRANSMISSION LINE THEORY 49 2.1 The Lumped-Element Circuit Model for a Transmission Line 49 Wave Propagation on a Transmission Line 51 The Lossless Line 52 2.2 Field Analysis of Transmission Lines 52 Transmission Line Parameters 52 The Telegrapher Equations Derived from Field Analysis of a Coaxial Line 55 Propagation Constant, Impedance, and Power Flow for the Lossless Coaxial Line 57 2.3 The Terminated Lossless Transmission Line 57 Special Cases of Lossless Terminated Lines 60 2-4 The Smith Chart 64 The Combined Impedance-Admittance Smith Chart 68 The Slotted Line 69 2.5 The Quarter-Wave Transformer 73 The Impedance Viewpoint 73 The Multiple Reflection Viewpoint 75 2.6 Generator and Load Mismatches 77 Load Matched to Line 78 Generator Matched to Loaded Line 78 Conjugate Matching 78 2.7 Lossy Transmission Lines 79 The Low-Loss Line 79 The Distortionless Line 81 The Terminated Lossy Line 82 The Perturbation Method for Calculating Attenuation 83 The Wheeler Incremental Inductance Rule 84 TRANSMISSION LINES AND WAVEGUIDES 91 3.1 General Solutions for TEM, TE, and TM Waves 92 TEM Waves 94 TE Waves 96 TM Waves 96 Attenuation Due to Dielectric Loss 97 3.2 Parallel Plate Waveguide 98 TEM Modes 99 TM Modes 100 TE Modes 103 3.3 Rectangular Waveguide 106 TE Modes 106 TM Modes 111 TEra0 Modes of a Partially Loaded Waveguide 115 3.4 Circular Waveguide 117 TE Modes 118 TM Modes 121 3.5 Coaxial Line 126 TEM Modes 126 Higher Order Modes 127 3.6 Surface Waves on a Grounded Dielectric Slab 131 TM Modes 131 TE Modes 134 Contents xi 3.7 Stripline 137 Formulas for Propagation Constant, Characteristic Impedance, and Attenuation 138 An Approximate Electrostatic Solution 140 3.8 Microstrip 143 Formulas for Effective Dielectric Constant, Characteristic Impedance, and Attenuation 144 An Approximate Electrostatic Solution 146 3.9 The Transverse Resonance Technique 149 TEow Modes of a Partially Loaded Rectangular Waveguide 150 3.10 Wave Velocities and Dispersion 151 Group Velocity 151 3.11 Summary of Transmission Linesand Waveguides 154 Other Types of Lines and Guides 154 MICROWAVE NETWORK ANALYSIS 161 4.1 Impedance and Equivalent Voltages and Currents 162 Equivalent Voltages and Currents 162 The Concept of Impedance 166 Even and Odd Properties of Z(w) and 169 4.2 Impedance and Admittance Matrices 170 Reciprocal Networks 171 Lossless Networks 173 43 The Scattering Matrix 174 Reciprocal Networks and Lossless Networks 177 A Shift in Reference Planes 180 Generalized Scattering Parameters 181 4.4 The Transmission tABCD) Matrix 183 Relation to Impedance Matrix 185 Equivalent Circuits for Two-Port Networks 186 4.5 Signal Flow Graphs 189 Decomposition of Signal Flow Graphs 190 Application to TRL Network Analyzer Calibration 193 4.6 Discontinuities and Modal Analysis 197 Modal Analysis of an -Plane Step in Rectangular Waveguide 199 4.7 Excitation of Waveguides—Electric and Magnetic Currents 204 Current Sheets That Excite Only One Waveguide Mode 204 Mode Excitation from an Arbitrary Electric or Magnetic Current Source 206 4.8 Excitation of Waveguides—Aperture Coupling 209 Coupling Through an Aperture in a Transverse Waveguide Wall 212 Coupling Through an Aperture in the Broad Wall of a Waveguide 214 xli Contents §_ IMPEDANCE MATCHING AND TUNING 222 5.1 Matching with Lumped Elements (L Networks) 223 Analytic Solutions 224 Smith Chart Solutions 225 5.2 Single-Stub Tuning 228 Shunt Stubs 228 Series Stubs 232 5.3 Double-Stub Tuning 235 Smith Chan Solution 235 Analytic Solution 238 5.4 The Quarter-Wave Transformer 240 5.5 The Theory of Small Reflections 244 Single-Section Transformer 244 Multisection Transformer 245 5*6 Binomial Multisection Matching Transformers 246 5.7 Chebyshev Multisection Matching Transformers 250 Chebyshev Polynomials 251 Design of Chebyshev Transformers 252 Sit Tapered Lines 255 Exponential Taper 257 Triangular Taper 258 Klopfenstein Taper 258 5.9 The Bode-Fano Criterion 261 MICROWAVE RESONATORS 266 6.1 Series and Parallel Resonant Circuits 266 Series Resonant Circuit 266 Parallel Resonanl Circuit 269 Loaded and Unloaded Q 271 6.2 Transmission Line Resonators 272 Short-Circuited A/2 Line 272 Short-Circuited A/4 Line 275 Open-Circuited A/2 Line 276 6.3 Rectangular Waveguide Cavities 278 Resonant Frequencies 27S Q of the TEio* Mode 279 6.4 Circular Waveguide Cavities 282 Resonant Frequencies 282 Q of the lEnmt Mode 284 6.5 Dielectric Resonators 287 Resonant Frequencies of TEqu Mode 287 6.6 Excitation of Resonators 291 Critical Coupling 291 A Gap-Coupled Microstrip Resonator 292 An Aperture-Coupled Cavity 296 6.7 Cavity Perturbations 298 Material Perturbations 298 Shape Perturbations 300 POWER DIVIDERS AND DIRECTIONAL COUPLERS 308 7*1 Basic Properties of Dividers and Couplers 308 Three-Port Networks (T-Junctions) 309 Four-Port Networks (Directional Couplers) 311 7.2 The T- Junction Power Divider 315 Lossless Divider 316 Resistive Divider 317 7.3 The Wilkinson Power Divider 318 Even-Odd Mode Analysis 319 Unequal Power Division and //-Way Wilkinson Dividers 322 7.4 Waveguide Directional Couplers 323 Bethe Hole Coupler 324 Design of Muldhole Couplers 327 7.5 The Quadrature (90°) Hybrid 333 Even-Odd Mode Analysis 333 7.6 Coupled Line Directional Couplers 337 Coupled Line Theory 337 Design of Coupled Line Couplers 341 Design of Multisection Coupled Line Couplers 345 7.7 The Lange Coupler 349 7.8 The 180° Hybrid 352 Even-Odd Mode Analysis of the Ring Hybrid 354 Even-Odd Mode Analysis of the Tapered Coupled Line Hybrid 357 Waveguide Magic-T 361 7.9 Other Couplers 361 MICROWAVE FILTERS 370 8.1 Periodic Structures 371 Analysis of Infinite Periodic Structures 372 Terminated Periodic Structures 374 k-fi Diagrams and Wave Velocities 375 8.2 Filter Design by the Image Parameter Method 378 Image Impedances and Transfer Functions for Two-Port Networks 378 Constant-A Filter Sections 380 m-Derived Filter Sections 383 Composite Filters 386 8.3 Filter Design by the Insertion Loss Method 389 Characterization by Power Loss Ratio 389 Maximally Flat Low-Pass Filter Prototype 392 Equal-Ripple Low-Pass Filter Prototype 394 Linear Phase Low-Pass Filter Prototypes 396 8.4 Filter Transformations 398 Impedance and Frequency Scaling 398 Bandpass and Bandstop Transformations 401 xlv Contents 8.5 Filter Implementation 405 Richard's Transfonnation 406 Kuroda's Identities 406 Impedance and Admittance Inverters 411 8.6 Stepped-Impedance Low-Pass Filters 412 Approximate Equivalent Circuits for Short Transmission Line Sections 412 8.7 Coupled Line Filters 416 Filter Properties of a Coupled Line Section 416 Design of Coupled Line Bandpass Filters 420 8.8 Filters Using Coupled Resonators 427 Bandstop and Bandpass Filters Using Quarter-Wave Resonators 427 Bandpass Filters Using Capacitively Coupled Series Resonators 431 Bandpass Filters Using Capacitively Coupled Shunt Resonators 433 THEORY AND DESIGN OF FERRIMAGNETIC COMPONENTS 441 9.1 Basic Properties of Ferrimagnetic Materials 442 The Permeability Tensor 442 Circularly Polarized Fields 447 Effect of Loss 449 Demagnetization Factors 451 9.2 Plane Wave Propagation in a Ferrite Medium 454 Propagation in Direction of Bias (Faraday Rotation) 455 Propagation Transverse to Bias (Birefringence) 458 93 Propagation in a Ferrite-Loaded Rectangular Waveguide 460 TEmo Modes of Waveguide with a Single Ferrite Slab 460 TEmo Modes of Waveguide with Two Symmetrical Ferrite Slabs 464 9.4 Ferrite Isolators 465 Resonance Isolators 465 The Field Displacement Isolator 469 9.5 Ferrite Phase Shifters 471 Nonreciprocal Latching Phase Shifter 471 Other Types of Ferrite Phase Shiners 474 TheGyrator 475 9.6 Ferrite Circulators 476 Properties of a Mismatched Circulator 476 Junction Circulator 478 NOISE AND ACTIVE RF COMPONENTS 486 10.1 Noise in Microwave Circuits 487 Dynamic Range and Sources of Noise 487 Noise Power and Equivalent Noise Temperature 489 Measurement of Noise Temperature 492 Noise Figure 493 Noise Figure of a Cascaded System 495 Noi se Figure of a Passive Two-Port Network 497 Noise Figure of a Mismatched Lossy Line 498 Contents 10.2 Dynamic Range and Intermodulation Distortion 500 Gain Compression 501 Intermodulation Distortion 502 Third-Order Intercept Point 504 Dynamic Range 505 Intercept Point of a Cascaded System 507 Passive Intermodulation 509 10.3 RF Diode Characteristics 509 Schottky Diodes and Detectors 509 PIN Diodes and Control Circuits 514 Varactor Diodes 520 Other Diodes 521 10.4 RF Transistor Characteristics 522 Field Effect Transistors (FETs) 523 Bipolar Junction Transistors (BJTs) 5 10.5 Microwave Integrated Circuits 526 Hybrid Microwave Integrated Circuits 527 Monolithic Microwave Integrated Circuits 528 MICROWAVE AMPLIFIER DESIGN 536 11.1 Two-Port Power Gains 536 Definitions of Two-Port Power Gains 537 Further Discussion of Two-Port Power Gains 540 11.2 Stability 542 Stability Circles 543 Tests for Unconditional Stability 545 11.3 Single-Stage Transistor Amplifier Design 548 Design for Maximum Gain (Conjugate Matching) 548 Constant Gain Circles and Design for Specified Gain 553 Low-Noise Amplifier Design 557 11.4 Broadband Transistor Amplifier Design 561 Balanced Amplifiers 562 Distributed Amplifiers 565 11.5 Power Amplifiers 570 Characteri sties of Power Ampli fi er s and Amplifier Classes 57 0 Large-Signal Characterization of Transistors 571 Design of Class A Power Amplifi ers 5 72 OSCILLATORS AND MIXERS 577 12.1 RF Oscillators 578 General Analysis 578 Oscillators Using a Common Emitter BJT 579 Oscillators Using a Common Gate FET 581 Practical Considerations 582 Crystal Oscillators 584 12.2 Microwave Oscillators 585 Transistor Oscillators 587 Dielectric Resonator Oscillators 590 12.3 Oscillator Phase Noise 594 Representation of Phase Noise 594 Leeson' s Model for Oscill ator Phase Noise 595 12.4 Frequency Multipliers 599 Reactive Diode Multipliers (Manley-Rowe Relations) 600 Resistive Diode Multipliers 602 Transistor Multipliers 604 12.5 Overview of Microwave Sources 608 Solid-State Sources 609 Microwave Tubes 612 12.6 Mixers 615 Mixer Characteristics 616 Single-Ended Diode Mixer 620 Single-Ended FET Mixer 622 Balanced Mixer 625 Image Reject Mixer 627 Other Mixers 629 INTRODUCTION TO MICROWAVE SYSTEMS 633 13.1 System Aspects of Antennas 633 Fields and Power Radiated by an Antenna 635 Antenna Pattern Characteristics 637 Antenna Gain and Efficiency 639 Aperture Efficiency and Effective Area 640 Background and Brightness Temperature 641 Antenna Noise Temperature and GIT 643 13X Wireless Communication Systems 646 The Friis Formula 647 Radio Receiver Architectures 650 Noise Characterization of a Microwave Receiver 652 Wireless Systems 655 13.3 Radar Systems 659 The Radar Equation 660 Pulse Radar 662 Doppler Radar 663 Radar Cross Section 664 13.4 Radiometer Systems 665 Theory and Applications of Radiome try 665 Total Power Radiometer 667 The Dicke Radiometer 669 13.5 Microwave Propagation 670 Atmospheric Effects 670 Ground Effects 672 Plasma Effects 673 13.6 Other Applications and Topics 674 Microwave Heating 674 Power Transfer 675 Biological Effects and Safely 675 APPENDICES 680 A Prefixes 681 B Vector Analysis 681 C Bessel Functions 683 D Other Mathematical Results 686 E Physical Constants 686 F Conductivities for Some Materials 687 G Dielectric Constants and Loss Tangents for Some Materials 687 Chapter O n e Electromagnetic Theory We begin our study of microwave engineering with a brie!' overview of the history and major applications of microwave technology, followed by a review of the fundamental topics in electromagnetic theory that we will need throughout the book. The interested reader will find further discussion of these topics in references [l]-[9], INTRODUCTION TO MICROWAVE ENGINEERING The term microwaves refers to alternating current signals with frequencies between 300 MHz (3 x 108 Hz) and 300 GHz (3 x 10"), with a corresponding electrical wavelength between k = c/f = lm and k — 1 mm, respectively. Signals with wavelengths on the order of millimeters are called millimeter waves. Figure 1.1 shows the location of the microwave frequency band in the electromagnetic spectrum. Because of the high frequencies (and short wavelengths), standard circuit theory generally cannot be used directly to solve microwave network problems. In a sense, standard circuit theory is an approximation or special use of the broader theory of electromagnetics as described by MaxwelTs equations. This is due to the fact that, in general, the lumped circuit element approximations of circuit theory are not valid at microwave frequencies. Microwave components are often distributed elements, where the phase of a voltage or current changes significantly over the physical extent of the device, because the device dimensions are on the order of the microwave wavelength. At much lower frequencies, the wavelength is large enough that there is insignificant phase variation across the dimensions of a component. The other extreme of frequency can be identified as optical engineering, in which the wavelength is much shorter than the dimensions of the component. In this case Maxwell's equations can be simplified to the geometrical optics regime, and optical systems can be designed with the theory of geometrical optics. Such techniques are sometimes applicable to millimeter wave systems, where they are referred to as quasioptical. In microwave engineering, then, one must often begin with Maxwell's equations and their solutions. It is in the nature of these equations that mathematical complexity arises, 1 2 Chapter 1: Electromagnetic Theory Frequency (Hz) 3 x 10s 3 x 106 3x107 3 x 10s 3xl09 ^-,-r-l-1- 3xl010 3xl0n 3xl01J 3xJ013 3xl0H J_I_I_I_ 9 M JH 1 3 iS I I 1 I -9 Microwaves JZ , HI ' IE 10-1 1 10 i I Wavelength (m) 10" 10"* 10 i-S iff* Typical Frequencies Approximate Band Designations AM broadcast band 535-1605 kHz Medium frequency 300 kHz to 3 MH?. Short wave radio band 3-30 MHz High frequency (HF) 3 MHz to 30 MHz FM broadcast band 8S-10S MHz Very high frequency (VHF) 30 MHz to 300 MHz VHFTV (2-4) 54-72 MHz Ultra higfi frequency (UHF) 300 MH2 to 3 GHz VHFTV (5-6) 76-88 MHz L band 1-2 GHz UHFTV(7-LV) 174-216 MHz S band 2-4 GHz LfHF TV (14-33) 470-890 MHz Cband 4-8 GHz US cellular telephone «24-849 MHz Xband 8-12 GHz $69-894 MHz Ku band 12-18 GHz European GSM cellular 880-915 MHz Kband 18-26 GHz 925-960 MHz Ka band 26-40 GHz GPS 157542 MHz U band 40-60 GHz L 227.60 MHz Vband 50-75 GHz Microwave ovens 245 GHz E band 60-90 GHz US DBS 11.7-12.5 GHz Wband 75-110 GHz US ISM bands 902-928 MHz Fband 90-140 GHz 2.400-2.484 GHz 5.725-5.850 GHz US UWB radio 3.1-10.6 GHz FIGURE 1.1 The electromagnetic spectrum. since Maxwell's equations involve vector differential or integral operations on vector field quantities, and these fields are functions of spatial coordinates. One of the goals of this book, however, is to try to reduce the complexity of a field theory solution to a result that can be expressed in terms of simpler circuit theory. A field theory solution generally provides a complete description of the electromagnetic field at every point in space, which is usually much more information than we really need for most practical purposes. We are typically more interested in terminal quantities such as power, impedance, voltage, and current, which can often be expressed in terms of circuit theory concepts. It is this complexity that adds to the challenge, as well as the rewards, of microwave engineering. Applications of Microwave Engineering Just as the high frequencies and short wavelengths of microwave energy make for difficulties in analysis and design of microwave components and systems, these same factors provide unique opportunities for the application of microwave systems. This is because of the following considerations: • Antenna gain is proportional to the electrical size of the antenna. At higher frequencies, more antenna gain is therefore possible for a given physical antenna size, which has important consequences for implementing miniaturized microwave systems. • More bandwidth (information-carrying capacity) can be realized at higher frequencies. A I % bandwidth at 600 MHz is 6 MHz (the bandwidth of a single television 1.1 Introduction to Microwave Engineering 3 channel), and at 60 GHz a 1% bandwidth is 600 MHz (100 television channels). Bandwidth is critically important because available frequency bands in the electromagnetic spectrum are being rapidly depleted, • Microwave signals travel by line of sight and are not bent by the ionosphere as are lower frequency signals. Satellite and terrestrial communication links with very high capacities are thus possible, with frequency reuse at minimally distant locations. • The effective reflection area (radar cross section) of a radar target is usually proportional to the target's electrical size. This fact, coupled with the frequency characteristics of antenna gain, generally makes microwave frequencies preferred for radar systems. • Various molecular, atomic, and nuclear resonances occur at microwave frequencies, creating a variety of unique applications in the areas of basic science, remote sensing, medical diagnostics and treatment, and heating methods. The majority of applications of today's microwave technology are to communications systems, radar systems, environmental remote sensing, and medical systems. As the frequency allocations listed in Figure 1.1 show, RF and microwave communications systems are pervasive, especially today when wireless connectivity promises to provide voice and data access to "everyone, anywhere, at any time." Probably the most ubiquitous use of microwave technology is in cellular telephone systems, which were first proposed in the 1970s. By 1997 there were more than 200 million cellular subscribers worldwide, and the number of subscribers and the capabilities of this service continue to grow. Satellite systems have been developed to provide cellular (voice), video, and data connections worldwide. Large satellite telephony systems, such as Iridium and Global star, unfortunately suffered from both technical drawbacks and weak business models, and have failed with losses of several billion dollars each. But smaller satellite systems, such as the Global Positioning Satellite (GPS) system and the Direct Broadcast Satellite (DBS) system, have been extremely successful. Wireless Local Area Networks (WLANs) provide high-speed networking between computers over short distances, and the demand for this capability is growing very fast. The newest wireless communications technology is Ultra Wide Band (UWB) radio, where the broadcast signal occupies a very wide frequency band but with a very low power level to avoid interference with other systems. Radar systems find application in military, commercial, and scientific systems. Radar is used for detecting and locating air, ground, and seagoing targets, as well as for missile guidance and fire control. In the commercial sector, radar technology is used for air traffic control, motion detectors (door openers and security alarms), vehicle collision avoidance, and distance measurement. Scientific applications of radar include weather prediction, remote sensing of the atmosphere, the oceans, and the ground, and medical diagnostics and therapy. Microwave radiometry, which is the passive sensing of microwave energy emitted from an object, is used for remote sensing of the atmosphere and the earth, as well as medical diagnostics and imaging for security applications. A Short History of Microwave Engineering The field of microwave engineering is often considered a fairly mature discipline because the fundamental concepts of electromagnetics were developed over 100 years ago, and probably because radar, being the first major application of microwave technology, was intensively developed as far back as World War n. But even though microwave engineering had its beginnings in the last century, significant developments in high-frequency solid-state devices, microwave integrated circuits, and the ever-widening applications of modern microsystems have kept the field active and vibrant. 4 Chapter 1: Electromagnetic Theory The foundations of modern electromagnetic theory were formulated in 1873 by James Clerk Maxwell [1], who hypothesized, solely from mathematical considerations, electromagnetic wave propagation and the notion that light was a form of electromagnetic energy. Maxwell's formulation was cast in its modern form by Oliver Heaviside, during the period from 1885 to 1887. Heaviside was a reclusive genius whose efforts removed many of the mathematical complexities of Maxwell's theory, introduced vector notation, and provided a foundation for practical applications of guided waves and transmission lines. Heinrich Hertz, a German professor of physics and a gifted experimentalist who also understood the theory published by Maxwell, carried out a set of experiments during the period 1887-1891 that completely validated Maxwell's theory of electromagnetic waves. Figure 1.2 shows a photograph of the original equipment used by Hertz in his experiments. FIGURE 1.2 Original apparatus used by Hertz for his electromagnetics experiments. (1) 50 MHz transmitter spark gap and loaded dipole antenna. (2) Parallel wire grid for polarization experiments. (3) Vacuum apparatus for cathode ray experiments. (4) Hot-wire galvanometer. (5) Reiss or Knochenhauer spirals. (6) RolJed-paper galvanometer. (7) Metal sphere probe. (8) Reiss spark micrometer. (9) Coaxial transmission line. (10-12) Equipment to demonstrate dielectric polarization effects. (13) Mercury induction coil interrupter. (14) Meidingercell. (15) Vacuum bell jar. (16) High-voltage induction coil. (17) Bun sen cells. (18) Large-area conductor for charge storage. (19) Circular loop receiving antenna, (20) Eight-sided receiver detector. (21) Rotating mirror and mercury interrupter. (22) Square loop receiving antenna. (23) Equipment for refraction and dielectric constant measurement. (24) Two square loop receiving antennas. (25) Square loop receiving antenna. (26) Transmitter dipole. (27) High-voltage induction coil. (28) Coaxial line. (29) High-voltage discharger. (30) Cylindrical parabolic reflector/receiver. (31) Cylindrical parabolic reflector/transmitter. (32) Circular loop receiving antenna. (33) Planar reflector. (34, 35) Battery of accumulators. Photographed on October 1, 1913 at the Bavarian Academy of Science, Munich, Germany, with Hertz's assistant, Julius Amman. Photograph and identification courtesy of J. K. Bryant, University of Michigan, 1.2 Maxwell's Equations 5 It is interesting to observe that this is an instance of a discovery occurring after a prediction has been made on theoretical grounds—a characteristic of many of the major discoveries throughout the history of science. All of the practical applications of electromagnetic the-ory, including radio, television, and radar, owe their existence to the theoretical work of Maxwell. Because of the lack of reliable microwave sources and other components, the rapid growth of radio technology in the early 1900s occurred primarily in the high frequency (HP) to very high frequency (VHF) range. It was not until the 1940s and the advent of radar development during World War II that microwave theory and technology received substantial interest. In the United States, the Radiation Laboratory was established at the Massachusetts Institute of Technology (MIT) to develop radar theory and practice. A number of top scientists, including N. Marcuvitz, 1.1. Rabi, J. S. Schwinger, H. A. Bethe, E. M. Purcell, C. G. Montgomery, and R. H. Dicke, among others, were gathered for what turned out to be a very intensive period of development in the microwave field. Their work included the theoretical and experimental treatment of waveguide components, microwave antennas, small aperture coupling theory, and the beginnings of microwave network theory. Many of these researchers were physicists who went back to physics research after the war (many later received Nobel Prizes), but their microwave work is summarized in the classic 28-volume Radiation Laboratory Series of books that still finds application today. Communications systems using microwave technology began to be developed soon after the birth of radar, benefitting from much of the work that was originally done for radar systems. The advantages offered by microwave systems, including wide band widths and n'ne-of-sight propagation, have proved to be critical for both terrestrial and satellite communications systems and have thus provided an impetus for the continuing development of low-cost miniaturized microwave components. We refer the interested reader to the special Centennial Issue of the IEEE Transactions on Microwave Theory and Techniques [2] for further historical perspectives on the field of microwave engineering. MAXWELL'S EQUATIONS Electric and magnetic phenomena at the macroscopic level are described by Maxwell's equations, as published by Maxwell in 1873 [1]. This work summarized the state of electromagnetic science at that time and hypothesized from theoretical considerations the existence of the electrical displacement current, which led to the discovery by Hertz and Marconi of electromagnetic wave propagation. Maxwell's work was based on a large body of empirical and theoretical knowledge developed by Gauss, Ampere, Faraday, and others. A first course in electromagnetics usually follows this historical (or deductive) approach, and it is assumed that the reader has had such a course as a prerequisite to the present material. Several books are available, [3]-[9], that provide a good treatment of electromagnetic theory at the undergraduate or graduate level. This chapter will outline the fundamental concepts of electromagnetic theory that we will require for the rest of the book. Maxwell's equations will be presented, and boundary conditions and the effect of dielectric and magnetic materials will be discussed. Wave phenomena are of essential importance in microwave engineering, so much of the chapter is spent on plane wave topics. Plane waves are the simplest form of electromagnetic waves and so serve to illustrate a number of basic properties associated with wave propagation. Although it is assumed that the reader has studied plane waves before, the present material should help to reinforce many of the basic principles in the reader's mind and perhaps to introduce some concepts that the reader has not seen previously. This material will also serve as a useful reference for later chapters. 6 Chapter 1: Electromagnetic Theory With an awareness of the historical perspective, it is usually advantageous from a pedagogical point of view to present electromagnetic theory from the "inductive," or axiomatic, approach by beginning with Maxwell's equations. The general form of time-varying Maxwell equations, then, can be written in "point," or differential, form as V x £ =--Al (1.1a) at dt> ?xH = - +J. (1.1b) dt VV = pi (1.1c) V-i5 = 0. (l.ld) The MKS system of units is used throughout this book. The script quantities represent time-varying vector fields and are real functions of spatial coordinates x,y,z, and the time variable t. These quantities are defined as follows: £ is the electric field intensity, in V/m. H is the magnetic field intensity, in A/m. V is the electric flux density, in Coul/m2. B is the magnetic flux density, in Wb/m2. M is the (fictitious) magnetic current density, in V7m2. J is the electric current density, in A/m2. p is the electric charge density, in Coul/m\ The sources of the electromagnetic field are the currents M and J, and the electric charge density p. The magnetic current M is a fictitious source in the sense that it is only a mathematical convenience; the real source of a magnetic current is always a loop of electric current or some similar type of magnetic dipole, as opposed to the flow of an actual magnetic charge (magnetic monopole charges are not known to exist). The magnetic current is included here for completeness, as we will have occasion to use it in Chapter 4 when dealing with apertures. Since electric current is really the flow of charge, it can be said that the electric charge density p is the ultimate source of the electromagnetic field. In free-space, the following simple relations hold between the electric and magnetic field intensities and flux densities: B = txQH, (1.2a) V = €o£, (1.2b) where fa = 4jt x 10"7 Henry/mis the permeability of free-space, and e0 = 8.854 x 10"12 farad/m is the permittivity of free-space. We will see in the next section how media other than free-space affect these constitutive relations. Equations (1.1 a>—(l.ld) are linear but are not independent of each other. For instance, consider the divergence of (1.1a), Since the divergence of the curl of any vector is zero [vector identity (B.12), from Appendix B], we have V ■ V x £ = 0 = -—(V - B) - V * M. at Since there is no free magnetic charge, V ■ Jd = 0, which leads to V ■ B = 0, or (l.ld). 1.2 Maxwell's Equations 7 The continuity equation can be similarly derived by taking the divergence of (1,1b), giving V-J + ^ = 0, (1.3) dt where (1. 1c) was used. This equation states that charge is conserved, or that current is continuous, since V ■ J represents the outflow of current at a point, and dp/dt represents the charge buildup with time at the same point. It is this result that led Maxwell to the conclusion that the displacement current density 'df>/'dt was necessary in (1. lb), which can be seen by taking the divergence of this equation. The foregoing differential equations can be converted to integral form through the use of various vector integral theorems. Thus, applying the divergence theorem (B. 15) to (1 .lc) and (lid) yields j>^V-ds = J pdv = Q, (1.4) (jf 5-^ = 0, (1.5) where Q in (1.4) represents the total charge contained in the closed volume V (enclosed by a closed surface S). Applying Stokes' theorem (B.16) to (1 .la) gives j> £-dl = -~ j B-ds-j^M-ds, (1,6) which, without the M. term, is the usual form of Faraday's law and forms the basis for Kirchhoff's voltage law. In (1.6), C represents a closed contour around the surface S, as shown in Figure 1.3. Ampere's law can be derived by applying Stokes' theorem to (1. lb): Sn-dl = — [V-ds+ [ J ds = — [v-ds + I, (1.7) Jc U Js Js & Js where J = fs J ■ ds is the total electric current flow through the surface S. Equations (1.4)-(l .7) constitute the integral forms of Maxwell's equations. The foregoing equations are valid for arbitrary time dependence, but most of our work will be involved with fields having a sinusoidal, or harmonic, time dependence, with steady-state conditions assumed. In this case phasor notation is very convenient, and so all field quantities will be assumed to be complex vectors with an implied e)iM time dependence and written with roman (rather than script) letters. Thus, a sinusoidal electric field in the x direction of the form Six, y, z, t) = xA(x, y, z) co$(), (1.8) where A is the (real) amplitude, o> is the radian frequency, and 0 is the phase reference of FIGURE 1.3 The closed contour C and surface S associated with Faraday's law. Chapter 1: Electromagnetic Theory the wave at r = 0, has the phasor form Ě(xiy,z) = xA{x,yiz)ei'1>. (L9) We will assume cosine-based phasors in this book, so the conversion from phasor quantities to real time-varying quantities is accomplished by multiplying the phasor by e}tů! and taking the real part: €{x, y, z, ť) = Re[Ě(x, y, 7)e^% (1.10) as substituting (1.9) into (1.10) to obtain (1.8) demonstrates. When working in phasor notation, it is customary to suppress the common eii factor on all terms. When dealing with power and energy, we will often be interested in the time average of a quadratic quantity. This can be found very easily for time harmonic fields. For example, the average of the square of the magnitude of an electric field given by £ = xEv cosM + 0i) + yE2 cos(o>r + 2) + zE2 cos(w/ + (111) which has the phasor form Ě = xEteJ^ + yE2e]r + 4>{) + E\ cos2(w; + fa) + E] cos2(ftjf + €' (1-21) which is seen to be the ratio of the real to the imaginary part of the total displacement current. Microwave materials are usually characterized by specifying the real permittivity, ef = fj-Co. and the loss tangent at a certain frequency. These constants are listed in Appendix G for several types of materials, It is useful to note that, after a problem has been solved assuming a lossless dielectric, loss can easily be introduced by replacing the real e with a complex € = €f — J€" = t'(l — j tan 5) = €a€r{\ — j tan 5). In the preceding discussion it was assumed that Pf was a vector in the same direction as E. Such materials are called isotropic materials, but not all materials have this property. Some materials are anisotropic and are characterized by a more complicated relation between Pe and E, or D and E. The most general linear relation between these vectors takes the form of a tensor of rank two (a dyad), which can be written in matrix form as Ey = [*] Ey J leJ (1.22) It is thus seen that a given vector component of E gives rise, in general, to three components of D. Crystal structures and ionized gases are examples of anisotropic dielectrics. For a 1.3 Fields in Media and Boundary Conditions 11 linear isotropic material, the matrix of (1.22) would reduce to a diagonal matrix with elements i. An analogous situation occurs for magnetic materials. An applied magnetic field may align magnetic dipole moments in a magnetic materia] to produce a magnetic polarization (or magnetization) Pm . Then, B = fji^H + Pm). For a linear magnetic material, pm is linearly related to H as where Xm is a complex magnetic susceptibility. From (1.23) and (1.24), (1.23) (1.24) (1.25) where /x = fj,o(\ + %m) —1*>' - j>" is the permeability of the medium. Again, the imaginary part of Xm or fi accounts for loss due to damping forces; there is no magnetic conductivity, since there is no real magnetic current. As in the electric case, magnetic materials may be anisotropic, in which case a tensor permeability can be written as ~Bx-\ rn BA = U .bA lu xx ř4 f^zy rHx-\ mm Hy _ IhA (1.26) An important example of anisotropic magnetic materials in microwave engineering is the class of ferrimagnetic materials known as femtes; these materials and their applications will be discussed further in Chapter 9. If linear media are assumed (€, \i not depending on £ or H), then Maxwell's equations can be written in phasor form as The constitutive relations are V x Ě = -júJiíH - M, V x H = jcoiM + /, V ■ B = 0. B = fiff, (1.27a) (1.27b) (1.27c) (1.27d) (1.28a) (1.28b) where ť and ti may be complex and may be tensors. Note that relations like (1.28a) and (1.28b) generally cannot be written in time domain form, even for linear media, because of the possible phase shift between Ď and Ě, or B and H. The phasor representation accounts for this phase shift by the complex form of e and tx. Maxwell's equations (1.27a)-( 1.27d) in differential form require known boundary values for a complete and unique solution. A general method used throughout this book is to solve the source-free Maxwell's equations in a certain region to obtain solutions with unknown coefficients, and then apply boundary conditions to solve for these coefficients. A number of specific cases of boundary conditions arise, as discussed below. Fields at a General Material Interface Consider a plane interface between two media, as shown in Figure 1.5. Maxwell's equations in integral form can be used to deduce conditions involving the normal and tangential fields 12 Chapter 1: Electromagnetic Theory Medium 2 Medium 1: e,, jij FIGURE 1.5 Fields, currents, and surface charge at a general interface between two media. at this interface. The time-harmonic version of (1.4), where S is the closed "pUlbox"-shaped surface shown in Figure 1.6, can be written as p dv. (1,29) In the limit as h 0, the contribution of Aan through the sidewalls goes to zero, so (1.29) reduces to ASDjn - ASDiri = ASA. Or £>2rt - £>in - Ps, (1.30) where ps is the surface charge density on the interface. In vector form, we can write n-02- D}) = Ps. (1-31) A similar argument for B leads to the result that h B2 = nB}, (1.32) since there is no free magnetic charge. For the tangential components of the electric field we use the phasor form of (1.6), j> E'dl= -jto J B ds — j^M ds, (1.33) in connection with the closed contour C shown in Figure 1.7, In the limit as h 0, the surface integral of B vanishes (since S = h At vanishes). The contribution from the surface integral of M, however, may be nonzero if a magnetic surface current density Ms exists on the surface. The Dirac delta function can then be used to write M = MsS(k), (1.34) where A is a coordinate measured normal from the interface. Equation (1.33) then gives A(Ed - ME& = -AfA/j, or Elt- El2 = -Ms, (1.35) Medium 2 ■VV k Medium 1 \^ i FIGURE 1.6 Closed surface 5 for equation {1.29). 1.3 Fields in Media and Boundary Conditions 13 4i T A/- FIGURE 1.7 Closed contour C for Equation {1.33). Medium 2 "> f" " " n Mm Medium 1 which can be generalized in vector form as A similar argument for the magnetic field leads to (1.36) (1.37) where /j is an electric surface current density that may exist at the interface. Equations (1.31), (1.32), (1.36), and (1.37) are the most general expressions for the boundary conditions at an arbitrary interface of materials and/or surface currents. Fields at a Dielectric Interface At an interface between two lossless dielectric materials, no charge or surface current densities will ordinarily exist. Equations (1.31), (1.32), (1.36), and (1,37) then reduce to nĎi^ňĎz, (1.38a) a- Si (1.38b) ň x £] = ň x Ěi, (1.38c) fix H] - ň x H2. (L38d) In words, these equations state that the normal components of Ď and B are continuous across the interface, and the tangential components of £ and H are continuous across the interface. Because Maxwell's equations are not all linearly independent, the six boundary conditions contained in the above equations are not all linearly independent. Thus, the enforcement of (1.38c) and (1.38d) for the four tangential field components, for example, will automatically force the satisfaction of the equations for the continuity of the normal components. Fields at the Interface with a Perfect Conductor (Electric Wall) Many problems in microwave engineering involve boundaries with good conductors (e.g.f metals), which can often be assumed as lossless (a -s- oo). In this case of a perfect conductor, all field components must be zero inside the conducting region. This result can be seen by considering a conductor with finite conductivity ( oo. (Such an analysis will be performed in Section 1.7.) If we also assume here that Ms — 0, which would be the case if the perfect conductor filled all the space on one side of the boundary, then (1.31), (1.32), (1.36), and (1.37) reduce to the following; ň-Ď = Pr, (1.39a) «-5=0, (I.39b) ítx£ = 0, (1.39c) (tx ft = /JT (1.39d) Chapter 1; Electromagnetic Theory where f>s and Js are the electric surface charge density and current density, respectively, on the interface, and n is the normal unit vector pointing out of the perfect conductor. Such a boundary is also known as an electric wait, because the tangential components of E are "shorted out," as seen from {1.39c), and must vanish at the surface of the conductor. The Magnetic Wall Boundary Condition Dual to the preceding boundary condition is the magnetic wall boundary condition, where the tangential components of H must vanish. Such a boundary does not really exist in practice, but may be approximated by a corrugated surface, or in certain planar transmission line problems. In addition, the idealization that h x H = 0 at an interface is often a convenient simplification, as we will see in later chapters. We will also see that the magnetic wall boundary condition is analogous to the relations between the voltage and current at the end of an open-circuited transmission line, while the electric wall boundary condition is analogous to the voltage and current at the end of a short-circuited transmission line. The magnetic wall condition, then, provides a degree of completeness in our formulation of boundary conditions and is a useful approximation in several cases of practical interest. The fields at a magnetic wall satisfy the following conditions: hD=0. (1.40a) n • B = 0, (1.40b) n x Ě = -Ms, (1.40c) ňxH=0, (1.40d) where h is the normal unit vector pointing out of the magnetic wall region. The Radiation Condition When dealing with problems that have one or more infinite boundaries, such as plane waves in an infinite medium, or infinitely long transmission lines, a condition on the fields at infinity must be enforced. This boundary condition is known as the radiation condition, and is essentially a statement of energy conservation. It states that, at an infinite distance from a source, the fields must either be vanishingly small (i.e., zero) or propagating in an outward direction. This result can easily be seen by allowing the infinite medium to contain a small loss factor (as any physical medium would have). Incoming waves (from infinity) of finite amplitude would then require an infinite source at infinity, and so are disallowed. THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS The Kelmhottz Equation In a source-free, linear, isotropic, homogeneous region, Maxwell's curl equations in phasor form are V x £ = -ja>fJLH, (1.41a) V x H = jtotE, (1.41b) and constitute two equations for the two unknowns, Ě and H. As such, they can be solved for either Ě or H. Thus, taking the curl of (1.41a) and using (1.41b) gives V x V x £ = —jíOfjíV x H = ú?fi€Ě, 1.4 The Wave Equation and Basic Plane Wave Solutions 15 which is an equation for E. This result can be simplified through the use of vector identity (B. 14), V x V x A = V(V - A) - V2A, which is valid for the rectangular components of an arbitrary vector A. Then, V2E + wV.)] = 2tt, so. a = — =--=-j- (1.48) k i± As with the lossless case, a wave impedance can be defined to relate the electric and magnetic fields: » =-. (1.57) Y Then (1.56) can be rewritten as Hy = -(E+e~yz - (1.58) n Chapter 1: Electromagnetic Theory Note that 77 is, in general, complex and reduces to the lossless case of n = vWe when y = Plane Waves in a Good Conductor Many problems of practical interest involve loss or attenuation due to good (but not perfect) conductors, A good conductor is a special case of the preceding analysis, where the conductive current is much greater than the displacement current, which means a ^ cot. Most metals can be categorized as good conductors. In terms of a complex €, rather than conductivity, this condition is equivalent to e" 3> £'. The propagation constant of (1.52) can then be adequately approximated by ignoring the displacement current term, to give y = or + jfi H jto^j-T- = (1 + j)J^- (1-59) The skin depth, or characteristic depth of penetration, is defined as k■ = - = ./—■ a y wfia Then the amplitude of the fields in the conductor decay by an amount 1 je or 36.8%, after traveling a distance of one skin depth, since e~az = e~ais = e~]. At microwave frequencies, for a good conductor, this distance is very small. The practical importance of this result is that only a thin plating of a good conductor (e.g., silver or gold) is necessary for low-loss microwave components. EXAMPLE 1.2 SKIN DEPTH AT MICROWAVE FREQUENCIES Compute the skin depth of aluminum, copper, gold, and silver at a frequency of lOGHz. Solution The conductivities for these metals are listed in Appendix F. Equation (1.60) gives the skin depths as ** V a>i.to Jnfnoo V 7r(10m)(4x x 10-7)vC = 5.03 x ID-3 For aluminum: & = 5.03 x 10~\/--1-T = 8.14 x 10"7 m. 3.816 y 107 For copper: S, = 5.03 x 10-3,/-= = 6.60 x 10~7 m. l/ 5.813 x 107 For gold: 5; - 5.03 x lO-3,/---- = 7.86 x 10~7 m. * V 4.098 x 107 For silver: 5, a 5.03 x 10_3t/--—— = 6.40 x 10~7 m. 6.173 x 107 1.5 General Plane Wave Solutions 19 TABLE 1.1 Summary of Results for Plane Wave Propagation in Various Media Type of Medium Lossless General Good Conductor Quantity (ť" = (7=0) Lossy e" » e' orrr » m' Complex propagation y = jat^/jn y = jwsfßf V = (1 + constant Phase constant = i = w^fjii ß = my) P = Im(y) ~ Vw//r/72 (wavenumber) Attenuation constant a = 0 a = Re(y) oi = Re(y) = *Ja>fJiIV"*} = E0 cos(fc ■ r - cot), (1.77) assuming that the amplitude constants A, B, and C contained in Eo are real. If these constants are not real, their phases should be included inside the cosine term of (1.77). It 22 Chapter t: Electromagnetic Theory FIGURE 1.8 Orientation of the E, W, and k = kQn vectors for a general plane wave. is easy to show that the wavelength and phase velocity for this solution are the same as obtained in Section 1.4. _ EXAMPLE 1.3 CURRENT SHEETS AS SOURCES OF PLANE WAVES An infinite sheet of surface current can be considered as a source for plane waves, If an electric surface current density Js = J0x exists on the z = 0 plane in free-space, find the resulting fields by assuming plane waves on either side of the current sheet and enforcing boundary conditions. Solution Since the source does not vary with x or y, the fields will not vary with x or y but will propagate away from the source in the ±z direction. The boundary conditions to be satisfied at % = 0 are n x (E2 - £i) = z x (E2 - Ei) = 0, «x(ff2-^) = ?x (H2 ~ Hi) = J<£, where E i, H t are the fields for z < 0, and £2, Hi are the fields for z > 0. To satisfy the second condition, H must have a y component. Then for £ to be orthogonal to H and z, E must have an x component, Thus the fields will have the following form: for z < 0, £1 Hx = -yAem\ for z > 0, E2 H2 = yBe~Jk6\ where A and B are arbitrary amplitude constants. The first boundary condition, that Ex is continuous at z = 0, yields A = B, while the boundary condition for H yields the equation -B- A = J0. Solving for A, B gives A = B = -JJ2, which completes the solution. ■ 1.5 General Plane Wave Solutions 23 Circularly Polarized Plane Waves The plane waves discussed above all had (heir electric field vector pointing in a fixed direction and so are called linearly polarized waves. In general, the polarization of a plane wave refers to the orientation of the electric field vector, which may be in a fixed direction or may change with time. Consider the superposition of an x linearly polarized wave with amplitude E\ and a y linearly polarized wave with amplitude both traveling in the positive z direction. The total electric field can be written as E = (£,* + E2y)e-Jk°z. (1-78) A number of possibilities now arise. If E\ ^ 0 and Ei — 0, we have a plane wave linearly polarized in the x direction. Similarly, if Ei =0 and E2 ^ 0, we have a plane wave linearly polarized in the ý direction. If E\ and E2 are both real and nonzero, we have a plane wave linearly polarized at the angle £i For example, if Et — E2 = £0, we have É = E0(jt-!-ý)e-^, which represents an electric field vector at a 455angle from the *-axis. Now consider the case in which E\ = jE2 = Eg, where Eq is real, so that E = EQ(x-jf)e-J^z. (1,79) The time domain form of this field is £(z, t) - E0{x cos((u; - k0z) + y cos(a>r - ktf. - n/2)\. (1.80) This expression shows that the electric field vector changes with time or, equivalently, with distance along the z-axis. To see this, pick a fixed position, say z = 0. Equation (1 ,S0) then reduces to £(0, ŕ) = £0(i cos cut + y sin ait], (1.81) so as ait increases from zero, the electric field vector rotates counterclockwise from the *-axis. The resulting angle from the jc-axis of the electric field vector at time /, at z = 0, is then 4> — tan~' / sin ait \ - = (ot, \COS Oil} which shows that the polarization rotates at the uniform angular velocity o>. Since the fingers of the right hand point in the direction of rotation when the thumb points in the direction of propagation, this type of wave is referred to as a right hand circularly polarized (RHCP) wave. Similarly, a field of the form £ = EQ(jt + j (1.82) constitutes a left hand circularly polarized (LHCP) wave, where the electric field vector rotates in the opposite direction. See Figure 1.9 for a sketch of the polarization vectors for RHCP and LHCP plane waves. 24 Chapter 1; Electromagnetic Theory The magnetic field associated with a circularly polarized wave may be found from Maxwell's equations, or by using the wave impedance applied to each component of the electric field. For example, applying (1.76) to the electric field of a RHCP wave as given in (1.79) yields H = -ix(Jc- }y)e-jk,>z = — B + }Z)e-}kttZ = — (# - jy)e~^\ T)a no m which is also seen to represent a vector rotating in the RHCP sense. 1.6 ENERGY AND POWER In general, a source of electromagnetic energy sets up fields that store electric and magnetic energy and carry power that may be transmitted or dissipated as loss. In the sinusoidal steady-state case, the time-average stored electric energy in a volume V is given by. We = ^Re / E D*dvt (1.83) 4 Jv which in the case of simple lossless isotropic, homogeneous, linear media, where e is a real scalar constant, reduces to Wt = | f E-E*dv. (L84) 4 Jv Similarly, the time-average magnetic energy stored in the volume V is Wm = jRe I H ■ B* dv, (1.85) 4 Jv which becomes = - f Hřťdv, (1.86) 4 h for a real, constant, scalar We can now derive Poynting's theorem, which leads to energy conservation for electromagnetic fields and sources. If we have an electric source current, Js, and a conduction current aE, as defined in (1.19), then the total electric current density is J = Js + aE. 1.6 Energy and Power 25 Then multiplying (1.27a) by H*, and multiplying the conjugate of (1.27b) by E, yields ^'•(Vx£)= -) E x H* ds = -ct / \E\2dv + j(o((€*\E\2-n\H\2)dv- j (E-J* + H* .Ms)dv, (1.87) Jy Jv Jv where S is a closed surface enclosing the volume V, as shown in Figure 1,10. Allowing e = €' — je" and jjl = pi' — to be complex to allow for loss, and rewriting (1.87) gives -\r ((£■/• + &* • M,)dv = \ (Wm — We). (1.93) In words, this complex power balance equation states that the power delivered by the sources (Ps) is equal to the sum of the power transmitted through the surface (P0), the power lost to heat in the volume (Pi), and 2co times the net reactive energy stored in the volume. Power Absorbed by a Good Conductor To calculate attenuation and loss due to an imperfect conductor, one must find the power dissipated in the conductor. We will show that this can be done using only the fields at the surface of the conductor, which is a very helpful simplification when calculating attenuation. Consider the geometry of Figure 1.11, which shows the interface between a lossless medium and a good conductor. We assume that a field is incident from z < 0 and that the field penetrates into the conducting region z > 0, The real average power entering the conductor volume defined by the cross-sectional surface So at the interface and the surface S is given from (1.91) as 1 JSn+S E x H* -ndst n.94) where ft is a unit normal vector pointing into the closed surface S$ + St and E. H are the fields at this surface. The contribution to the integral in (1.94) from the surface S can be made zero by proper selection of this surface. For example, if the field is a normally incident plane wave, the Poynting vector S = E x H* will be in the z direction, and so Pi, f -M <7 » tli€ Ttt ■ft =1$ ! —p s S FIGURE 1.11 An interface between a lossless medium and a good conductor with a closed surface Sq + S for computing the power dissipated in the conductor. 1.7 Plane Wave Reflection from a Media Interface 27 tangential to the top, bottom, front, and back of S, if these walls are made parallel to the Z-axis. If the wave is obliquely incident, these walls can be slanted to obtain the same result. And, if the conductor is good, the decay of the fields from the interface at z = 0 will be very rapid, so that the right-hand end of iS can be made far enough away from z = 0 so that there is negligible contribution to the integral from this part of the surface 5. The time-average power entering the conductor through So can then be written as P3V ffipe ( £ x H* ids. (1.95) From vector identity (B.3) we have z ■(£ x H*) = (Z x £)- H* = 7)H • H\ (1.96) since H = n x Ě/t), as generalized from (1.76) for conductive media, where tj is the intrinsic wave impedance of the conductor. Equation (1.95) can then be written as P3V = ^f \H\2ds, (1.97) where R, = Re(ří) = Re[o + j)^ = ^ = X (1.98) is called the surface resistivity of the conductor. The magnetic field H in (1.97) is tangential to the conductor surface and needs only to be evaluated at the surface of the conductor; since H, is continuous at z = 0, it doesn't matter whether this field is evaluated just outside the conductor or just inside the conductor. In the next section we will show how (1.97) can be evaluated in terms of a surface current density flowing on the surface of the conductor, where the conductor can be assumed to be perfect. PLANE WAVE REFLECTION FROM A MEDIA INTERFACE A number of problems to be considered in later chapters involve the behavior of electromagnetic fields at the interface of a lossy or conducting medium, and so it is beneficial at this lime to study the reflection of a plane wave normally incident from free-space onto the surface of a conducting half-space. The geometry is shown in Figure 1.12 where the lossy half-space z > 0 is characterized by the parameters and a. FIGURE 1.12 Plane wave reflection from a lossy medium; normal incidence. Chapter!: Bectrom agnaticTheory General Medium With no loss of generality, we can assume that the incident plane wave has an electric field vector oriented along the jc-axis and is propagating along the positive z-axis. The incident fields can then be written, for g < 0, as Hi = y—EQe^t i?0 (1.99a) (1.99b) where is the wave impedance of free-space, and Eq is an arbitrary amplitude. Also in the region z < 0, a reflected wave may exist with the form Ěr = xrE0e+Jk,>t, (1.100a) Hr = -y— Eoe+J*a\ (1.100b) where T is the unknown reflection coefficient of the reflected electric field. Note that in (1.100), the sign in the exponential terms has been chosen as positive, to represent waves traveling in the — z direction of propagation, as derived in (1.46). This is also consistent with the Poynting vector Šr = Ěr x H* = -\T]2\E(,\2z/i}o^ which shows power to be traveling in the —z direction for the reflected wave. As shown in Section 1.4, from equations (1.54) and (1.58), the transmitted fields for z > 0 in the lossy medium can be written as B^xTEae-r*, (1.101a) H, = tIEle-y^ (1.101b) n where T is the transmission coefficient of the transmitted electric field and rj is the intrinsic impedance of the lossy medium in the region z > 0. From (1.57) and (1.52) the intrinsic impedance is (1.102) and the propagation constant is y = a + j$ = jcOy/JAfyf\ - ja/toe. (L103) We now have a boundary value problem where the general form of the fields are known via (1.99)—<1,101) on either side of the material discontinuity at z = 0. The two unknown constants, T and T, are found by applying two boundary conditions on Ex and Hy at 2 = 0. Since these tangential field components must be continuous at z = 0, we arrive at the following two equations: (i + n = t, i - r _ t (1.104a) (1.104b) 1.7 Plane Wave Reflection from a Media Interface 29 Solving these equations for the reflection and transmission coefficients gives r = ^, (1.105a) T = l + r = ^—. (1.105b) This is a general solution for reflection and transmission of a normally incident wave at the interface of a lossy material, where tj is the impedance of the material. We now consider three special cases of the above result Lossless Medium If the region for z > 0 is a lossless dielectric, then a = 0, and fi and e are real quantities. The propagation constant in this case is purely imaginary and can be written as Y ~ Jfi= }<»-Jv* = jko^/fXt^r, (1.106) where fc0 = o>^ffZo€ii is the wavenumberof a plane wave in free-space. The wavelength in the dielectric is Hf the phase velocity is (slower than the speed of light in free-space) and the wave impedance of the dielectric is (1.109) In the lossless case, rj is real, so both f and T from (1.105) are real, and E and H are in phase with each other in both media. Power conservation for the incident, reflected, and transmitted waves can be demonstrated by computing the Poynting vectors in the two regions. Thus, for z < 0, the complex Poynting vector is ft = E x H* = (£,- + Er) x (Hi + HT) = ziE^-ie-^ + rejkaZ)(e-Jko1 - rV*°T = z\E0\2—(1 - |r|2 + Pe2Jki,z - r**_2JfcK) m = z|£o|2-a-|r|2 + 2/T sin 2*o2), (1.110a) m since T is real. For z > 0, the complex Poynting vector is V 2n Xq 1 (1.107) (1.108) Chapter 1: Electromagnetic Theory which can be rewritten, using (1.105), as S+ = z\E»\\ p = z\Ei)\2-d ~ iri2). (1.110b) (n + mr m Now observe that at z — 0, S~ = S+, so that complex power flow is conserved across the interface. Now consider the time-average power flow in the two regions. For z < 0, the time-average power flow through a 1-m2 cross section is p-^l-Re(S~- z)= ^|£ol2-(l-|r|2). (1.111a) and for z > 0, the time-average power flow through a 1-m2 cross section is P+ = iRe(5+ ■ z) = l-\E{)\2-(l - in2) = (LIlib) 2 2 »7o so real power flow is conserved. We now note a subtle point. When computing the complex Poynting vector for z < 0 in (1.110a), we used the total E and H fields. If we compute separately the Poynting vectors for the incident and reflected waves, we obtain Si = EjX #J = z^^, (1.112a) m ST = Er x H; = -g|£°|2|r|2, <1.112b) m and we see that §i + Sr ^ S~ of (1.110a). The missing cross-product terms account for stored reactive energy in the standing wave in the z < 0 region. Thus, the decomposition of a Poynting vector into incident and reflected components is not, in general, meaningful. Some books define a time-average Poynting.vector as (1 /2)Re(E x H*), and in this case such a definition applied to the individual incident and reflected components will give the correct result,sinceP{ = (l/2)Re|£0l3/^of andPr = -([/2)\EQ\2\r\2/r) 0 is a good (but not perfect) conductor, the propagation constant can be written as discussed in Section 1.4: I (OLIO 1 y = a + JP = (1 + J)J~Y- = U +J)Y- U-113) Similarly, the intrinsic impedance of the conductor simplifies to V = (l + j\[^ = (l+j)-^r- (1-114) V 2(7 ads Now the impedance is complex, with a phase angle of 45°, so E and H will be 45° out of phase, and T and t will be complex. In (1.113) and (1.114), % = 1 fa is the skin depth, as defined in (1.60). For z < 0, the complex Poynting vector can be evaluated at z — 0 to give S-(_z = 0) = z\E<>\2—(1 - |rf + T - T*). (1.115a) m 1.7 Plane Wave Reflection from a Media Interface 31 For z > 0, the complex Poynting vector is S+ = ElxH* = z\E(i\2\T\1-xe-2a\ and using (1.105) for T and F gives S+ = ||£0|2- 4T? ..g-2" = z|£0|2-(l - |F|2 + r - f'VH (1.115b) So at the interface at z = 0, S~ = 5+, and complex power is conserved. Observe that if we were to compute the separate incident and reflected Poynting vectors for z < 0 as Si = Ei xH* = z-^, (1.116a) no Sr = ErxH; = -z1**1"'1^, (1.116b) no we do not obtain Si + Sr — S~ of (1.115a), even for z = 0. It is possible, however, to consider real power flow in terms of the individual traveling wave components. Thus, the time-average power flows through a 1-m2 cross section are P~ = Ut(S~ ■ z> = if£0|2—(1 - \r\\ (1.117a) 2 2 jjo p+ = \ms+ ■ z) = Ue0\2-(\ - trpy-^, (i.inb) 2 2 i)o which shows power balance at z = 0. In addition, = |£oi2/27fo> and Pr = — |£ol2 |T|2/ 2rfo so that F, + Pr = P~, showing that the real power flow for z < 0 can be decomposed into incident and reflected wave components. Notice that S+, the power density in the lossy conductor, decays exponentially according to the e~^z attenuation factor. This means that power is being dissipated in the lossy material as the wave propagates into the medium in the +z direction. The power, and also the fields, decay to a negligibly small value within a few skin depths of the material, which for a reasonably good conductor is an extremely small distance at microwave frequencies. The electric volume current density flowing in the conducting region is given as /, =a£, = *Z>~y:A/m2, (1.118) and so the average power dissipated in (or transmitted into) a 1 m2 cross-sectional volume of the conductor can be calculated from the conductor loss term of (1.92) (Joule's law) as P< = I / E, ■ Jt*dv = i / [ f (xEaTe~yz) ■ {xa EnTe^1)* dz dy dx ~\o\E^\T\2 P e-^dz=°^P^, (1.119) 2 /;=0 4a Since 1/tj = £t5^/(1 + j) = (cr/2a)(l — y), the real power entering the conductor through a 1-m2 cross section (as given by (1 /2)Re(5+ ■ z) at z = 0) can be expressed using (1.115b) as P' = |£0|2|r|V/4a), which is in agreement with (1.119). Chapter 1: Electromagnetic Theory Perfect Conductor Now assume that the region z > 0 contains a perfect conductor. The above results can be specialized to this case by allowing oo; from (1.114) n 0; from (1.60) Ss -» 0; and from (l.l05a,b) 7 — 0, and l—>- -1. The fields for z > 0 thus decay infinitely fast, and are identically zero in the perfect conductor, The perfect conductor can be thought of as "shorting out" the incident electric field For z < 0, from (1.99) and (1.100), the total E and H fields are, since r = -1, E = Ei + ET = xEvie-J^ _ = ~x2jEQ sin *0z, (1.120a) H = Hi + Hr = y — £0(e"^ + eJ'*°J) = * — EQ cos k<>z■ (1.120b) Observe that at z = 0, E = 0 and H = y(2//?o)£o- The Poynting vector for z < Ois S~ = E x H* = zj — |£o!: sin kQz cos (1-121) which has a zero real part and thus indicates that no real power is delivered to the perfect conductor. The volume current density of (1.118) for the lossy conductor reduces to an infinitely thin sheet of surface current in the limit of infinite conductivity: Js = n x H = —% x (y—Eo cos koz] \ m } = x—£0A/m. (1.122) The Surface Impedance Concept In many problems, particularly those in which the effect of attenuation or conductor loss is needed, the presence of an imperfect conductor must be taken into account. The surface impedance concept allows us to do this in a very convenient way. We will develop this method from the theory presented in the previous sections. Consider a good conductor in the region z > 0. As we have seen, a plane wave normally incident on this conductor is mostly reflected, and the power that is transmitted into the conductor is dissipated as heat within a very short distance from the surface. There are three ways to compute this power. First, we can use Joule's law, as in (1.119). For a 1 m2 area of conductor surface, the power transmitted through this surface and dissipated as heat is given by (1.119). Using (1.105b) for T, (1.114) for n, and the fact that a = l/Ss, gives the following result: o-\t\2 mmř s s/?, (1 where we have assumed n <$C fft, which is true for a good conductor. Then the power of (1.119) can be written as (1.124) where fi,=Re<,) = Re(i±i) = 4 = M ,1.125) \ (Töj / oös V 2cr is the surface resistance of the metal. Another way to find the power loss is to compute the power flow into the conductor using the Poynting vector, since all power entering the conductor at z = 0 is dissipated. As 1.7 Plane Wave Reflection from a Media Interface 33 in (1.115b), we have which for large conductivity becomes, since n Sít so that the total current flow is the same. Then use Joule's law to find the power lost: * \J~sl2 R* I ...» l2 2|£0|2Ä, «3 (1.130) where /5 denotes a surface integral over the conductor surface, in this case chosen as 1 m2. The result of (1.130) agrees with our previous results for P1 in (1.126) and (1.124), and shows that the power loss can be accurately and simply calculated as (1.131) in terms of the surface resistance Rs and the surface current Js, or tangential magnetic field H(. It is important to realize that the surface current can be found from fa = ft x as if the metal were a perfect conductor. This method is very general, applying to fields 34 Chapter 1: Electromagnetic Theory other than plane waves, and to conductors of arbitrary shape, as long as bends or corners have radii on the order of a skin depth or larger. The method is also quite accurate, as the only approximation in the above was that n <§C no. which is a good approximation. As an example, copper at 1 GHz has \ = 0.012 £2, which is indeed much less than no = 377 £2. EXAMPLE 1A PLANE WAVE REFLECTION FROM A CONDUCTOR Consider a plane wave normally incident on a half-space of copper. If / = 1 GHz, compute the propagation constant, impedance, and skin depth for the conductor. Also compute the reflection and transmission coefficients. Solution For copper, a = 5.813 x IQ1 S/m, so from (1.60) the skin depth is 2 a, - ,/-= 2.088 x IQ'* m, and the propagation constant is, from (1.113), Y = lí^ = (4.789 + /4.789) x 105 m_I. The intrinsic impedance is, from (1.114), i] = = (8.239 + j8.239) x 10"3 £2, which is quite small relative to the impedance of free-space (?jo = 377 £2), The reflection coefficient is then r = 2—2* = 1479.99° i 1 + *}» (practically that of an ideal short circuit), and the transmission coefficient is t = ^— = 6.181 xlO-W. OBLIQUE INCIDENCE AT A DIELECTRIC INTERFACE We continue our discussion of plane waves by considering the problem of a plane wave obliquely incident on a plane interface between two lossless dielectric regions, as shown in Figure 1.13. There are two canonical cases of this problem: the electric field is either in the xz plane (parallel polarization), or normal to the xz plane (perpendicular polarization). An arbitrary incident plane wave, of course, may have a polarization that is neither of these, but it can be expressed as a linear combination of these two individual cases. The general method of solution is similar to the problem of normal incidence: we will write expressions for the incident, reflected, and transmitted fields in each region and match boundary conditions to find the unknown amplitude coefficients and angles. 1.8 Oblique Incidence at a Dielectric Interface 35 Parallel Polarization In this case, the electric field vector lies in the xz plane, and the incident fields can be written as Ei = EQ(x cos 4 - z sin %)e~^ixsia$i+z(1.132a) Rf = StaHSi^frw**, (1.132b) where k\ = o^/ju-o^i, and /j] = Vmo7^T arc the wavenumber and wave impedance of region 1, The reflected and transmitted fields can be written as Er = EQ r(i cos 9r + z sin $r (1,133a) ffr = ^i^laM^ (1.133b) E, = E0r(Jf cos £ - z sin 0,)f>-^tjU * 6>+iC™ #, (1.134a) = ^■pe-A<**ta4+"™*>i (LI 34b) In the above, V and 7" are the reflection and transmission coefficients, and kz, m are the wavenumber and wave impedance of region 2, defined as At this point, we have T, T, 9r, and 0, as unknowns. We can obtain two complex equations for these unknowns by enforcing the continuity of Ex and Hyt the tangential field components, at the interface at z = 0. We then obtain cos $ x sin* + r cos 0r e-jkl* ®& = T cos % e-Jk2XSinS\ (1.135a) i r t e-jkix&mfy _ ^_£-jkixsin^ _ sin#( (LI35b) m n\ m Both sides of (1.135a) and (1.135b) are functions of the coordinate x. If Ex and Hy are to be continuous at the interface z — 0 for all xt then this x variation must be the same on both Chapter 1: Electromagnetic Theory sides of the equations, leading to the following condition: fc] sin $i = k\ sin Br = k2 sin 0,, which results in the well-known Snell's laws of reflection and refraction: Bi=$f, (1.136a) *i sin $t = k2 sin $,. (1.136b) The above argument ensures that the phase terms in (1.135) vary with x at the same rate on both sides of the interface, and so is often called the phase matching condition. Using (1.136) in (L135) allows us to solve for the reflection and transmission coefficients as 772 cos 9, + r}\ cos 6{ 2rt2 cos 6j T =---■-, (Ll37b) T]2 cos 0t + IJ] CCS Bi Observe that for normal incidence, we have — 6r = $t = 0, so then r =- and T — in agreement with the results of Section 1.7. For this polarization, a special angle of incidence, called the Brewster angle, exists where T = 0. This occurs when the numerator of (1.137a) goes to zero (#,- = #&): T)2 cos 6{ = tji cos 8t, which can be reduced using cos 6, to give i = 71 - sin2 $, = ^1 - H sin2 fe, sin 4 = 5 ' - (1.138) Perpendicular Polarization In this case, the electric field vector is perpendicular to the xz plane. The incident field can be written as Ei =E0ye-jk^^+zcos8i\ (1.139a) ^ = —(-xCOS0» +1 sin ^-i^Ä***^- (1.139b) where fci = ü)^fji^[ and 771 = VMoAi are the wavenumber and wave impedance for region 1, as before. The reflected and transmitted fields can be expressed as ET = EQrye-jk,ix^-<™e'\ (1.140a) Hr = — (x cos $r + z sin $, )e-Jk>(* ™*(1.140b) Ii 1.8 Oblique Incidence at a Dielectric Interface 37 Ei = EoTye~jhix *n (1.14 la) Ht = —cos ft + z sin ftj^^^******* (U4lb) fa withfo = €2. As $i increases, the refraction angle 6t will increase, but at a faster rate than 0, increases. The incidence angle 0, for which 8, = 90° is called the critical angle, 0C7 thus sin$c= S. (1.145) V «3 At this angle and beyond, the incident wave will be totally reflected, as the transmitted wave is not propagating into region 2. Let us look at this situation more closely, for the case of &i > 6C with parallel polarization. _ When &j > &c (1.144) shows that sin 9r > I, so that cos 9S = y/1 — sin2 Bt must be imaginary, and the angle B( loses its physical significance. At this point, it is better to 1.8 Oblique incidence at a Dielectric Interface 39 replace the expressions for the transmitted fields in region 2 with the following: Et = äffe» - Li^e-iß*e-°^ (U46a) Hs = ^L$e-rt*e-az. (1.146b) The form of these fields is derived from (1.134) after noting that - jk$ sin 0, is still imaginary for sin Qt > 1, but - jk2 cos $t is real, so we can replace sin 0, by ß/fa, and cos 0, by joe/ Substituting (1.146b) into the Helmhottz wave equation for H gives -ß2 +a2 + k%=0. (1.147) Matching Ex and Hy of (L146) with the x~ and y components of the incident and reflected fields of (1.132) and (1.133) at z = 0 gives cos 0, e-jklXsilL9' + rcos9re-jk,x™°' = T]—e-}ßx. (1.148a) h j_e-jk,xSw9, _ ^-jktxswv, _ J_e-m (1.148b) To obtain phase matching at the z = 0 boundary, we must have k\ sin &i = k\ sin 0r = ß, which leads again to Snell's law for reflection, 0/ = 0M and to ß = k\ sin 0,. Then a is determined from (1.147) as a = - k\ = yfk2 sin2 ^ — Af. (1.149) which is seen to be a positive real number, since sin2 0f > ei/e|. The reflection and transmission coefficients can be obtained from (1.148) as r = pSA - mcos 1, (1.150a) (jafh)i}2 + m cos 9t _ 2n2 cos ft T =---■--. (1.150b) (jtt/k2)V2 + m cos &i Since T is of the form (a — jb)/(a + jb), its magnitude is unity, indicating that all incident power is reflected. The transmitted fields of (1.146) show propagation in the x direction, along the interface, but exponential decay in the z direction. Such a field is known as a surface wave* since it is tighdy bound to the interface. A surface wave is an example of a nonuniform plane wave, so called because it has an amplitude variation in the z direction, apart from the propagation factor in the x direction. r Some authors argue that the term "surface wave" should not be used for a field of this type, since it exists only when plane wave fields exist in the z < 0 region, and so prefer to call it a "surface wave-like" field, or a "forced surface wave," 40 Chapter 1: Electromagnetic Theory Finally, it is of interest to calculate the complex Poynting vector for the surface wave fields of (LI46): St = E, x Ht =-[ z-—Ir.JSr- \e (L151) This shows that no real power flow occurs in the z direction. The real power flow in the x direction is that of the surface wave field, and decays exponentially with distance into region 2. So even though no real power is transmitted into region 2, a nonzero field does exist there, in order to satisfy the boundary conditions at the interface. SOME USEFUL THEOREMS Finally, we discuss several theorems in electromagnetics that we will find useful for later discussions. The Reciprocity Theorem Reciprocity is a general concept that occurs in many areas of physics and engineering, and the reader may already be familiar with the reciprocity theorem of circuit theory. Here we will derive the Lorentz reciprocity theorem for electromagnetic fields in two different forms. This theorem will be used later in the book to obtain general properties of network matrices representing microwave circuits and to evaluate the coupling of waveguides from current probes and loops, and the coupling of waveguides through apertures. There are a number of other important uses of this powerful concept. Consider the two separate sets of sources, J\, Mi and 3%,M^ which generate the fields Ei, Hi, and E2, H2, respectively, in the volume V enclosed by the closed surface S, as shown in Figure 1.15. Maxwell's equations are satisfied individually for these two sets of sources and fields, so we can write Vx£] = ->/iH,-i,, (1.152a) V x Hi = jaxEi + /], (1.152b) V x E2 = -jtotiHz- M2, (1.153a) V x H2 = Joj€E2 + h- (1.153b) Now consider the the quantity V ■ (£1 x H2 — E2 x Hi), which can be expanded using FIGURE 1.15 Geometry for the Lorentz reciprocity theorem. 1.9 Some Useful Theorems 41 vector identity (B.8) to give V - (£, x H2 - E2 x Hi) = Ji • £2 - Ji • Ej + M2 ■ #i - Mi ■ H2. (U54) Integrating over the volume V, and applying the divergence theorem (B.15), gives | V.(£ixff2-£2x = ^(£i xr/2-£2x ■ rfj (1,155) = y (£2 ■ /, - £, • /2 + 5, • M2 - H2 ■ Mi)dv Equation (1.155) represents a general form of the reciprocity theorem, but in practice a number of special situations often occur leading to some simplification. We will consider three cases. S encloses no sources. Then Jt — J~, — M \ — A~/: — 0, and the fields £ 1, # 1, and E2, H2 are source-free fields. In this case, the right-hand side of (1.155) vanishes with the result that I EixH2-ds = & E2x Hi-ds. (1.156) This result will be used in Chapter 4, when demonstrating the symmetry of the impedance matrix for a reciprocal microwave network. S bounds a perfect conductor. For example, 5 may be the inner surface of a closed, perfectly conducting cavity. Then the surface integral of (1.155) vanishes, since £\ x H\ h = {h x £|) - r?2 (by vector identity B.3), and h x E\ is zero on the surface of a perfect conductor (similarly for £2). The result is j (£1 J2-H^ M2)dv = J (£2 ■ /1 - H2 Mi)dv. (1.157) This result is analogous to the reciprocity theorem of circuit theory. In words, this result states that the system response E\ or E2 is not changed when the source and observation points are interchanged. That is, £2 (caused by J2) at /i is the same as £1 (caused by J\) at J2. S is a sphere at infinity. In this case, the fields evaluated on S are very far from the sources andsocan be considered locally as plane waves. T/hen me impedance relation f? == n x Eft) applies to (1.155) to give (£, x H2 - E2 x H\) ■ n = (n x E\) -H2-{hx E2) - r?i = —H\ • fi2--H2 • H\ — 0. n n so that the result of (1.157) is again obtained. This result can also be obtained for the case of a closed surface S where the surface impedance boundary condition applies. 42 Chapter 1: Electromagnetic Theory Image Theory In many problems a current source is located in the vicinity of a conducting ground plane. Image theory permits the removal of the ground plane by placing a virtual image source on the other side of the ground plane. The reader should be familiar with this concept from electrostatics, so we will prove the result for an infinite current sheet next to an infinite ground plane and then summarize the other possible cases. Consider the surface current density Js = JsqX parallel to a ground plane, as shown in Figure 1.16a. Because the current source is of infinite extent and is uniform in the x, y directions, it will excite plane waves traveling outward from it. The negatively traveling wave will reflect from the ground plane at z = 0, and then travel in the positive direction. Thus, there will be a standing wave field in the region 0 < z < d and a positively traveling wave for z > d. The forms of the fields in these two regions can thus be written as Esx = Mdm -e~jknZ), for 0 < z < d, (1.158a) H* = — (e}kbl + e-jh>% for 0 < z < d, (1.158b) forz>d, (1.159a) for z>d, (1.159b) = Be where j)0 is the wave impedance of free-space. Note that the standing wave fields of (1.158) have been constructed to satisfy the boundary condition that Ex = 0 at z = 0. The remaining boundary conditions to satisfy are the continuity of E at z = dt and the discontinuity in the H Ground PJane A v / / / / / / -- ■ Source ■ Image (a) Source -d (b) FIGURE 1.16 Illustration of image theory as applied to an electric current source next to a ground plane, (a) An electric surface current density parallel (o a ground plane, (b) The ground plane of (a) replaced with image current at z = —d. 1.9 Some Useful Theorems 43 field at z = d due to the current sheet From (1.36), since Ms = 0, K — EJ b=0' while from (1.37) we have Js = zx9{h?-h;)\z=0. Using (1.158) and (1.159) then gives 2jA sin kQd = Be~jhid and Jso = -—e~'M - — cos M, which can be solved for A and B. 2 S = — j/^o^sin^o^- So the total fields are El = -jJsds.mk^)z, forO < z < d, Jsae~ik(id cosfo)Z, for 0 < z < = -jJ$(ir}Q$mkode~jk9Z' for g > —;/ 7jo sin k0de~jk (1.160a) (1.160b) (1.161a) (1.161b) (1.162a) (1.162b) Now consider the application of image theory to this problem. As shown in Figure 1.16b, the ground plane is removed and an image source of - Js is placed at z = —d< By superposition, the total fields for z > 0 can be found by combining the fields from the two sources individually. These fields can be derived by a procedure similar to that above, with the following results: Fields due to source at z = d\ Et = J^riMz-di 2 for z > d for z < d, (1.163a) 2 I 2 Fields due to source aiz = —d: JjOTO -jMz+d) 2 ior z > d for z —d forz < —d, (1.163b) (1.164a) h„ ---- Js(i c-iMz+d) -I ?fLeJh(z+d) forz > —d for z < —d, (1.164b) 44 Chapter 1: Electromagnetic Theory Original Geometry I f I I" I M I (a) 00 Image Equivalent I lit w FIGURE 1.17 Electric and magnetic current images, (a) An electric current parallel to a ground plane, (b) An electric current normal to a ground plane, (c) A magnetic current parallel to a ground plane, (d) A magnetic current normal to a ground plane. The reader can verify that the solution is identical to that of (1.161) for 0 < z < d, and to (1.162) forz > thus verifying the validity of the image theory solution. Note that image theory only gives the correct fields to the right of the conducting plane. Figure 1.17 shows more general image theory results for electric and magnetic dipoles. REFERENCES [1] J. C. Maxwell, A Treatise on Electricity and Magnetism, Dover, N.Y., 1954, [2] A. A. Oliner, "Historical Perspectives on Microwave Field Theory," IEEE Trans. Microwave Theory and Techniques, vol. MTT-32, pp. 1022-1045, September 1984 (this special issue contains other articles on the history of microwave engineering). [3] J. D. Kraus and D. A. Fletsch, Electromagnetics, Fifth Edition, McGraw-Hill, N.Y., 1999. [4J C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, N.Y., 1989. [5] R. E. Collin, Foundations for Microwave Engineering, Second Edition, McGraw-Hill, N.Y-, 1992. [61 D. K. Cheng, Fundamentals of Engineering Electromagnetics, Addison-Wesley, Reading, Mass., 1992. [7] S, Ramo, T R. Whinnery, add T. van Duzer, Fields and Waves in Communication Electronics, Third Edition, John Wiley & Sons, N.Y., 1994. Problems 45 {8] F. Ulaby, Fundamentals of Applied Electromagnetics, Second Edition, Prentice-Hall, N.J., 2001-[9] D. M, Pozar, Microwave and RF Design of Wireless Systems, Wiley, NJ.T 2001. PROBLEMS 1.1 Assume that an infinite sheet of electric surface current density /, = J0x A/m is placed od the z = 0 plane between free-space for z < 0, and a dielectric with « = *rt0 for z > 0, as shown below. Find the resulting E and H fields in the two regions. HINT: Assume plane wave solutions propagating away from the current sheet, and match boundary conditions to find the amplitudes, as in Example 1,3. 1.2 Let E = Efifi + E^fy + E.£ be an electric field vector in cylindrical coordinates. Demonstrate that it is incorrect to interpret the expression V-E in cylindrical coordinates as pV2Efi + $VJ£^, + iV2Ez by evaluating both sides of the vector identity V x V x E = V(V ■ E) — V2E for the given electric field. 1.3 Consider a permanent magnet with a steady magnetic field H = Hoy, and a parallel plate capacitor with an electric field E = Eqx, arranged as shown in the figure below. Calculate the Poynting vector at a point between both the magnet poles and the capacitor plates. This nonzero result seems to imply real power flow in the z direction, but clearly there is no wave propagation or power delivered from the sources. How do you explain this apparent paradox? L4 A plane wave traveling along the z-axis in a dielectric medium with tt = 2.55 has an electric field given by £y = £0 cos(cor - jtz). The frequency is 2.4 GHz, and Eq = 30 V/m. (a) Find the amplitude Chapter 1: Electromagnetic Theory and direction of the magnetic field, (b) Find the phase velocity and wavelength, (c) Find the phase shift between the positions z\ =0,5m and z2 = 1.7 m. 1.5 Show that a linearly polarized plane wave of the form E = Ea(x + 2y)e-J*l,,I can be represented as the sum of an RHCP and an LHCP wave. 1.6 Compute the Poynting vector for the general plane wave field of (1.76). 1.7 A plane wave is normally incident on a dielectric slab of permittivity and thickness d, where d = Ao/(4A/^7), and kQ is the free-space wavelength of the incident wave, as shown below, If free-space exists on both sides of the slab, find the reflection coefficient of the wave reflected from the front of the slab. 1JJ Consider an RHCP plane wave normally incident from free-space {z < 0) onto a half-space (z > 0) consisting of a good conductor. Let the incident electric field be of the form and find the electric and magnetic fields in the region z > 0. Compute the Poynting vectors for z < 0 and z > 0, and show that complex power is conserved. What is the polarization of the reflected wave? 1,9 Consider a plane wave propagating in a lossy dielectric medium for t < 0, with a perfecdy conducting plate at z = 0. Assume that the lossy medium is characterized by e = (5 — j2)^o, \i — Hq, and that the frequency of the plane wave is 1.0 GHz, and let the amplitude of the incident electric field be 4 V/m at z = 0. Find the reflected electric field for z < 0, and plot the magnitude of the total electric field for -0.5 < z < 0. 1.10 A. plane wave at 1 GHz Ls normally incident on a thin copper sheet of thickness t. (a) Compute the transmission losses, in dB, of the wave at the air-copper and the copper-air interfaces, (b) If the sheet is to be used as a shield to reduce the level of the transmitted wave by 150 dB, what is the minimum sheet thickness? 1.11 A uniform lossy medium with tr = 3.0, tan S — 0.1, and }i = fio fills the region between z = 0 and z = 20 cm, with a ground plane at z = 20 cm, as shown below, An incident plane wave with an electric field, Ei = ílOOť"^ V/m. is present at z = 0 and propagates in the +z direction. The frequency is / = 3.0 GHz. Problems 47 (a) Compute Pt, the power density of the incident wave, and Pr, the power density of the reflected wave, at z = 0- (b) Compute the input power density, Pin, at z = 0, from the total fields at z = 0, Does 1.12 Redo Problem Ll; but with an electric surface current density of Js = JJce ^*A/m, where < fco- 1.13 A parallel polarized plane wave is obliquely incident from free-space onto a magnetic material with permittivity e0 and permeability fiopir. Find the reflection and transmission coefficients. Does a Brewster angle exist for this case, where the reflection coefficient vanishes for a particular angle of incidence? 1.14 Repeat Problem 1.13 for the perpendicularly polarized case. 1.15 An anisotropic material has a tensor permittivity [e} as given below. At a certain point in the material, the electric field is known to be £ = 2£ + 3y + 4z. What is D at this point? P: u I = 20 cm z - 1 -2j 0-[ť]=í0 2j 3 0 . 0 0 4. 1.16 Consider the gyrotropic permittivity tensor shown below: The D and E fields are related as Show that the transformations Et - JEy, D+ = Ds-jDy, £_ - D_ = Dx+ jD allow the relation between £ and D to XT ° h a P t e r T w o Transmission Line Theory In many ways transmission line theory bridges the gap between field analysis and basic circuit theory, and so is of significant importance in microwave network analysis. As we will see, the phenomenon of wave propagation on transmission lines can be approached from an extension of circuit theory or from a specialization of Maxwell's equations; we shall present both viewpoints and show how this wave propagation is described by equations very similar to those used in Chapter 1 for plane wave propagation. THE LUMPED-ELEMENT CIRCUIT MODEL FOR A TRANSMISSION LINE The key difference between circuit theory and transmission line theory is electrical size. Circuit analysis assumes that the physical dimensions of a network are much smaller than the electrical wavelength, while transmission lines may be a considerable fraction of a wavelength, or many wavelengths, in size. Thus a transmission line is a distributed-parameter network, where voltages and currents can vary in magnitude and phase over its length. As shown in Figure 2.1a, a transmission line is often schematically represented as a two-wire line, since transmission lines (for TEM wave propagation) always have at least two conductors. The piece of line of infinitesimal length of Figure 2. la can be modeled as a lumped-element circuit, as shown in Figure 2. lb, where R, L, G, C are per unit length quantities defined as follows: R = series resistance per unit length, for both conductors, in ß/m. L = series inductance per unit length, for both conductors, in H/m. G = shunt conductance per unit length, in S/m. C = shunt capacitance per unit length, in F/m. The series inductance L represents the total self-inductance of the two conductors, and the shunt capacitance C is due to the close proximity of the two conductors. The series 49 50 Chapter 2: Transmission Line Theory Hz, l) + v(.z, 1) Az- iz) id. 0 o-WvV LA; i{Z +Az, 1) GAz > =F C&t vfz + A;, i) -*-Az--fr- m FIGURE 2.1 Voltage and cuirent definitions and equivalent circuit for an incremental length of transmission line, (a) Voltage and current definitions, (b) Lumped-element equivalent circuit. resistance R represents the resistance due to the finite conductivity of the conductors, and the shunt conductance G is due to dielectric loss in the material between the conductors. ft and G, therefore, represent loss, A finite length of transmission line can be viewed as a cascade of sections of the form shown in Figure 2.1b. From the circuit of Figure 2. lb, Kirchhoff's voltage law can be applied to give i>(z, 0 - RAzi(z, t) - LAz3'(M) - v(z + Az, t) = 0, (2.1a) dt and Kirchhoff's current law leads to i(z, *) - GAzu(z + Az, f) - CAzdViZ'tAZKt) -i(z + Az, t) = 0. (2.1b) dt Dividing (2.1a) and (2.1b) by Az and taking the limit as Az 0 gives the following differential equations: —--= -Ri(7.,M - L—-—, (2.2a) dz dt —--= -Gv(z, t)-C———, (2,2b) dz dt These equations are the time-domain form of the transmission line, or telegrapher, equations. For the sinusoidal steady-state condition, with cosine-based phasors, (2.2) simplify to ° =-(* + jvDHz), (2.3a) dz dm dz = -(G + jaC)V(z). (2.3b) Note die similarity in the form of (2.3) and Maxwell's curl equations of (1.41a) and (1.41b). 2.1 The Lumped-Element Circuit Model for a Transmission Line 51 Wave Propagation on a Transmission Line The two equations of (2.3) can be solved simultaneously to give wave equations for Viz) and Hz): d2V{z) 2 dz2 d2I{z) -y*V(z) = 0, (2.4a) - y2l(z) = 0, (2.4b) dz2 where y = a -I- jfi = j(R + ]o)L)(G + jioC) (2.5) is the complex propagation constant, which is a function of frequency. Traveling wave solutions to (2,4) can be found as V{z) = V+e-** + így* (2.6a) Hz) = Ce~Yl + (2.6b) where the e~YZ term represents wave propagation in the +z direction, and the eYZ term represents wave propagation in the -z direction. Applying (2.3a) to the voltage of (2.6a) gives the current on the line: Comparison with (2.6b) shows that a characteristic impedance, Z0, can be defined as y y G -f joiC to relate the voltage and current on the line as Y+ -V-j+-Zo- — - Then (2.6b) can be rewritten in the following form: HZ) = I",«r* ~ SjnPS (2.8) Converting back to the time domain, the voltage waveform can be expressed as viz, t) = i v+i mm +)e~az a 9) + \V~\ cos(wr + Bz + where 0* is the phase angle of the complex voltage V*. Using arguments similar to those in Section 1.4, we find that the wavelength on the line is (2.10) P and the phase velocity is ft vp = ^=Xf. (2.11) Chapter 2: Transmission Line Theory The Lossless Line The above solution was for a general transmission line, including loss effects, and it was seen that the propagation constant and characteristic impedance were complex. In many practical cases, however, the loss of the line is very small and so can be neglected, resulting in a simplification of the above results. Setting if — G = 0 in (2.5) gives the propagation constant as Y = a + j8 - jojVlČ, or 0 = <ú\fLČ, (2.12a) «=0. (2.12b) As expected for the lossless case, the attenuation constant a is zero. The characteristic impedance of (2.7) reduces to Zo = y|. (2.13) which is now a real number. The general solutions for voltage and current on a lossless transmission line can then be written as V(z) = V+e~m + V;eiSi\ (2.14a) m = - dub) The wavelength is and the phase velocity is k = — ^ —g=i (2.15) vp = - = -P=. (2.16) FIELD ANALYSIS OF TRANSMISSION LINES In this section we will rederive the time-harmonic form of the telegrapher's equations, starting with Maxwell's equations. We will begin by deriving the transmission line parameters (R, L, G, C) in terms of the electric and magnetic fields of the transmission line and then derive the telegrapher equations using these parameters for the specific case of a coaxial line. Transmission Line Parameters Consider aim section of a uniform transmission line with fields E and H, as shown in Figure 2.2, where S is the cross-sectional surface area of the line. Let the voltage between the conductors be V0e±j&z and the current be I0e±JfSz. The time-average stored magnetic energy for this 1 m section of line can be written, from (1.86), as Wm = ^ I h * h*ds, 2.2 Field Analysis of Transmission Lines 53 FIGURE 2,2 Field lines on an arbitrary TEM transmission line. and circuit theory gives Wm = L\I0\2/4, in terms of the current on the line. We can thus identify the self-inductance per unit length as L = -j-z [ HH*dsWm. (2.17) IA>I Js Similarly, the time-average stored electric energy per unit length can be found from (1.84) =11 j EE* ds, and circuit theory gives We = C\ V0\2/4, resulting in the following expression for the capacitance per unit length: C = ttttť j Ě-Ě*ds F/m. (2.18) |ÍTlJ From (1.130), the power loss per unit length due to the finite conductivity of the metallic conductors is dt (assuming H is tangential to 5), and circuit theory gives Pc = R\f0\2/2t so the series resistance if per unit length of line is R-^~ [ HH'dlU/m. (2.19) In (2.19), Rs = 1/crSj is the surface resistance of the conductors, and Ci + Cj represent integration paths over the conductor boundaries. From (1.92), the time-average power dissipated per unit length in a lossy dielectric is where is the imaginary part of the complex dielectric constant € = e' — jt" = e'(I — j tan 5), Circuit theory gives P=0 Jfi=w Pl 2jt C= , - / / -jpdpdf^ —— F/m, (In bjay J^o Jp=a p2 In b/a (2jt)2 1^=0 a2 v J^b2 r| 2jt \a b) iff? f2* fb 1 t , 2jT«€" (In d/a)2 J0=o 'p=(1 />2 In fc/a ■ G = Table 2.1 summarizes the parameters for coaxial, two-wire, and parallel plate Unes. As we will see in the next chapter, the propagation constant, characteristic impedance, and attenuation of most transmission lines are derived directly from a field theory solution; the approach here of first finding die equivalent circuit parameters (L, C, Rt G) is useful only for relatively simple lines. Nevertheless, it provides a helpful intuitive concept, and relates a transmission line to its equivalent circuit model. TABLE 2.1 2,2 Field Analysis of Transmission Lines Transmission Line Parameters for Some Common Lines 55 COAX TWO-WIRE r D L 0 0 PARALLEL PLATE L C R G fi b —In - 2k a 2rt(' InA/a 2tz 271QX" ]nb/a — cosh" 7t (I cash-[{Df 2a) cosh"1 (D/2a) w if it! The Telegrapher Equations Derived from Field Analysis of a Coaxial Line We now show that the telegrapher equations of (2.3), derived using circuit theory, can also be obtained from Maxwell's equations. We will consider the specific geometry of the coaxial line of Figure 2.3. Although we will treat TEM wave propagation more generally in the next chapter, the present discussion should provide some insight into the relationship of circuit and field quantities. A TEM wave on the coaxial line of Figure 2.3 will be characterized by Ez—Hz= 0; furthermore, due to azimuthal symmetry, the fields will have no ^-variation, and so 3/30 = 0. The fields inside the coaxial line will satisfy Maxwell's curl equations, V x E = —jtofiH, V x H = jaxE, (2.21a) (2.21b) where t =e' — je" may be complex to allow for a lossy dielectric filling. Conductor loss will be ignored here. A rigorous field analysis of conductor loss can be carried out, but at this point would tend to obscure our purpose; the interested reader is referred to references [I.] or [2]. Expanding (2.21a) and (2.21b) then gives the following vector equations: -P -p- dz dz + 0 + 0- 3E i a —--hZ--- dz l 3 + Z-—(pH$) = juxifiEp + fiEf). P dp (2.22a) (2.22b) Since the I components of these two equations must vanish, it is seen that and H$ must have the forms Es — Ha = m p six) p (2.23a) (2.23b) Chapter 2: Transmission Line Theory To satisfy the boundary condition that = 0 at p = a, b, we must have E$ = 0 everywhere, due to the form of E^ in (2.23a). Then from the p component of (2.22a), it is seen that Hp = 0. With these results, (2.22) can be reduced to 0 = -jwiH*. (2.24a) dz = -jo*Ep. (2.24b) From the form of H$ in (2.23b) and (2.24a), Ep must be of the form Ep = —, (2.25) Using (2.23b) and (2.25) in (2.24) gives dh(z) dz dg(z) = -ja>ng(z), (2.26a) = -jaxhiz), (2.26b) Now the voltage between the two conductors can be evaluated as Cb [h dp b V(z) = / Ep(p, z)dp = h(z) / — = h(z) In (2.27a) Jfi—a J p=a P M and the total current on the inner conductor at p = a can be evaluated using (2,23b) as I(Z)= I H(a,z)ad4> = 27tg(z). (2.27b) J=0 Then h(z) and g(z) can be eliminated from (2.26) by using (2.27) to give 9 V(z) _ j cop. In bja - —— = -jcoW - )<')-—— -dz lab/a Finally, using the results for L, G, and C for a coaxial line as derived above, we obtain the telegrapher equations as ) = -jaLI(z), (2.28a) dz am dz = -(G + >C)V(z) (2.28b) (excluding R, the series resistance, since the conductors were assumed to have perfect conductivity). A similar analysis can be carried out for other simple transmission lines. 2.3 The Terminated Lossless Transmission Line 57 Propagation Constant, Impedance, and Power Flow for the Lossless Coaxial Line Equations (2.24a) and (2.24b) for EP and can be simultaneously solved to yield a wave equation for Ep (or H^): ^+^V£p=0, (2.29) from which it is seen that the propagation constant is y2 = -co2pie, which, for lossless media, reduces to jS - ai^Jie = wsfLC, (2.30) where the last result is from (2.12). Observe that this propagation constant is of the same form as that for plane waves in a lossless dielectric medium. This is a general result for TEM transmission lines. The wave impedance is defined as Zw = EpjH$, which can be calculated from (2.24a) assuming an dependence to give 2. = ^ = ^=^=^. (2.31) "0 P This wave impedance is then seen to be identical to the intrinsic impedance of the medium, n, and again is a general result for TEM transmission lines. The characteristic impedance of the coaxial line is defined as 7 _Vo _ Ep In bja _ qlnb/a [JThib/a where the forms for Ep and from Example 2.1 have been used. The characteristic impedance is geometry dependent and will be different for other transmission line configurations. Finally, the power flow (in the z direction) on the coaxial line may be computed from the Poynting vector as 1 / 1 t2* tb V I* 1 P = - ExH*ds = - / *\ pdpd* = -V0i;t (2.33) 2 X 2 Jp=a Inp2 In b/a 2 a result that is in clear agreement with circuit theory. This shows that the flow of power in a transmission line takes place entirely via the electric and magnetic fields between the two conductors; power is not transmitted through the conductors themselves. As we will see later, for the case of finite conductivity, power may enter the conductors, but this power is then lost as heat and is not delivered to the load. THE TERMINATED LOSSLESS TRANSMISSION LINE Figure 2.4 shows a lossless transmission line terminated in an arbitrary load impedance Zt. This problem will illustrate wave reflection on transmission lines, a fundamental property of distributed systems. Assume that an incident wave of the form V^e~^z is generated from a source at 2 < 0. We have seen that the ratio of voltage to current for such a traveling wave is Zo, the characteristic impedance. But when the line is terminated in an arbitrary load %L 7^ Zo, the ratio of voltage to current at the load must be ZL. Thus, a reflected wave must 59 Chapter 2; Transmisston Line Theory Vizi Hz) . ----"-*7y ---- .____i_.. ' 0 FIGURE 2A A transmission line terminated in a load impedance Zi. be excited with the appropriate amplitude to satisfy this condition. The total voltage on the line can then be written as in (2.14a), as a sum of incident and reflected waves: V(z) = V^e^z + V~e^. (2.34a) Similarly, the total current on the line is described by (2.14b): Hz) = ^- = — I, and is given by Vmin = |V+|(1 - III). (2.40b) As |T| increases, the ratio of to Vmin increases, so a measure of the mismatch of a line, called the standing wave ratio (SWR), can be defined as ^=ljf_|Tl This quantity is also known as the voltage standing wave ratio, and is sometimes identified as VSWR. From (2.41) it is seen that SWR is a real number such that 1 < SWR < oo, where SWR = 1 implies a matched load. From (2.39), it is seen that the distance between two successive voltage maxima (or minima; is £ = 2n/2p — nX/ln — X/2, while the distance between a maximum and a minimum is I = 7r/2fi = X/A, where A is the wavelength on the transmission line. The reflection coefficient of (2.35) was defined as the ratio of the reflected to the incident voltage wave amplitudes at the load (I = 0), but this quantity can be generalized to any point i on the line as follows. From (2.34a), with z — the ratio of the reflected component to the incident component is TO = ^Jft- = r<0)-^] 0 ~ l-Te-VPt^ UAi) where (2.36a,b) have been used for V(z) and I(z), A more usable form may be obtained by using (2,35) for T in (2.43): 9 7 J_L Z, = 0 -J 0 FIGURE ZJS A transmission line terminated in a short circuit. 2.3 The Terminated Lossless Transmission Line 61 ta) (0 FIGURE 2.6 {a) Voltage, (b) current, and (c) impedance (Rlň = 0 or oo) variation along a short-circuited transmission line. (open circuit). Equation (2.45c) also shows that the impedance is periodic in £, repeating for multiples of X/2. The voltage, current, and input reactance for the short-circuited line are plotted in Figure 2.6. Next consider the open-circuited line shown in Figure 2.7, where Zi = oo. Dividing the numerator and denominator of (2.35) by ZL and allowing Z{ — oo shows that the reflection coefficient for this case is F = 1, and the standing wave ratio is again infinite. From (2.36) the voltage and current on the line are V(z) = V} [e~m + eJfil] = 2Vf cos fiz, = -2- IW^ - — sin Bz, (2.46a) (2.46b) m. Kz> /, =0 ~l 0 FIGURE 2.7 A transmission line terminated in an open circuit. 62 Chapter 2: Transmission Line Theory which shows that now / = 0 at the load, as expected for an open circuit, while the voltage is a maximum. The input impedance is Zm = -jZ0 cot Bt, (2,46c) which is also purely imaginary for any length, t. The voltage, current, and input reactance of the open-circuited line are plotted in Figure 2.8, Now consider terminated transmission lines with some special lengths. If t = k/2, (2.44) shows that Zisi = ZL, (2.47) meaning that a half-wavelength line (or any multiple of kj 2) does not alter or transform the load impedance, regardless of the characteristic impedance. If the line is a quarter-wavelength long or, more generally, I — k/4 + tik/2, for n = 1,2, 3,..., (2.44) shows that the input impedance is given by (2.48) Such a line is known as a quarter-wave transformer because it has the effect of transforming the load impedance, in an inverse manner, depending on the characteristic impedance of the line. We will study this case more thoroughly in Section 2.5. FIGURE 2.8 (a) Voltage, (b) current, and (c) impedance (Rltl = 0 or eo) variation along an open-circuited transmission line. 2.3 The Terminated Lossless Transmission Line 63 0 FIGURE 2.9 Reflection and transmission at the junction of two transmission lines with differect characteristic impedances. Now consider a transmission line of characteristic impedance Zq feeding a line of different characteristic impedance, Z\, as shown in Figure 2.9. If the load line is infinitely long, or if it is terminated in its own characteristic impedance, so that there are no reflections from its end, then the input impedance seen by the feed line is Zj, so that the reflection coefficient T is F = &5& (2.49) L\ ■+- Z0 Not all of the incident wave is reflected; some of it is transmitted onto the second line with a voltage amplitude given by a transmission coefficient, T. From (2.36a), the voltage for z < 0 is V(z) = V+{e-j?z + Ve^ť), z < 0, (2.50a) where V* is the amplitude of the incident voltage wave on the feed line. The voltage wave for z > 0, in the absence of reflections, is outgoing only, and can be written as V(z) = V+Te-j/iz, for z > 0. (2.50b) Equating these voltages at z = 0 gives the transmission coefficient, T, as r = i + r = i + |i—^ = -^-. (2.51) Z| + Zrj Zi + Zq The transmission coefficient between two points in a circuit is often expressed in dB as the insertion loss^ /L, /X = -201ogj7"| dB. (2.52) POINT OF INTEREST: Decibels and Nepers Often the ratio of two power levels, Pi and Pi, in a microwave system is expressed in decibels (dB) as 10 log — dB. Pi Thus, a power ratio of 2 is equivalent to 3 dB, while a power ratio of 0.1 is equivalent to — 10 dB. Using power ratios in dB makes it easy to calculate power loss or gain through a series of components, since multiplicative loss or gain factors can he accounted for by adding the loss or gain in dB for each stage. For example, a signal passing through a 6 dB attenuator followed by a 23 dB amplifier will have an overall gain of 23 — 6 = 17 dB. Chapter 2: Transmission Lina Theory Decibels are used only to represent power ratios, but if P] = VfiR^ and P2 = V22/R2, then the resulting power ratio in terms of voltage ratios is ,0iog4^=201og^ J^dB, 5 vat, vj v Äi where /?], are the load resistances and V\, ^ are the voltages appearing across these loads. If the load resistances are equal, then this formula simplifies to 20 log ^ dB. The ratio of voltages across equal load resistances can also be expressed in terms of nepers (Np) as In^Np. The corresponding expression in terms of powers is since voltage is proportional to the square root of power. Transmission line attenuation is sometimes expressed in nepers. Since 1 Np corresponds to a power ratio of e2, the conversion between nepers and decibels is INp = 10 log e2 = 8.686 dB. Absolute powers can also be expressed in decibel notation if a reference power level is assumed. If we let P2 — l mW, then the power P\ can be expressed in dBm as 10 log —!— dBm. I do W Thus a power of 1 mW is 0 dBm, while a power of J W is 30 dBm, etc. THE SMITH CHART The Smith chart, shown in Figure 2.10, is a graphical aid that is very useful when solving transmission line problems. Although there are a number of other impedance and reflection coefficient charts that can be used for such problems [3], the Smith chart is probably the best known and most widely used. It was developed in 1939 by P. Smith at the Bell Telephone Laboratories [4]. The reader may feel that, in this day of scientific calculators and powerful computers, graphical solutions have no place in modem engineering. The Smith chart, however, is more than just a graphical technique. Besides being an integral part of much of the current computer-aided design (CAD) software and test equipment for microwave design, the Smith chart provides an extremely useful way of visualizing transmission tine phenomenon and so is also important for pedagogical reasons. A microwave engineer can develop intuition about transmission line and impedance-matching problems by learning to think in terms of the Smith chart. At first glance the Smith chart may seem intimidating, but the key to its understanding is to realize that it is essentially a polar plot of the voltage reflection coefficient, r. Let the reflection coefficient be expressed in magnitude and phase (polar) form as F = | F\e*$. Then the magnitude | T | is plotted as a radius (| T| < 1) from the center of the chart, and the angle #(—180° < 9 < 180*) is measured from the right-hand side of the horizontal diameter. Any 2.4 The Smřth Chart 65 FIGURE 2.10 The Smith chart. passively realizable (|r| < 1) reflection coefficient can then be plotted as a unique point on the Smith chart, The real utility of the Smith chart, however, lies in the fact that it can be used to convert from reflection coefficients to normalized impedances (or admittances), and vice versa, using the impedance (or admittance) circles printed on the chart. When dealing with impedances on a Smith chart, normalized quantities are generally used, which we wi II denote by lowercase letters. The normalization constant is usually the characteristic impedance of the line. Thus, z = Z/Zo represents the normalized version of the impedance Z. If a lossless line of characteristic impedance Zo is terminated with a load impedance ZLi the reflection coefficient at the load can be written from (2.35) as T = ^{ = |r|^, (2.53) where zl = Zl/Zq is the normalized load impedance. This relation can be solved for zl in terms of T to give (or, from (2.43) with t = 0) 1 + in** ... This complex equation can be reduced to two real equations by writing T and zl in terms Chapter 2: Transmission Line Theory of their real and imaginary parts. Let T = Tr + jTj, and zi = rL + jxL. Then, (l + rr) + jrv (i-rrwiy The real and imaginary parts of this equation can be found by multiplying the numerator and denominator by the complex conjugate of the denominator to give i _ r2 - r2 2t\ (1 - Vr)r + rf H = „ „J . „2- (2.55b) Rearranging (2.55) gives (rr - D2 + (r, - j-j = f—J , (2.56b) which are seen to represent two families of circles in the Vr, F; plane. Resistance circles are defined by (2.56a), and reactance circles are defined by (2,56b). For example, the ri = 1 circle has its center at ľr — 0.5, Vs = 0, and has a radius of 0.5, and so passes through the center of the Smith chart. All of the resistance circles of (2,56a) have centers on the horizontal r; = 0 axis, and pass through the V = 1 point on the right-hand side of the chart. The centers of all of the reactance circles of (2.56b) lie on the vertical Tr = 1 line (off the chart), and these circles also pass through the T = 1 point. The resistance and reactance circles are orthogonal. The Smith chart can also be used to graphically solve the transmission line impedance equation of (2.44), since this can be written in terms of the generalized reflection coefficient as 1 + Ye-2W Zm = Z6i_re_2m, (2.57) where F is the reflection coefficient at the load, and ť is the (positive) length of transmission line. We then see that (2.57) is of the same form as (2.54), differing only by the phase angles of the ľ terms. Thus, if we have plotted the reflection coefficient \r\^e at the load, the normalized input impedance seen looking into a length í of transmission line terminated with zl can be found by rotating the point clockwise an amount 28£ (subtracting 2j8r from 0) around the center of the chart. The radius stays the same, since the magnitude of F does not change with position along the line. To facilitate such rotations, the Smith chart has scales around its periphery calibrated in electrical wavelengths, toward and away from the "generator" (which just means the direction away from the load). These scales are relative, so only the difference in wavelengths between two points on the Smith chart is meaningful. The scales cover a range of 0 to 0,5 wavelengths, which reflects the fact that the Smith chart automatically includes the periodicity of transmission line phenomenon. Thus, a line of length X/2 (or any multiple) requires a rotation of 26£ — 2n around the center of the chart, bringing the point back to its original position, showing that the input impedance of a load seen through a X/2 line is unchanged. We will now illustrate the use of the Smith chart for a variety of typical transmission line problems through examples. 2.4 The Smith Chart 67 EXAMPLE 2,2 BASIC SMITH CHART OPERATIONS A load impedance of 40 + jlO £2 terminates a 100 Q transmission line that is 0.3X long, Find the reflection coefficient at the load, the reflection coefficient at the input to the line, the input impedance, the SWR on the line, and the return loss. Solution The normalized load impedance is zL = # = 0.4 + yo.7, which can be plotted on the Smith chart as shown in Figure 2.11. Using a compass and the voltage coefficient scale below die chart, the reflection coefficient magnitude at the load can be read as |T| = 0.59. This same compass setting can then be applied to the standing wave ratio (SWR) scale to read SWR = 3.87, and to the return loss (in dB) scale to read RL = 4.6 dB. Now draw a radial line through the load impedance point, and read the angle of the reflection coefficient at the load from the outer scale of the chart as 104°. We now draw an SWR circle through the load impedance point. Reading the reference position of the load on the wavelengths-toward-generator (WTG) scale gives a value of 0.106A. Moving down the fine $3\ toward the generator brings FIGURE 2.11 Smiui chart for Example 2.2. Chapter 2: Transmission Line Theory us to 0.406A on the WTG scale, which is equivalent to 0.020A. Drawing a radial line at this position gives the normalized input impedance at the intersection with SWR circle of Zw = 0.365 — j'0.61 L Then the input impedance of the line is Zin = = 36.5- j61.1 ft The reflection coefficient at the input still has a magnitude of |r | = 0.59; the phase is read from the radial line at the phase scale as 248°. ■ The Combined Impedance-Admittance Smith Chart The Smith chart can be used for normalized admittance in the same way that it is used for normalized impedances, and it can be used to convert between impedance and admittance. The latter technique is based on the fact that, in normalized form, the input impedance of a load zl connected to a X/4 line is, from (2.44), Zm = l/zL, which has the effect of converting a normalized impedance to a normalized admittance. Since a complete revolution around the Smith chart corresponds to a length of A/2, a X/4 transformation is equivalent to rotating the chart by 180°; this is also equivalent to imaging a given impedance (or admittance) point across the center of the chart to obtain the corresponding admittance (or impedance) point. Thus, the same Smith chart can be used for both impedance and admittance calculations during the solution of a given problem. At different stages of the solution, then, the chart may be either an impedance Smith chart or an admittance Smith chart. This procedure can be made less confusing by using a Smith chart that has a superposition of the scales for a regular Smith chart and the scales of a Smith chart which has been rotated 180°, as shown in Figure 2.12. Such a chart is referred to as an impedance and admittance Smith chart and usually has different-colored scales for impedance and admittance. EXAMPLE 23 SMITH CHART OPERATIONS USING ADMITTANCES A load of Zl = 100 +■ j 50 Í2 terminates a 50 Í2 line. What are the load admittance and the input admittance if the line is 0. \5k long? Solution The normalized load impedance isz^ = 2 + /LA standard Smith chart can be used for this problem by initially considering it as an impedance chart and plotting zl and the SWR circle. Conversion to admittance can be accomplished with a X/4 rotation of zi (easily obtained by drawing a straight line through zl and the center of the chart to intersect the SWR circle). The chart can now be considered as an admittance chart, and the input admittance can be found by rotating 0.15X from yi. Alternatively, we can use the combined zy chart of Figure 2.12, where conversion between impedance and admittance is accomplished merely by reading the appropriate scales. Plotting zL on the impedances scales and reading the admittance scales at this same point give jj, = 0,40 - 7*0.20. The actual load admittance 2.4 The Smith Chart 69 FIGURE 2.12 Z Y Smith chart with solution for Example 2.3. is then YL = yL Yq = H = 0.0080 - J0.0040 S. Then, on the WTG scale, the load admittance is seen to have a reference position of 0214k, Moving 0.15A. past this point brings us to 0.364*. A radial line at this point on the WTG scale intersects the SWR circle at an admittance of y = 0.61 + ;0.66, The actual input admittance is then Y = 0.0122 + j0.0132 S. ■ The Slotted Line A slotted line is a transmission line configuration (usually waveguide or coax) that allows the sampling of the electric field amplitude of a standing wave on a terminated line. With this device the SWR and the distance of the first voltage minimum from the load can be measured, and from this data the load impedance can be determined. Note that because the load impedance is in general a complex number (with two degrees of freedom), two distinct quantities must be measured with the slotted line to uniquely determine this impedance. A typical waveguide slotted line is shown in Figure 2.13. Although the slotted line used to be the principal way of measuring an unknown impedance at microwave frequencies, it has been largely superseded by the modern vector 70 Chapter 2: Transmission Line Theory FIGURE 2.13 An X-band waveguide slotted line. Courtesy of Agilent Technologies, Santa Rosa, Calif. network analyzer in terms of accuracy, versatility, and convenience. The slotted line is still of some use, however, in certain applications such as high-millimeter wave frequencies, or where it is desired to avoid connector mismatches by connecting the unknown load directly to the slotted line, thus avoiding the use of imperfect transitions. Another reason for studying the slotted line is that it provides an unexcelled tool for learning basic concepts of standing waves and mismatched transmission lines. We will derive expressions for finding the unknown load impedance from slotted line measurements and also show how the Smith chart can be used for the same purpose. Assume that, for a certain terminated line, we have measured the SWR on the line and ^mint the distance from the load to the first voltage minimum on the line. The load impedance ZL can then be determined as follows. From (2.41) the magnitude of the reflection coefficient on the line is found from the standing wave ratio as 1 SWR +1 From Section 23, we know that a voltage minimum occurs when e^~2^ = — 1, where 9 is the phase angle of the reflection coefficient, T = |r|e^. The phase of the reflection coefficient is then 0 - it + Iptum, (2.59) where £mjn is the distance from the load to the first voltage minimum. Actually, since the voltage minimums repeat every k/2, where X is the wavelength on the line, any multiple of k/2 can be added to i^n without changing the result in (2.59), because this just amounts to adding 28nk/2 = 2nn\o9, which will not change F. Thus, the two quantities SWR and £mj„ can be used to find the complex reflection coefficient T at the load. It is then 2.4 The Smith Chart 71 straightforward to use (2.43) with (. = 0 to find the load impedance from Tr i + r Zl = Zoy^-p (2.60) The use of the Smith chart in solving this problem is best illustrated by an example. EXAMPLE 2.4 IMPEDANCE MEASUREMENT WITH A SLOTTED LINE The following two-step procedure has been carried out with a 50 Q coaxial slotted line to determine an unknown load impedance: 1. A short circuit is placed at the load plane, resulting in a standing wave on the line with infinite SWR, and sharply defined voltage minima, as shown in Figure 2.14a. On the arbitrarily positioned scale on the slotted fine, voltage minima are recorded at z = 0.2 cm, 2.2 cm, 4.2 cm. 2. The short circuit is removed, and replaced with the unknown load. The standing wave ratio is measured as SWR = 1.5, and voltage minima, which are not as sharply defined as those in step 1, are recorded at z - 0.72 cm, 2.72 cm. 4.72 cm, as shown in Figure 2.14b. Find the load impedance. Solution Knowing that voltage minima repeat every A./2, we have from the data of step 1 above that k — 4.0 cm In addition, because the reflection coefficient and input impedance also repeat every A/2, we can consider the load terminals to be effectively located at any of the voltage minima locations listed in step 1. Thus, if we say the load is at 4.2 cm, then the data from step 2 shows that the next voltage minimum away from the load occurs at 2.72 cm. giving — 4.2 — 2.72 = 1.48 cm = 031k. ■ ■ K ■ I L I H I I M I J I I I I J I I I I J 0 12 3 4 5 (a) lmll""l " " 111111 Nlil 0 12 3 4 5 IVI Short circuit IVI - vm Unknown load FIG L" RE 2.14 Voltage standing wave patterns for Example 2.4. (a) S tanding wave for short-circuit load, (b) Standing wave for unknown load. 72 Chapter 2: Transmission Line Theory FIGURE 2.15 Smith chart for Example 2.4. Applying (2.58M2.60) to this data gives 6 = 71 + ^(1.48) = 86.4°, 4.0 so f« 0.2ťj86> = 0.0126 + jG. 1996. The load impedance is then Zi = 5o(^I)=47.3 + yT9.7£}. For the Smith chart version of the solution, we begin by drawing the SWR circle for SWR = 1.5, as shown in Figure 2.15; the unknown normalized load impedance must lie on this circle. The reference that we have is that the load is 0.37A away from the first voltage minimum. On the Smith chart, the position of a voltage mini mum corresponds to the minimum impedance point (minimum voltage, maximum current), which is the horizontal axis (zero reactance) to the left of the origin. Thus, we begin at the voltage minimum point and move 0.37A toward the load 2.5 The Quarter-Wave Transformer 73 (counterclockwise), to the normalized load impedance point, zl — 0.95 -f j0.4, as shown in Figure 2,15. The actual load impedance is then ZL = 47.5 H- j20 R, in close agreement with the above result using the equations. Note that, in principle, voltage maxima locations could be used as well as voltage minima positions, but voltage minima are more sharply defined than voltage maxima, and so usually result in greater accuracy. ■ THE QUARTER-WAVE TRANSFORMER The quarter-wave transformer is a useful and practical circuit for impedance matching and also provides a simple transmission hue circuit that further illustrates the properties of standing waves on a mismatched line. Although we will study the design and performance of quarter-wave matching transformers more extensively in Chapters, the main purpose here is the application of the previously developed transmission line theory to a basic transmission line circuit. We will first approach the problem from the impedance viewpoint, and then show how this result can also be interpreted in terms of an infinite set of multiple reflections on the matching section. The Impedance Viewpoint Figure 2.16 shows a circuit employing a quarter-wave transformer. The load resistance RL, and the feedfine characteristic impedance Zo, are both real and assumed to be given. These two components are connected with a lossless piece of transmission line of (unknown) characteristic impedance Z\ and length X/4. It is desired to match the load to the Zo line, by using the X/4 piece of line, and so make T — 0 looking into the X/4 matching section. From (2.44) the input impedance Z^ can be found as K^yZ^tan^ 'Zi+jRLtanfit y ; To evaluate this for pi — (2tt/X){X/4) = jr/2, we can divide the numerator and denominator by tan fit and take the limit as fit tz/2 to get Z]n = ft, (2.62) In order for T = 0, we must have Zm — Z0, which yields the characteristic impedance Z| as Z, = LXG + jwC), (2.82) which can be rearranged as y = ^iXy»c)(i + ^)(1 + JL) If the line is low-loss we can assume that R C, which means that both the conductor loss and dielectric loss are small. Then, RG ^ co2LCy and (2.83) reduces to (2.84) If we were to ignore the (R/cdL + G/tt>C) term, we would obtain the result that y was purely imaginary (no loss), so we will instead use the first two terms of the Taylor series expansion for a/1 + x ~ 1 + x/2 H----, to give the first higher order real term for y: so that *4f^sl§4(l*a4 <285a) p 0*/LG (2,85b) where Z0 = +JL/C is the characteristic impedance of the line in the absence of loss, Note from (2.85b) that the propagation constant ft is the same as the lossless case of (2.12). By the same order of approximation, the characteristic impedance Zo can be approximated as a real quantity; / R + jtoL [L Equations (2.85)-(2.86) are known as the high-frequency, low-loss approximations for transmission fines, and are important because they show that the propagation constant and characteristic impedance for a low-loss fine can be closely approximated by considering the line as lossless. 2.7 Lossy Transmission Lines 81 EXAMPLE 2.6 ATTENUATION CONSTANT OF THE COAXIAL LINE In Example 2.1 the L, C, /?, and G parameters were derived for a lossy coaxial line. Assuming the loss is small, derive the attenuation constant from (2.85a) and the results of Example 2.1. Solution From (2.85a), Using the results derived in Example 2.1 gives 2 lub/a \a b) J where ri = \[\lJ^ is the intrinsic impedance of the dielectric material filling the coaxial line. Also, jff = oj-JLC = (o^/JuT, and Zq = -JhJC = (r)/2n) In b/a. The above method for the calculation of attenuation requires that the line parameters L, C, R, and G be known. These can often be derived vising the formulas of (2.17)-(2.20), but a more direct and versatile procedure is to use the perturbation method, to be discussed shortly. The Distortionless Line As can be seen from the exact equations (2.82) and (2.83) for the propagation constant of a lossy line, the phase term is generally a complicated function of frequency, o, when loss is present. In particular, we note that is generally not exactly a linear function of frequency, as in (2.85b), unless the line is lossless. If 0 is not a linear function of frequency (of the form /J = aw), then the phase velocity vp = id/ft will be different for different frequencies w. The implication is that the various frequency components of a wideband signal will travel with different phase velocities, and so arrive at the receiver end of the transmission line at slightly different times. This will lead to dispersion, a distortion of the signal, and is generally an undesirable effect. Granted, as we have argued above, the departure of ft from a linear function may be quite small, but the effect can be significant if the line is very long. This effect leads to the concept of group velocity, which we will address in detail in Section 3.10. There is a special case, however, of a lossy line that has a linear phase factor as a function of frequency. Such a line is called a distortionless line, and is characterized by line parameters that satisfy the relation M From (2.83) the exact complex propagation constant, under the condition specified by (2,87), reduces to R R2 y = ja>SLCjl-2j- (2.88) 82 Chapter 2: Transmission Line Theory which shows that = t 2tlpZa 8 = JS*^** Chapter 2: Transmission Line Theory where Zq = (tj/lrc) In bja is the characteristic impedance of the coaxial line and V0 is the voltage across the line at z = 0. The first step is to find P0, the power flowing on the lossless line: 2 Js 2Z0 ]p=a 70=o 2np2 In bja 2Z0 as expected from basic circuit theory. The loss per unit length, Pl, comes from conductor loss {Ptc) and dielectric loss (Ptd)' From (1,131), the conductor loss in a 1 m length of line can be found as Pu- = § / \Ht\2ds = & f f = a)|2ad0 + ( l/^(p = fc)l26J0}^ A \a + b)' The dielectric loss in a 1 m length of line is, from (1.92), 2 Jv 2 Jp=4 j$=q /z=o in &/a where is the imaginary part of the complex dielectric constant, e = e' - jf*T, Finally, applying (2.96) gives 2PC 4ttZ0 I* 5/ mfc, A/a 2r)\x\b/a \a b} + 2 ' where i} = -Jixf*'* This result is seen to agree with that of Example 2.6. The Wheeler Incremental Inductance Rule Another useful technique for the practical evaluation of attenuation due to conductor loss for TEM or quasi-TEM lines is the Wheeler incremental inductance rule [6J. This method is based on the similarity of the equations for the inductance per unit length and resistance per unit length of a transmission line, as given by (2.17) and (2.19), respectively. In other words, the conductor loss of a line is due to current flow inside the conductor which, as was shown in Section 1.7, is related to the tangential magnetic field at the surface of the conductor, and thus to the inductance of the line. From (1.131), the power loss into a cross section S of a good (but not perfect) conductor is Pl = TJslJAlds " T l^fds W/m2, (2.97) so the power loss per unit length of a uniform transmission line is (2.98) 2.7 Lossy Transmission Lines 85 where the Ľne integral of (2.98) is over the cross-sectional contours of both conductors. Now, from (2,17), the inductance per unit length of the line is I'ds, (2.99) which is computed assuming the conductors are lossless. When the conductors have a small loss, the H field in the conductor is no longer zero, and this field contributes a small additional "incremental" inductance, AL, to that of (2.99). As discussed in Chapter 1, the fields inside the conductor decay exponentially so that the integration into the conductor dimension can be evaluated as AL = ^rf \Ht\2dt, (2.100) since f™ e~2z^' dz = 8,/L (The skin depth is = JTJT^IB.) Then Pt from (2.98) can be written in terms of A L as since Rs — v^Mo/2/2, where Z$ is the characteristic impedance of the line. In (2.102), AL is evaluated as the change in inductance when all conductor wails are receded by an amount 6^/2. Equation (2.102) can also be written in terms of the change in characteristic impedance, since Zo = /f=vfe=iv (2-,03) so that 2Zn (2.104) where AZo is the change in characteristic impedance when all conductor walls are receded by an amount Ss /2. Yet another form of the incremental inductance rule can be obtained by using the first two terms of a Taylor series expansion for Z0. Thus, so that Sf dZ$ AZ0 = Zo l^y^ - Zrj = 2 dt where Zo (5^/2) refers to the characteristic impedance of the line when the walls are receded by &s/2, and I refers to a distance into the conductors. Then (2.104) can be written as fiSs dZij _ Rs dZp (2 106) Uc ~ 4Z0 dt ~ 2Zt$ dt' ^ ' } Chapter 2: Transmission Line Theory where jj = Vm-oAis the intrinsic impedance of the dielectric, and Rs is the surface resistivity of the conductor. Equation (2.106) is one of the most practical forms of the incremental inductance rule, because the characteristic impedance is known for a wide variety of transmission lines. EXAMPLE 2.8 USING THE WHEELER INCREMENTAL INDUCTANCE RULE TO FIND THE ATTENUATION CONSTANT Calculate the attenuation due to conductor loss of a coaxial line using the incremental inductance rule. Solution From (2.32) the characteristic impedance of the coaxial line is Zb = f k*. 2tt a Then, nsing the incremental inductance rule of the form in (2.106), the attenuation due to conductor loss is Rs dZ a,- —-- 2Z0r) dt o Rs ídlnb/a 4 Inb/a 1 Rs /l 1\ : ~4jrZt,| db Ta J ~ 4jrZ0 \b + a)' which is seen to be in agreement with the result of Example 2.7. The negative sign on the second differentiation in the above equation is because the derivative for the inner conductor is in the —p direction (receding wall). ■ Regardless of how attenuation is calculated, measured attenuation constants for practical lines are usually higher. The main reason for this discrepancy is the fact that realistic transmission lines have metallic surfaces that are somewhat rough, which increases the loss, while our theoretical calculations assume perfectly smooth conductors. A quasiempirical formula that can be used to correct for surface roughness for any transmission line is [7] a = tan^1 - (2-107) where ac is the attenuation due to perfectly smooth conductors, a'v is the attenuation corrected for surface roughness, A is the rms surface roughness, and Sj is the skin depth of the conductors. REFERENCES [1] S. Ramo, J. R, Wionery, and T. Van Duzer, Fields and Waves in Communication Electronics, Third Edition, John Wiley & Sons, N-Y, 1994. [2] J. A. Stratton, Electromagnetic Theory, McGraw-Hill, N.Y., 1941. [3] H. A. Wheeler, "Reflection Charts Relating io Impedance Matching," IEEE Trans. Microwave Theory and Techniques, vol. MTT-32, pp. 10O8-IO21, September 1984. [4] P. H. Smith, 'Transmission Line Calculator," Electronics, vol. 12, No, 1. pp. 29-31, January 1939. [5] P. J. Nahin, Oliver Heaviside: Sage in Solitude, IEEE Press, N.Y., 1988. [6] H. A. Wheeler, "Formulas for the Skin Effect," Proc. IRE, vol. 30, pp. 412-424, September 1942. [7] T. C. Edwards, Foundations for Microstrip Circuit Design, John Wiley & Sons, N-Y„ 1987. Problems 87 PROBLEMS 2.1 The current on a transmission line is given as i(r) — 1.2cos(1.5l x 1010/ — 80,3s) A. Determine (a) the frequency, d. 25 For (he parallel plate line of Problem 2.3, derive the telegrapher equations using the field theory approach. 2.6 RG-402U semi-rigid coaxial cable has an inner conductor diameter of 0.91 mm, and a dielectric diameter (equal to the inner diameter of the outer conductor) of 3.02 mm. Both conductors are copper, and the dielectric material is Teflon. Compute the R, L, G, and C parameters of this line at 1 GHz, and use these results to find the characteristic impedance and attenuation of the line at 1 GHz. Compare your results to the manufacturer's specifications of 50 SI and 0.43 dB/m, and discuss reasons for the difference. 2.7 Compute and plot the attenuation of the coaxial line of Problem 2.6, in dB/m, over a frequency range of 1 MHz to 100 GHz. Use log-log graph paper. 2.8 A lossless transmission line of electrical length t = 0.3A. is terminated with a complex load impedance as shown below. Find the reflection coefficient at the load, the SWR on the line, the reflection coefficient at the input of the line, and the input impedance to the line. o— / = 0.3A Z« = 75 a zl = 30 - j2q a Chapter 2: Transmission Line Theory 2.9 A lossless transmission line is terminated with a 100 Q load. If the SWR on the line is 1.5, find the two possible values for the characteristic impedance of the line. 2.10 Let Zsc be the input impedance of a length of coaxial line when one end is short-circuited, and let Zqc be the input impedance of the line when one end is open-circuited. Derive an expression for the characteristic impedance of the cable in terms of and Zx. 2.11 A 100 Q transmission line has an effective dielectric constant of 1.65. Find the shortest open-circuited length of this fine that appears at its input as a capacitor of 5 pF at 2.5 GHz. Repeat for an inductance of 5 nH. 2.12 A radio transmitter is connected to an antenna having an impedance 80 + J40Q. with a 50 Q. coaxial cable. If the 50 £2 transmitter can deliver 30 W when connected to a 50 Q load, how much power is delivered to the antenna? 2.13 A 75 £2 coaxial transmission line has a Length of 2.0 cm and is terminated with a load impedance of 37,5 + j75 £2. If the dielectric constant of the line is 2.56 and the frequency is 3.0 GHz, find the input impedance to the line, the reflection coefficient at the load, the reflection coefficient at the input, and the SWR on the fine. 2.14 Calculate SWR> reflection coefficient magnitude, and return loss values to complete the entries in the following table: SWR 1.00 t.01 1.05 1.10 1.20 1.50 2.00 2.50 RLiöB) oo iri o.oo 0.01 — 30.0 0.10 — 10.0 2.15 The transmission line circuit shown below has Vg = 15 V rms, Zs = 75 £2, Z0 = 75 £2, Zt = 60 j4Q £2, and I = 0.7X. Compute the power delivered to the load using three different techniques: (a) find f and compute (b) find Zj„ and compute (c) find VL and compute Pr. = Zg + Z„ Re{Zj(1); and Pi = Problems 88 Discuss the rationale for each of these methods. Which of these methods can be used if the line is not lossless? 2.16 For a purely reactive load impedance of the form ZL = jXt show that the reflection coefficient magnitude |r| is always unity. Assume the characteristic impedance Zo is real. 2.17 Consider the transmission line circuit shown below. Compute the incident power, the reflected power, and the power transmitted into the infinite 75 £2 line. Show that power conservation is satisfied. 50 ft -a/2 AVA-□- 10V(~) Z0=50ft Z,=75ft Pre! 2.18 A generator is connected to a transmission line as shown below. Find the voltage as a function of z along the transmission line. Plot the magnitude of this voltage for — t < z < 0. 100 tl AVvV 10v ZD = 100 Si ZL = 80-;40£1 -i 2.19 Use the Smith chart to find the following quantities for the transmission line circuit below: (a) The SWR on the line. (b) The reflection coefficient at the load. (c) The load admittance. (d) The input impedance of the line. (e) The distance from the load to die first voltage minimum. (f) The distance from the load to the first voltage maximum. / = 0.4A Z„=^> z0«5on ZL = 60 +JÍ0Í2 2.20 2.21 Repeat problem 2.19 for ZL = 40 - ;30 fi. Repeat problem 2.19 for i = 1 .SX. Chapter 2: Transmission Line Theory 2.22 Use the Smith chart to find the shortest lengths of a short-circuited 75 ft line to give the following input impedance: (a) Zin = 0. (b) Z;„ = oo. (c) Zjn = ;75£l (d) Zin = -j50ft. <<0 Zin = ;t0ft. 2.23 Repeat Problem 2,22 for an open-circuited length of 75 ft line. 2.24 A slotted-line experiment is performed with the following res ults: distance between successive minima = 2,1 cm; distance of first voltage minimum from load = 0.9 cm; SWR of load = 2.5. If ZQ = 50 ft, find the load impedance. 2.25 Design a quarter-wave matching transformer to match a 40 ft load to a 75 ft line. Plot the SWR for 0.5 < //f0 < 2.0, where f0 is the frequency at which the hue is k/4 long. 2.26 Consider the quarter-wave matching transformer circuit shown below. Derive expressions for V+ and V', the amplitudes of the forward and reverse traveling waves on the quarter-wave line section, in terms of V1, the incident voltage amplitude. 2.27 Derive Equation (2.71) from (2.70). 2.28 In Example 2.7, the attenuation of a coaxial line due to finite conductivity is a Rs P M Show that ajj is minimized for conductor radii such that x In x = I -j- x, where x = bfa. Solve this equation for Jt, and show that the corresponding characteristic impedance for tf = 1 is 77 ft. 2.29 Compute and plot the factor by which attenuation is increased due to surface roughness, for rms roughness ranging from zero to 0.01 mm. Assume copper conductors at 10 GHz. 2JO A 50 ft transmission line is matched to a 10 V source and feeds a load ZL = 100 ft. If the line is 2.31 long and has an attenuation constant a = 0,5 dB/k, find the powers that are delivered by the source, lost in the line, and delivered to the load. 231 Consider a nonreciprocal transmission line having different propagation constants, and fi~t for propagation in the forward and reverse directions, with corresponding characteristic impedances Z£ and Zy. (An example of such a line could be a microstrip transmission line on a magnetized ferrite substrate.) If the line is terminated as shown belowt derive expressions for the reflection coefficient and impedance seen at the input of the line. Chapter Three Transmission Lines and Waveguides i One of the early milestones in microwave engineering was the development of waveguide and other transmission lines for the low-loss tranmission of microwave power. Although Heav-iside considered the possibility of propagation of electromagnetic waves inside a closed hollow tube in 1893, he rejected the idea because he believed that two conductors were necessary for the transfer of electromagnetic energy [1]. In 1897, Lord Rayleigh (John William Strutt) [2] mathematically proved that wave propagation in waveguides was possible, for both circular and rectangular cross sections, Rayleigh also noted the infinite set of modes of the TE and TM type that were possible and the existence of a cutoff frequency, but no experimental verification was made at the time. The waveguide was essentially forgotten until it was rediscovered independently in 1936 by two men [3]. After preliminary experiments in 1932, George C. South worth of the AT&T Company in New York presented a paper on the waveguide in 1936. At the same meeting, W. L, Barrow of MIT presented a paper on the circular waveguide, with experimental confirmation of propagation. Early microwave systems relied on waveguide and coaxial lines for transmission line media. Waveguide has the advantage of high power-handling capability and low loss but is bulky and expensive. Coaxial line has very high bandwidth and is convenient for test applications, but is a difficult medium in which to fabricate complex microwave components. Planar transmission lines provide an alternative, in the form of stripline, microstrip, slotline, coplanar waveguide, and many other types of related geometries. Such transmission lines are compact, low in cost, and are capable of being easily integrated with active devices such as diodes and transistors to form microwave integrated circuits. The first planar transmission line may have been a flat-strip coaxial line, similar to stripline, used in a production power divider network in World War II [4]. But planar lines did not receive intensive development until the 1950s. Microstrip line was developed at ITT laboratories [5] and was a competitor of stripline. The first microstrip lines used a relatively thick dielectric substrate, which accentuated the non-TEM mode behavior and frequency dispersion of the line. This characteristic made it less desirable than stripline until the 1960s, when much thinner substrates began to be used. This reduced the frequency dependence of the line, and now microstrip is often the preferred medium for microwave integrated circuits. 91 92 Chapter 3: Transmission Lines and Waveguides In this chapter we will study the properties of several types of transmission lines and waveguides that are in common use today. As we know from Chapter 2, a transmission line is characterized by a propagation constant and a characteristic impedance; if the line is lossy, attenuation is also of interest. These quantities will be derived by a field theory analysis for the various lines and waveguides treated here. We will begin with a general discussion of the different types of wave propagation and modes that can exist on transmission lines and waveguides. Transmission lines that consist of two or more conductors may support transverse electromagnetic (TEM) waves, characterized by the lack of longitudinal field components, TEM waves have a uniquely defined voltage, current, and characteristic impedance. Waveguides, often consisting of a single conductor, support transverse electric (TE) and/or transverse magnetic (TM) waves, characterized by the presence of longitudinal magnetic or electric, respectively, field components. As we will see in Chapter 4, a unique definition of characteristic impedance is not possible for such waves, although definitions can be chosen so that the characteristic impedance concept can be used for waveguides with meaningful results. GENERAL SOLUTIONS FOR TEM, TE, AND TM WAVES In this section we will find general solutions to Maxwell's equations for the specific cases of TEM, TE, and TM wave propagation in cylindrical transmission lines or waveguides. The geometry of an arbitrary transmission line or waveguide is shown in Figure 3.1, and is characterized by conductor boundaries that are parallel to the z-axis. These structures are assumed to be uniform in the z direction and infinitely long. The conductors will initially be assumed to be perfecdy conducting, but attenuation can be found by the perturbation method discussed in Chapter 2. 3.1 General Solutions for TEM.TE, and TM Waves 93 We assume time-harmonic fields with an eJiai dependence and wave propagation along the z-axis. The electric and magnetic fields can then be written as E(x, y, z) = [e(x, y) + zez(x, y)]e~j^, (3.1a) H(x, y, z) - [h{x, y) + zhz(x, y)]e~jfiz, (3.1b) where e(x, y) and h(x, y) represent the transverse (i, y) electric and magnetic field components, while ez and hz are the longitudinal electric and magnetic field components. In the above, the wave is propagating in the +z direction; — z propagation can be obtained by replacing 0 with — fj. Also, if conductor or dielectric loss is present, the propagation constant will be complex; jp should then be replaced with y = a + jfi. Assuming that the transmission line or waveguide region is source free, MaxwelTs equations can be written as Vx£ = -ja>nHt (3.2a) V x H = joxE. (3.2b) With an e~^z z dependence, the three components of each of the above vector equations can be reduced to the following: 1y~ + jpEy = -ja>fiHx, (3.3a) 9 E -j$Ex - —& = -/frfitiy, (3.3b) dx dEy dEx -Jq>]jlHz, (3.3c) dx dy + jpHy = ja>€EX! (3.4a) dH, -jpHx ~—±= joxEy, (3.4b) dx Sx By °AC) The above six equations can be solved for the four transverse field components in terms of Ez and Hz (for example, Hx can be derived by eliminating Ey from (3.3a) and (3.4b)) as follows: where kl€=k2- p2 (3.6) has been defined as the cutoff wavenumber; the reason for this terminology will become Chapter 3: Transmission Lines and Waveguides clear later. As in previous chapters, k = o)JJi€ = 2ji/k (3.7) is the wavenumber of the material filling the transmission line or waveguide region. If dielectric loss is present, e can be made complex by using e = £<>€r(\ — } tan&), where tan & is the loss tangent of the material. Equations (3,5a-d) are very useful general results that can be applied to a variety of waveguiding systems. We will now specialize these results to specific wave types. TEM Waves Transverse electromagnetic (TEM) waves are characterized by Ez — Hz = 0. Observe from (3.5) that if Ez — Hz = 0, then the transverse fields are also all zero, unless k2 = 0 {k2 = 02)t in which case we have an indeterminate result. Thus, we can return to (3.3)-(3.4) and apply the condition that Ez = Hz= 0. Then from (3.3a) and (3.4b), we can eliminate Hx to obtain jí2Ey — (ti2flčEy, or jň = to^jn = k, (3.8) as noted earlier. (This result can also be obtained from (3.3b) and (3.4a).) The cutoff wavenumber, kc = iihTž = 0. 3.1 General Solutions for TEM, TE, and TM Waves 95 Using the fact that V-Z) = eVr-g = 0 with (3.13) shows that *(*, y) also satisfies Laplace's equation, V24>(j:,y) = 0, (3.14) as expected from electrostatics. The voltage between two conductors can be found as Vn = *i - *2 = j E dl, (3.15) where and 4>2 represent the potential at conductors 1 and 2, respectively. The current flow on a conductor can be found from Ampere's law as ■dl, (3.16) where C is the cross-sectional contour of the conductor TEM waves can exist when two or more conductors are present. Plane waves are also examples of TEM waves, since there are no field components in the direction of propagation; in this case the transmission line conductors may be considered to be two infinitely large plates separated to infinity. The above results show that a closed conductor (such as a rectangular waveguide) cannot support TEM waves, since the corresponding static potential in such a region would be zero (or possibly a constant), leading to e = 0, The wave impedance of a TEM mode can be found as the ratio of the transverse electric and magnetic fields: ex ay, nr Ztem = tt = -r = ,/ - = >?. (3.17a) where (3.4a) was used. The other pair of transverse field components, from (3.3a), give 2™=^ = ^ = *. (3.17b) Combining the results of (3.17a) and (3.17b) gives a general expression for the transverse fields as 1 h(x, y) = —Hj x e{x, y). (3,18) Ztem Note that the wave impedance is the same as that for a plane wave in a lossless medium, as derived in Chapter 1; the reader should not confuse this impedance with the characteristic impedance, Zo, of a transmission line. The latter relates an incident voltage and current and is a function of the line geometry as well as the material filling the line, while the wave impedance relates transverse field components and is dependent only on the material constants. From (2.32), the characteristic impedance of the TEM line is Zo = V/I, where V and / are the amplitudes of the incident voltage and current waves. The procedure for analyzing a TEM fine can be summarized as follows; 1. Solve Laplace's equation, (3.14), for 4>C*,y), The solution will contain several unknown constants. 2. Find these constants by applying the boundary conditions for the known voltages on the conductors. 3. Compute e and E from (3.13), (3.1a). Compute k and H from (3.18), (3,1b). 4. Compute V from (3.15), / from (3.16). 5. The propagation constant is given by (3.8), and the characteristic impedance is given by Z0 = V/I. Chapter 3: Transmission Lines and Waveguides TE Waves Transverse electric (TE) waves, (also referred to as if-waves) are characterized by Ez — 0 and Ht $fi 0. Equations (3.5) then reduce to fix = (3.19a) k\ 5x -iß 1y ~ | 3.V 1 ■jco/jl dři. -jß dH, H, = A (3.19b) = (3.19c) In this case, ^ 0, and the propagation constant /J — ^fc2 — k2 is generally a function of frequency and the geometry of the tine or guide. To apply (3.19), one must first find Ht from the Helmholtz wave equation, / d2 a2 a3 a which, since H^(x, y, z) = ht{x, y)e~j^zt can be reduced to a two-dimensional wave equa- tion for hz: (é+é+*?)A:=0' <32,) since M = k2 — fi2. This equation must be solved subject to the boundary conditions of the specific guide geometry. The TE wave impedance can be found as z^-wy-iir-T-T (3,22) which is seen to be frequency dependent. TE waves can be supported inside closed conductors, as well as between two or more conductors. TM Waves Transverse magnetic (TM) waves (also referred to as £-waves) are characterized by Ez ^ 0 and Hz = 0. Equations (3.5) then reduce to j(t>€ dE? Hx = J——±, (3.23a) Ul dx ' k2 dy • —J0J€ dE, Hy = ——^, (3.23b) -jßdEz mq&A E.x = —p;—r—, (3.23c) -jß dE. 3.1 General Solutions for TEM, TE, and TM Waves 97 As in the TE case, kc ^ 0, and the propagation constant £ = yfk2 — k2 is a function of frequency and the geometry of the line or guide. Ez is found from the Helmholtz wave equation, / a3 d2 a2 A which, since E^(x, y, z) = ez{x, y)e~^z, can be reduced to a two-dimensional wave equa- tion for ez: ( 32 92 A {w+d?+k;h=0' (3-25) since k2 = k2 — B2. This equation must be solved subject to the boundary conditions of the specific guide geometry. The TM wave impedance can be found a* Z™ = £^ = i- = ^, (3.26, Hy Hx ok k which is frequency dependent. As for TE waves, TM waves can be supported inside closed conductors, as well as between two or more conductors. The procedure for analyzing TE and TM waveguides can be summarized as follows: 1. Solve the reduced Helmholtz equation, (3.21) or (3.25), for hz or ez. The solution will contain several unknown constants, and the unknown cutoff wavenumber, kc. 1. Use (3.19) or (3.23) to find the transverse fields from h, or ez. 3. Apply the boundary conditions to the appropriate field components to find the unknown constants and kc> 4. The propagation constant is given by (3,6), and the wave impedance by (3.22) or (3.26). Attenuation Due to Dielectric Loss Attenuation in a ti^stnission line or waveguide can be caused by either dielectric loss or conductor loss. If Vo^oer is the (real) wavenumber in the absence of loss. Equation (3.28) shows that when the loss is small the phase constant, fi, is unchanged, while the attenuation constant due to dielectric loss is given by k2 taná ad = ——— Np/m (TE or TM waves). (3.29) 2p This result applies to any TE or TM wave, as long as the guide is completely filled with the dielectric. It can also be used for TEM lines, where kc = 0, by letting = k: k tan 5 ctj = —— Np/m (TEM waves). (3.30) PARALLEL PLATE WAVEGUIDE The parallel plate waveguide is probably the simplest type of guide that can support TM and TE modes; it can also support a TEM mode, since it is formed from two flat plates, or strips, as shown in Figure 3.2. Although an idealization, this guide is also important for practical reasons, since its operation is quite similar to that of a variety of other waveguides, and models the propagation of higher order modes in stripline. In the geometry of the parallel plate waveguide in Figure 3.2, the strip width W is assumed to be much greater than the separation, d, so that fringing fields and any x variation vj FIGURE 3.2 Geometry of a parallel plate waveguide. 3.2 Parallel Plate Waveguide 99 can be ignored. A material with permittivity € and permeability fi is assumed to fill the region between the two plates. We will discuss solutions for TEM, TM, and TE waves. TEM Modes As discussed in Section 3.1, the TEM mode solution can be obtained by solving Laplace's equation, (314), for the electrostatic potential <&(x, y) between the two plates. Thus> V?(x, y) = 0. forO < x < VV. 0 < y < d. (3.31) If we assume that the bottom plate is at ground (zero) potential and the top plate at a potential of Vot then the boundary conditions for (jc, y) is 4>0c,y) = A + By, and the constants A, B can be evaluated from the boundary conditions of (3.32) to give the final solution as y) = V0y/d. (3.33) The transverse electric field is, from (3.13), B& y) = -V,*(x, y) = -A (3.34) d so that the total electric field is E{x, y, z) = e(x, y)e~^ = -yj^jkz, 0-35) where k = ta^fjn is the propagation constant of the TEM wave, as in (3,8). The magnetic field, from (3.18), is R{x, y, z)=-zx fflh y, z) = x%~jk\ (3.36) « rid where n = vW^ *s the intrinsic impedance of the medium between the parallel plates. Note that Ez = Hz = 0 and that the fields are similar in form to a plane wave in a homogeneous region. The voltage of the top plate with respect to the bottom plate can be calculated from (3.15) and (3.35) as V = - / Eydy = V<,e~Jk\ (3.37) />-=o as expected. The total current on the top plate can be found from Ampere's law or the 100 Chapter 3: Transmission Lines and Waveguides surface current density: I = Jtzdx= (-yxH)- zdx = / Hxdx = -^-e~}kz (3.38) Thus the characteristic impedance can be found as Z0 = - = 2-, (3.39) / w which is seen to be a constant dependent only on the geometry and material parameters of the guide. The phase velocity is also a constant: oj 1 Vp = s = (3-40) which is the speed of hght in the material medium. Attenuation due to dielectric loss is given by (3.30). The formula for conductor attenuation will be derived in the next subsection, as a special case of TM mode attenuation. TM Modes As discussed in Section 3.1, TM waves are characterized by Hz = 0 and a nonzero Ez field that satisfies the reduced wave equation of (3.25), with d/dx — 0: ^2+k^jez(x,y) = Q7 (3.41) where k2 = k1 — 01 is the cutoff wavenumber, and Ez{x, y, z) = ez(x, y)e~~&z. The general solution to (3.41) is of the form y) = A sin kcy + B cos kcy, (3.42) subject to the boundary conditions that ez(x, y) = 0, at y = 0, (3.43) This implies that 5=0 and kcd = n7r, fern = 0,1,2,3..., or kc=t^-i «=0,1,2,3.... (3.44) a Thus the cutoff wavenumber kc is constrained to discrete values as given by (3.44); this implies that the propagation constant $ is given by ß = - Jfc2 = ^jt2 - (/itt/íi)2, (3.45) The solution for e^Ot, y) is then thus, tiny ez{x, y) = A„ sin —(3.46) Ez(x, y, z) = An sin £§j£H&> (3.47) d 3.2 Parallel Plate Waveguide 101 The transverse field components can be found, using (3.23), to be Hx = JJHAn cos nJHe-^, (3.48a) kc d Ev = ^A„ cos (3.48b) Ex = Hy = 0. (3.48c) Observe that for n = 0, ft — k — o)^/JJL€, and that Ez = 0. The Ey and Hx fields are then constant in y, so that the TMo mode is actually identical to the TEM mode. For n > 1, however, the situation is different. Each value of n corresponds to a different TM mode, denoted as the TM„ mode, and each mode has its own propagation constant given by (3.45), and field expressions as given by (3.48). From (3.45) it can be seen that 0 is real only when k > kc. Since k = a>^/jl€ is proportional to frequency, the TM„ modes (for n > 0) exhibit a cutoff phenomenon, whereby no propagation will occur until the frequency is such that k > kc. The cutoff frequency of the TM,. mode can then be deduced as ^ = ^W = m7=- (3A9) Thus, the TM mode that propagates at the lowest frequency is the TMj mode, with a cutoff frequency of fc = l/2d^/JU\ the TM2 mode has a cutoff frequency equal to twice this value, and so on. At frequencies below the cutoff frequency of a given mode, the propagation constant is purely imaginary, corresponding to a rapid exponential decay of the fields. Such modes are referred to as cutoff, or evanescent, modes. TM„ mode propagation is analogous to a high-pass filter response. The wave impedance of the TM modes, from (3,26), is a function of frequency: z™ = z£i = £ = (3.50) Hx oj€ k which we see is pure real for / > /c, but pure imaginary for / < fc. The phase velocity is also a function of frequency: vP = ^, (3.51) P and is seen to be greater than 1 i^fpTe = w/k, the speed of light in the medium, since ft X — 27?/k, the wavelength of a plane wave in the material. The phase velocity and guide wavelength are defined only for a propagating mode, for which 0 is real. One may also define a cutoff wavelength for the TM„ mode as 2d X, = —. (3.53) n 102 Chapter 3: Transmission Lines and Waveguides It is instructive to compute the Poynting vector to see how power propagates in the TM„ mode. From (1.91), the time-average power passing a transverse cross section of the parallel plate guide is Pp=-Rej / Ě x H* ■ zdy dx =--Re / EyH* dy dx wRe(j})oj€ 2 fd a nity \A„\2 [ cos- —— dy = 2k2 Jy={) d wRt(0)oj€d 2 -—-\Ati\ torn > 0 -||-\AA2 for« = 0 (3.54) where (3.48a,b) were used for Eyi Hx* Thus, P0 is positive and nonzero when £ is real, which occurs for / > fc. When the mode is below cutoff, $ is imaginary and so Po = Q- The TM (or TE) waveguide mode propagation has an interesting interpretation when viewed as a pair of bouncing plane waves. For example, consider the dominant TMi mode, which has a propagation constant, h * y/k2 - (Ti/df, (3.55) and Ez field, which can be rewritten as Ez = A\ sin Sr^S d 2j This result is in the form of two plane waves traveling obliquely, in the —y, +z and +y, +z directions, respectively, as shown in Figure 3.3. By comparison with the phase factor of (1.132), the angle 0 that each plane wave makes with the z-axis satisfies the relations ksmO = 4 (3,57a) d kcosO-^i, (3.57b) so that {n/dy2 + j62 = k2t as in (3.55). For / > fc, is real and less than £i, so 6 is some angle between 0C and 90°, and the mode can be thought of as two plane waves alternately bouncing off of the top and bottom plates. 0 z FIGURE 3.3 Bouncing plane wave interpretation of the TMi parallel plate waveguide mode. 3.2 Parallel Plate Waveguide 103 The phase velocity of each plane wave along its direction of propagation (0 direction) is iofk = 1 /^/W> which is the speed of light in the material filling the guide. But the phase velocity of the plane waves in the z direction is (o/0\ = l/.v//I?cos#, which is greater than the speed of light in the material. (This situation is analogous to ocean waves hitting a shoreline: the intersection point of the shore and an obliquely incident wave crest moves faster than the wave crest itself.) The superposition of the two plane wave fields is such that complete cancellation occurs at y = 0 and y = dt to satisfy the boundary condition that Ez — 0 at these planes. As / decreases to fc, fi\ approaches zero so that, by (3.57b), 9 approaches 90°. The two plane waves are then bouncing up and down with no motion in the +z direction, and no real power flow occurs in the z direction. Attenuation due to dielectric loss can be found from (3.29). Conductor loss can be treated using the perturbation method. Thus, <3-58) where Pv is the power flow down the guide in the absence of conductor loss, as given by (3.54). Pt is the power dissipated per unit length in the two lossy conductors and can be found from (2.97) as /RAT <* a to2€2R,w t where Rs is the surface resistivity of the conductors. Using (3.54) and (3.59) in (3.58) gives the attenuation due to conductor loss as 2a)€Rs 2k Rs <*c = ——- = —-f Np/m, for n > 0. (3.60) pa pr}d As discussed previously, the TEM mode is identical to the TMo mode for the parallel plate waveguide, so the above attenuation results for the TM„ mode can be used to obtain the TEM mode attenuation by letting n = 0. For this case, the n = 0 result of (3.54) must be used in (3.58), to obtain a( = & Np/m. (3.61) nd TE Modes TE modes, characterized by Ez = 0, can also propagate on the parallel plate waveguide. From (3.21), with dfdx = 0, must satisfy the reduced wave equation, (^+^)fcz(*,y) = 0, (3'62) where k2 = k2 — 01 is the cutoff wavenumber and Hz(xt y, z) = hz{x, y)e~^z. The general solution to (3.62) is hz{x, y) = A sini^y -f B cos kcy. (3.63) The boundary conditions are that Ex = 0 at y = 0, d; Ez is identically zero for TE modes. From (3.19c), we have Ex = ^y^- [A coskcy - B sin kcy] e~m, (3.64) 104 Chapter 3: Transmission Lines and Waveguides and applying the boundary conditions shows that A = 0 and tin kc = —, « = 1,2,3..., (3.65) a as for the TM case. The final solution for H. is then Hz(x, y) = BH cos S^T**'. (3.66) a The transverse fields can be computed from (3.19) as Ex = J—^Bn sin —fe-v\ (3,67a) Hy = J^B^mn-^e-^% (3.67b) Kc d Ey = Hx = 0. (3.67c) The propagation constant of the TE„ mode is thus, which is the same as the propagation constant of the TM„ mode. The cutoff frequency of the TE„ mode is fc = TT^r (3-69) which is also identical to that of the TM„ mode. The wave impedance of the TE„ mode is, from (3,22), _ Es _ m _ kn which is seen to be real for propagating modes and imaginary for uonpropagating, or cutoff, modes. The phase velocity, guide wavelength, and cutoff wavelength are similar to the results for the TM modes. The power flow down the guide for a TE„ mode can be calculated as P0 = -Re/ / Ex H* zdy dx = -Re/ / EXH; dy dx 2 Jx=b Jy=0 2 JX=Q Jy=Q = -^-|BJ2Re^), for«>0, (3.71) which is zero if the operating frequency is below the cutoff frequency (fi imaginary). Note that if n = 0, then Ex = Hy = 0 from (3.67), and thus P0 = 0, implying that there is no TEo mode. Attenuation can be calculated in the same way as for the TM modes. The attenuation due to dielectric loss is given by (3.29). It is left as a problem to show that the attenuation due to conductor loss for TE modes is given by 2k2cRs 2k2 Rs 3.2 Parallel Plate Waveguide 105 «tVd Q v v ___TMr - ' TEM -t v cuioff ^ 1 1 1 TEj r—i—-i—- ■ 01 2.1 456789 10 k_ = kd kc it FIGURE 3.4 Attenuation due to conductor loss for the TEM, TMj, and TE) modes of a parallel plate waveguide, Figure 3.4 shows the attenuation due to conductor loss for the TEM, TM], and TE( modes. Observe that ctc -+ oo as cutoff is approached for the TM and TE modes. Table 3.1 summarizes a number of useful results for TEM, TM, and TE mode propagation on parallel plate waveguides. Field lines for the TEM, TMi, and TE| modes are shown in Figure 3,5. TABLE 3.1 Summary of Results for Parallel Plate Waveguide Quantity TEM Mode TM„ Mode TE„ Mode k »Vi" 0 Ml j d nn/d ß oo In IK - 2d i n 2.71 j kf = 2d f n k iTT/k 2nfß 2x/ß to/k = XlyfJTž co/ß ü)/ß m (k tanS)/2 (k2 tanS)/2ß (k2taü8)/2ß ac IkRJßtid 2k]RJkßr\d E, 0 0 H- 0 0 BK cos (u7ty/d)e~m Ex 0 0 {j(oß/kc)Bn sin (nny/d)e~iß-' Ey {-VJd)e-^ (-jß/k<)Ans(nxy/d)e-jf>t 0 Ht (V0/j}d)e-JK Uax/kt)An cos (nny/d)e~-'^ 0 Hv 0 0 (Jß/kc)Bnua(n7zy/d)e-^ Z Ztem = qd/w Z™ = ßrj/k Zte = kn/ß 106 Chapter 3: Transmission Lines and Waveguides 3.3 (a) t u u u t h u n, i i 1 r i i l i t i i i i i FIGURE 3.5 Field lines for the (a) TEM, (b)TMi, and (c) TE, modes of a parallel plate waveguide. There is no variation across the width of the waveguide. RECTANGULAR WAVEGUIDE Rectangular waveguides were one of the earliest types of transmission lines used to transport microwave signals and are still used today for many applications. A large variety of components such as couplers, detectors, isolators, attenuators, and slotted lines are commercially available for various standard waveguide bands from 1 GHz to over 220 GHz. Figure 3.6 shows some of the standard rectangular waveguide components that are available. Because of the recent trend toward miniaturization and integration, a lot of microwave circuitry is currently fabricated using planar transmission lines, such as microstrip and stripline, rather than waveguide. There is, however, still a need for waveguides in many applications such as high-power systems, niillimeter wave systems, and in some precision test applications. The hollow rectangular waveguide can propagate TM and TE modes, but not TEM waves, since only one conductor is present. We will see that the TM and TE modes of a rectangular waveguide have cutoff frequencies below which propagation is not possible, similar to the TM and TE modes of the parallel plate guide. TE Modes The geometry of a rectangular waveguide is shown in Figure 3.7, where it is assumed that the guide is filled with a material of permittivity € and permeability t*~ It is standard convention to have the longest side of the waveguide along the *-axis, so that a > b. The TE modes are characterized by fields with Ez — 0, while Hz must satisfy the reduced wave equation of (3.21): / a2 d2 -A with Hz(x, y,z) = hz(x, y)e~^z, and fc2 = k2 - 01 is the cutoff wavenumber, The partial differential equation of (3.73) can be solved by the method of separation of variables by letting hz(x, y) = X(x)Y(y), (3.74) and substituting into (3.73) to obtain 1 d2X 1 d2Y X dx2 Y dy2 3.3 Rectangular Waveguide 107 FIGURE 3,6 Photograph of Ka-band (WR-28) rectangulai waveguide components. Clockwise from top: a variable attenuator, an E-H (magic) tee junction, a directional coupler, an adaptor to ridge waveguide, an E-plane swept bend, an adjustable short, and a sliding matched load. Courtesy of Agilent Technologies, Santa Rosa, Calif, Then, by the usual separation of variables argument, each of the terms in (3.75) must be equal to a constant, so we define separation constants kx and ky, such that i+k2:X = Q, (3.76a) dx2 ^+k2yY = 0, (3.76b) and k2 + k2 = k2, (3.77) 108 Chapter 3: Transmission Lines and Waveguides The general solution for hz can then be written aa hz(x, y) = {Acoskxx + B sin kxx)(C coskyy + Dsinkyy). (3.78) To evaluate the constants in (3.78) we must apply the boundary conditions on the electric field components tangential to the waveguide walls. That is, ex(x, y)-0. aty= 0, b. (3.79a) ey(x, y) = 0, at x = 0, a. (3.79b) We thus cannot use hz of (3.78) directiy, but must first use (3.19c) and (3.19d) to find ex and ey from hz: ex = ——ky (A cos kxx + B sin kx x)(- C $mkyy + D cos kyy), (3.80a) ey = ^^kxi—A sinkxx + B cos kxx)(C cos kyy + Dsmkyy). (3.80b) Then from (3.79a) and (3.80a), we see that D = 0, and ky = nixjb for n — 0,1,2.... From (3.79b) and (3.80b) we have that B = 0 and kx - mnja for m - 0, 1, 2.... The final solution for Hz is then mux mzy #,(*, y, z) = cos-cos —^e~jftz. (3.81) a b where Am„ is an arbitrary amplitude constant composed of the remaining constants A and C of (3.78). The transverse field components of the TEm„ mode can be found using (3.19) and (3.81): % = —rn-A™ cos-sin ~-€~^, (3.82a) Kfb a b Ev = —^-Am„ sm —— cos ——e "% (3.82b) Hx = k}a a b jBmTT „ mnx nny iR, - „^ , J-^— Amn sin-cos ~e m. (3.82c) kta a b Hv = ^r-Amn cos-sin ——e'^. (3.82d) k-b a b The propagation constant is ,_^_y*_(^)'_(£)*. which is seen to be real, corresponding to a propagating mode, when **Wr?y +(t)- Each mode (combination of m and n) thus has a cutoff frequency fCmn given by 3h3 Rectangular Waveguide 109 The mode with the lowest cutoff frequency is called the dominant mode; since we have assumed a > b, the lowest fc occurs for the TEjo (m — 1, n = 0) mode: Thus the TE(0 mode is the dominant TE mode and, as we will see, the overall dominant mode of the rectangular waveguide. Observe that the field expressions for E and H in (3,82) are all zero if bom m =. n = 0; thus there is no TEoo mode. At a given operating frequency f, only those modes having fc / will lead to an imaginary j8 (or real a), meaning that all field components will decay exponentially away from the source of excitation. Such modes are referred to as cutoff, or evanescent, modes. If more than one mode is propagating, the waveguide is said to be overmoded. From (3.22) the wave impedance that relates the transverse electric and magnetic fields is ZjE = ^ = ^ = ^ (3.86) where rj = vW* is the intrinsic impedance of the material filling the waveguide. Note that Zte is real when fi is real (a propagating mode), but is imaginary when $ is imaginary (an evanescent mode). The guide wavelength is defined as the distance between two equal phase planes along the waveguide, and is equal to 2tf In ~J > T % = — > — = h (3.87) which is thus greater than k, the wavelength of a plane wave in the filling medium. The phase velocity is a> to , , ,_ Up = — > — = 1/v7^ (3.88) p K which is greater than 1 the speed of light (plane wave) in the filling material. In the vast majority of applications the operating frequency and guide dimensions are chosen so that only the dominant TEio mode will propagate. Because of the practical importance of the TEjo mode, we will list the field components and derive the attenuation due to conductor loss for this case. Specializing (3.81) and (3.82) to the m = 1, n = 0 case gives the following results for the TEin mode fields: Hz = A jo cos —e-JP\ (3.89a) a Ey = zlW±A^—e-*\ (3.89b) it a Hx = ^A,0sin™«-'* (3.89c) n a Ex = Ez = Hy = 0. (3.89d) Chapter 3: Transmission Lines and Waveguides In addition, for the TEjq mode, kc = 7t/a, (3.90) and B = yfk2 ~ in I a)2, (3.91) The power flow down the gnide for the TEfo mode is calculated as Pw = ]-Rtf f ExH*-zdydx 2 Jx=(j Jy=0 = f EyH*xdydx f Cb 7ZX Re(^)lA10l2 / / sinJ —dydx Re(j8). (3.92) lit2 cofjia^\A\Q\2b 4x2 Note that this result gives nonzero real power only when £ is real, corresponding to a propagating mode. Attenuation in a rectangular waveguide can occur because of dielectric loss or conductor loss. Dielectric loss can be treated by making € complex and using a Taylor series approximation, with the general result given in (3.29). Conductor loss is best treated using the perturbation method. The power lost per unit length due to finite wall conductivity is, from (1.131), (3.93) where Rs is the wall surface resistance, and the integration contour C encloses the perimeter of the guide walls. There are surface currents on all four walls, but from symmetry the currents on the top and bottom walls are identical, as are the currents on the left and right side walls. So we can compute the power lost in the walls at x = 0 and y — 0 and double their sum to obtain the total power loss. The surface current on the x =0 (left) wall is J, = nxH\x=i)=xx zHz\x^ = -yHz= -yA0^, (3.94a) while the surface current on the y = 0 (bottom) wall is Js-hx H\y=0 = % x (xHx\y=(j + ZHt\y=G) = -zJ—A]0sm—e-^ +JfA10cos— e~^. (3.94b) tt a a Substituting (3.94) into (3.93) gives Pe = rsf 1&, Uy + rJ [I in I2 + I ^rl2] dx Jy=0 J 1=0 = rs\Aio\2{b+j + ^y (3.95) 3.3 Rectangular Waveguide 111 The attenuation due to conductor loss for the TEio mode is then Pi 2jr2Rs{b + a/2 + P2a3/27T2) a.. — 2^10 (jOixa}bp Rs ilbn2 + a3k2) Np/m. (3,96) TM Modes The TM modes are characterized by fields with Hz = 0, while Ez must satisfy the reduced wave equation of (3.25): / a2 d2 A W + Iy-2+VeziX'y) = 0' °'91) with Ez(x, y,z) = ez{xy y)e~^z and k2 = k2 — Bl. Equation (3.97) can be solved by the separation of variables procedure that was used for the TE modes. The general solution is (hen *z(xi y) = (A coskxx + B smkxx)(C coskyy + D sin kyy). (3.98) The boundary conditions can be applied directly to et: ez(x, y) = 0, at* = 0, a, (3.99a) ez(x,y)-0, aty=0,fc. (3.99b) We will see that satisfaction of the above conditions on ez will lead to satisfaction of the boundary conditions by ex and ey. Applying (3.99a) to (3.98) shows that A = 0 and kx = mx/a, for m = 1, 2, 3.... Similarly, applying (3,99b) to (3.98) shows that C = 0 and ky = n7i/b. for n = 1,2.3.... The solution for Ez then reduces to EJx, y, z) = Bmn sin — sin ^-e~^\ (3.100) a b where Bmn is an arbitrary amplitude constant. The transverse field components for the TMm„ mode can be computed from (3.23) and (3.100) as -jBmn rrmx mry .„ Ex--T^—Bmn cos-sin —-e ipz, (3.101a) akt a b Ey - -■■ Bmn sin-cos ——e ;p\ (3.101b) bkj a b iaitnn „ . mizx mry , ~ tlr>—sm-cos-^■ (3.101c) bkf a b Hy = Bmncos-sm -re~^z. (3.10Id) akj a b As for the TE modes, the propagation constant is 112 Chapter 3: Transmission Lines and Waveguides Frequency (GHz) FIGURE 3,8 Attenuation of various modes in a rectangular brass waveguide with a = 2.0 cm. and is real for propagating modes, and imaginary for evanescent modes. The cutoff frequency for the TMm„ modes is also the same as that of the TEmrt modes, as given in (3.84). The guide wavelength and phase velocity for TM modes are also the same as those for TE modes. Observe that the field expressions for E and H in (3.101) are identically zero if either m or n is zero. Thus there are no TMm, TMoi. or TM|q modes, and the lowest order TM mode to propagate (lowest fc) is the TM| i mode, having a cutoff frequency of which is seen to be larger than fCl0 for the cutoff frequency of the TEio mode. The wave impedance relating the transverse electric and magnetic fields is, from (3.26), Ex -Ey fin Attenuation due to dielectric loss is computed in the same way as for the TE modes, with the same result. The calculation of attenuation due to conductor loss is left as a problem; Figure 3.8 shows the attenuation versus frequency for some TE and TM modes in a rectangular waveguide. Table 3.2 summarizes results for TE and TM wave propagation in rectangular waveguides, and Figure 3.9 shows the field lines for several of the lowest order TE and TM modes. EXAMPLE 3.1 CHARACTERISTICS OF A RECTANGULAR WAVEGUIDE Consider a length of Teflon-filled copper A"-band rectangular waveguide, having dimensions a — 1.07 cm and b = 0,43 cm. Find the cutoff frequencies of the first five propagating modes. If the operating frequency is 15 GHz, find the attenuation due to dielectric and conductor losses. 3.3 Rectangular Waveguide 113 TABLE 3.2 Summary of Results for Rectangular Waveguide Quantity TEm„ Mode TUmn Mode yj(mn/a)2 + (nn/b)* T % Altana 2ß . mnx , nny Sl>7 B„,n sm-sin -rftr»1 a b k ß J(mn/a)2 + {nnfb)2 k ß co ß k2tanS ~W 0 Affl„ cos-cos —— e Jpz jcüßnn , m;rx . nny — jßmn „ mjr.r n;ry —£r-Amn cos ■—- sin -rf-e^1 —3-fimn cos-sin a b klb -jüffimn k2a k}a jßmn , mnx nny ik, Am„ sm-cos —— e~JP>- a b nny ifS, —jßnn „ . mnx wry C jßnn . mitx A,xn sin-cos b a mnx nny _m, -cos---€ }p~ a b mnx . nny Amncos-sin-^e jpz k2b a b Zte = 7 k2b jcoenn _ Bmnsm k;b —jüxmn _ mnx nny -~-Bmn cos--sm -r^e-'ßz kfa a b z -ßT} Solution From Appendix G, for Teflon, €r = 2.08 and tan 5 = 0.0004. From (3.84) the cutoff frequencies are given by Computing fc for the first few values of m and n gives: Mode m n /■(GHz) TE 1 0 9.72 TE 2 0 19.44 TE 0 1 24.19 TE, TM 1 1 26.07 TE, TM 2 1 31.03 .12 3 2 3 2 FIGURE 3.9 Field lines for some of the lower order modes of a rectangular waveguide. Reprinted with permission from Fields and Waves it) Communication Electjnnks, S. Ramo, J.R. Whinnery, and T. Van Duzer. Copyright © 1965 by John Wiley &. Sons, Inc. Table 8.02. 3.3 Rectangular Waveguide 115 Thus the TEiot TE2o, TEoi, TEn, and TMn modes will be the first five modes to propagate. At 15 GHz, k = 345.1 nT1, and the propagation constant for the TEio mode is \ (^)-(^-vR!) 2 = 345.1 m_t. From (3.29), the attenuation due to dielectric loss is ad = - t3nS = 0.119 np/m = 1.03 dB/m. The surface resistivity of the copper walls is (a = 5.8 x 107 S/m) R, = ^ = 0.032 n, and the attenuation due to conductor loss, from (3.96), is <*ť = {Ibn1 + a^k1) = 0.050 np/m = 0.434 dB/m. ďbpky] ■ TE,„o Modes of a Partially Loaded Waveguide The above results also apply for a rectangular waveguide filled with a homogeneous dielectric or magnetic material, but in many cases of practical interest (such as impedance matching or phase-shifting sections) a waveguide is used with only a partial filling. Then an additional set of boundary conditions are introduced at the material interface, necessitating a new analysis. To illustrate the technique we will consider the lEmo modes of a rectangular waveguide that is partially loaded with a dielectric slab, as shown in Figure 3.10. The analysis still follows the basic procedure outlined at the end of Section 3.1. Since the geometry is uniform in the y direction and n = 0, the TEfflo modes have no y dependence. Then the wave equation of (3.21) for hz can be written separately for the dielectric and air regions as (it* + k^)hl = °' f°r ° ~ * ~ (3.105a) (£y-f^^ =0, forr<*, z) = hz(pt 4>)e~&z, (3.111) can be expressed in cylindrical coordinates as Again, a solution can be derived using the method of separation of variables. Thus, we let hz(p, 0) = R(p)P(), (3.113) and substitute into (3.112) to obtain 1 d2R 1 dR 1 d2P ,2 _ „ A R dp1 pR dp p2P d), while the right side depends only on 0. Thus, each side must be equal to a constant, which we will call &£, Then, -ld2P P dip2 P d^ -k2 or ^r+klP=0. (3.115) Also, o2 dp1 dp "2S+^ + (A2-^)»=0. (3.116) 3.4 Circular Waveguide 119 The general solution to (3.115) is P(4>) = Asinfc^ + Bcosk^. (3.117) Since the solution to hz must be periodic in 4> (that is, hz(p,4>) = $ ±2m;r)), ^ must he an integer, n. Thus (3.117) becomes f(^) = Asin/i0 + flcosn0T (3.118) while (3.116) becomes which is recognized as Bessel's differentia] equation. The solution is R(p) = CJn(krp) + DYn(krp), (3.120) where Jn(x) and Yn(x) are the Bessel functions of first and second kinds, respectively. Since Y„(kcp) becomes infinite at p = 0, this term is physically unacceptable for the circular waveguide problem, so that D = 0. The solution for hz can then be written as hz{p, ,z)= ^p(Asinw^ + Bcosn)j;[(kcp)e-Jffz, (3.123) where the notation J'„(kcp) refers to the derivative of Jn with respect to its argument. For E$ to vanish at p = a, we must have J^kect) = G. (3.124) If the roots of J'n(x) are defined as p'nm, so that J'„(p'„m) = 0, where p'nm is the /nth root of J„, then £c must have the value Values of p'nm are given in mathematical tables; the first few values are listed in Table 3.3, The TE„m modes are thus denned by the cutoff wavenumber, kCnm = p'^ja, where n refers to the number of circumferential (4>) variations, and m refers to the number of radial TABLE 33 Values of p'^ for TE Modes of a Circular Waveguide t) 4i P„2 P'„3 0 3.832 7.016 10.174 1 1.841 5.331 8.536 2 3.054 6.706 9.970 120 Chapter 3: Transmission Lines and Waveguides (p) variations. The propagation constant of the TE„„, mode is ßnm = y*a - ft2 = ^ - (3-126) with a cutoff frequency of fc = ' = Pntn . (3.127) The first TE mode to propagate is the mode with the smallest p'ntn, which from Table 3.3 is seen to be the TEi i mode. This mode is then the dominant circular waveguide mode, and the one most frequently used. Because m > 1, there is no TELn mode, but there is a TEnt mode. The transverse field components are, from (3.110) and (3.121), Ep = zJj^l (A cos«0 - B smnz. (3,12Sd) kf.p The wave impedance is 11$ tip p In the above solutions there are two remaining arbitrary amplitude constants, A and B. These constants control the amplitude of the sin/10 and cos ntp terms, which are independent. That is, because of the azimuthal symmetry of the circular waveguide, both the sin n and cos/10 terms are valid solutions, and can be present in a specific problem to any degree. The actual amplitudes of these terms will be dependent on the excitation of the waveguide. From a different viewpoint, the coordinate system can be rotated about the z-axis to obtain an hi with either A = 0 or B = 0. Now consider the dominant TEi i mode with an excitation such that B = 0. The fields can be written as Hz = Asinh (kcp)e-J'ß\ (3.130b) Ea = J^A sin 4>r{{kcp)e-tf\ (3.130c) ** Hp = —r-A sin J[(kcp)e-jß\ (3.130d) Hj, = A cos tf>7E (krp)e-jßl, (3.130e) kjp Ez=0. (3.1300 3.4 Circular Waveguide 121 The power flow down the guide can be computed as r^ = -Re/ / E x H* - Ip dip dp =^ r r [eph; - EtHfipd^dp a>ti\A\2Re(B) m & 2k* c / f \\ cos2 $4(h& + H ^ $f?tkf*)] P d ^ Jf>=oJ=o LP J 7l(t>ll\A\ =-^-(Pii - 1) Iw* (3.131) which is seen to be nonzero only when 8 is real, corresponding to a propagating mode. (The required integral for this result is given in Appendix C.) Attenuation due to dielectric loss is given by (3,29). The attenuation due to a lossy waveguide conductor can be found by computing the power loss per unit length of guide: 2 L Ik \Js\2ad r/> = 0 2jt = ^/*[|«*|2 + |HIl2]o^ ?7 [ k^a2 C°s2 * + Sin2 * J = -L^(1 + j^2J-/.^ (3-132) The attenuation constant is then ** 2Pe vkMpfi - i) akn& \ pg - I} TM Modes For the TM modes of the circular waveguide, we must solve for ez from the wave equation in cylindrical coordinates: where Ez(p, $t z) = ez(p, 4})e~^zt and k2 = k2 — 62. Since this equation is identical to (3.107), the general solutions are the same. Thus, from (3.121), ez(p,4>) = (A sin ncp + B cos nip) J^p). (3.135) 122 Chapter 3: Transmission Lines and Waveguides TABLE 3.4 Values of pam for TM Modes of a Circular Waveguide 1 Pnl Pn2 P„i 0 2.405 5.520 8.654 1 3.832 7.016 10.174 2 5.135 8.417 11.620 The difference between the TE solution and the present solution is that the boundary conditions can now be applied directly to ez of (3.135), since Ez(p, 1, so there is no TMjo mode. From (3.110), the transverse fields can be derived as Ep = ^^-(A sinn4> + B cosntb)Jj^j^n^ (3.141a) ft t -^-(Acosn - B smn&Mk^e'^ (3.141b) Kc p % = Í^-(A cosn

)Jn(kcp)e~m, (3.141c) k-p = ^^(Asinm^ + B cos n(j})J^kcp)e~m. (3.14ld) The wave impedance is Z™=|^ = ^ = f. (3.142) Calculation of the attenuation for TM modes is left as a problem. Figure 3.12 shows the attenuation due to conductor loss versus frequency for various modes of a circular waveguide. Observe that the attenuation of the TEoi mode decreases to a very small value with increasing frequency, This property makes the TEoi mode of interest for low-loss transmission over long distances. Unfortunately, this mode is not the dominant mode of the circular 3.4 Circular Waveguide 123 0.07 0.06 0.05 0.04 I pa §1 0.03 0.02 0.01 0 1 3 5 7 9 11 13 15 Frequency (GHz) FIGURE 3.12 Attenuation of various modes in a circular copper waveguide with a = 2.54 cm. waveguide, so in practice power can be lost from the TEoi mode to lower-order propagating modes. Figure 3.13 shows the relative cutoff frequencies of the TE and TM modes, and Table 3.5 summarizes results for wave propagation in circular waveguide. Field lines for some of the lowest order TE and TM modes are shown in Figure 3.14. EXAMPLE 3.2 CHARACTERISTICS OF A CIRCULAR WAVEGUIDE Find the cutoff frequencies of the first two propagating modes of a Teflon-filled circular waveguide with a = 0.5 cm. If the interior of the guide is gold plated, calculate the overall loss in dB for a 30 cm length operating at 14 GHz. Solution From Figure 3.13, the first two propagating modes of a circular waveguide are the TEj i and TMoi modes. The cutoff frequencies can be found using (3.127) and (3.140): TEu: fc TMoi: £ --gjl^--1841(3 X ^ = 12.19 GHz, InaJTr 2tt (0.005) ^fTM = J^c = 2.405(3 x.0^ = 1592 GHz TE, TE2, TEq, TE3]TE4iTE 12 J_I fc fdJE, ™01 1 f TM|| TM2i TMa; FIGURE 3.13 Cutoff frequencies of the first few TE and TM modes of a circular waveguide, relative to the cutoff frequency of the dominant TE| | mode. 124 Chapter 3: Transmission Lines and Waveguides TABLE 3.5 Summary of Results for Circular Waveguide Quantity TE„m Mode TM„r„ Mode k a>^/p* kc Pnm a a * siv - 4 2jt K In K In 0 at J Jfc^tanS 2B *2tan)Jtt{kcp}e~ifc 0 EP ~r^( A cos n - 8 sin n)JAk + B<^n4>)J',l{k\p)e-i^ ——(Acos«# — B sin n)Jn(kcp)e kjp Hp -~ (A sin n + B cos ntp)J^(kcp)e~j0z -—;—(Aco$n# - #$inn^)J',,(fct.p)e ^^{Aco&tup - Buan4>)Ja{kcp)e^i^ k-p ^€ (A sinw^ -J- ff cosn^)J^(kcp)e z - ky ZjE~J ''i'm — — So only the TEn mode is propagating at 14 GHz. The wavenurober is , InfJTr 2;r(14x lOWfOS k =- - =--——z-= 422.9 m V c 3 x 10s and the propagation constant of the TEn mode is The attenuation due to dielectric loss is calculated from (3,29) as £2tan3 (422.9)^(0.0004) . - , ad = —— = -—- ' ' £—- =0.172 np/m - 1.49 dB/m. 2p 2(208.0) p The conductivity of gold is a =4.1 x 107 S/m, so the surface resistance is Rs = j^= 0.0367 ft. FIGURE 3.14 Field lines for some of the lower order modes of a circular waveguide. Reprinted with permission from Fields and Waves in Communication Electronics, S. Ramo, J.R. Whinnery, and T. Van Duzer. Copyright © 1965 by John Wiley & Sons, Inc. Table 8.04. 126 Chapter 3: Transmission Lines and Waveguides Then from (3.133) the attenuation due to conductor loss is a, = A- (kl + ,f ) = 0.0672 np/m = 0.583 dB/m. The total attenuation is a = a(y>,0) i__3_/ 9$(A4*n r i P dp\ dp } P2 3(/>, ) subject to the boundary conditions that $(a,)=V0, (3.144a) *(&, $) = 0. (3.144b) Using the method of separation of variables, we Jet 2 By the usual separation of variables argument, the two terms in (3.146) must be equal to constants, so that pdf dR\ , 1 d2P = -kl (3.148) P dtf and k2p+kj=0. (3.149) The general solution to (3.148) is P(^) = Acostuf> + B&iatup, (3.150) where k^ = n must be an integer, since increasing by a multiple of 2n should not change the result. Now, because of the fact that the boundary conditions of (3,144) do not vary with 0, the potential 4>(p, 4>) should not vary with 0. Thus, n must be zero. By (3.149), this implies that kfi must also be zero, so that the equation for R(p) in (3.147) reduces to BpVdp) 0. The solution for R(p) is then R(p) = Clnp + £>, and so <$>(p, $) = Clnp + D. (3.151) Applying the boundary conditions of (3.144) gives two equations for the constants C and D: = % = CIna + D, (3.152a) Q(b,(p, $>) can be written as The E and H fields can then be found using (3.13) and (3.18). Then the voltage, current, and characteristic impedance can be determined as in Chapter 2. Attenuation due to dielectric or conductor loss has already been treated in Chapter 2. Higher Order Modes The coaxial line, like the parallel plate waveguide, can also support TE and TM waveguide modes in addition to a TEM mode. In practice, these modes are usually cutoff (evanescent), and so have only a reactive effect near discontinuities or sources, where they are excited. It is important in practice, however, to be aware of die cutoff frequency of the lowest order 128 Chapter 3: Transmission Lines and Waveguides waveguide-type modes, to avoid the propagation of these modes. Deleterious effects may otherwise occur, due to the superposition of two or more propagating modes with different propagation constants. Avoiding the propagation of higher order modes sets an upper limit on the size of a coaxial cable; this ultimately limits the power handling capacity of a coaxial line (see the Point of Interest on power capacity of transmission lines). We will derive the solution for the TE modes of the coaxial line; the TEn mode is the dominant waveguide mode of the coaxial line, and so is of primary importance. For TE modes, Ez = 0, and Hz satisfies the wave equation of (3.112): /a2 13 l a2 A W + WP + 7^klY'M) = 0- (3154) where Hz(p, 0, z) = hz{p, ^>)e~^z, and k1 = k1 — 82. The general solution to this equation, as derived in Section 3,4, is given by the product of (3.1 IS) and (3.120): kz{p, 4>) = (A sinrttf. + B cos ti$)(CJ„{kcp) + DY„{kcp)). (3-155) In this case, a < p < b, so we have no reason to discard the Y„ term. The boundary conditions are that Efip, r z) = 0, for p = arb. (3.156) Using (3.110b) to find E^ from Hz gives Ef = J-^(A sinn0 + B cos ntpKCJ^p) + DY^kcp))e~j?z. (3.157) Applying (3.156) to (3.157) gives two equations: Cr„(k,a) + DY^Ka) = 0, (3.158a) CJ'n(kcb) + DY'nikcb) = 0. (3.158b) Since this is a homogeneous set of equations, the only nontrivial (C $ 0. D 0) solution occurs when the determinant is zero. Thus we must have J^a)Y^(kcb) = rR{kcb)Y'n{kca). (3.159) This is a characteristic (or eigenvalue) equation for kc. The values of kt, that satisfy (3.159) then define the TEnm modes of the coaxial line. Equation (3.159) is a transcendental equation, which must be solved numerically for Jtc. Figure 3.16 shows the result of such a solution for n = 1, for various b/a ratios. An approximate solution that is often used in practice is k ~ — a + b Once kc is known, the propagation constant or cutoff frequency can be determined. Solutions for the TM modes can be found in a similar manner; the required determinant^] equation is the same as (3.159), except for the derivatives. Field lines for the TEM and TEn modes of the coaxial line are shown in Figure 3.17. EXAMPLE 3.3 HIGHER ORDER MODE OF A COAXIAL LINE Consider a piece of RG-142 coaxial cable, with a = 0.035" and b = 0.116", and a dielectric with er = 2.2. What is the highest usable frequency, before the TEU waveguide mode starts to propagate? 3.5 Coaxial Line 129 LO 11 12 FIGURE 3.16 Normalized cutoff frequency of the dominant TEj | waveguide mode for a coaxial line. Solution We have b 0.116 a 0.035 = 3.3. From Figure 3.16, this value of b/a gives kca = 0.47 (the approximate result is kca = 2/(1 + b/a) = 0.465). Thus, the cutoff frequency of the TEn mode is ckc 2ll^~r = 17 GHz. In practice, a 5% safety margin is usually recommended, so /m, = 0.95( 17 GHz) = 16 GHz. ■ (a) Cb) FIGURE 3.17 Field lines for the (a) TEM and (t>) TEt, modes of a coaxial line. Chapter 3: Transmission Lines and Waveguides an exception being the 75 £2 coax used in television systems. The reasoning behind these choices is that an air-filled coaxial line has niinimum attenuation for a characteristic impedance of 77 Q (Problem 2.28), while maximum power capacity occurs for a characteristic impedance of 30 £l (Problem 3.28). A 50 characteristic impedance thus represents a compromise between minimum attenuation and maximum power capacity. Requirements for coaxial connectors include low SWR, higher-order-mode-free operation at a high frequency, high repeatability after a connect-disconnect cycle, and mechanical strength. Connectors are used in pairs, with a male end and a female end (or plug and jack). The photo above shows several types of commonly used coaxial connectors and adapters. From top left; Type-N, TNC. SMA, APC-7, 2,4 mm. Type-N: This connector was developed in 1942 and named after its inventor, P, Neil, of Bell Labs, The outer diameter of the female end is about 0.625 in. The recommended upper frequency limit ranges from II to 18 GHz, depending on cable size. This rugged but large connector is often found on older equipment. TNC: This is a threaded version of the very common BNC connector. Its usage is limited to frequencies below 1 GHz. SMA: The need for smaller and lighter connectors led to the development of this connector in the 1960s. The outer diameter of the female end is about 0.250 in. It can be used up to frequencies in the range of 18-25 GHz, and is probably the most commonly used microwave connector today. APC-7: This is a precision connector (Amphenol precision connector) that can repeatedly achieve an SWR less than 1.04 at frequencies up to 1S GHz. The connectors are "sexless," with butt contact between both inner conductors and outer conductors. This connector is used most commonly for measurement and instrutnentarion applications. 2.4 mm: The need for connectors at millimeter wave frequencies led to the development of two variations of the SMA connector; the K connector is useful to about 40 GHz, while the 2.4 mm connector is useful to about 50 GHz. The size of these connectors is similar to the SMA connector. 3.6 Surface Waves on a Grounded Dielectric Slab 131 SURFACE WAVES ON A GROUNDED DIELECTRIC SLAB We briefly discussed surface waves in Chapter 1, in connection with the field of a plane wave totally reflected from a dielectric interface. In general, surface waves can exist in a variety of geometries involving dielectric interfaces. Here we consider the TM and TE surface waves that can be excited along a grounded dielectric slab. Other geometries that can be used as surface waveguides include an ungrounded dielectric slab, a dielectric rod, a corrugated conductor, or a dielectric coated conducting rod. Surface waves are typified by a field that decays exponentially away from the dielectric surface, with most of the field contained in or near the dielectric. At higher frequencies the field generally becomes more tightly bound to the dielectric, making such waveguides practical. Because of the presence of the dielectric, the phase velocity of a surface wave is less than the velocity of light in a vacuum. Another reason for studying surface waves is that they may be excited on some types of planar transmission tines, such as microstrip and slotline. TM Modes Figure 3.18 shows the geometry of a grounded dielectric slab waveguide. The dielectric slab, of thickness d and relative dielectric constant er, is assumed to be of infinite extent in the y and z directions. We will assume propagation in the +z direction with an e~^z propagation factor, and no variation in the y direction (dfdy = 0). Because there are two distinct regions, with and without a dielectric, we must separately consider the field in these regions, and then match tangential fields across the interface. Ez must satisfy the wave equation of (3.25) in each region: f |P + '> = °> for 0 < jc < rf, (£2 +*S-02^(*,xl = O, for d < x < co, where Ez(x, y, z) = e,{x, y)e~^z. Now define the cutoff wavenumbers for the two regions as k2 = evAj — B2, h2 = B2- k2, (3.160a) (3.160b) (3.161a) (3.161b) where the sign on h2 has been selected in anticipation of an exponentially decaying result for x > d. Observe that the same propagation constant /? has been used for both regions. This Dielectric / Ground plane FIGURE 3.18 Geometry of a grounded dielectric slab. 132 Chapter 3: Transmission Lines and Waveguides must be the case to achieve phase matching of the tangential fields at the x = d interface for all values of z- The general solutions to (3,160) are then ez(x, y) — A sin kcx + B cos kcxr for 0 < x < d, (3.162a) ez(x,y) = Cehx + De~h*, ford"<* which is the equation of a circle m(hekcdthd plane, as shown in Figure 3.19. The radius of the circle is - lkod, which is proportional to the electrical thickness of the dielectric 3.6 Surface Waves on a Grounded Dielectric Slab 133 sJab. Multiplying (3.165) by d gives kcd tanked = €rhd, which is also plotted in Figure 3.19. The intersection of these curves implies a solution to both (3.165) and (3.166), Observe that kc may be positive or negative; from (3.162a) this is seen to merely change the sign of the constant A. As ye^ ~ 1M becomes larger, the circle may intersect more than one branch of the tangent function, implying that more than one TM mode can propagate. Solutions for negative ht however, must be excluded since we assumed h was positive real when applying boundary condition (3.163b). For any nonzero thickness slab, with a permittivity greater than unity, there is at least one propagating TM mode, which we will call the TMo mode. This is the dominant mode of the dielectric slab waveguide, and has a zero cutoff frequency. (Although for £o = 0, k<, = h = 0 and all fields vanish.) From Figure 3.19, it can be seen that the next TM mode, the TM| mode, will not turn on until the radius of the circle becomes greater than jr. The cutoff frequency of the TM„ mode can then be derived as n =0, 1,2, (3.167) Once kc and h have been found for a particular surface wave mode, the field expressions can be found as £;U, y, z) = ffy{x, y, z) = A sin kcxe Asmkcde~b{x~d)e-rtz -J? A coskrxe~^z ^Asinkcde-h(x-d)e-^ h -j(0€d€T A coskcxe~jf>z -J^Asixikrde-hi*-d)e-^ h for 0 < x < d for d < x < oo, f or 0 < x < d for d < .v < co, for 0 < .v < d ford < x < oo. (3.168a) (3.168b) (3,168c) FIGURE 3,19 Graphical solution of the transcendental equation for the cutoff frequency of a TM surface wave mode of the grounded dielectric slab. Chapter 3: Transmission Lines and Waveguides TEModes TE modes can also be supported by the grounded dielectric slab. The Hz field satisfies the wave equations Qp + lg\ k0. y) = 0, for 0 < x < dt (3.169a) - h2^ hz{x, y) = 0, fordl it J \l / ft j , \ J t kcd \ Invalid / \ solutions I FIGURE 3.20 Graphical solution of the transcendental equation for the cutoff frequency of a TE surface wave mode. Figure depicts a mode below cutoff. After finding the constants kc and h, the field expressions can be derived as Hz(.x, y, z) = Hx(x,y.z) = EJx, y,z) = Bcoskcxe~^z —— sin kcxe~}P^ h ---sm kcxe JPZ for 0 < x < d for d < x < oo, for 0 < x < d for d < x < oo, f or 0 < x < d for d < x < oo. (3.175a) (3.175b) (3.175c) EXAMPLE 3.4 SURFACE WAVE PROPAGATION CONSTANTS Calculate and plot the propagation constants of the first three propagating surface wave modes of a grounded dielectric sheet with €r =- 2.55, for d/\o = 0 to 1.2. Solution The first three propagating surface wave modes are the TMo, TK>, and TM i modes. The cutoff frequencies for these modes can be found from (3.167) and (3.174) as TMo: f< = 0 TE]: fc = TMi: fr = s=4 — =0. 1 c d 1 The propagation constants must be found from the numerical solution of (3.165) and (3,166) for the TM modes, and (3.172) and (3.173) for the TE modes. This can 136 Chapter 3: Transmission Lines and Waveguides FIGURE 3.21 Surface wave propagation constants for a grounded dielectric slab with er = 2,55. be done with a relatively simple root-finding algorithm (see the Point of Interest on root-finding algorithms); the results are shown in Figure 3,21. ■ POINT OF INTEREST: Root-Finding Algorithms In several examples throughout this book we will need to numerically find the root of a transcendental equation, so it may be useful to review two relatively simple but effective algorithms for doing this. Both methods can be easily programmed. In the interval-halving method the root of fix) = 0 is first bracketed between the values x^ and x^- These values can often be estimated from the problem under consideration. If a single root lies between -Ti and x%i men f{x \ )f(x2) < 0. An estimate, of the root is made by halving the interval between xt and x2. Thus, *3 = If f{Xi)f(Xf) < 0, then the root must lie in the interval jcj < x <: xy, if f(x^)fix2) < 0i then the root must be in the interval x$ < x < *3. A new estimate, Xi, can be made by halving the appropriate interval, and this process repeated until the location of the root has been determined with the desired accuracy. The figure below illustrates this algorithm for several iterations. The Newton-Raphson method begins with an estimate, jci , of the root of f(x) = 0. Then a new estimate, x2, is obtained from the formula x2 = x, - where f\x\) is the derivative of f(x) alx\. This result is easily derived from a two-term Taylor series expansion of /{x) near x = = f(x^) + (x - x{)f'{x\). It can also be interpreted geometrically as fitting a straight line at x — x\ with the same slope as fix) at this point; this line then intercepts the .t-axis at x = x2, as shown in the figure below. Reapplying the above formula 3.7 Stripline 137 gives improved estimates of the root, Convergence is generally much faster than with the interval halving method, but a disadvantage is that the derivative of f(x) is required; this can often be computed numerically. The Newton-Raphson technique can easily be applied to the case where the root is complex, (a situation that occurs, for example, when finding the propagation constant of a line or guide with loss). Interval halving Newton-Raphson Reference: R. W. Hombeck, Numerical Methods, Quantum Publishers, New York, 1975. STRIPLINE We now consider stripline, a planar-type of transmission line that lends itself well to microwave integrated circuitry and photolithographic fabrication. The geometry of a stripline is shown in Figure 3.22a, A thin conducting strip of width W is centered between two wide conducting ground planes of separation b, and the entire region between the ground planes is filled with a dielectric. In practice, stripline is usually constructed by etching the center conductor on a grounded substrate of thickness b/2, and then covering with another grounded substrate of the same thickness. An example of a stripline circuit is shown in Figure 3.23, Since stripline has two conductors and a homogeneous dielectric, it can support a TEM wave, and this is the usual mode of operation, Like the parallel plate guide and coaxial lines, however, the stripline can also support higher order TM and TE modes, but these are usually avoided in practice (such modes can be suppressed with shorting screws between the ground £- H---- FIGURE 3.22 Stripline transmission line, fa) Geometry, (b) Electric and magnetic field lines. 138 Chapter 3: Transmission Lines and Waveguides FIGURE 3.23 Photograph of a stripline circuit assembly, showing four quadrature hybrids, open-circuit tuning stubs, and coaxial transitions. Courtesy of Harlan Howe, Jr., M/A-COM Inc. planes and by restricting the ground plane spacing to less than A/4). Intuitively, one can think of stripline as a sort of "flattened out" coax—both have a center conductor completely enclosed by an outer conductor and are uniformly filled with a dielectric medium. A sketch of the field lines for stripline is shown in Figure 3.22b. The main difficulty we will have with stripline is that it does not lend itself to a simple analysis, as did the transmission lines and waveguides that we have previously discussed. Since we will be concerned primarily with die TEM mode of the stripline, an electrostatic analysis is sufficient to give the propagation constant and characteristic impedance. An exact solution of Laplace's equation is possible by a conformal mapping approach [6], but the procedure and results are cumbersome. Thus, we will present closed-form expressions that give good approximations to the exact results and then discuss an approximate numerical technique for solving Laplace's equation for a geometry similar to stripline; this technique will also be applied to microstrip line in the following section. Formulas for Propagation Constant, Characteristic Impedance, and Attenuation From Section 3.1 we know that the phase velocity of a TEM mode is given by (3.176) 3.7 Stripline 139 thus the propagation constant of the stripline is ců (3.177) In (3.176), c = 3 x 10s m/sec is the speed of light in free-space. The characteristic impedance of a transmission line is given by v'LC l vpC (3.178) where L and C are the inductance and capacitance per unit length of the line. Thus, we can find Zo if we know C. As mentioned above, Laplace's equation can be solved by conformal mapping to find the capacitance per unit length of the stripline. The resulting solution, however, involves complicated special functions [6], so for practical computations simple formulas have been developed by curve fitting to the exact solution [6], [7]. The resulting formula for characteristic impedance is 30* b Z0 = jTr W£+0Mlb where We is the effective width of the center conductor given by 0 (3.179a) 5* b W ~b (0.35 - W/bý w for — > 0.35 b W for — < 0.35. b (3.179b) These formulas assume a zero strip thickness, and are quoted as being accurate to about 1% of the exact results. It is seen from (3.179) that the characteristic impedance decreases as the strip width W increases. When designing stripline circuits, one usually needs to find the strip width, given the characteristic impedance (and height b and permittivity er), which requires the inverse of the formulas in (3.179). Such formulas have been derived as for JTr Z0 < 120 V0.6 - x for jTr Zo > 120, where 30jt W _ I v b ~ |0 85 (3.180a) .v — -0.441. (3.180b) Since stripline is a TEM type of line, the attenuation due to dielectric loss is of the same form as that for other TEM lines and is given in (3.30). The attenuation due to conductor loss can be found by the perturbation method or Wheeler's incremental inductance rule. An approximate result is 2.7 x 10-3Rs€rZQ 30k (b - t) 0A6RS Z0/> B for7?7Zo < 120 for ,/ě7Z0 > 120 Np/m, (3.181) with , 2W I b + t, 2b-t\ A = 1 + ---+ ---In - , b-t k b-t \ t } { 0.414/ 1 , 4jtW\ —— 0.5 +-+ — In-], 0.7/) V W 2tt t } B = 1 + -- (0.5 W + where t is the thickness of the strip 140 Chapter 3: Transmission Lines and Waveguides EXAMPLE 3.5 STRIPLINE DESIGN Find the width for a 50 ft copper stripline conductor, with b = 0.32 cm and er = 2.20. If the dielectric loss tangent is 0.001 and the operating frequency is 10 GHz, calculate the attenuation in dB/X. Assume a conductor thickness of / = 0.01 mm* Solution Since T^Zq = ^22(50) = 74.2 < 120,and* a 30tt/(^€;Zo) - 0.441 = 0.830, (3.180) gives the width as W = bx = (0.32)(0.830) = 0.266 cm. At 10 GHz, the wavenumber is c From (3.30) the dielectric attenuation is *tan<5 (310.6X0.001) ad = ~y~ =----=0.155 Np/m. The surface resistance of copper at 10 GHz is Rs = 0.026 ft. Then from (3181) the conductor attenuation is 2.7 x IQ-iRst,.Z0A . 307r(o — /) since A = 4.74. The total attenuation constant is a = ad + ac - 0.277 Np/m. IndB, a(dB) = 20 log e" = 2.41 dB/m. At 10 GHz, the wavelength on the stripline is k = *L = 2.02 cm, so in terms of wavelength the attenuation is cr(dB) = (2.41 )(0.0202) = 0.049 dB/X. ■ An Approximate Electrostatic Solution Many practical problems in microwave engineering are very complicated and do not lend themselves to straightforward analytic solutions, but require some sort of numerical approach. Thus it is useful for the student to become aware of such techniques; we will introduce such methods when appropriate throughout this book, beginning with a numerical solution for the characteristic impedance of stripline. We know that the fields of the TEM mode on a stripline must satisfy Laplace's equation, (3.11), in the region between the two parallel plates. The actual stripline geometry of Figure 3.22a extends to ±oo, which makes the analysis more difficult. Since we suspect, from the field line drawing of Figure 3.22b, that the field lines do not extend very far away from the center conductor, we can simplify the geometry by truncating the plates beyond some distance, say \x\ > a/2, and placing metal walls on the sides. Thus, the geometry we will 3.7 Stripline 141 analyze looks like that shown in Figure 3.24, where a 3> b so that the fields around the center conductor are not r^rturbed by the sidewalls. We then have a closed, finite region in which the potential {x, y) will have a slope discontinuity there, because D = — €o*rVt<$? is discontinuous at y = b/2. So separate solutions for 4>(x, y) must be found for 0 < y < b/2, and b/2 < y < b. The general solutions for <&(,#, y) in these two regions can be written as *(*, y) = 2_, A„ cos-sinh- 7t=i a a odd V Bh cos-sum —(b — y) odd for 0 < y < b/2 toib/2 /3y, we have ™ /nnx nnx . iiTcy -■ V An I — I cos-cosh — „_! V a f a a (.Hill ~ , ?nX\ nnx .nit V A„ I — ) cos-cosh — (b - y) B=1 \ a f a a odd for 0 < y < b/2 for b/2 < y < b, (3.186) 142 Chapter 3: Transmission Lines and Waveguides The surface charge density on the strip at y = b/2 is A = y = - y = i>l2~) = ta€r[Ey(x7 y = b/2+) - Es{x, y = b/2~)] = 2 An [ — ) cos-cosh-, (3.187) V a / a 2a ^' which is seen to be a Fourier series in x for the surface charge density, fa. If we know the surface charge density, we could easily find the unknown constants, An, and then the capacitance. We do not know (he exact surface charge density, but we can make a good guess by approximating it as a constant over the width of the strip PÄ*) = 1 for|x|W/2. Equating this to (3.187) and using the orthogonality properties of the cos(mjt.c /a) functions gives the constants A„ as y^y^, (3.189) The voltage of the center strip relative to the bottom conductor is ^2 . nub - - / Ey(x = 0, y)dy = A„ sinh 2a n=l odd (3.190) The total charge, per unit length, on the center conductor is -Wf2 so that the capacitance per unit length of the stripline is Q = / ps(x)dx - W C/m, (3.191) _ Q _ W C = V = ^ 2a sin(nff W/2a)sinh(n jrfr/2a) Fd/m' (3-192) „=1 (nn)2^r cosh(njrb/2a) odd The characteristic impedance is then found as [Z _ J~LC 1 0 ~ V C ~ C ~ vpC~ cC where c = 3 x 10s m/sec. EXAMPLE 3.6 NUMERICAL CALCULATION OF STRIPLINE PARAMETERS Evaluate the above expressions for a stripline having = 2.55 and a = 100b, to find the characteristic impedance for W/b = 0.25 to 5.0. Compare with the results from (3.179). 3.8 Microstrip 143 Solution A Computer program was written to evaluate (3,192). The series was truncated after 500 terms, and the results are shown below. Numerical Formula W/b Eq. (3.192) Eq. (3.179) 0.25 98.S ß 86.6 £2 0.50 73.3 62.7 1.0 49.0 41.0 2.0 28.4 24.2 3.5 16.8 15.0 5.0 11.8 10.8 We see that the results are in reasonable agreement with the closed-form equations of (3.179), particularly for wider strips. Better results could be obtained if more sophisticated estimates were used for the charge density, ps. ■ MICROSTRIP Microstrip line is one of the most popular types of planar transmission lines, primarily because it can be fabricated by photolithographic processes and is easily integrated with other passive and active microwave devices. The geometry of a microstrip line is shown in Figure 3.25a. A conductor of width W is printed on a thin, grounded dielectric substrate of thickness d and relative permittivity e>; a sketch of the field lines is shown in Figure 3.25 b. If the dielectric were not present (€r = 1), we could think of the line as a two-wire line consisting of two flat strip conductors of width W, separated by a distance 2d (the FIGURE 3.25 Microstrip transmission line, (a) Geometry, (b) Electric and magnetic field lines. Chapter 3: Transmission Lines and Waveguides ground plane can be removed via image theory). In this case we would have a simple TEM transmission line, with vp = c and B — k$. The presence of the dielectric, and particularly the fact that the dielectric does not fill the ah region above the strip (y > d), complicates the behavior and analysis of mi-crostrip line, Unlike stripline, where all the fields are contained within a homogeneous dielectric region, microstrip has some (usually most) of its field lines in the dielectric region, concentrated between the strip conductor and the ground plane, and some fraction in the air region above the substrate. For this reason the microstrip line cannot support a pure TEM wave, since the phase velocity of TEM fields in the dielectric region would be c/y^T, but the phase velocity of TEM fields in the air region would be c. Thus, a phase match at the dielectric-air interface would be impossible to attain for a TEM-type wave, Ln actuality, the exact fields of a microstrip line constitute a hybrid TM-TE wave, and require more advanced analysis techniques than we are prepared to deal with here. In most practical applications, however, the dielectric substrate is electrically very thin {d A.), and so the fields are quasi-TEM. In other words, the fields are essentially the same as those of the static case. Thus, good approximations for the phase velocity, propagation constant, and characteristic impedance can be obtained from static or quasi-static solutions. Then the phase velocity and propagation constant can be expressed as iV = (3.193) B=k^, (3.194) where es is the effective dielectric constant of the microstrip line. Since some of the field lines are in the dielectric region and some are in air, the effective dielectric constant satisfies the relation 1 < «e < er, and is dependent on the substrate thickness, d, and conductor width, W» We will first present design formulas for the effective dielectric constant and characteristic impedance of microstrip line; these results are curve-fit approximations to rigorous quasi-static solutions [8], [9]. Then we will outline a numerical method of solution (similar to that used in the previous section for stripline) for the capacitance per unit length of microstrip line. Formulas for Effective Dielectric Constant, Characteristic Impedance, and Attenuation The effective dielectric constant of a microsuip line is given approximately by 2 2 sfTTTMJW The effective dielectric constant can be interpreted as the dielectric constant of a homogeneous medium that replaces the air and dielectric regions of the rnicrostrip, as shown in Figure 3.26. The phase velocity and propagation constant are then given by (3.1.93) and (3.194). 3.8 Microstrip 145 FIGURE 3.26 Equivalent geometry of quasi-TEM microstrip line, where the dielectric slab of thickness d and relative permittivity has been replaced with a homogeneous medium of effective relative permittiviry, ee. Given the dimensions of the microstrip line, the characteristic impedance can be calculated as 60 (%d W\ 1207T Te [W/d + 1.393 + 0.667 In (W/d + 1.444)] for W/d < 1 for W/d > 1, (3.196) For a given characteristic impedance Zo and dielectric constant e7t the W/d ratio can be found as W_ d where I pS - 1 -in(2B - 1) + ^- Jln(5 - D + 0.39- ^Jj 60 V 2 er + 1 V & / for W/d < 2 for W/d > 2, (3.197) B = 377tt Considering microstrip as a quasi-TEM line, the attenuation due to dielectric loss can be determined as Mrfe - 1)tan 5 &d = ^ r_f_--— Np/m, (3.198) 2^r ~ 1) where tan h is the loss tangent of the dielectric. This result is derived from (3.30) by multiplying by a "filling factor" ^(6,-1)' which accounts for the fact that the fields around the microstrip line are partly in air (lossless) and partly in the dielectric. The attenuation due to conductor loss is given approximately by [8] ZoW Np/m, (3.199) where Rs = ^\i§/2a is the surface resistivity of the conductor. For most microstrip 146 Chapter 3: Transmission Lines and Waveguides substrates, conductor loss is much more significant than dielectric loss; exceptions may occur with some semiconductor substrates, however. EXAMPLE 3.7 MICROSTRIP DESIGN Calculate the width and length of a microstrip line for a 50 £2 characteristic impedance and a 90° phase shift at 2.5 GHz. The substrate thickness is d = 0.127 cm, with €r = 2.20, Solution We first find W/d for Z0 = 50 Q, and initially guess that W/d > 2. From (3.197), B = 7.985, W/d = 3.081. So W/d > 2; otherwise we would use the expression for W/d < 2. Then W = 3.08 It/ = 0.391 cm. From (3.195) the effective dielectric constant is <-V = 1.87. The line length, £, for a 90° phase shift is found as 4> = 90° = $t = y^M, *o = — =52.35 m An Approximate Electrostatic Solution We now look at an approximate quasi-static solution for the microstrip line, so that the appearance of design equations like those of (3.195)-(3.197) is not a complete mystery. This analysis is very similar to that carried out for stripline in the previous section. As in that analysis, it is again convenient to place conducting sidewalls on the microstrip line, as shown in Figure 327, The sidewalls are placed at x = ±a/2, where a > d, so that the walls should not perturb the field lines localized around the strip conductor. We then can solve Laplace's equation in the region between the sidewalls; y) = 0, for |*l < a/2, 0 < y < co, (3.200) FIGURE 3.27 Geometry of a microstrip line with conducting sidewalls. 3.8 Microstrip 147 with boundary conditions, (x, y) in these regions. Solving (3,200) by the method of separation of variables and applying the boundary conditions of (3.201a,b) gives the general solutions as fflfo y) = « . nnx . - nny > A„ cos —— sjnh- odd Ior 0 < y < d £ B„ cos-e-K*y/a for d < y < oo. (3.202) odd Now the potential must be continuous at y = d, so from (3,202) we have that nnd ,, 4,, sinh-= Bne-H*J/a, a (3.203) so $>(x> y) can be written as y) = for 0 < v < d ,22, /jjtx nny 2, A„ cos-sinh- odd fj A„ cos ®£ sinh — «T"^"^ for < y < oo. n=l ö CJ odd (3.204) The remaining constants, A„, can be found by considering the surface charge density on the strip. We first find Ey - -d

tV/2. (3.207) 148 Chapter 3; Transmission Lines and Waveguides Equating (3.207) to (3,206) and using the orthogonality of the cosnnx/a functions gives the constants A„ as 4a sin nit W/2a An =-=-------—, (3.208) (njr)2eo[sinh(njrd/(j) + €T co$h(nnd/a)] The voltage of the strip relative to the ground plane is V = - / EY(x = 0, y)dy = V An sinh — (3,209) odd The total charge, per unit length, on the center strip is Q = j ps(x)dx = W C/m, (3.210) so the static capacitance per unit length of the microstrip line is C ~ V ~ °° 4a sin(«7r W/2a) sinh(ttjr^/a) ' (3-211) {tmfiWeoiswhbwdfa) + ercosh(«7rd/«) odd Now to find the effective dielectric constant, we consider two cases of capacitance: Let C = capacitance per unit length of the microstrip line with a dielectric substrate Ov i) Let C0 = capacitance per unit length of the microstrip line with an air dielectric (er = 1) Since capacitance is proportional to the dielectric constant of the material homogeneously filling the region around the conductors, we have that (3.212) So (3.212) can be evaluated by computing (3.211) twice; once with tr equal to the dielectric constant of the substrate (for C), and then with = 1 (for C0). The characteristic impedance is then Z0 = -L = (3.213) vpC cC where c = 3x 10s m/sec. EXAMPLE 3.8 NUMERICAL CALCULATION OF MICROSTRIP PARAMETERS Evaluate the above expressions for a microstrip line on a substrate with €r = 2.55. Calculate the effective dielectric constant and characteristic impedance for W/d = 0.5 to 10.0, and compare with the results from (3.195) and (3.196). Let a = lOOrf. Solution A computer program was written to evaluate (3.211) for e — e0 and then e = e>£o-Then (3.212) was used to evaluate the effective dielectric constant, and (3.213) 3.9 The Transverse Resonance Technique 149 to evaluate the characteristic impedance, Z0. The series was truncated after 50 terms, and the results are shown in the following table. Numerical Solutions Formulas W/d 20{Q) 0.5 1.977 100.9 1.938 119.8 1.0 1.989 94.9 1.990 89.8 2.0 2.036 75.8 2.068 62.2 4.0 2.179 45.0 2.163 39.3 7.0 2.287 29.5 2.245 25,6 10.0 2.351 21.7 2.198 19.1 The comparison is reasonably good, although better results could be obtained from the approximate numerical solution by using a better estimate of the charge density on the strip. ■ THE TRANSVERSE RESONANCE TECHNIQUE According to the general solutions to Maxwell's equations for TE or TM waves given in Section 3.1, a uniform waveguide structure always has a propagation constant of the form B = Jk2-k2 = Jk? - k\ - kjr (3.214) where kc = ^Jk2 + k2 is the cutoff wavenumber of the guide and, for a given mode, is a fixed function of the cross-sectional geometry of the guide. Thus, if we know kc we can determine the propagation constant of the guide. In previous sections we determined kc by solving the wave equation in the guide, subject to the appropriate boundary conditions; this technique is very powerful and general, but can be complicated for complex waveguides, especially if dielectric layers are present. In addition, the wave equation solution gives a complete field description inside the waveguide, which is much more information than we really need if we are only interested in the propagation constant of the guide. The transverse resonance technique employs a transmission line model of the transverse cross section of the waveguide, and gives a much simpler and more direct solution for the cutoff frequency. This is another example where circuit and transmission line theory can be used to simplify the field theory solution. The transverse resonance procedure is based on the fact that in a waveguide at cutoff, the fields form standing waves in the transverse plane of the guide, as can be inferred from the "bouncing plane wave" interpretation of waveguide modes discussed in Section 3.2. This situation can be modeled with an equivalent transmission line circuit operating at resonance. One of the conditions of such a resonant line is the fact that, at any point on the line, the sum of the input impedances seen looking to either side must be zero. That is, 4^) + Z£0O = 0, for all (3.215) where Zrm(x) and Z*n{x) are the input impedances seen looking to the right and left, respectively, at the point x on the resonant line. 150 Chapter 3: Transmission Lines and Waveguides The transverse resonance technique only gives results for the cutoff frequency of the guide. If fields or attenuation due to conductor loss are needed, the complete field theory solution will be required. The procedure will now be illustrated with an example. TE0n Modes of a Partially Loaded Rectangular Waveguide The transverse resonance technique is particularly useful when the guide contains dielectric layers because the boundary conditions at the dielectric interfaces, which require the solution of simultaneous algebraic equations in the field theory approach, can be easily handled as junctions of different transmission lines. As an example, consider the rectangular waveguide partially filled with dielectric, as shown in Figure 3.2S. To find the cutoff frequencies for the TEqw modes, the equivalent transverse resonance circuit shown in the figure can be used. The line for 0 < y < t represents the dielectric-filled part of the guide, and has a transverse propagation constant kyij and a characteristic impedance for TE modes given by Zd = — — ——, (3.216a) where ka = ío^/io^q, tjo — V/W^o- For t < y < b, the guide is air filled and has a transverse propagation constant kya and an equivalent characteristic impedance given by Za = !», (3.216b) kya Applying condition (3.215) yields kya tan kyi{t 4- kyd tan kya{b -0 = 0. (3.217) This equation contains two unknowns, kyil and kyj. An additional equation is obtained from the fact that the longitudinal propagation constant, ft, must be the same in both regions, for phase matching of the tangential fields at the dielectric interface. Thus, with kx = 0, 0 — y^ti - kyd — \JH — k^a i or /8) The speed of light in a medium is the velocity at which a plane wave would propagate in that medium, while the phase velocity is the speed at which a constant phase point travels. For a TEM plane wave, these two velocities are identical, but for other types of guided wave propagation the phase velocity may be greater or less than the speed of light. If the phase velocity and attenuation of a line or guide are constants that do not change with frequency, then the phase of a signal that contains more than one frequency component will not be distorted. If the phase velocity is different for different frequencies, then the individual frequency components will not maintain their original phase relationships as they propagate down the transmission line or waveguide, and signal distortion will occur. Such an effect is called dispersion, since different phase velocities allow the "faster" waves to lead in phase relative to the "slower" waves, and the original phase relationships will gradually be dispersed as the signal propagates down the line. In such a case, there is no single phase velocity that can be attributed to the signal as a whole. However, if the bandwidth of the signal is relatively small, or if the dispersion is not too severe, a group velocity can be defined in a meaningful way. This velocity then can be used to describe the speed at which the signal propagates. Group Velocity As discussed above, the physical interpretation of group velocity is the velocity at which a narrow band signal propagates. We will derive the relation of group velocity to the propagation constant by considering a signal f(t) in the time domain. The Fourier transform of this signal is defined as Now consider the transmission line or waveguide on which the signal f(t) is propagating as a linear system, with a transfer function Z(io) that relates the output, F^fcy), of the fine to the input, F(u>), of the fine, as shown in Figure 3.29. Thus, (3.219a) and the inverse transform is then (3.219b) F0do) = Z(o>)FUo), (3.220) FIGURE 3.29 A transmission line or waveguide represented as a linear system with transfer function Z() can be expressed as Z(a>) = Ae~Jfiz = \Z(oj)\e~jt, (3.221) where A is a constant and B is the propagation constant of the line or guide. The time-domain representation of the output signal, f0(t), can then be written as Ut) = ^-T Fiumw^-^do). (3.222) Now if \Z(co)\ = A is a constant, and the phase \fr of Z{co) is a linear function of o), say $ =a) = Ae'jaM does not distort the input signal. A lossless TEM wave has a propagation constant B =m, where wm 0. The Fourier transform, S( - w„), (3.225) where we have used the complex form of the input signal as expressed in (3.224). We will then need to take the real part of the output inverse transform to obtain the time-domain output signal. The spectrums of F(o>) and S(a>) are depicted in Figure 3.30, The output signal spectrum is SM = AF(a>- i3%$r&% (3.226) -üj„, Q a>m at 0 ü)a «i (a) (b> FIGURE 3.30 Fourier spectrums of the signals (a) /(t) and (b) s(t). 3.10 Wave Velocities and Dispersion 153 and id the time domain, = ^-Re f SMejo*d-a)0)ei(ü,t-ßz)da>. In general, the propagation constant jff may be a complicated function of o>. But if F(cn) is narrowband {„), then $ can be linearized by using a Taylor series expansion about co0: dß_ do> ld2ß 2da>2 (d> - i0of + ür=a>,, Retaining the first two terms gives where ßo =ß«h % "to (3.228) (3.229) Then after a change of variables to y — (o — co0, the expression for s(At) becomes F(y)ej(l~&z)y dy\ ■•»>,„ J = AR^\f(t-ß'0z)ej^'-ß^} = Af(t - ß'0z) cos(üU - ßcz), (3.230) which is a time-shifted replica of the original modulation envelope, f(t), of (3.224). The velocity of this envelope is the group velocity, vs: 1 (dß\ -i (3.231) EXAMPLE 3.9 WAVEGUIDE WAVE VELOCITIES Calculate the group velocity for a waveguide mode propagating in an air-filled guide. Compare this velocity to the phase velocity and speed of light. Solution The propagation constant for a mode in an air-filled waveguide is ?=fate?-**. Taking the derivative with respect to frequency gives dp oijc1 k0 so from (3.234) the group velocity is Chapter 3: Transmission Lines and Waveguides The phase velocity is vp = ai/B = (fcoc)/0. Since B < fr0, we have thai vs < c < vp, which indicates that the phase velocity of a waveguide mode may be greater than the speed of light, but the group velocity (the velocity of a narrowband signal) will be less than the speed of light. ■ SUMMARY OF TRANSMISSION LINES AND WAVEGUIDES In this chapter we have discussed a variety of transmission lines and waveguides; here we will summarize some of the basic properties of these transmission media and their relative advantages in a broader context. In the beginning of this chapter we made the distinction between TEM, TM, and TE waves and saw that transmission lines and waveguides can be categorized according to which type of waves they can support. We have seen that TEM waves are nondispersive, with no cutoff frequency, whereas TM and TE waves exhibit dispersion and generally have nonzero cutoff frequencies. Other electrical considerations include bandwidth, attenuation, and power handling capacity. Mechanical factors are also very important, however, and include such considerations as physical size (volume and weight), ease of fabrication (cost), and the ability to be integrated with other devices (active or passive), Table 3.6 compares several types of transmission media with regard to the above considerations; this table only gives general guidelines, as specific cases may give better or worse results than those indicated. Other Types of Lines and Guides While we have discussed the most common types of waveguides and transmission lines, there are many other guides and lines (and variations) that we have not discussed. A few of the more popular types are briefly mentioned here. Ridge waveguide. The bandwidth of a rectangular waveguide is, for practical purposes, less than an octave (a 2:1 frequency range). This is because the TE20 mode begins to propagate at a frequency equal to twice the cutoff frequency of the TE)0 mode. The ridge waveguide, shown in Figure 3.31, consists of a rectangular waveguide loaded with conducting ridges on TABLE 3.6 Comparison of Common Transmission Lines and Waveguides Characteristic Coax Waveguide Stripline Microstrip Modes: Preferred TEM TE10 TEM Quasi-TEM Other TM,TE TM,TE TM.'I'E Hybrid TM.TE Dispersion None Medium None Low Bandwidth High Low High High Loss Medium Low High High Power capacity Medium High Low Low Physical size Large Large Medium Small Ease of fabrication Medium Medium Easy Easy Integration with Hard Hard Fair Easy other components 3.11 Summary of Transmission Lines and Waveguides 155 FIGURE 3.31 Cross section of a ridge waveguide. the top an d/or bottom walls. This loading tends to lower the cutoff frequency of the dominant mode, leading to increased bandwidth and better impedance characteristics. Such a guide is often used for impedance matching purposes, where the ridge may be tapered along the length of the guide. The presence of the ridge, however, reduces the power-handling capacity of the waveguide. Dielectric waveguide. As we have seen from our study of surface waves, metallic conductors are not necessary to confine and support a propagating electromagnetic field. The dielectric waveguide shown in Figure 3.32 is another example of such a guide, where er2t the dielectric constant of the ridge, is usually greater than eri, the dielectric constant of the substrate. The fields are thus mostly confined to the area around the dielectric ridge. This type of guide supports TM and TE modes, and is convenient for integration with active devices. Its small size makes it useful for millimeter wave to optical frequencies, although it can be very lossy at bends or junctions in the ridge line. Many variations in this basic geometry are possible. Slotline. Of the many types of planar 1 ines that have been proposed, slotline probably ranks next, behind microstrip and stripline, in terms of popularity. The geometry of a slotline is shown in Figure 3.33. It consists of a thin slot in the ground plane on one side of a dielectric substrate. Thus, like microstrip, the two conductors of slotline lead to a quasi-TEM type of mode. Changing the width of the slot changes the characteristic impedance of the line. Coplanar waveguide. A structure similar to slotline is coplanar waveguide, shown in Figure 3.34. Coplanar waveguide can be thought of as a slotline with a third conductor centered in the slot region. Because of the presence of this additional conductor, this type of line can support even or odd quasi-TEM modes, depending on whether the £-fields in the two slots are in the opposite direction, or the same direction. Coplanar waveguide is particularly useful for fabricating active circuitry, due to the presence of the center conductor and the close proximity of the ground planes. Covered microstrip. Many variations of the basic microstrip geometry are possible, but one of the more common is covered microstrip, shown in Figure 3.35. The metallic cover plate is often used for electrical shielding and physical protection of the microstrip circuit FIGURE 3.32 Dielectric waveguide geometry. 156 Chapter 3: Transmission Unas and Waveguides FIGURE 3.33 Geometry of a printed slotline. and is usually situated several substrate thicknesses away from the circuit. Its presence can, however, perturb the operation of the circuit enough so that its effect must be taken into account during design. POINT OF INTEREST: Power Capacity of Transmission Lines The power handling capacity of an air-filled transmission line or waveguide is limited by voltage breakdown, which occurs at a field strength of about E4 = 3 x 10* V/m for room temperature air at sea level pressure. In an air-filled coaxial line, the electric field varies as = V0/(plnb/a), which has a maximum at p = a. Thus the maximum voltage before breakdown is Km> = Eda In -, (peak-to-peak), and the maximum power capacity is then 2Z0 p _ 'max _ in As might be expected, this result shows that power capacity can be increased by using a larger coaxial cable (largera, b with fixed bja for the same characteristic impedance). But propagation of higher order modes limits the maximum operating frequency for a given cable size. Thus, (here is an upper limit on the power capacity of a coaxial line for a given maximum operating frequency, which can be shown to be given by Pmai — 0.025 no V JrasX / \ /ma* / As an example, at 10 GHz the maximum peak power capacity of any coaxial line with no higher order modes is about 520 kW. In an air-filled rectangular waveguide, the electric field varies as Ey = E0 sin(jrx/a), which has a maximum value of £„ at Jt = a/2. Thus the maximum power capacity before breakdown is abEl abEl AZju 4ZT„ which shows that power capacity increases with guide size. For most waveguides, b £2 la. To FIGURE 3,34 Coplanar waveguide geometry. Problems 157 FIGURE 335 Covered microstxip line. avoid propagation of the TE20 mode, we must have a < o//ma^, where is the maximum operating frequency. Then the maximum power capacity of the guide can be shown to be 'Jo Vim**/ V/mas/ As an example, at 10 GHz the maximum peak power capacity of a rectangular waveguide operating in the TEi0 mode is about 2300 kW, which is considerably higher than the power capacity of a coaxial cable at the same frequency. Because arcing and voltage breakdown are very high-speed effects, the above voltage and power hmits are peak quantities. In addition, it is good engineering practice to provide a safety factor of at least two, so the maximum powers which can be safely transmitted should be limited 10 about half of the above values. If there are reflections on the line or guide, die power capacity is further reduced. In the worsi case, a reflection coefficient magnitude of unity will double the maximum voltage on the line, so the power capacity will be reduced by a factor of four. The power capacity of a line can be increased by pressurizing the line with air or an inert gas, or by using a dielectric. The dielectric strength (Ed) of most dielectrics is greater than that of air, but the power capacity may be primarily limited by the heating of the dielectric due to ohmic loss. Reference: P. A. Rizzi, Microwave Engineering—Passive Circuits, Prentice-Hail, Mew Jersey, 1988. REFERENCES [1] O. Heaviside, Electromagnetic Theory, vol. 1,1893. Reprinted by Dover, New York, 1950. [2] Lord Rayleigh, "On the Passage of Electric Waves Through Tubes," Philos. Mag., vol. 43,pp, 125-132, 1897. Reprinted in Collected Papers, Cambridge Univ. Press, 1903. [3 ] K- S. Packard, "The Origin of Waveguides: A Case of Multiple Rediscovery,'' IEEE Trans. Microwave Theory and Techniques, vol. MTT-32, pp. 961-969, September 1984. [4] R. M. Barrett, "Microwave Printed Circuits—An Historical Perspective," IEEE Trans. Microwave Theory and Techniques, vol. MTT-32, pp. 983-990, September 1984. [5] D. D. Grieg and H. F. Englemann, "Microstrip—A New Transmission Technique for the Kiloroega- cycle Range," Proc. IRE, vol. 40, pp. 1644-1650, December 1952. [6] H. Howe, Jr., Stripline Circuit Design, Artech House, Dedham, Mass., 1974. [7] I. J. Bahl and R. Garg, "A Designer's Guide to Stripline Circuits," Microwaves, January 1978, pp. 90- 96. [8] I. J. Bahl and D. K. Trivedi, "A Designer's Guide to Microstrip Line," Microwaves. May 1977, pp. 174-182. [9] K. C. Gupta, R. Garg, and 1. J. Bahl, Microstrip Lines and Slotlines, Artech House, Dedham, Mass., 1979. PROBLEMS 3.1 Derive Equations (3.5a-d) from equations (3.3) and (3.4). 3.2 Calculate the attenuation due to conductor loss for the TEH mode of a parallel plate waveguide. Chapter 3: Transmission Lines and Waveguides 3.3 Consider a section of air-filled K-bmá waveguide. From the dimensions given in Appendix I, determine die cutoff frequencies of the first two propagating modes. From the recommended operating range given in Appendix I for this guide, determine the percentage reduction in bandwidth that this operating range represents, relative to the theoretical bandwidth for a single propagating mode. 3.4 Compute the TEio mode attenuation, in dB/m, for /ř-band waveguide operating at / = 20 GHz. The waveguide is made from brass, and is filled with a dielectric material having €r = 2,2 and taní = 0.002. 3.5 An attenuator can be made using a section of waveguide operating below cutoff, as shown below. If a = 2.286 cm and the operating frequency is 12 GHz, detennine the required length of the below-culoff section of waveguide to achieve an attenuation of 100 dB between the input and output guides. Ignore the effect of reflections at the step discontinuities. 3.6 Find expressions for the electric surface current density on the walls of a rectangular waveguide for a TEio mode. Why can a narrow slot be cut along the centerline of die broad wall of a rectangular waveguide without perturbing the operation of the guide? (Such a slot is often used in a slotted line for a probe to sample the standing wave field inside the guide.) 3.7 Derive the expression for the attenuation of the TMmn mode of a rectangular waveguide, due to imperfectly conducting walls. 3.8 For the partially loaded rectangular waveguide shown on the next page, solve (3.109) with = 0 to find the cutoff frequency of the TEio mode. Assume a — 2.286 cm, ( = a/2, and e> = 2.25. 3.9 Consider the partially filled parallel plate waveguide shown below. Derive the solution (fields and cutoff frequency) for the lowest order TE mode of this structure. Assume die metal plates are infinitely wide. Can a TEM wave propagate on this structure? Problems 159 3.10 Consider the partially filled parallel plate waveguide shown below. Derive the solution (fields and cutoff frequency) for the TE modes. Can a TEM wave exist in this structure? Ignore fringing fields at the sides, and assume no x dependence. 3.11 Derive Equations (3.1 lOa-d) for the transverse field components in terms of longitudinal fields, in cylindrical coordinates. 3.12 Derive the expression for the attenuation of the TMnm mode in a circular waveguide with finite conductivity. 3.13 Consider a circular waveguide with a = 0.8 cm, and filled with a dielectric material having €r = 2.3. Compute the cutoff frequencies and identify the first four propagating modes. 3.14 Derive the E and H fields of a coaxial line from the expression for the potential given in (3.153). Also find expressions for the voltage and current on the line and the characteristic impedance. 3.15 Derive a transcendental equation for the cutoff frequency of the TM modes of a coaxial waveguide. Using tables, obtain an approximate value of kca for the TM01 mode, if bja = 2. 3.16 Derive an expression for the attenuation of a TE surface wave on a grounded dielectric slab, when the ground plane has finite conductivity, 3.17 Consider the grounded magnetic slab shown below. Derive a solution for the TM surface waves that can propagate on this structure. 3,18 Consider the partially filled coaxial line shown below. Can a TEM wave propagate on this fine? Derive the solution for the TMo^ (no azimuthal variation) modes of this geometry. 160 Chapter 3: Transmission Lines and Waveguides 3.19 Design a stripline transmission line for a 70 £2 characteristic impedance. The ground plane separation is 0.316 cm, and the dielectric constant of the filling material is 2.20. What is the guide wavelength on this transmission line if the frequency is 3,0 GHz? 3.20 Design a microstrip transmission line for a 100 £2 characteristic impedance. The substrate thickness is 0.158 cm, with er = 2.20. What is the guide wavelength on this transmission line if the frequency is 4.0 GHz? 3.21 A 100 Si microstrip line is printed on a substrate of thickness 0.0762 cm, with a dielectric constant of 2.2. Ignoring losses and fringing fields, find the shortest length of this line that appears at its input as a capacitor of 5 pF at 2.5 GHz. Repeat for an inductance of 5 nH. Using a microwave CAD package with a physical model for the microstrip line, compute the actual input impedance seen when losses are included {assume copper conductors and tan S = 0.001). 3.22 A microwave antenna feed network operating at 5 GHz requires a 50 Í2 printed transmission line that is 16A. long. Possible choices are (1) copper microstrip, with d = 0.16 cm, er = 2,20, and tan S = 0.001, or (2) copper stripline, with b = 0.32 cm, er = 2.20, £ = 0.01 mm, and taná = 0.001, Which line should be used, if attenuation is to be minimized? 3.23 Consider the TE modes of an arbitrary uniform waveguiding structure, where the transverse fields are related to řf, as in (3.19). If fft is of the form Hz(x, y, z) = ht(x, y)e~J^, where h^x, y) is a real function, compute the Poynting vector and show that real power flow occurs only in the z direction. Assume that 0 is real, corresponding to a propagating mode. 3.24 A piece of rectangular waveguide is air filled for % < 0 and dielec trie filled for z > 0. Assume that both regions can support only the dominant TE|0 mode, and that a TEio mode is incident on the interface from z < 0. Using a field analysis, write general expressions for the transverse field components of the incident, reflected, and transmitted waves in the two regions, and enforce the boundary conditions at the dielectric interface to find the reflection and transmission coefficients. Compare these results to those obtained with an impedance approach, using ZTE for each region. 3.25 Use the transverse resonance technique to derive a transcendental equation for the propagation constant of the TM modes of a rectangular waveguide thai is air filled for 0 < x < d and dielectric filled for d < x < a. 3.26 Apply the transverse resonance technique to find the propagation constants for the TE surface waves that can be supported by the structure of Problem 3.17. 3.27 An X-band waveguide filled with Teflon is operating at 9.5 GHz. Calculate the speed of light in this material and the phase and group velocities in the waveguide. 3.28 As discussed in the Point of Interest on the power handling capacity of transmission lines, the maximum power capacity of a coaxial line is limited by voltage breakdown, and is given by na~Ei . b ■Qm =-- lo - i/o a where Ed is the field strength at breakdown. Find the value of b/a that maximizes the maximum power capacity and show that the corresponding characteristic impedance is about 30 ÍÍ, Chapter Four Microwave Network Analysis Circuits operating at low frequencies, for which the circuit dimensions are small relative to the wavelength, can be treated as an interconnection of lumped passive or active components with unique voltages and currents defined at any point in the circuit. In this situation the circuit dimensions are small enough so that there is negligible phase change from one point in the circuit to another. In addition, the fields can be considered as TEM fields supported by two or more conductors. This leads to a quasi-static type of solution to Maxwell's equations, and to the well-known Kirchhoff voltage and current laws and impedance concepts of circuit theory [1], As the reader is aware, there exists a powerful and useful set of techniques for analyzing low-frequency circuits. In general, these techniques cannot be directly applied to microwave circuits. Ft is the purpose of the present chapter, however, to show how circuit and network concepts can be extended to handle many microwave analysis and design problems of practical interest. The main reason for doing this is that it is usually much easier to apply the simple and intuitive ideas of circuit analysis to a microwave problem than it is to solve Maxwell's equations for the same problem. In a way, field analysis gives us much more information about the particular problem under consideration than we really want or need. That is, because the solution to Maxwell's equations for a given problem is complete, it gives the electric and magnetic fields at all points in space. But usually we are interested in only the voltage or current at a set of terminals, the power flow through a device, or some other type of "global" quantity, as opposed to a minute description of the response at all points in space. Another reason for using circuit or network analysis is that it is then very easy to modify the original problem, or combine several elements together and find the response, without having to analyze in detail the behavior of each element in combination with its neighbors. A field analysis using Maxwell's equations for such problems would be hopelessly difficult. There are situations, however, where such circuit analysis techniques are an oversimplification, leading to erroneous results. In such cases one must resort to a field analysis approach, using Maxwell's equations. It is part of the education of a microwave engineer to be able to determine when circuit analysis concepts apply, and when they should be cast aside. The basic procedure for microwave network analysis is as follows. We first treat a set of basic, canonical problems rigorously, using field analysis and Maxwell's equations (as we have 161 162 Chapters Microwave Network Analysis done in Chapters 2 and 3, for a variety of transmission line and waveguide problems). When so doing, we try to obtain quantities that can be directly related to a circuit or transmission line parameter. For example, when we treated various transmission lines and waveguides in Chapter 3 we derived the propagation constant and characteristic impedance of the line. This allowed the transmission line or waveguide to be treated as a distributed component characterized by its length, propagation constant, and characteristic impedance. At this point, we can interconnect various components and use network and/or transmission line theory to analyze the behavior of the entire system of components, including effects such as multiple reflections, loss, impedance transformations, and transitions from one type of transmission medium to another (e.g., coax to microstrip). As we will see, a transition between different transmission lines, or a discontinuity on a transmission line, generally cannot be treated as a simple junction between two transmission lines, but must be augmented with some type of equivalent circuit to account for reactances associated with the transition or discontinuity. Microwave network theory was originally developed in the service of radar system and component development at the MIT Radiation Lab in the 1940s, This work was continued and extended at the Polytechnic Institute of Brooklyn by researchers such as E. Weber, N. Marcuvitz, A. A. Oliner, L. B. Felsen, A. Hessel, and others [2]. IMPEDANCE AND EQUIVALENT VOLTAGES AND CURRENTS Equivalent Voltages and Currents At microwave frequencies the measurement of voltage or current is difficult (or impossible), unless a clearly defined terminal pair is available. Such a terminal pair may be present in the case of TEM-type lines (such as coaxial cable, microstrip, or stripline), but does not strictly exist for non-TEM lines (such as rectangular, circular, or surface waveguides). Figure 4.1 shows the electric and magnetic field lines for an arbitrary two-conductor TEM transmission line. As in Chapter 3, the voltage, V, of the + conductor relative to the — conductor can be found as where the integration path begins on the + conductor and ends on the - conductor. It is important to realize that, because of the electrostatic nature of the transverse fields between the two conductors, the voltage defined in (4.1) is unique and does not depend on the shape of the integration path. The total current flowing on the + conductor can be determined from an application of Ampere's law as where the integration contour is any closed path enclosing the 4- conductor (but not 4.1 (4.1) (4.2) 4.1 Impedance and Equivalent Voltages and Currents 163 ----H £ FIGURE 4.1 Electric and magnetic field lines for an arbitrary two-conductor TEM line. the - conductor), A characteristic impedance Zo can then be defined for traveling waves as At this point, after having defined and determined a voltage, current, and characteristic impedance (and assuming we know the propagation constant for the line), we can proceed to apply the circuit theory for transmission lines developed in Chapter 2 to characterize this line as a circuit element. The situation is more difficult for waveguides. To see why, we will look at the case of a rectangular waveguide, as shown in Figure 4.2. For the dominant TEio mode, the transverse fields can be written, from Table 3.2, as (4.3) Ey{x, y, z) j (opia A sin —e~^z = My(x. y)e~^z, (4.4a) 71 a Hxix,y,z) ^—A sin — e-V* = Ahx(x, y)e~^. (4.4b) -v+ b 0 FIGURE 4,2 Electric field lines for the TEio mode of a rectangular waveguide. 164 Chapter 4: Microwave Network Analysis Applying (4.1) to the electric field of (4.4a) gives V = fm I dy. (4.5) 71 y Thus it is seen that this voltage depends on the position, x, as well as the length of the integration contour along the y direction. Integrating from y = 0 to b for x = a/2 gives a voltage that is quite different from that obtained by integrating from y = 0 to b for x = 0, for example. What, then, is the correct voltage? The answer is that there is no "correct" voltage in the sense of being unique or pertinent for all applications. A similar problem arises with current, and also impedance. We will now show how we can define voltages, currents, and impedances that can be useful for non-TEM lines. There are many ways to define equivalent voltage, current, and impedance for waveguides, since these quantities are not unique for non-TEM lines, but the following considerations usually lead to the most useful results [I], [3], [4]: • Voltage and current are defined only for a particular waveguide mode, and are defined so that the voltage is proportional to the transverse electric field, and the current is proportional to the transverse magnetic field, • In order to be used in a manner similar to voltages and currents of circuit theory, the equivalent voltages and currents should be defined so that their product gives the power flow of the mode. • The ratio of the voltage to the current for a single traveling wave should be equal to the characteristic impedance of the line. This impedance may be chosen arbitrarily, but is usually selected as equal to the wave impedance of the line, or else normalized to unity. For an arbitrary waveguide mode with both positively and negatively traveling waves, the transverse fields can be written as Rt(x, y,z) = hxiy)(A+e-^ - A V^) = (l+e~Jßl - / V*), (4.6b) where e and h are the transverse field variations of the mode, and A+, A~ are the field amplitudes of the traveling waves. Since £, and Ht are related by the wave impedance, Zw, according to (3.22) or (3.26), we also have that Ět(x, y, z) = efje, y)(A+e~Jßt + A~ejßz) ě(x, y) C, (4.6a) h(xt y) = z x ěf>, y) (4.7) Equation (4.6) also defines equivalent voltage and current waves as (4.8a) (4.8b) with V+/I+ = V // = Zrj. This definition embodies the idea of making the equivalent voltage and current proportional to the transverse electric and magnetic fields, respectively. 4.1 Impedance and Equivalent Voltages and Currents 165 The proportionality constants for this relationship are C\ = V+/A+ = V~ jA" and C% = I+/A+ = I~/A~, and can be deterrniiied from the remaining two conditions for power and impedance. The complex power flow for the incident wave is given by P+ = \\A+\2fj jjexh*.zds. (4.9) 5 2 S Since we want this power to be equal to (I/2)V+/+*, we have the result that C,CJ = jje xh* -ids, (4.10) where the surface integration is over the cross section of the waveguide. The characteristic impedance is V+ v- Cj 4 zo = — -- -{_ =;. since v+ — C\ A and 1+ = C2A, from (4.6a,b). If it is desired to have Zq = Zwt the wave impedance (Zte or ZTM) of the mode, then -1 = Zw (Zte or Z™). (4.12a) Alternatively, it may be desirable to normalize the characteristic impedance to unity (Zo = 1), in which case we have (4.12b) So for a given waveguide mode, (4.10) and (4.12) can be solved for the constants, C\ and Ci, and equivalent voltages and currents defined. Higher order modes can be treated in the same way, so that a general field in a waveguide can be expressed in the following form: Ět{x, y,z) = T (^-e~^ + ^ej^A ě„U, y), (4.13a) y, z) = JT fM*~%& - JM^A hlt(x, y), (4.13b) where V* and 1* are the equivalent voltages and currents for the rath mode, and C\n and Cin are the proportionality constants for each mode. EXAMPLE 4.1 EQUIVALENT VOLTAGE AND CURRENT FOR A RECTANGULAR WAVEGUIDE Find the equivalent voltages and currents for a TEio mode in a rectangular waveguide. 166 Chapter 4: Microwave Network Analysis Solution The transverse field components and power flow of the TEio rectangular waveguide mode and the equivalent transmission line model of this mode can be written as follows: Waveguide Fields Transmission Line Model Ey = (A+e-Vi + A V^) sin {nx/a) V{z) = V+e~^ + V~e^ Hx = ^-(A+e-^- - A~em) $m(7zx/a) Hz) = I+e~Jfit - I'e^ ZTE V f We now find the constants Cy = V+/A+ = V/A- and C2 = f+/A+ = r/A~ that relate the equivalent voltages V± and currents /± to the field amplimdes, A±. Equating incident powers gives 4Zte 2 2 2 If we choose Z0 = Zte> then we also have that V+ C, Solving for Ci, C2 gives Ci c2 ab i fas Zte V 2 which completes the transmission line equivalence for the TEjo mode. The Concept of Impedance We have used the idea of impedance in several different applications, so it may be useful at this point to summarize this important concept. The term impedance was first used by Oliver Heaviside in the nineteenth century to describe the complex ratio V/ / in AC circuits consisting of resistors, inductors, and capacitors; the impedance concept quickly became indispensable in the analysis of AC circuits. It was then applied to transmission lines, in terms of lumped-element equivalent circuits and the distributed series impedance and shunt admittance of the line. In the 1930s, Schelkunoff recognized that the impedance concept could be extended to electromagnetic fields in a systematic way, and noted that impedance should be regarded as characteristic of the type of field, as well as the medium [2]. And, in relation to the analogy between transmission lines and plane wave propagation, impedance may even be dependent on direction. The concept of impedance, then, forms an important link between field theory and transmission line or circuit theory. 4.1 Impedance and Equivalent Voltages and Currents 167 Below we summarize the various types of impedance we have used so far and their notation: * 7] — vWe — intrinsic impedance of the medium. This impedance is dependent only on the material parameters of the medium, and is equal to the wave impedance for plane waves. * 2W = EtJHt = \/Yw ~ wave impedance. This impedance is a characteristic of the particular type of wave, TEM, TM, and TE waves each have different wave impedances (Ztem, ZTm. Zte), which may depend on the type of line or guide, the material, and the operating frequency. * Zo = 1/Yo = ^JL/C = characteristic impedance. Characteristic impedance is the ratio of voltage to current for a traveling wave on a transmission line. Since voltage and current are uniquely defined for TEM waves, the characteristic impedance of a TEM wave is unique. TE and TM waves, however, do not have a uniquely defined voltage and current, so the characteristic impedance for such waves may be defined in various ways. EXAMPLE 42 APPLICATION OF WAVEGUIDE IMPEDANCE Consider a rectangular waveguide with a = 2.286 cm and b — 1.016 cm (X-band guide), air filled for z < 0 and Rexolite filled (€r = 2.54) for z > 0, as shown in Figure 4.3. If the operating frequency is 10 GHz, use an equivalent transmission line model to compute the reflection coefficient of a TEio wave incident on the interface from z < 0. Solution The propagation constants in the air (z < 0) and the dielectric (z > 0) regions are A =y*S-(^)2 = 158.0 m-'. whereto = 209.4 m-1. The reader may verify that the TEk> mode is the only propagating mode in either waveguide region. Now we can set up an equivalent transmission line for the TEio mode in each waveguide and treat the problem as the reflection of an incident voltage wave at the junction of two infinite transmission lines. TE„, OJlfW ^0 Zoo r-.—> e FIGURE 4.3 Geometry of a partially filled waveguide and its transmission line equivalent for Example 4.2. 168 Chapter 4: Microwave Network Analysis By Example 4.1 and Table 3.2, the equivalent characteristic impedances for the two lines are jp (209.4X377) & 158.0 * & $i 304.1 The reflection coefficient seen looking into the dielectric filled region is then T = Zů" ~ Z± = -0.316. With this result, expressions for the incident, reflected, and transmitted waves can be written in terms of fields, or in terms of equivalent voltages and currents. ■ We now consider the arbitrary one-port network shown in Figure 4.4, and derive a genera] relation between its impedance properties and electromagnetic energy stored ia, and the power dissipated by, the network. The complex power delivered to this network is given by (191): = -£ěx H* 2 Ts ■ dš = P{ + 2ja>(Wm - W& (4.14) where Pt is real and represents the average power dissipated by the network, and Wm and We represent the stored magnetic and electric energy, respectively. Note that the unit normal vector in Figure 4.4 is pointing into the volume. If we define real transverse modal fields, e and A, over the terminal plane of the network such that Mtp* y, z) = V(z)e(x, y)e~m, (4.15a) Ht(x. y, z) = KzMx, y)e-tf\ (4.15b) with a normalization such that e x h • ds = 1, then (4.14) can be expressed in terms of the terminal voltage and current: P= \ í Vrěxhdš= - vr 2Js 2 (4.16) FIGURE 4.4 An arbitrary one-port network. 4.1 Impedance and Equivalent Voltages and Currents 169 Then the input impedance is % RliX-V Vr P te!pCl (417) Zm-R + JX- } - |/|2 - Jj/|2_ p . (4.17) Thus we see that the real part, R, of the input impedance is related to the dissipated power, while the imaginary part, Xy is related to the net energy stored in the network. If the network is lossless, then P£ = 0 and R = 0. Then Zjj, is purely imaginary, with a reactance 4MW,-W,) |/|2 which is positive for an inductive load (Wm > We), and negative for a capacitive load {Wm < W£). Even and Odd Properties of Z{u>) and r(w) Consider the driving point impedance, Z(ca), at the input port of an electrical network. The voltage and current at this port are related as V{co) = Z(a))I(o>). For an arbitrary frequency dependence, we can find the time-domain voltage by taking the inverse Fourier transform of V(o): 1 v{t) = — / V{a))e}™da>. (4.19) lit Since v(t) must be real, we have that u(() = v*{t)f or %Pfcm dto = / V*(a>)e-jmdo>= I V'{-a>)eja" dto, where the last term was obtained by a change of variable from 0 to -co. This shows that V() = V*(co), (4.20) which means that Re{ V{co)) is even in )} is odd in w. Similar results hold for 7(tt>), and for Z(o>) since V*{-oj) = Z*(~tu)J*(-a>) = Z*(-to)Ha>) = V((o) - Z(m)I(vj). Thus, if Z((t>) = R(g>) + j X(a>), then R(a>) is even in co and X((o) is odd in co. These results can also be inferred from (4.17). Now consider the reflection coefficient at the input port; Z(oj) - Z0 R() =-=-. (4.21) Z(u>) + Z0 R(to) + Z0 + jX(co) ^ n-) = \r(-a>)\2, (4.23) which shows that |r(o>)|2 and |r(w)| are even functions of co. This result implies that only even series of the form a + bto2 + ceo4 H----can be used to represent |r(w)| or j r(to)\2. Chapter 4: Microwave Network Analysis IMPEDANCE AND ADMITTANCE MATRICES In the previous section we have seen how equivalent voltages and currents can be defined for TEM and non-TEM waves. Once such voltages and currents have been defined at various points in a microwave network, we can use the impedance and/or admittance matrices of circuit theory to relate these terminal or "port" quantities to each other, and thus to essentially arrive at a matrix description of the network. This type of representation lends itself to the development of equivalent circuits of arbitrary networks, which will be quite useful when we discuss die design of passive components such as couplers and filters. We begin by considering an arbitrary AT-port microwave network, as depicted in Figure 4.5. The ports in Figure 4.5 may be any type of transmission line or transmission line equivalent of a single propagating waveguide mode. (The term port was introduced by H. A. Wheeler in the 1950s to replace the less descriptive and more cumbersome phrase, "two-terminal pair" [3], [2].) If one of the physical ports of the network is a waveguide supporting more than one propagating mode, additional electrical ports can be added to account for these modes. At a specific point on the nth port, a terminal plane, tn, is defined along with equivalent voltages and currents for the incident (V+, /+) and reflected (V~, I~) waves. The terminal planes are important in providing a phase reference for the voltage and current phasors. Now at the nth terminal plane, the total voltage and current is given by V = v+ + v~ (4.24a) (4.24b) as seen from (4,8) when z = 0, The impedance matrix [Z] of the microwave network then relates these voltages and currents: -vr - z2l 12 Z\n ~\ r h~i h Z/vjv - L*n J Vit 3 -+- Jinn UU /- FFGURE 4JÍ An arbitrary N-port microwave network. 4.2 Impedance and Admittance Matrices 171 or in matrix form as [V] = [Z][/]. Similarly, we can define an admittance matrix [Y] as " Yi[ - ■ - Yin Y21 I (4.25) h h l/jvJ YNN\ LVft-J v2 or in matrix form as [/] = mm Of course, the [Z] and [Y] matrices are the inverses of each other: [Y] = [Z]-K Note that both the [Z] and [Y] matrices relate the total port voltages and currents. From (4.25), we see that Z,-; can be found as (4.26) (4.27) Z- (4.28) 4=0 forJt^j In words, (4.28) states that Zl} can be found by driving port j with the current open-circuiting all other ports (so 7* = 0 for k j), and measuring the open-circuit voltage at port t. Thus, Za is the input impedance seen looking into port i when all other ports are open-circuited, and Z;j is the transfer impedance between ports i and j when all other ports are open-circuited. Similarly, from (4.26), Yi} can be found as (4.29) Vt=0 for Jt# j which slates that Yij can be detennined by driving port j with the voltage Vj, short-circuiting all other ports (so V* = 0 for k £ j), and measuring the short-circuit current at port i. In general, each Zjj or element may be complex. For an arbitrary N-port network, the impedance and admittance matrices are N x N in size, so there are 2N2 independent quantities or degrees of freedom. In practice, however, many networks are either reciprocal or lossless, or both. If the network is reciprocal (not containing any nonreciprocal media such as ferrites or plasmas, or active devices), we will show that the impedance and admittance matrices are symmetric, so that Z^ = Z^, and Y^ = Ifp. If the network is lossless, we can show that all the Zij or Y^- elements are purely imaginary. Either of these special cases serves to reduce the number of independent quantities or degrees of freedom that an N-port network may have. We now derive the above characteristics for reciprocal and lossless networks. Reciprocal Networks Consider the arbitrary network of Figure 4.5 to be reciprocal (no active devices, ferrites, or plasmas), with short circuits placed at all terminal planes except those of ports I and 2. Now let £„, Ha and Ef,, H\7 be the fields anywhere in the network due to two independent sources, a and b, located somewhere in the network. Then the reciprocity theorem of (1.156) 172 Chapters Microwave Network Analysis states that j)Ea x Hbds = jtEbx Ha ■ ds, (430) where we will take S as the closed surface along the boundaries of the network and through the terminal planes of the ports. If the boundary walls of the network and transmission lines are metal, then = 0 on these walls (assuming perfect conductors). If the network or the transmission lines are open structures, like microstrip or slotline, the boundaries of the network can be taken arbitrarily far from the lines so that E^ is negligible. Then the only nonzero contribution to the integrals of (4.30) come from the cross-sectional areas of ports 1 and 2. From Section 4.1, the fields due to sources a and b can be evaluated at the terminal planes t\ and t2 as E\a = = E\b = ViKi Ho, = hbhi Ela = Ěm - liJii Ě2b = Hzb - hbhi, (4.31) where ě|, h \ and e2, h2 are the transverse modal fields of ports 1 and 2, respectively, and the Vs and /s are the equivalent total voltages and currents. (For instance, E\b is the transverse electric field at terminal plane t\ of port 1 due to source b.) Substituting the fields of (4.31) into (4.30) gives (Vja/a, - VWi„) / ěi x hj ■ dš + - V2l>I2a) f ě2 K h ■ dš = 0, (4.32) where S\,S2 are the cross-sectional areas at the terminal planes of ports 1 and 2. As in Section 4.1, the equivalent voltages and currents have been defined so that the power through a given port can be expressed as VT/l; then comparing (4.31) to (4.6) implies that C\ = C2 ~ I for each port, so that / ěi x h\ • dš = I ě2 x h2 - dš — 1. (4,33) JS, J Si This reduces (4.32) to Vi0 fy - VihILa + Via lib ~ Vib lia = 0. (4.34) Now use the 2 x 2 admittance matrix of the (effectively) two-port network to eliminate the /s: h = Yl]V] + Yl2V2r l2 ~ Y2\V\ + Y22Vi, Substitution into (4.34) gives - VuV^KYn - Y2l) = 0. (4.35) Since the sources a and b are independent, the voltages V^, Vib, V2a, and can take on arbitrary values. So in order for (4.35) to be satisfied for any choice of sources, we must have T|2 = Y2\, and since Oie choice of which ports are labeled as 1 and 2 is arbitrary, we have the general result that (4.36) Then if [Y\ is a symmetric matrix, its inverse, [Z], is also symmetric. 4.2 Impedance and Admittance Matrices 173 Lossless Networks Now consider a reciprocal lossless Alport junction; we will show that the elements of the impedance and admittance matrices must be pure imaginary. If the network is lossless, then the net real power delivered to the network must be zero. Thus, Re{Fav) = 0, where Pav = \ivwr = \az][iwr = \u Y[z][i r (We have used the result from matrix algebra that ([A][S])' = [B]'[A]'.) Since the 7„s are independent, we must have the real part of each self term (/„ Z„„/*) equal to zero, since we could set all port currents equal to zero except for the nth current. So, Re{I„Zml*} = |/„|2Re{{Zflfl} = 0, or Re{2ww} = 0. (4.38) Now let all port currents be zero except for Im and /„. Then (4.37) reduces to Re{(/„/; + /„/;)Zmnj = 0, since Zmrl = Znm. But (/„ /* + lm I*) is a purely real quantity which is, in general, nonzero. Thus we must have that Re(Z,,„) =0 (4.39) Then (4.38) and (4.39) imply that Ref Zm„} = 0 for any m, n. The reader can verify that this also leads to an imaginary [Y] matrix. EXAMPLE 4 J EVALUATION OF IMPEDANCE PARAMETERS Find the Z parameters of the two-port T-network shown in Figure 4.6. Solution From (4.28), Zj j can be found as the input impedance of port l when port 2 is open-circuited: Z -V] = zA + zc. -WW -WW + Port 2 FIGURE 4.6 A two-port T-nerwork. 174 Chapter 4: Microwave Network Analysis The transfer impedance Z\i can be found measuring the open-circuit voltage at port 1 when a current li is applied at port 2. By voltage division, 7 -Vx % Zc /,=o h ZB 4- Zc = Zr. The reader can verify that Z21 = Z\%, indicating that the circuit is reciprocal. Finally, Z22 is found as Vi Z11 — — = ZB + Zc- 4.3 THE SCATTERING MATRIX We have already discussed the difficulty in denning voltages and currents for non-TEM Unes. In addition, a practical problem exists when trying to measure voltages and currents at microwave frequencies because direct measurements usually involve the magnitude (inferred from power) and phase of a wave traveling in a given direction, or of a standing wave. Thus, equivalent voltages and currents, and the related impedance and admittance matrices, become somewhat of an abstraction when dealing with high-frequency networks. A representation more in accord with direct measurements, and with the ideas of incident, reflected, and transmitted waves, is given by the scattering matrix. Like the impedance or admittance matrix for an iV-port network, the scattering matrix provides a complete description of the network as seen at its N ports. While the impedance and admittance matrices relate the total voltages and currents at the ports, the scattering matrix relates the voltage waves incident on the ports to those reflected from the ports. For some components and circuits, the scattering parameters can be calculated using network analysis techniques. Otherwise, the scattering parameters can be measured directly with a vector network analyzer; a photograph of a modem network analyzer is shown in Figure 4.7. Once the scattering parameters of the network are known, conversion to other matrix parameters can be performed, if needed. Consider the /V-port network shown in Figure 4.5, where V+ is the amplitude of the voltage wave incident on port «, and V~ is the amplitude of the voltage wave reflected from port n. The scattering matrix, or [S] matrix, is defined in relation to these incident and reflected voltage waves as ni- -vf - S|2 = • m -V,v- [V "] = [S][V+]. A specific element of the [S] matrix can be determined as % - 77+ (4.40) (4.41) Vf =0 for Jt#; In words, (4.41) says that 5,-; is found by driving port j with an incident wave of voltage Vf, and measuring the reflected wave amplitude, V~, coming out of port i. The incident waves on all ports except the jth port are set to zero, which means that all ports should be terminated in matched loads to avoid reflections. Thus, Su is the reflection coefficient seen looking 4.3 The Scattering Matrix 175 FIGURE 4,7 A photograph of the Hewlett-Packard HP8510B Network Analyzer, This test instrument is used to measure the scattering parameters (magnitude and phase) of a one- or two-port microwave network from 0.05 GHz to 26.5 GHz. Built-in microprocessors provide error correction, a high degree of accuracy, and a wide choice of display formats. This analyzer can also perform a fast Fourier transform of the frequency domain data to provide a time domain response of the network under test. Courtesy of Agilent Technologies, Santa Rosa, Calif. into port i when all other ports are terminated in matched loads, and Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads. EXAMPLE 4.4 EVALUATION OF SCATTERING PARAMETERS Find the S parameters of the 3 dB attenuator circuit shown in Figure 4.8. Solution From (4.41), S\\ can be found as the reflection coefficient seen at port 1 when port 2 is terminated in a matched load (Zo = 50 fi): aáP - Zo v?=a z£ + zo IT) Zo on port 2 Chapter 4: Microwave Network Analysis 8.5<5 Ü 8.56 Ü -A/Wv-o Pan i4i.itn Pon o- FIGURE 4.8 A matched 3 dB attenuator with a 50 Si characteristic impedance (Example 4.4). but, Z|J = 8.56+ [141.8(8.56 + 50)]/(141.8 + 8.56 + 50) = 50 Q, so Sn = 0. Because of the symmetry of the circuit, 522 = 0. S21 can be found by applying an incident wave at port 1, V,+, and measuring the outcoming wave at port 2, V*2_. This is equivalent to the transmission coefficient from port 1 to port 2: ~ TFT From the fact that Si 1 = 5^2 — 0, we know that Vj~ = 0 when port 2 is terminated in Zo = 50 £2, and that V2+ = 0. In this case we then have that V,+ = Vi and V{ = V2 So by applying a voltage Vi at port 1 and using voltage division twice we find V2~ = V2 as the voltage across the 50 Q load resistor at port 2: where 41.44 = 141.8(58.56)/(141.8 + 5S.56) is the resistance of the parallel combination of the 50 £1 load and the 8,56 Q resistor with the 141.8 Q resistor. Thus, Si2 = S21 = 0.707. If the input power is |Vj+|2/2Zo, then the output power is {Vf^/ZZo = |S2l^+|V2Zo = \S21\2/2Z0\V+\2 = |V,+ |2/4Z0, which is one-half (-3 dB) of the input power. ■ We now show how the [S] matrix can be determined from the [Z] (or [YJ) matrix, and vice versa. First, we must assume that the characteristic impedances, Zcm* of all the ports are identical. (This restriction will be removed when we discuss generalized scattering parameters.) Then for convenience, we can set Zoi, = 1. From (4,24) the total voltage and current at the nth port can be written as % = V+ + V". (4,42a) * = # - K * Vn+ - V-, (4.42b) Using the definition of [ZJ from (4.25) with (4.42) gives mm = [ZW+] - [Z][V-] = [VI = [V+] + [V"], which can be rewritten as ([Z] + [UUV-] = <[Z] - [t/])[V+], (4.43) 4.3 The Scattering Matrix 177 where [I/] is the unit, or identity, matrix defined as 1 0 •■■ 0' o i o Comparing (4.43) to (4.40) suggests that [S] = ([Z] + [£/]r'([Z] -[£/]), (4.44) giving the scattering matrix in terms of the impedance matrix. Note that for a one-port network (4.44) reduces to Su = zu - 1 in agreement with the result for the reflection coefficient seen looking into a load with a normalized input impedance of Zu■ To find [Z] in terms of [5], rewrite (4.44) as [Z][S] + [U][S] = [Z] - [U\ and solve for [Z] to give [Z} = ([u] + \m[u]-[S])-1. (4.45) Reciprocal Networks and Lossless Networks As we discussed in Section 4.2, the impedance and admittance matrices are symmetric for reciprocal networks, and purely imaginary for lossless networks. Similarly, the scattering matrices for these types of networks have special properties. We will show that the [5] matrix for a reciprocal network is symmetric, and that the [S] matrix for a lossless network is unitary. By adding (4.42a) and (4.42b) we obtain vn+ = \{vn + 45, or [V+}=^[Z] + [U})[I\. (4.46a) By subtracting (4.42a) and (4.42b) we obtain V-=i(VW„), or [V-J = i([Z] - [£/])[/]. (4.46b) Eliminating [I ] from (4.46a) and (4.46b) gives [v-] = ([z] - muz] + [U]rl[v+i so that [S] = ([Z] - [U])(\Z] + [[/])-1. (4.47) Taking the transpose of (4.47) gives [£]' = {([Z] + [[/])-' )'([zi -[[/])'. Now [(/] is diagonal, so [U]' = [U], and if the network is reciprocal, [Z] is symmetric 178 Chapter 4: Microwave Network Analysis so that [Z]' = [Z]. The above then reduces to [S]! =([Z] + [U])~l([Z] -[£/]), which is equivalent to (4.44). We have thus shown that [51 = [5]', (4.48) for reciprocal networks. If the network is lossless, then no real power can be delivered to the network. Thus, if the characteristic impedances of all the ports are identical and assumed to be unity, the average power delivered to the network is Pav = ^Re{[v-J'[/f} = ^ReKtV+j' + [V-VXIV+T - [V]*)} = l[v+rtv4r - l-iv-y[v-r = o, (4.49) since the terms -[V+]'[V~T + [^~]'[V+]+ are of the form A - A*, and so are purely imaginary. Of the remaining terms in (4.49), (l/2)[ V+]'[ V"]* represents the total incident power, while (1 /2)[ V~]f[V"]* represents the total reflected power, So for a lossless junction, we have the intuitive result that the incident and reflected powers are equal; [V+]r[V+r = [V-]'[V-r. (4.50) Using [V] = [S][V+] in (4.50) gives [V+Y[V+]* = [V+]'[SY[Sf[V+]\ so that, for nonzero m'[sr = [ui or t^r = U5]'}"'. (4.51) A matrix that satisfies the condition of (4.51) is called a unitary matrix. The matrix equation of (4.51) can be written in summation form as J^SkiS^^iij, for all i, j, (4.52) where &ij = 1 if i = j and % = 0 if i ^ j is the Kronecker delta symbol. Thus, if i = j (4.52)reduces to £^5** =1, (4.53a) t-i while if i # j (4.52) reduces to £st(^. = 0, fori^y. (4.53b) In words, (4.53a) states that the dot product of any column of \S] with the conjugate of that column gives unity, while (4.53b) states that the dot product of any column with the 4.3 The Scattering Matrix 179 conjugate of a different column gives zero (orthogonal). If the network is reciprocal, then [S] is symmetric, and the same statements can be made about the rows of the scattering matrix. EXAMPLE 4.5 APPLICATION OF SCATTERING PARAMETERS A two-port network is known to have the following scattering matrix: .15Z0° 0.85Z-450 Lo.s ".85/45° 0.2^0° Determine if the network is reciprocal, and lossless. If port two is terminated with a matched load, what is the return loss seen at port 1? If port two is terminated with a short circuit, what is the return loss seen at port 1? Solution Since [S] is not symmetric, the network is not reciprocal. To be lossless, the S parameters must satisfy (4.53). Taking the first column (r = 1 in (4.53a)) gives |Snl2 + |%I2 = (0.15)2 + (0.85)2 = 0.745 ? 1, so the network is not lossless. When port 2 is terminated with a matched load, the reflection coefficient seen at port 1 is T = 5n — 0.15. So the return loss is RL = -20log |r| = -201og(0.15) = 16.5 dB. When port 2 is terminated with a short circuit, the reflection coefficient seen at port 1 can be found as follows. From the definition of the scattering matrix and the fact that V2+ = - V2~ (for a short circuit at port 2), we can write V- = 5,, V? + 5,2V+ = SUV+ - 5,2V~, Vj = S22 F,+ + S22 V? = S21V+ - S22 V2~. The second equation gives y-= S2i y+ Dividing the first equation by V* and using the above result gives the reflection coefficient seen at port 1 as = Qi5_(0.85.-45-)(0.85Z45°)=_Q452 I +0.2 So the return loss is RL = -20 log | T| = -20 log(0.452) = 6.9 dB. ■ An important point to understand about 5 parameters is that the reflection coefficient looking into port n is not equal to Snn, unless all other ports are matched (this is illustrated in the above example). Similarly, the transmission coefficient from port m 10 port n is not equal to S„m, unless all other ports are matched. The S parameters of a network are properties only of the network itself (assuming the network is linear), and are defined under the condition 180 Chapter 4: Microwave Network Analysis vyIJUV Pon ! iV-pon network [SI [SI I Port n FIGURE 49 Shifting reference planes for an W-port network. that all ports are matched. Changing the terminations or excitations of a network does not change its 5 parameters, but may change the reflection coefficient seen at a given port, or the transmission coefficient between two ports. A Shift in Reference Planes Because the S parameters relate amplitudes (magnitude and phase) of traveling waves incident on and reflected from a microwave network, phase reference planes must be specified for each port of the network. We now show how die S parameters are transformed when the reference planes are moved from their original locations. Consider the Alport microwave network shown in Figure 4.9, where the original terminal planes are assumed to be located at z„ — 0 for the nth port, and where z„ is an arbitrary coordinate measured along the transmission line feeding the nth port. The scattering matrix for the network with this set of terminal planes is denoted by [51. Now consider a new set of reference planes defined at zn — for the nth port, and let the new scattering matrix be denoted as [5']. Then in terms of the incident and reflected port voltages we have that =[S][V+], (4.54a) [V'~] = [ (4.54b) where die unprimed quantities are referenced to the original terminal planes at zn — 0, and the primed quantities are referenced to the new terminal planes at zn = 4- Now from the theory of traveling waves on lossless transmission lines we can relate the new wave amplitudes to the original ones as C = V„V\ (4.55a) v;~ = V-e-^. (4.55b) where 6n = fi^^ is the electrical length of the outward shift of the reference plane of port n. 4.3 The Scattering Matrix 181 Writing (4.55) in matrix form and substituting into (4.54a) gives 0 0 [V-] = [S] u -j92 iv'+] Multiplying by the inverse of the first matrix on the left gives r 0 e-jh 0 e Comparing with (4.54b) shows that IS) = 0 0 0 [S] . 0 0 [V"+], (4.56) which is the desired result. Note that S^,„ = e~2^"Smt, meaning that the phase of S„„ is shifted by twice the electrical length of the shift in terminal plane n, because the wave travels twice over this length upon incidence and reflection. Generalized Scattering Parameters So far we have considered the scattering parameters for networks with the same characteristic impedance for all ports. This is the case in many practical situations, where the characteristic impedance is often 50 ÍÍ. In other cases, however, the characteristic impedances of a multiport network may be different, which requires a generalization of the scattering parameters as defined up to this point. Consider the A7-port network shown in Figure 4.10, where Z^ is the (real) characteristic impedance of the nth port, and and V~, respectively, represent the incident and reflected voltage waves at port n. In order to obtain physically meaningful power relations in terms TAP— vt,ai y Port) *-W m\ Alport network FIGURE 4.10 An JV-port network with different characteristic impedances, 182 Chapters Microwave Network Analysis of wave amplitudes, we must define a new set of wave amplitudes as a„ = V+fy/ŽZ, (4.57a) bn = V-/^, (4.57b) where a„ represents an incident wave at the nth port, and bn represents a reflected wave from that port [1], [5], Then from (442a,b) we have that % = V+ + V- = + bn), (4.5Sa) fn = ^(V; - V-) = -~] m iS][al (4.60) where the í , jdi element of the scattering matrix is given by h (4.61) fft=0 for k&j and is analogous to the result of (4,41) for networks with identical characteristic impedance at all ports. Using (4.57) in (4.61) gives (4.62) which shows how the S parameters of a network with equal characteristic impedance (Vf J V t with = 0 for k ^ j) can be converted to a network connected to transmission lines with unequal characteristic impedances. POINT OF INTEREST: The Vector Network Analyzer The S parameters of passive and active networks can be measured with a vector network analyzer, which is a two- (or four-) channel microwave receiver designed to process the magnitude and phase of the transmitted and reflected waves from the network. A simplified block diagram of a network analyzer similar to the HP8510 system is shown below. In operation, the RF source is usually set to sweep over a specified bandwidth. A four-port reflectometer samples the incident, reflected, and transmitted RF waves; a switch allows the network to be driven from either port 1 or port 2. Four dual-conversion channels convert these signals to 100 kHz IF frequencies, which are then detected and converted to digital form, A powerful internal computer is used to calculate 4.4 The Transmission (ABCD) Matrix 183 20 MHz 1ST IF 100 kHz 2ND IF X v Sample i ajid hold X A/D Y can\. Computer processing and error correctioD SE Display RF source and test set IF processing Digital processing and display the magnitude and phase of the S parameters, or other quantities that can be derived from the S parameters, such as SWR, return loss, group delay, impedance, etc. An important feature of this network analyzer is the substantial improvement in accuracy made possible with error correcting software. Errors caused by directional coupler mismatch, imperfect directivity, loss, and variations in the frequency response of the analyzer system are accounted for by using a twelve-term error model and a calibration procedure. Another useful feature is the capability to determine the time domain response of the network by calculating the inverse Fourier transform of the frequency domain data. THE TRANSMISSION (ABCD) MATRIX The Z, Y, and S parameter representations can be used to characterize a microwave network with an arbitrary number of ports, but in practice many microwave networks consist of a cascade connection of two or more two-port networks. In this case it is convenient to define a 2 x 2 transmission, or ABCD matrix, for each two-port network. We will then see that the ABCD matrix of the cascade connection of two or more two-port networks can be easily found by multiplying thtABCD matrices of the individual two-ports. The ABCD matrix is defined for a two-port network in terms of the total voltages and currents as shown in Figure 4.1 la and the following: Vi =AV2 + Bh, h = CV2 + £>/2, or in matrix form as (4.63) It is important to note from Figure 4.1 la that a change in the sign convention of Ii has been made from our previous definitions, which had h as the current flowing into port 2. 184 Chapters Microwave Network Analysis Port l A B C D ° i (a) FIGURE 4.11 (a) A two-port network; (b) a cascade connection of two-port networks. The convention that h flows out of port 2 will be used when dealing with ABCD matrices so that in a cascade network h will be the same current that flows into the adjacent network, as shown in Figure 4.1 lb. Then the left-hand side of (4.63) represents the voltage and current at port 1 of the network, while the column on the right-hand side of (4.63) represents the voltage and current at pott 2. In the cascade connection of two two-port networks shown in Figure 4.1 lb, we have that RHS Wit} [l|=K lift} Substituting (4.64b) into (4,64a) gives which shows that the ABCD matrix of the cascade connection of the two networks is equal to the product of the ABCD matrices representing the individual two-ports. Note that the order of multiplication of the matrix must be the same as the order in which the networks are arranged, since matrix multiplication is not, in general, commutative. The usefulness of the ABCD matrix representation lies in the fact that a library of ABCD matrices for elementary two-port networks can be built up, and applied in building-block fashion to more complicated microwave networks that consist of cascades of these simpler two-ports. Table 4.1 lists a number of useful two-port networks and their ABCD matrices. EXAMPLE 4.6 EVALUATION OF ABCD PARAMETERS Find the ABCD parameters of a two-port network consisting of a series impedance Z between ports 1 and 2 (the first entry in Table 4.1). Solution From the defining relations of (4.63), we have that v2\hJ which indicates that A is found by applying a voltage V\ at port 1, and measuring TABLE 4.1 4.4 The Transmission (ABCD) Matrix The ABCD Parameters of Some Useful Two-Port Circuits 185 Circuit ABCD Parameters —o —o -o A = 1 C = 0 A = 1 C = Y A = cos j8£ C= /Tosin^f A = iV C = 0 0 = Z D= 1 5 = 0 B = jZ0sin/3£ D = cos ßt B D 0 1 N o-=E 3 A = A C c = Yt + r2 + i Zi B D B D 1 = Z, + Z2 + z,z2 die open-circuit voltage V2 at port 2. Thus, A = 1. Similarly, -I D = r = 0, Relation to Impedance Matrix Knowing the Z parameters of a network, one can determine the ABCD parameters. Thus, from the definition of the ABCD parameters in (4,63), and from the defining relations for the Z parameters of (4.25) for a two-port network with I2 to be consistent with the sign convention used witii ABCD parameters, Vj = l\Z\\ — hZ\2, Vi — I[Z2\ — /2Z22, (4.66a) (4.66b) 186 Chapter 4: Microwave Network Analysis we have that A = - B = D = -- /iZ„ v2 /iZ2l Vi /[Zu - j h Zu /] Zn z„ - ; z,]2 — II h h=o /,Z2I /l /2Z22/Z2 v:=o /2 ~ Zn/Z2], = Zi.V- - Z 12 vi=o -21 = Z22/Z21. (4.67a) (4.67b) (4.67c) (4.67d) If the network is reciprocal, then Zn = Z2i and (4.67) can be used to show that AD - BC = 1. Equivalent Circuits for Two-Port Networks The special case of a two-port microwave network occurs so frequently in practice that it deserves further attention. Here we will discuss the use of equivalent circuits to represent an arbitrary two-port network. Useful conversions for two-port network parameters are given in Table 4.2. Figure 4.12a shows a transition between a coaxial line and a microstrip line, and serves as an example of a two-port network. Terminal planes can be defined at arbitrary points on the two transmission lines; a convenient choice might be as shown in the figure. But because of the physical discontinuity in the transition from a coaxial line to a microstrip line, electric and/or magnetic energy can be stored in the vicinity of the junction, leading to reactive effects. Characterization of such effects can be obtained by measurement or by theoretical analysis (although such an analysis may be quite complicated), and represented by the two-port "black box" shown in Figure 4.12b. The properties of the transition can then be expressed in terms of the network parameters (Z, Y, S, or ABCD) of the two-port network. This type of treatment can be applied to a variety of two-port junctions, such as transitions from one type of transmission line to another, transmission line discontinuities such as step changes in width, or bends, etc. When modeling a microwave junction in this way, it is often useful to replace the two-port "black box" with an equivalent circuit containing a few idealized components, as shown in Figure 4.12c. (This is particularly useful if the component values can be related to some physical features of the actual junction.) There is an unlimited number of ways in which such equivalent circuits can be defined; we will discuss some of the most common and useful types below. As we have seen from the previous sections, an arbitrary two-port network can be described in terms of impedance parameters as V, =Zn/, + Z12/2, V2 = Z2,/i + Z22/2, or in terms of admittance parameters as h ~ YjyVx + YnVi- (4.68a) (4.68b) TABLE 4.2 Conversions Between Two-Port Network Parameters 5 Z y AßCD Si. (Zjj — Z0XZ22 + Z[i) — ZijZji (ro-hiX^+h^ + i^hi /t + fl/Zo - CZ« - Z> AZ AT A + BjZa + CZo + D 2Z|jZö -2f|2ro 2S.AD-BC) A2 AK A + B/Zo + CZo + ß ft 2Zii Zo 2 AZ AK A + fl/Zo + CZc 4- D $12 & (Zi, + ZflXZu - Zo) - ZuZj, a0 + i'LiXJ'o-y22) + /iifji -A + B/Za - CZ(t + D AZ Ay A + 3/Zv + CZo + D Zu „ (1 + 5,,X1 - Sik>-|-^i25^ ^(1-511X1-52J>-S1SS3I 7., ha A ^11 in C Z|2 z,2 AD - BC ü-SriHl-S^-StsSzi m C *(l - J„)(l -Sx.)-Sl2Si, Zai in ] C % ■ ;'Ü -S;iXl - fc) - SiiJfy Z22 y,, O ifl c h. (1 -5„)(l + S2j) + 5|2Sji Z22 |Z| hi D B h2 -Zu & BC - AD |Z| B °n + S,1Kl+5^)-5,2S2l -Z2I |Z| rii -] 3 r (I + SmXI -Sn) + SnSn "u + SiiXl+SyJ-Sis^i in IZI hj A a (t+S-.i.W SlU+ $,!$! Zu -h. A 1 Zii h! ft 5 „ (I +5,,XI + SK)-SnS2i m z2i -1. hi B C 1 (1 -£„XI 1 -Ifl C Zu 25; ] ZjT hi +5i2) + 5|-S,| Z21 -hi D Zji hi |Z| = Z|,ZH — ZijZji; |f 1 = Y\ t hi - hshi ; Ar = ifn +y0)(i',j +r0)_ rlir"n; AZ = (Zu +Z0KZ1J + Z0)-ZljZii; ft = 1/Zo 188 Chapter 4: Microwave Network Analysis Micro&trip /line IS] Coaxial line Micmsmp line (b) §fc -T- £j .A'Y-y'Y. (0 FIGURE 4.12 A coax-to-nucrosrrip transition and equivalent circuit representations, (a) Geometry of the transition, (b) Representation of the transition by a "black box." (c) A possible equivalent circuit for the transition [6]. tf the network is reciprocal, then Zl2 = Z2i and Yu = Yi\. These representations lead naturally to the T and jt equivalent circuits shown in Figure 4.13a and 4.13b. The relations in Table 4.2 can be used to relate the component values to other network parameters. Other equivalent circuits can also be used to represent a two-port network. If the network is reciprocal, there are six degrees of freedom (the real and imaginary parts of three matrix elements), so the equivalent circuit should have six independent parameters. Anonreciprocal network cannot be represented by a passive equivalent circuit using reciprocal elements. Z|] - Zl2 (a] Tii + fia '\2 + ^12 (b) FIGURE 4.13 Equivalent circuits for a reciprocal two-port network, (a) T equivalent, (b) ti equivalent. 4.5 Signal Flow Graphs 189 If the network is lossless, which is a good approximation for many practical two-port junctions, some simplifications can be made in the equivalent circuit. As was shown in Section 4,2, the impedance or admittance matrix elements are purely imaginary for a lossless network. This reduces the degrees of freedom for such a network to three, and implies that the T and tt equivalent circuits of Figure 4.13 can be constructed from purely reactive elements. SIGNAL FLOW GRAPHS We have seen how transmitted and reflected waves can be represented by scattering parameters, and how the interconnection of sources, networks, and loads can be treated with various matrix representations. In this section we discuss the signal flow graph, which is an additional technique that is very useful for the analysis of microwave networks in terms of transmitted and reflected waves. We first discuss the features and the construction of the flow graph itself, and then present a technique for the reduction, or solution, of the flow graph. The primary components of a signal flow graph are nodes and branches: • Nodes: Each port, i, of a microwave network has two nodes, a{ and b{. Node a, is identified with a wave entering port i, while node bi is identified with a wave reflected from port i. The voltage at a node is equal to the sum of all signals entering that node. • Branches: A branch is a directed path between two nodes, representing signal flow from one node to another. Every branch has an associated S parameter or reflection coefficient. At this point it is useful to consider the flow graph of an arbitrary two-part network, as shown in Figure 4.14. Figure 4.14a shows a two-port network with incident and reflected waves at each port, and Figure 4.14b shows the corresponding signal flow graph representation. The flow graph gives an intuitive graphical illustration of the network behavior. For example, a wave of amplitude a\ incident at port 1 is split, with part going through Sii and out port 1 as a reflected wave and part transmitted through S21 to node b2. At node f>2, the wave goes out port 2; if a load with nonzero reflection coefficient is connected at port 2, this wave will be at least partly reflected and reenter the two-port network at node fl2 Part of the wave can be reflected back out port 2 via ^d part can be transmitted out port 1 through Sl2. & id illlr w FIGURE 4.14 The signal flow graph representation of a two-port network, (a) Definition of incident and reflected waves, (b) Signal flow graph. 190 Chapter 4: Microwave Network Analysis (a) -o- FIGURE 4.15 The signal flow graph representations of a one-port network and a source, (a) A one-port network and its flow graph, (b) A source and its flow graph. Two other special networks, a one-port network and a voltage source, are shown in Figure 4.15 along with their signal flow graph representations. Once a microwave network has been represented in signal flow graph form, it is a relatively easy matter to solve for the ratio of any combination of wave amplitudes. We will discuss how this can be done using four basic decomposition rules, but the same results can also be obtained using Mason's rule from control system theory. Decomposition of Signal Flow Graphs A signal flow graph can be reduced to a single branch between two nodes using the four basic decomposition rules below, to obtain any desired wave amplitude ratio, • Rule 1 (Series Rule). Two branches, whose common node has only one incoming and one outgoing wave (branches in series), may be combined to form a single branch whose coefficient is the product of the coefficients of the original branches. Figure 4.16a shows the flow graphs for this rule. Its derivation follows from the basic relation that V^ = 5^ = 5^2^. (4.69) • Rule 2 (Parallel Rule), Two branches from one common node to another common node (branches in parallel) may be combined into a single branch whose coefficient is the sum of the coefficients of the original branches. Figure 4.16b shows the flow graphs for this rule. The derivation follows from the obvious relation that fi = SaVy + Sb Vi = (Sa + StWi. (4.70) • Rule 3 (Self-Loop Rule). When a node has a self-loop (a branch that begins and ends on the same node) of coefficient S, the self-loop can be eliminatedby multiplying coefficients of the branches feeding that node by 1/(1 - 5). Figure 4.16c shows the flow graphs fortius rule, which can be derived as follows. From the original network we have that ^-52,^ + 5,2^, (4,71a) V3 = S32V2. (4.71b) 4,5 Signal Flow Graphs 191 *2I J32 ->— 0-V. S2\Syj V, V, V, v", V3 V, FIGURE 4.16 Decomposition rules, (a) Series rule, (b) Parallel rule, (c) Self-loop rule, (d) Splitting rule. Eliminating V2 gives Vi = -^-Vu (4.72) 1 — 022 which is seen to be the transfer function for the reduced graph of Figure 4,16c, * Rule 4 (Splitting Rule). A node may be split into two separate nodes as long as the resulting flow graph contains, once and only once, each combination of separate (not self loops) input and output branches that connect to the original node. This rule is illustrated in Figure 4,16d, and follows from the observation that V4 = 042V2 = 521542 Vis (4-73) in both the original flow graph and the flow graph with the split node. We now illustrate the use of each of these rules with an example. EXAMPLE 4.7 APPLICATION OF SIGNAL FLOW GRAPH Use signal flow graphs to derive expressions for and roul for the two-port network shown in Figure 4.17. 192 Chapter 4: Microwave Network Analysis FIGURE 4.17 A tenninared two-port network. V,o-► FIGURE 4.18 Signal flow path for the two-port network with general source and load impedances of Figure 4.17. Solution The signal flow graph for the circuit of Figure 4.17 is shown in Figure 4.18, In terms of node voltages, rin is given by the ratio b\fa\. The first two steps of the required decomposition of the flow graph are shown in Figures 4.19a,b, from which the desired result follows by inspection: - b\ S\2S2\Tt 1 m = — = on +--„ _ ■ &\ 1 - SuTt Next, Fou, is given by the ratio 62/^2 ■ The first two steps for this decomposition are shown in Figures 4.19c,d. The desired result is 1 out — — — i>2l + ~-7. 77" • _ ai 1-iSnT, ■ M J2I S11 \ bt (a) >2\ (I I I — SiJFf ■o-*--a (b) a-*-9- FIGURE 4.19 Decompositions of the flow graph of Figure 4.18 to rind rin = b\fa\ and T^, = ki/ai- (a) Using Rule 4 on node ai, (b) Using Rule 3 for the self-loop at node b2, (c) Using Rule 4 on node bv. (d) Using Rule 3 for the self-loop at node at. 4.5 Signal Flow Graphs 193 I I Measurement plane for port 1 Reference plane for device port 1 Am Cm Error box Device under IS) I2 ] (a) >12 -I -HIT83 FIGURE 4.21a Block diagram and signal flow graph for the Thru connection. for ports 1 and 2, since they are symmetrically connected as shown in the figure. To avoid confusion in notation we will denote the measured 5-parameters for the Thru, Reflect, and Line connections as the jT], [R], and [L] matrices, respectively. Figure 4,21a shows the arrangement for the Thru connection and the corresponding signal flow graph. Observe that we have made use of the fact that 521 = Sn and that the error boxes are identical and symmetrically arranged. The signal flow graph can be easily reduced using the decomposition rules to give the measured 5-parameters at the measurement planes in terms of the 5-parameters of the error boxes as Tu = = $ti 4- «2=0 ]2 (h) FIGURE 4.21b Block diagram and signal flow graph for the Reflect connection. 4,5 Signal Flow Graphs 196 hup box AB CD 5,2 I I -1-r- Error box Reference plane for DUT -»-fc. - FIGURE 4Jlc Block diagram and signal flow graph for the Line connection. The Reflect connection is shown in Figure 4,21b, with the corresponding signal flow graph. Note that this arrangement effectively decouples the two measurement ports, so Rn = Rzi — 0, The signal flow graph can be easily reduced to show that D . »i "ii= - = SU+ S*Tl (4.75) By symmetry we have R22 ~ R\\> The Line connection is shown in Figure 4.21c, with its corresponding signal flow graph. A reduction similar to that used for the Thru case gives L -> = 5„ + Ll2= ~ a2 a2=0 a,=0 1 - sy-w Ji2e_ 1 - sy-w (4.76a) (4.76b) By symmetry and reciprocity we have L22 = L\\ and L2\ = L\2, We now have five equations (4.74)-(4.76) for the five unknowns Su, S\2, S22, TLi and e~yi; the solution is straightforward but lengthy. Since (4.75) is the only equation that contains Tt, we can first solve the four equations in (4.74) and (4.76) for the other four unknowns. Equation (4.74b) can be used to eliminate S[2 from (4.74a) and (4.76), and then $1 ] can be eliminated from (4.74a) and (4.76a). This leaves two equations for and e-yt; Lne-W - Lr25f2 = Tl2e~?c - ¥j$&ar&- (4,77a) e~lyl (T„ - SvJx2) - = L„ {e~^ - S?2) - S22T{2. (4.77b) Equation (4.77a) can now be solved for S22 and substituted into (4.77b) to give a quadratic equation for e~Yi. Application of the quadratic formula then gives the solution for e~yi in terms of the measured TRL S -parameters as L]2 + 2§ - (T„ - Lnf ± J[L\2 + r,| -(7,, - Lu?]2 - 4L\2T?2 (4.78) Chapter 4: Microwave Network Analysis The choice of sign can be determined by the requirement that the real and imaginary parts of y be positive, or by knowing the phase of Ti (as determined from (4.83)) to within 180°. Next we multiply (4.74b) by S22 and subtract from (4.74a) to get Til =#1 + S22Tn, (4.79a) and similarly multiply (4.76b) by $22 and subtract from (4.76a) to get Lu =5,i + 522L[2^£. (4.79b) Eliminating 5] j from these two equations gives S22 in terms of e~yi as % = TT"~LLl-yr (4-80) in - Li2e yi Solving (4.79a) for S\\ gives 5,i = r11- 522r12, (4.81) and solving (4.74b) for S\2 gives S}2 = T12(l - S|2), (4.82) Finally, (4.75) can be solved for p£ to give + S22 (/tn — 5u) Equations (4.78) and (4.80M4.83) give the S-parameters for the error boxes, as well as the unknown reflection coefncent, VL (to within the sign), and the propagation factor, e~yl. This completes the calibration procedure for the TRL method. The S-parameters of the DUT can now be measured at the measurement reference planes shown in Figure 4.20, and corrected using the above TRL error box parameters to give the S-parameters at the reference planes of the DUT. Since we are now working with a cascade of three two-port networks, it is convenient to use abcd parameters. Thus, we convert the error box S-parameters to the corresponding abcd parameters, and convert the measured S-parameters of the cascade to the corresponding ambmcmdm parameters. If we use a'b'c'd' to denote the parameters for the DUT, then we have that \am bm~\_\a Bit a' £' "I I" A B]"' [cm dm\ [c d\[C D'\[c Dj from which we can determine the abcd parameters for the DUT as \af B'lfA fll^rA"1 Bml\a b1 [a d'\~[c d] [cm d"'\[c d] l®m* 4.6 Discontinuities and Modal Analysis 197 POINT OF INTEREST: Computer-Aided Design for Microwave Circuits Computer-aided design (CAD) software packages have become essential tools for the analysis, design, and optimization of microwave circuits and systems. Several microwave CAD products are commercially available, including SERENADE (Ansoft Corp.), Microwave Office (Applied Wave Research, Inc.), ADS (Agilent Technologies, Inc.), and others. These packages are capable of treating microwave circuits consisting of lumped elements, distributed elements, discontinuities, coupled lines, waveguides, and active devices. Both linear and nonlinear modeling, as well as circuit optimization, are generally possible. Although such computer programs can be fast, powerful, and accurate, they cannot serve as a substitute for an experienced engineer with a good understanding of microwave design. A typical design process will usually begin with specifications or design goals for the circuit. Based on previous designs and his or her own experience, the engineer can develop an initial design, including specific components and a circuit layout. CAD can then be used to model and analyze the design, using data for each of the components and including effects such as loss and discontinuities. The CAD software can be used to optimize the design by adjusting some of the circuit parameters to achieve the best performance, If the specifications are not met, the design may have to be revised. The CAD analysis can also be used to study the effects of component tolerances and errors, to improve circuit reliability and robustness. When the design meets the specifications, an engineering prototype can be built and tested. If the measured results satisfy the specifications, the design process is completed. Otherwise the design will need to be revised, and the procedure repeated. Without CAD tools, the design process would require the construction and measurement of a laboratory prototype at each iteration, which would be expensive and time consuming. Thus, CAD can gready decrease the time and cost of a design, while enhancing its quality. The simulation and optimization process is especially important for monolithic microwave integrated circuits (MMICs) because these circuits cannot easily be tuned or trimmed after fabrication. CAD techniques are not without limitations, however. Of primary importance is the fact that a computer model is only an approximation to a "real-world" circuit, and cannot completely account for the inevitable effects of component and fabricational tolerances, surface roughness, spurious coupling, higher order modes, junction discontinuities, and thermal effects. These limitations generally become most serious at frequencies above 10 GHz. DISCONTINUITIES AND MODAL ANALYSIS By either necessity or design, microwave networks often consist of transmission lines with various types of transmission line discontinuities. In some cases discontinuities are an unavoidable result of mechanical or electrical transitions from one medium to another (e.g., a junction between two waveguides, or a coax-to-microstrip transition), and the discontinuity effect is unwanted but may be significant enough to warrant characterization. In other cases discontinuities may be deliberately introduced into the circuit to perform a certain electrical function (e.g., reactive diaphragms in waveguide or stubs in microstrip line for matching or filter circuits). In any event, a transmission line discontinuity can be represented as an equivalent circuit at some point on the transmission line. Depending on the type of discontinuity, the equivalent circuit may be a simple shunt or series element across the line or, in the more general case, a T- or ix -equivalent circuit may be required. The component values of an equivalent circuit depend on the parameters of the line and the discontinuity, as well as the frequency of operation. In some cases the equivalent circuit involves a shift in the phase reference planes on the transmission lines. Once the equivalent circuit of a given discontinuity is known, its effect can be incorporated into the analysis or design of the network using the theory developed previously in this chapter. 198 Chapter 4: Microwave Network Analysis The purpose of the present section is to discuss how equivalent circuits are obtained for transmission line discontinuities; we will see that the basic procedure is to start with a field theory solution to a canonical discontinuity problem and develop a circuit model, with component values. This is thus another example of our objective of replacing complicated field analyses with circuit concepts, Figures 4.22 and 4.23 show some common transmission line discontinuities and their equivalent circuits. As shown in Figures 4.22a-c, thin metallic diaphragms (or "irises") can be placed in the cross section of a waveguide to yield equivalent shunt inductance, capacitance, or a resonant combination. Similar effects occur with step changes in the height or width of the waveguide, as shown in Figures 4.22d,e. Similar discontinuities can also be made in circular waveguide. The best reference for waveguide discontinuities and their equivalent circuits is The Waveguide Handbook [8]. Some typical microstrip discontinuities and transitions are shown in Figure 4,23; similar geometries exist for stripline and other printed transmission lines such as slotline, covered microstrip, coplanar waveguide, etc. Since printed transmission lines are newer, relative to waveguide, and much more difficult to analyze, more research work is needed to accurately Symmetrical inductive diaphragm r;;; Asymmetrical inductive diaphragm (a) Equivalent circuit Symmetrical caparitive diaphragm V/////////////Ä Asymmetrical capaciiive diaphragm (b) T X Equivalent circuit Rectangular resonant iris Change in height Circular resonant iris (c) id) T i Equivalent circuit ^oi Equivalent circuit A Change in width m. Zq2 Equivalent circuit FIGURE 4.22 Rectangular waveguide discontinuities. 4.6 DiscontinuMas and Modal Analysis 199 FIGURE 4.23 Some common microstrip discontinuities, (a) Open-ended rnicrostrip. (b) Gap in microstrip. (c) Change in width, (d) T-junction, „sin — e>K\ (4.88a) * —' a 00 H; = V^sin—(4.88b) 4.6 Discontinuities and Modal Analysis 201 where A„ is the unknown amplitude coefficient of the reflected TE^o mode in guide 1. The reflection coefficient of the incident TE10 mode is then A t. Similarly, the transmitted modes into guide 2 can he written, for z > 0, as PC n—\ ■ H* = -ll—c sin(4.8%) where the propagation constant in guide 2 is and the wave impedance in guide 2 is K = (4.91) ft Now at z = 0, the transverse fields (Ey, Hx) must be continuous for 0 < x < c; in addition, must be zero for c < x < a because of the step. Enforcing these boundary conditions leads to the following equations: XX t-^ . flTTX Ey = sin--h > Af, sin-= a a gs,,sin — for0<* ■— sin-= - > sm- for 0 < x < c. (4.92b) Z? a frf Z* a ^ Z<; c 1 n=\ » n=l i Equations (4.92a) and (4.92b) constitute a doubly infinite set of linear equations for the modal coefficients AR and B„. We will first eliminate the JS„s, and then truncate the resulting equation to a finite number of terms and solve for the A„s, Multiplying (4.92a) by sin (mTzxja), integrating from x — 0 to a, and using the orthogonality relations from Appendix D yields + -Am = Y, B*1** = Yl BkI™k> (4'93) * n=l A'=J where /m„ — J sin-sin-djr (4.94) J.x=q a V is an integral that can be easily evaluated, and (1 Um^n . is the Kronecker delta symbol. Now solve (4.92b) for Bk by multiplying (4.92b) by sm(k7txjc) and integrating from x = 0 to c. After using the orthogonality relations, we obtain 202 Chapter 4: Microwave Network Analysis Substituting Bk from (4.96) into (4.93) gives an infinite set of linear equations for the Ans, where m = 1,2,..., a a i 2Z(kImkIk„An _ ^ 2ZckImkIn at L „=1 k=l C**n *=l "l L For numerical calculation we can truncate the above summations to N terms, which will result in N linear equations for the first N coefficients, An. For example, let A' = 1. Then (4.97) reduces to a a 2ZU2 , 2Z\ll a ■ ^ + -ULAl = -Ul - -. (4.98) Solving for A i (the reflection coefficient of the incident TEio mode) gives A i = Z;~ Z}. for/V = l. (4.99) Zf+ Z* where Z( = 4Z\I2]/ac, which looks like an effective load impedance to guide 1. Accuracy is improved by using larger values of A', and leads to a set of equations which can be written in matrix form as [Q)[A] = [Pl (4.100) where [Q] is a square A' x A7 matrix of coefficients, [P] is an N x 1 column vector of coefficients given by Pm = f^^Lhi-aSmU (4.102) and [A] is an N x 1 column vector of the coefficients A„. After the A„s are found, the Bns can be calculated from (4.96), if desired. Equations (4.100)-(4.102) lend themselves well to computer implementation, Figure 4.25 shows the results of such a calculation. If the width, c, of guide 2 is such that all modes are cutoff (evanescent), then no real power can be transmitted into guide 2, and all the incident power is reflected back into guide 1. The evanescent fields on both sides of the discontinuity store reactive power, however, which implies that the step discontinuity and guide 2 beyond the discontinuity look like a reactance (in this case an inductive reactance) to an incident TEio mode in guide 1. Thus the equivalent circuit of the -plane step looks like an inductor at the z = 0 plane of guide 1, as shown in Figure 4.22e. The equivalent reactance can be found from the reflection coefficient A[ (after solving (4.100)) as X = -jZl\±^. (4.103) 1 - A\ Figure 4.25 shows the normalized equivalent inductance versus the ratio of the guide widths, cfa, for a free-space wavelength X = 1,4a and for N = 1,2, and 10 equations. The modal analysis results are compared to calculated data from reference [8]. Note that the solution converges very quickly (because of the fast exponential decay of the higher-order evanescent modes), and that the result using just two modes is very close to the data of reference [8]. The fact that the equivalent circuit of the H-plane step looks inductive is a result of the actual value of the reflection coefficient, A\, but we can verify this result by computing 4.6 Discontinuities and Modal Analysis 203 1.0 o.s 0.6 0.4 0.2 - -Modal analysis using jv equations. * • • Calculated data from Maicuvitz [8 J, A = 1.4a J—I—L_4 () 0.1 0.2 0.3 0.4 0.5 0.6 0.7 da FIGURE 4.25 Equivalent inductance of an H-plane asymmetric step. the complex power flow into the evanescent modes on either side of the discontinuity. For example, the complex power flow into guide 2 can be found as P= f f ExH = -b I EyH*dx Jj.=0 /c r «> X>si 2 ^ n = I n Z dx dy nnx jbc (4.104) where the orthogonality property of the sine functions was used, as well as (4.89H4.91). Equation (4.104) shows that the complex power flow into guide 2 is purely inductive. A similar result can be derived for the evanescent modes in guide 1; this is left as a problem. POINT OF INTEREST: Microstrip Discontinuity Compensation Because a microstrip circuit is easy to fabricate and allows the convenient integration of passive and active components, many types of microwave circuits and subsystems are made in microstrip form. One problem with microstrip circuits (and other planar circuits), however, is that (he inevitable discontinuities at bends, step changes in widths, and junctions can cause a degradation in circuit performance. This is because such discontinuities introduce parasitic reactances mat can lead to phase and amplitude errors, input and output mismatch, and possibly spurious coupling. One approach for eliminating such effects is to construct an equivalent circuit for the discontinuity (perhaps by measurement), including it in the design of the circuit, and compensating for its effect by adjusting other circuit parameters (such as line lengths and characteristic impedances, or tuning stubs). Another approach is to minimize the effect of a discontinuity by compensating the discontinuity directly, often by chamfering or mitering the conductor. 204 Chapter 4: Microwave Network Analysis Consider the case of a bend in a microstrip line. The straightforward right-angle bend shown below has a parasitic discontinuity capacitance caused by the increased conductor area near the bend. This effect could be eliminated by making a smooth, "swept" bend with a radius r > 3W, but this takes up more space. Alternatively, the right-angle bend can be compensated by mitering the comer, which has the effecr of reducing the excess capacitance at the bend. As shown below, this technique can be applied to bends of arbitrary angle. The optimum value of the miter length, rt, depends on the characteristic impedance and the bend angle, but a value of a = 1.8 W is often used in practice. The technique of mitering can also be used to compensate step and T-junction discontinuities, as shown below. Right-angle bend Swept bend Mitered bends MUered step Mitered 7-juoction Reference: T. C- Edwards, Foundations far Microwave Circuit Design, Wiley, New York, 198-1 - 4.7 EXCITATION OF WAVEGUIDES—ELECTRIC AND MAGNETIC CURRENTS So far we have considered the propagation, reflection, and transmission of guided waves in the absence of sources, but obviously the waveguide or transmission line must be coupled to a generator or some other source of power. For TEM or quasi-TEM lines, there is usually only one propagating mode that can be excited by a given source, although there may be reactance (stored energy) associated with a given feed. In the waveguide case, it may be possible for several propagating modes to be excited, along with evanescent modes that store energy. In this section we will develop a formalism for determining the excitation of a given waveguide mode due to an arbitrary electric or magnetic current source. This theory can then be used to find the excitation and input impedance of probe and loop feeds and, in the next section, to determine the excitation of waveguides by apertures. Current Sheets That Excite Only One Waveguide Mode Consider an infinitely long rectangular waveguide with a transverse sheet of electric surface current density at z = 0, as shown in Figure 4.26. First assume that this current has x and >■ components given as Ä y) = -x 2Atltn7X mux nny 2Atl„mn m jtx nny cos-sin—- + y——-sin-cos— h a (4.105) We will show that such a current excites a single TEra„ waveguide mode traveling away from the current source in both the +2 and —z directions. 4.7 Excitation of Waveguides—Etectric and Magnetic Currents 205 J,, or MK FIGURE 426 An infinitely long rectangular waveguide with surface current densities at z = 0. From Table 3.2, the transverse fields for positive and negative traveling TEm„ waveguide modes can be written as E± E± „ /nit \ 4. mnx . nny V a / ""' a b = ± (—) AÍ„ sin — cos -rp^^, TT+ . /nn\ , mnx . nny (4.106a) (4.106b) (4.106c) (4.106d) where the ± notation refers to waves traveling in the +z direction or -z direction, with amplitude coefficients A^n and A~„, respectively. From (1.36) and (1,37), the following boundary conditions must be satisfied at z = 0: (£+ -E-)xz=Q, žx(//+ -H-) = JS. (4.107a) (4,107b) Equation (4.107a) states that the transverse components of the electric field must be continuous at z = 0, which when applied to (4.106a) and (4.106b) gives 4+" — A: (4.108) Equation (4.107b) states that the discontinuity in the transverse magnetic field is equal to the electric surface current density. Thus, the surface current density at z = 0 must be I = y(H? - H-) - x{H? - Hy~) 2A*rtnjr mnx . nny ^lA^mn . mnx nny cos-sin —— + y- a b a sin-cos-, (4,109) a b where (4,108) was used. This current is seen to be the same as the current of (4.105), which shows, by the uniqueness theorem, that such a current will excite only the TE„in mode propagating in each direction, since Maxwell's equations and all boundary conditions are satisfied. The analogous electric current that excites only the TMmK mode can be shown to be 7™(*,y) = .r a b bab cos sin sin cos (4.110) It is left as a problem to verify that this current excites TMrtn modes that satisfy the appropriate boundary conditions. 206 Chapter 4: Microwave Network Analysis Similar results can be derived for magnetic surface current sheets. From (1.36) and (1.37) the appropriate boundary conditions are =M„ (4,111a) zx(H+~H~) = 0. (4.111b) For a magnetic current sheet at z = 0, the TEmn waveguide mode fields of (4.106) must now have continuous Hx and Hy field components, due to (4.11 lb). This results in the condition that K« = -K«- (4T12) Then applying (4.111a) gives the source current as ■ yg — x2Zj^A^nnm7z mitx rnxy ^Zj^A^njr _ mux nrry * ™~ a b b ~~~ a ~~ b M - -—^--sin —- cos —- - V— - ■ cos-sin o (4.113) The corresponding magnetic surface current that excites only the TMmn mode can be shown to be .-,™ —4&SS.fl3fc . mnx nifty y2B*imr mux . nny tm -■■ b aba a b These results show that a single waveguide mode can be selectively excited, to the exclusion of all other modes, by either an electric or magnetic current sheet of the appropriate form. In practice, however, such currents are very difficult to generate, and are usually only approximated with one or two probes or loops. In this case many modes may be excited, but usually most of these modes are evanescent Mode Excitation from an Arbitrary Electric or Magnetic Current Source We now consider the excitation of waveguide modes by an arbitrary electric or magnetic current source [4]. With reference to Figure 4.27, first consider an electric current source J located between two transverse planes at z\ and t%t which generates the fields E+, H+ traveling in the +z direction, and the fields E~, H~ traveling in the -z direction. These fields can be expressed in terms of the waveguide modes as follows; E+ = YlAtK = Y^At^+ie,n)e-^'\ z > z2i (4.115a) n a H+ = Y,A"R» = Y,A"{h»+ih™)e~^'Z< Z>Z2' (4115b) 1 1 UHF V| i tu FIGURE 4.27 An arbitrary electric or magnetic current source in an infinitely long waveguide. 4.7 Excitation of Waveguides— Electric and Magnetic Currents 207 (4,115c) n n h~ = J2 a;h- = J2 A« + z < Zl, (4.115d) H n where the single index n is used to represent any possible te or TM mode. For a given current /, we can determine die unknown amplitude A+ by using the Lorentz reciprocity theorem of (1.155) with M\ — M2 = 0 (since here we are only considering an electric current source), where S is a closed surface enclosing the volume V\ and H are the fields due to the current source 7j (for i = 1 or 2). To apply the reciprocity theorem to the present problem, we let the volume V be the region between the waveguide walls and the transverse cross-section planes at z i and zz-Then let Ey = £± and H\ = H±, depending on whether z >.z2, orz < zi, and let E2, H2 be the nth waveguide mode traveling in the negative z direction: Substitution into the above form of the reciprocity theorem gives, wiUi J\= J and J2 = 0, The portion of the surface integral over the waveguide walls vanishes because the tangential electric field is zero there; that is, E xH-z = H-(zxE) = 0 on the waveguide walls. This reduces the integration to the guide cross section, So, at the planes z\ and z2. In addition, the waveguide modes are orthogonal over the guide cross section: E2 = E- K % - zez„)e^\ H2 = H; =(-hn+zhzn)e^z. (4.116) = ± / em x hr, ■ ids = 0, form ^ n. (4.117) Using (4.115) and (4.117) then reduces (4.116) to / (£+ x 8- - e; x ds + a; / (e; x h~ - E;xHa~)-ds Since the second integral vanishes, this further reduces to [(e„ + zezn) x (-k„ + zhz„) - (e» - ze2n) x (hn + %hzn)\ ■ zds en x h„ ■ zds (4.118) 206 Chapters Microwave Network Analysis where °„ = 2 Í ěn x h„ ids. (4.119) is a normalization constant proportional to the power flow of the nth mode. By repeating the above procedure with Ei = E+ and = the amplitude of the negatively traveling waves can be derived as A* = TT f £" ' Jdv = "ET f + &z*) ' Je~jß°zdv. (4.120) The above results are quite general, being applicable to any type of waveguide (including planar lines such as striptine and microstrip), where modal fields can be defined. Example 4.8 applies this theory to the problem of a probe-fed rectangular waveguide. EXAMPLE 4.8 PROBE-FED RECTANGULAR WAVEGUIDE For the probe-fed rectangular waveguide shown in Figure 4.28, determine the amplitudes of the forward and backward traveling TE| o modes, and the input resi stance seen by the probe. Assume that the TEio mode is the only propagating mode. Solution If the current probe is assumed to have an infinitesimal diameter, the source volume current density / can be written as J(x, y, z) = k& (x - U &(z)y, for 0 < y < b. From Chapter 3 the TE10 modal fields can be written as e\ = y sin —, a - —x 71x h[ = — sin—, Zi a where Z] = k^/Bx is the TEi0 wave impedance. From (4.119) the normalization constant P\ is 2 fa fh j jix , ab /"i = — / / sir—dxdy = —. *>\ Jx-0 Jy=Q & ^1 Then from (4,118) the amplitude A* is ť\ Jy a \ 1' r\ a ■ r 1 X FIGURE 4J8 A uniform current probe in a rectangular waveguide. 4.8 Excitation of Waveguides—Aperture Coupling 209 Similarly, A1--. a If the TEjo mode is the only propagating mode in the waveguide, then this mode carries all of the average power, which can be calculated for real Z\ as p = | f £+ xH+* -ds + l- t E~ xH*ds 2 A, 2 4 La f f \A+ . 3 xx = f f ~— sin — ax dy = / E+ xH+*- ds _ ab\A+\2 ~~ 2Zi ' If the input resistance seen looking into the probe is and the terminal current is IG, then P = I^Rin/2, so that the input resistance is 2P ab\At\2 bZ\ R'm — -i — I2 I27, a which is real for real Z\ (corresponding to a propagating TEi0 mode). ■ A similar derivation can be carried out for a magnetic current source M. This source will also generate positively and negatively traveling waves which can be expressed as a superposition of waveguide modes, as in (4.115). For J{ = J2= 0, the reciprocity theorem of (1.155)reduces to (£i x H2-E2xHl)-ds = j {Hx- Mi- H2 M\)dv. (4.121) By following the same procedure as for the electric current case, the excitation coefficients of the nth waveguide mode can be derived as K = T-f K -Mdv = ^- f (-hn+zhin)Me^zdv, (4.122) A; = -i- f H?Mdv = ±- f (hn + thZtt) • Me-**"* dv, (4.123) *rt JV "n JV where P„ is defined in (4.119). EXCITATION OF WAVEGUIDES—APERTURE COUPLING Besides the probe and loop feeds of the previous section, waveguides and other transmission lines can also be coupled through small apertures. One common application of such coupling is in directional couplers and power dividers, where power from one guide is coupled to another guide through small apertures in a common wall. Figure 4.29 shows a variety of waveguide and other transmission line configurations where aperture coupling can be employed. We will first develop an intuitive explanation for the fact that a small aperture can be represented as an infinitesimal electric and/or an infinitesimal magnetic dipole, then we will use the results of Section 4.7 to find the fields generated by these equivalent currents. 210 Chapters Microwave Network Analysis Coupling aperture Coupling aperture FIGURE 4.29 Waveguide I Waveguide 2 Feed waveguide (a) '■^v™ . *■ ■;. .f:::; Microstrip 1 Ground plane Microstrip 2 Cavity J- (b) ~7 Waveguide T Slripline Id) Various waveguide and other transmission line configurations using aperture coupling, (a) Coupling between two waveguides via an aperture in the common broad wall, (b) Coupling to a waveguide cavity via an aperture in a transverse wall, (c) Coupling between two microstrip lines via an aperture in the common ground plane, (d) Coupling from a waveguide to a stripline via an aperture. Our analysis will be somewhat phenomenological [4], [10]; a more advanced theory of aperture coupling based on the equivalence theorem can be found in reference [11], Consider Figure 4.30a, which shows the normal electric field lines near a conducting wall (the tangential electric field is zero near the wall). If a small aperture is cut into the conductor the electric field lines will fringe through and around the aperture as shown in Figure 4.30b. Now consider Figure 4.30c, which shows the fringing field lines around FIGURE 4 JO (a) (c) 3 (d) (0 Illustrating the development of equivalent electric and magnetic polarization currents at an aperture in a conducting wall, (a) Normal electric field at a conducting wait, (b) Electric field lines around an aperture in a conducting wall, (c) Electric field Hues around electric polarization currents norma) to a conducting wall, (d) Magnetic field lines near a conducting wall, (e) Magnetic field lines near an aperture in a conducting wall, (f) Magnetic field lines near magnetic polarization currents parallel to a conducting wall. 4.8 Excitation of Waveguides—Aperture Coupling 211 iwo infinitesimal electric polarization currents, Pe, normal to a conducting wall (without an aperture). The similarity of the field lines of Figures 4.30c and 4.30b suggests that an aperture excited by a normal electric field can be represented by two oppositely directed infinitesimal electric polarization currents, Pgt normal to the closed conducting wall. The strength of this polarization current is proportional to the normal electric field, thus, Pe = €0aenE„8(jc - -v0)5(y - ya)&(z - zo), (4.124) where the proportionality constant a4 is defined as the electric polarizability of the aperture, and (jco, y0, zo) are the coordinates of the center of the aperture. Similarly, Figure 4,30e shows the fringing of tangential magnetic field lines (the normal magnetic field is zero at the conductor) near a small aperture. Since these field lines are similar to those produced by two magnetic polarization currents located parallel to the conducting wall (as shown in Figure 4.300, we can conclude that the aperture can be replaced by two oppositely directed infinitesimal polarization currents, Pmt where Pm = -amH,Hx - xQ)S(y - yo)8(.z - t& (4.125) In (4.125), am is defined as the magnetic polarizability of the aperture. The electric and magnetic polarizabilities are constants that depend on the size and shape of the aperture, and have been derived for a variety of simple shapes [3], [10], [11]. The polarizabilities for circular and rectangular apertures, which are probably the most commonly used shapes, are given in Table 4.3. We now show that the electric and magnetic polarization currents, Pe and Pm, can be related to electric and magnetic current sources, J and M, respectively. From Maxwell's equations (1.27a) and (1.27b) we have V x E st - j(op,H - M, (4.126a) v x H = jp,0Pm. (4.128b) These results then allow us to use the formulas of (4.118), (4.120), (4.122), and (4.123) to compute the fields from these currents. TABLE 4.3 Electric ami Magnetic Polarizations Aperture Shape Cm Round hole 3 3 Rectangular slot ntd2 Tied2 16 16 (H across slot) 212 Chapter 4: Microwave Network Analysis The above theory is approximate because of various assumptions involved in the evaluation of the polarizabilities, but generally gives reasonable results for apertures which are small (where the term smalt implies small relative to an electrical wavelength), and not located too close to edges or corners of the guide. In addition, it is important to realize that the equivalent dipoles given by (4.124) and (4.125) radiate in the presence of the conducting wall to give the fields transmitted through the aperture. The fields on the input side of the conducting wall are also affected by the presence of the aperture, and this effect is accounted for by the equivalent dipoles on the incident side of the conductor (which are the negative of those on the output side). In this way, continuity of tangential fields is preserved across the aperture. In both cases, the presence of the (closed) conducting wall can be accounted for by using image theory to remove the wall and double the strength of the dipoles. These details will be clarified by applying this theory to apertures in transverse and broad walls of waveguides, Coupling Through an Aperture in a Transverse Waveguide Wall Consider a small circular aperture centered in the transverse wall of a waveguide, as shown in Figure 4.31a. Assume that only the TEjq mode propagates in the guide, and that such a mode is incident on the transverse wall from z < 0. Then, if the aperture is assumed to be closed, as in Figure 4.31b, the standing wave fields in the region z < 0 can be written as £v = A(e~m - eiffz) sin —, (4.129a) Hx = —ie-** + ^)sin—, (4.129b) where 8 and Zio are the propagation constant and wave impedance of the TEio mode. From (4.124) and (4.125) we can determine the equivalent electric and magnetic polarization currents from the above fields as P£ = zweEzh(x - |js& - = 0, (4.130a) % = -xamrl,8 (x - - |W) = x 2Atxm '10 since £2 = 0 for a TE mode. Now, by (4.128b), the magnetic polarization current Pm is equivalent to a magnetic current density A = tmi» =12-J?^S(* - l)s(y - gftft (4.131, As shown in Figure 4.3Id, the fields scattered by the aperture are considered as being produced by the equivalent currents P^ and — Pm on either side of the closed wall. The presence of the conducting wall is easily accounted for using image theory, which has the effect of doubling the dipole strengths and removing the wall, as depicted in Figure 4.3le (forz < 0) and Figure 4,3 If (for z > 0). Thus the coefficients of the transmitted and reflected waves caused by the equivalent aperture currents can be found by using (4.131) in (4.122) and (4.123) to give io «{2}(o^Pm)dv =-—-=--—, (4.132a) abZm ah =a;/ Aio = T~ I "Jo ' (^jtofxaP^dv = -—-=--—, (4.132b) ap£,\a ab 4.8 Excitation of Waveguides—Aperture Coupling 213 i It) z y t E-.H -2P FIGURE 4.31 2P„ Applying small-hole coupling theory and image theory to the problem of an aperture in the transverse wall of a waveguide, (a) Geometry of a circular aperture in the transverse wall of a waveguide, (b) Fields with aperture closed, (c) Fields with aperture open, (d) Fields with aperture closed and replaced with equivalent dipoles. (e) Fields radiated by equivalent dipoles for z < 0; wall removed by image theory. (f) Fields radiated by equivalent dipoles for z > 0; wall removed by image theory. since hto = (—*/Z]o)sinOrjc/a), and Pio = ab/Z[Q. The magnetic polarizability am is given in Table 4.3. The complete fields can now be written as Ey = [Ae'ifc + (A;0 - A)eJfiz] sin for z < 0. (4.133a) Hx = — [-Ae~m + (A7Q-A1 sin —, for z < 0, (4.133b) Z10 lu a and Ey = At0e-^zsin —, forz > 0. (4.134a) a Hx = Z^k€-m sin ™i for j > 0. (4.134b) 214 Chapter 4: Microwave Network Analysis via FIGURE 4.32 Equivalent circuit of the aperture in a transverse waveguide wall. Then the reflection and transmission coefficients can be found as r = Al0 - A _ 4jßa„ ab -1. A ab (4.135 a) (4.135b) since Z\o = k$r]()/p. Note that |T| > 1; this physically unrealizable result (for a passive network) is an artifact of the approximations used in the above theory. An equivalent circuit for this problem can be obtained by comparing the reflection coefficient of (4.135a) with that of the transmission line with a normalized shunt susceptance,jB, shown in Figure 4.32, The reflection coefficient seen looking into this line is r = 1-yin l-O+JB) -JB 1+vl, l + (l+jB) 2 + jB If the shunt susceptance is very large (low impedance), T can be approximated as f = - -1 l + (Z/JB) JB Comparison with (4.135a) suggests that the aperture is equivalent to a normalized inductive susceptance. B = -ab 2K Coupling Through an Aperture in the Broad Wall of a Waveguide Another configuration for aperture coupling is shown in Figure 4.33, where two parallel waveguides share a common broad wall and are coupled with a small centered aperture. We will assume that a TEio mode is incident from z < 0 in the lower guide (guide 1), and 2b 0 z ail a x FIGURE 433 Two parallel waveguides coupled through an aperture in a common broad wall, 4.8 Excitation of Waveguides—Aperture Coupling 215 compute the fields coupled to the upper guide. The incident fields can be written as £y = Asin—e~m, (4.136a) a Hx=^sw—e~^. (4.136b) Zio a The excitation field at the center of the aperture at (x = a/2, y = b, z = 0) is then Ey = A, (4.137a) Hx = ™ (4.137b) (If the aperture were not centered at x = a/2, the Hz field would be nonzero and would have to be included.) Now from (4.124), (4.125), and (4.128), the equivalent electric and magnetic dipoles for coupling to the fields in the upper guide are Jy s= jtoeqa*A$(x - ^)S(y - b)S(z), (4.138a) z10 8{x-l)^y-b^- (4 0, (4.140a) y a — A+ irx H+ =-sin —e-#\ for z > 0, (4.140b) Z,0 a where A+, A" are the unknown amplitudes of the forward and backward traveling waves in the upper guide, respectively. By superposition, the total fields in the upper guide due to the electric and magnetic currents of (4.138) can be found from (4.118) and (4.122) for the forward wave as A+ = ^- f {E;Jy - H-M,)dv = f« - (4.141a) no Jv no \ zl0 / and from (4.120) and (4.123) for the backward wave as where Pi0 = ab/Z^. Note that the electric dipole excites the same fields in both directions, but the magnetic dipole excites oppositely polarized fields in the forward and backward directions. 216 Ch apter 4: Mi crowave N etwork Anal ys i s REFERENCES [1] S. Ramo, T. R. Whionery, and T. van Duzer, Fields and Waves in Communication Electronics, John Wiley & Sons, N.Y., 1965. [2] A. A. Oliner, "Historical Perspectives on Microwave Field Theory," IEEE Tram. Microwave Theory and Techniques, vol. MTT-32, pp. 1022-1045, September 1984. [3] C. G. Montgomery, R. H. Dicke, and £. M. Purcdl, Principles of Microwave Circuits, vol. 8 of MIT Rad. Lab. Series, McGraw-Hill, N.Y, 1948. 14] R. E. Collin, Foundations for Microwave Engineering, Second Edition, McGraw-Hill, N.Y., 1992. [5] G. Gonzalez, Microwave Transistor Amplifiers, Prentice-Hall, N.J., 1984. [6] J. S. Wright, O. P. Jain, W. J. Chudobiak, and V. Makios, "Equivalent Circuits of Microstrip Impedance Discontinuities and Launchers," IEEE Trans, Microwave Theory and Techniques, vol. MTT-22, pp. 48-52, January 1974. [7] G. F. Engen and C. A. Hoer, 'Thru-Ren'ect-Line: An Improved Technique for Calibrating the Dual Six-Port Automatic Network Analyzer," IEEE Trans. Microwave Theory and Techniques, vol. MTT-27, pp, 987-998, December 1979. [8] N. Marcuvitz, Waveguide Handbook, vol. 10 of MIT Rad. Lab. Series, McGraw-Hill, N.Y., 1948. [9) IC C. Gupta, R, Garg, and 1.1. Bahl, Microstrip Lines and Slotlines, Artech House, Dedham, Mass., 1979. [101 G- Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Chapter 5. Artech House, Dedham, Mass., 1980. [II] R. E. Collin, Field Theory of Guided Waves, McGraw-Hill, N.Y., 1960. PROBLEMS 4.1 Solve the problem of Example 4.2 by writing expressions for the itscident, reflected, and transmitted Ey and Hx fields for the regions g < 0 and z > 0, and applying the boundary conditions for these fields at the dielectric interlace at z = 0. 4.2 Consider the reflection of a TEio mode, incident from z < 0, at a step change in the height of a rectangular waveguide, as shown below. Show that if the method of Example 4.2 is used, the result T = 0 is obtained. Do you think this is (he correct solution? Why? (This problem shows that the one-mode impedance viewpoint does not always provide a correct analysis.) 43 Consider a series RLC circuit with a current, /. Calculate the power lost and the stored electric and magnetic energies, and show that the input impedance can be expressed as in (4.17). 4.4 Show that the input impedance, Z, of a parallel RLC circuit satisfies the condition that Z(-to) = Z"(w). 4.5 Show that the admittance matrix of a lossless A'-port network has purely imaginary elements. 4.6 Does a nonreciprocal lossless network always have a purely imaginary impedance matrix? 4.7 Derive the [Z] and [T] matrices for the following two-port networks: Problems 217 4 Ji Consider a two-port network, and let Z^, Z\J, Zoo be the input impedance seen when port 2 is short-circuited, when port 1 is short-circuited, when port 2 is open-circuited, and when port I is open-circuited, respectively. Show that the impedance matrix elements are given by jrt, _ 7U) sl. _ y2 _ 72 _ (70) _ 7t.v,\7a) *t1 — ^oc1 ^22 — ^-OC' ^12 — — (^DC ^sC J '-» 0- -0 Port 2 4.11 Consider two two-port networks with individual scattering matrices, [SA] and [SB]. Show thai the overall $21 parameter of the cascade of these networks is given by _ J21J21 21 _ 1 (A(« 4.12 Consider a lossless two-port network. (a) If the network is reciprocal, show that |S2\ \2 = 1 — |5| 1 i1. (b) If the network is nonreciprocal, show that it is impossible to have unidirectional transmission, where S,? = 0 and S2i # 0, 4wl3 Show that it is impossible to construct a three-port network that is lossless, reciprocal, and matched at all ports. Is it possible to construct a nonreciprocal three-port network that is lossless and matched at all ports? 4.14 Prove the following decoupling theorem: For any lossless reciprocal three-port network, one port (say port 3) can be terminated in a reactance so that the other two ports (say ports 1 and 2) are decoupled (no power flow from port 1 to port 2, or from port 2 to port 1). 4.15 A certain three-port network is lossless and reciprocal, and has Su = S2i and S\t = Show that if port 2 is terminated with a matched load, then port 1 can be matched by placing an appropriate reactance at port 3. 4.1. Port 1 Port Port 1 2 T Port 4.29 As shown in the figure below, a variable attenuator can be implemented using a four-port 90° hybrid coupler by tenninating ports 2 and 3 with equal but adjustable loads, (a) Using the given scattering matrix for the coupler, show that the transmission coefficient between the input (port 1 > and the output (port 4) is given as T = JF, where T is the reflection coefficient of the mismatch at ports 2 and 3, Also show that the input port is matched for all values of T. (b) Plot the attenuation, in dB, from the input to the output as a function of ZL/Zo, for 0 < ZL/Zo < 10 (let ZL be real), Port 2 In —- Port 1 90° Hvbrid Out-— Port 4 1-51 Port 3 E5l = vT 0 ; 1 0 j 0 ü 1 10 0; 0 \ j 0 4.30 Use signal flow graphs to find the power ratios P<>fP\ and Py/P] for the mismatched three-port network shown below. ] tT IA'I = q % 0 Pon 2 Porl 3 N 220 Chapter 4: Microwave Network Analysis 4,31 The end of an open-circuited microsrrip line has fringing fields that can be modeled as a shunt capacitor, Cf, at the end of the line, as shown below, This capacitance can be replaced with an additional length, A, of microstrip line. Derive an expression for the length extension in terms of ihe fringing capacitance. Evaluate the length extension for a 50 Q open-circuited microstrip line on a substrate with d = 0.158 cm and ef = 2.2 {w = 0.487 cm, ee — 1.894), if the fringing capacitance is known to be Cf = 0.075pF. Compare your result with the approximation given by Hammerstad and / sf + 0.3 \ /w + 0,262rf\ Bekkadal that A = 0.412rf ( —- ) [ ). \ee -0.258/ \w+ 0.8134/ 432 For the H-plane step analysis of Section 4.6, compute the complex power flow in the reflected modes in guide 1, and show that the reactive power is inductive. 4.33 For the //-plane step of Section 4.6, assume that k = \.2a and c = 0.8a, so that a TEi0 mode can propagate in each guide. Using N = 2 equations, compute the coefficients A{ and A2 from the modal analysis soluuon and draw the equivalent circuit of the discontinuity. 4.34 Derive the modal analysis equations for the symmetric //-plane step shown below. (HINT: Because of symmetry, only the TE„o modes, for n odd, will be excited.) 4.35 Find the transverse E and H fields excited by the current of (4.110} by postulating traveling TMm„ modes on either side of the source at z — 0, and applying the appropriate boundary conditions. 4.36 Show that the magnetic surface current density of (4.114) excites TMwn waves traveling away from the source. 437 An infinitely long rectangular waveguide is fed with a probe of length d, as shown below. The current on this probe can be approximated as I(y) = f0 sink{d — y)/ sinkd. If the TE|0 mode is the only propagating mode in the waveguide, compute the input resistance seen at the probe terminals. a/2 It" a x Problems 221 4.38 Consider the infinitely long waveguide fed with two probes driven 180° out of phase, as shown below. What are the resulting excitation coefficients for the TEio and TEjo modes? What other modes can be excited by this feeding arrangement? 439 Consider a small current loop on the side wall of a rectangular waveguide, as shown below. Find the TE|Q fields excited by this loop, if the loop is of radius rrj. y 4.40 A rectangular waveguide is shorted at z = 0, and has an electric current sheet, Jsy, located at z = d, where JSy = 2ltA , 7tx -— sin —. a a Find expressions for the fields generated by this current by assuming standing wave fields for 0 < z < d, and traveling wave fields for z > d> and applying boundary conditions at % = 0 and z = d. Now solve the problem using image theory, by placing a current sheet - Jsy at z = — d, and removing the shorting wall at z = 0. Use the results of Section 4.7 and superposition to find the fields radiated by these two currents, which should be the same as the first results for z > 0. Chapter Five Impedance Matching and Tuning This chapter marks a turning point in that we now begin to apply the theory and techniques of the previous chapters to practical problems in microwave engineering. We begin with the topic of impedance matching, which is often a part of the larger design process for a microwave component or system. The basic idea of impedance matching is illustrated in Figure 5.1, which shows an impedance matching network placed between a load impedance and a transmission line. The matching network is ideally lossless, to avoid unnecessary loss of power, and is usually designed so that the impedance seen looking into the matching network is Zq. Then reflections are eliminated on the trarismission line to the left of the matching network, although there wiU be multiple reflections between the matching network and the load. This procedure is also referred to as tuning. Impedance matching or tuning is important for the following reasons: • Maximum power is delivered when the load is matched to the line (assuming the generator is matched), and power loss in the feed line is luinimized. ♦ Impedance matching sensitive receiver components (antenna, low-noise amplifier, etc) improves the signal-to-noise ratio of the system. * Impedance matching in a power distribution network (such as an antenna array feed network) will reduce amplitude and phase errors. As long as the load impedance, Zt, has some nonzero real part, a matching network can always be found. Many choices are available, however, and we will discuss the design and performance of several types of practical matching networks. Factors that may be important in the selection of a particular matching network include the following: • Complexity—As with most engineering solutions, the simplest design that satisfies the required specifications is generally the most preferable. A simpler matching network is usually cheaper, more reliable, and less lossy than a more complex design, * Bandwidth—Any type of matching network can ideally give a perfect match (zero reflection) at a single frequency. In many applications, however, it is desirable to match a load over a band of frequencies. There are several ways of doing this with, of course, a corresponding increase in complexity, • Implementation—Depending on the type of transmission line or waveguide being used, one type of matching network may be preferable compared to another. For example, 222 5,1 Matching with Lumped Elements [L Networks) 223 Muti'luiii network FIGURE 5.1 A lossless network matching an arbitrary load impedance to a transmission line. tuning stubs are much easier to implement in waveguide than are multisection quarter-wave transformers. Adjustability—In some applications the matching network may require adjustment to match a variable load impedance. Some types of matching networks are more amenable than others in this regard. MATCHING WITH LUMPED ELEMENTS (L NETWORKS) Probably the simplest type of matching network is the ^-section, which uses two reactive elements to match an arbitrary load impedance to a transmission line. There are two possible configurations for this network, as shown in Figure 5.2. If the normalized load impedance, Zl = Zi/Zq, is inside the 1 + jx circle on the Smith chart, then the circuit of Figure 5.2a should be used. If the normalized load impedance is outside the 1 +■ jx circle on the Smith chart, the circuit of Figure 5.2b should be used. The 1 -I- jx circle is the resistance circle on the impedance Smith chart for which r = 1. In either of the configurations of Figure 5.2, the reactive elements may be either inductors or capacitors, depending on the load impedance. Thus, there are eight distinct possibilities for the matching circuit for various load impedances. If the frequency is low enough and/or the circuit size is small enough, actual lumped-element capacitors and inductors can be used. This may be feasible for frequencies up to about 1 GHz or so, although modern microwave integrated circuits may be small enough so that lumped elements can be used at higher frequencies as well. There is, however, a large range of frequencies and circuit sizes where lumped elements may not be realizable. This is a limitation of the L-section matching technique. We will now derive the analytic expressions for the matching network elements of the two cases in Figure 5.2, then illustrate an alternative design procedure using the Smith chart. Zu ß * 7*L ß (a) £b) FIGURE 5,2 L-section matching networks, (a) Network for zL inside the 1 + jx circle, (b) Network for zl outside the 1 + jx circle. 224 Chapter 5: Impedance Matching and Tuning Analytic Solutions Although we will discuss a simple graphical solution using the Smith chart, it may be useful to derive expressions for the L-section matching network, components. Such expressions would be useful in a computer-aided design program for L-section matching, or when it is necessary to have more accuracy than the Smith chart can provide. Consider first the circuit of Figure 5,2a, and let Zi = Rl + }Xl> We stated that this circuit would be used when zl = Zl/Z$ is inside the 1 -I- jx circle on the Smith chart, which implies that RL > Z0 for this case. The impedance seen looking into the matching network followed by the load impedance must be equal to Zo, for a match: Z0 = jX +---. (5.1) Rearranging and separating into real and imaginary parts gives two equations for the two unknowns, X and B: B{XRL-XLZ{)) = RL-Z,h (5.2a) X{l-BXL) = BZQRL-XL. (5.2b) Solving (5.2a) for X and substituting into (5.2b) gives a quadratic equation for B. The solution is xL ± ^rT/tJrI + x2l-Zq rl B =--f-=--. (5.3a) Note that since Rl > Zo, the argument of the second square root is always positive. Then the series reactance can be found as X = i + ^--^. (5.3b) B RL BRL Equation (5.3a) indicates that two solutions are possible for B and X. Both of these solutions are physically realizable, since both positive and negative values of B and X are possible (positive X implies an inductor, negative X implies a capacitor, while positive B implies a capacitor and negative B implies an inductor,) One solution, however, may result in significantly smaller values for the reactive components, and may be the preferred solution if the bandwidth of the match is better, or the SWR on the line between the matching network and the load is smaller. Now consider the circuit of Figure 5.2b, This circuit is to be used when zl is outside the 1 + jx circle on the Smith chart, which implies that Rl < Z(). The admittance seen looking into the matching network followed by the load impedance ZL = Rl + jXL must be equal to I/Zq, for a match: Rearranging and separating into real and imaginary parts gives two equations for the two unknowns, X and B: BZG(X + XL) = ZQ- RL> (5.5a) (X+XL) = BZoRL. (5.5b) 5.1 Matching with Lumped Elements [L Networks) 225 Solving for X and B gives X = iVtfiCZrj- Rl)-Xl, (5.6a) s = ±*JiZa-RL)fRL (5 6b) Since r?l < Zq, the arguments of the square roots are always positive. Again, note that two solutions are possible. In order to match an arbitrary complex load to a line of characteristic impedance Zo, the real part of the input impedance to the matching network must be Zo, while the imaginary part must be zero. This implies that a general matching network must have at least two degrees of freedom; in the L-section matching circuit these two degrees of freedom are provided by the values of the two reactive components. Smith Chart Solutions Instead of the above formulas, the Smith chart can be used to quickly and accurately design L-section matching networks, a procedure best illustrated by an example. EXAMPLE 5.1 L-SECTION IMPEDANCE MATCHING Design an L-section matching network to match a series RC load with an impedance ZL = 200 - j 100 £2, to a 100 Q line, at a frequency of 500 MHz. Solution The normalized load impedance is Zi = 1 — /l, which is plotted on the Smith chart of Figure 5.3a. This point is inside the 1 + jx circle, so we will use the matching circuit of Figure 5.2a. Since the first element from the load is a shunt susceptance, it makes sense to convert to admittance by drawing the SWR circle through the load, and a straight line from the load through the center of the chart, as shown in Figure 5.3a. Now, after we add the shunt susceptance and convert back to impedance, we want to be on the l+jx circle, so that we can add a series reactance to cancel the jx and match the load. This means that the shunt susceptance must move us from yL to the 1 + jx circle on the admittance Smith chart. Thus, we construct the rotated 1 + jx circle as shown in Figure 5.3a (center at r = 0.333). (A combined ZY chart is convenient to use here, if it is not too confusing.) Then we see that adding a susceptance of jb = j03 will move us along a constant conductance circle to y = 0.4 + j0.5 (this choice is the shortest distance from yj to the shifted l + jx circle). Converting back to impedance leaves us at z = 1 — jl .2, indicating that a series reactance x = jl.2 will bring us to the center of the chart. For comparison, the formulas of (5.3a,b) give the solution as b = 0.29, x = 1.22. This matching circuit consists of a shunt capacitor and a series inductor, as shown in Figure 5.3b. For a frequency of / = 500 MHz, the capacitor has a value of c=*r7zr092pF' and the inductor has a value of L = ^=38.8nH. 27zf 226 Chapter 5: Impedance Matching and Tuning It may also be interesting to look at the second solution to this matching problem. If instead of adding a shunt susceptance of b = 0,3, we use a shunt susceptance of b = -0.7, we will move to a point on the lower half of the shifted 1 + jx circle, to y = 0.4 — j0.5. Then converting to impedance and adding a series reactance of x = —1.2 leads to a match as well. The formulas of (5.3a,b) give this solution as b = —0.69, x = -1.22. This matching circuit is also shown in Figure 5.3b, and is seen to have the positions of the inductor and capacitor reversed from the first matching network. At a frequency of / = 500 MHz, die capacitor has a value of C = -1 2nfxZ{ = 2.61 pF, while the inductor has a value of L= -4^ =46.1nH. lllfb Figure 5.3c shows the reflection coefficient magnitude versus frequency for these two matching networks, assuming that the load impedance of Zt, = 200 — j 100 f2 at 500 MHz consists of a 200 £2 resistor and a 3.18 pF capacitor in series. There is not a substantial difference in bandwidth for these two solutions. ■ FIGURE 5.3 Solution to Example 5.1. (a) Smith chart for the L-section matching networks, 5.1 Matching with Lumped Elements (L Networks) 227 38.8 tiH Solution I 261 pF /(GHz) FIGURE 5.3 Continued, (b) The two possible L-section matching circuits, (c) Reflection coefficient magnitudes versus frequency for the matching circuits of (b). POINT OF INTEREST; Lumped Elements for Microwave Integrated Circuits Lumped if, L, and C elements can be practically realized at microwave frequencies if the length, £, of the component is very small relative to the operating wavelength. Over a limited range of values, such components can be used in hybrid and monolithic microwave integrated circuits (MICs) at frequencies up to 60 GHz, if the condition that £ < X/10 is satisfied, Usually, however, the characteristics of such an element are far from ideal, requiring that undesirable effects such as parasitic capacitance and/or inductance, spurious resonances, fringing fields, loss, and perturbations caused by a ground plane be incorporated in the design via a CAD model (see the Point of Interest concerning CAD). Resistors are fabricated with thin films of lossy material such as luchrome, tantalum nitride, or doped semiconductor material. In monolithic circuits such films can be deposited or grown, while chip resistors made from a lossy film deposited on a ceramic chip can be bonded or soldered in a hybrid circuit. Low resistances are hard to obtain. Small values of inductance can be realized with a short length or loop of transmission line, and larger values (up to about 10 nH) can be obtained with a spiral inductor, as shown in the following figures. Larger inductance values generally incur more loss, and more shunt capacitance; this leads to a resonance that limits the maximum operating frequency. Capacitors can be fabricated in several ways. A short transmission line stub can provide a shunt capacitance in the range of 0 to 0.1 pF. A single gap or interdigital set of gaps in a Chapters: Impedance Matching and Tuning Planar resistor Chip resistor Loop inductor Spiral inductor Interdigital Metal-insulator- Chip capacitor gap capacitor metal capacitor transmission line can provide a series capacitance up to about 0.5 pF. Greater values (up to about 25 pF) can be obtained using a metal-insulator-metal (MIM) sandwich, either in monolithic or chip (hybrid) form. SINGLE-STUB TUNING We next consider a matching technique that uses a single open-circuited or short-circuited length of transmission line (a "stub"), connected either in parallel or in series with the transmission feed line at a certain distance from the load, as shown in Figure 5.4. Such a tuning circuit is convenient from a microwave fabrication aspect, since lumped elements are not required. The shunt tuning stub is especially easy to fabricate in microstrip or striphne form. In single-stub tuning, the two adjustable parameters are the distance, J, from the load to the stub position, and the value of susceptance or reactance provided by the shunt or series stub. For the shunt-stub case, the basic idea is to select d so that the admittance, Y, seen looking into the line at distance d from the load is of the form Yq + j B, Then the stub susceptance is chosen as — j B, resulting in a matched condition. For the series stub case, the distance d is selected so that the impedance, Z, seen looking into the line at a distance d from the load, is of the form Zo + jX. Then the stub reactance is chosen as —jX, resulting in a matched condition. As discussed in Chapter 2, the proper length of open or shorted transmission line can provide any desired value of reactance or susceptance. For a given susceptance or reactance, the difference in lengths of an open- or short-circuited stub is A./4. For transmission line media such as microstrip or stripline, open-circuited stubs are easier to fabricate since a via hole through the substrate to the ground plane is not needed. For lines like coax or waveguide, however, short-circuited stubs are usually preferred, because the cross-sectional area of such an open-circuited line may be large enough (electrically) to radiate, in which case the stub is no longer purely reactive. Below we discuss both Smith chart and analytic solutions for shunt and series stub tuning. The Smith chart solutions are fast, intuitive, and usually accurate enough in practice. The analytic expressions are more accurate, and useful for computer analysis. Shunt Stubs The single-stub shunt tuning circuit is shown in Figure 5.4a. We will first discuss an example illustrating the Smith chart solution, and then derive formulas for d and t. 5.2 Single-Stub Tuning 229 3} O-----0 Open or snorted stub (b) FIGURE 5.4 Single-stub tuning circuits, (a) Shunt stub, (b) Series stub. EXAMPLE 5.2 SINGLE-STUB SHUNT TUNING For a load impedance Zj, = 60 — jSO £2, design two single-stub (short circuit) shunt tuning networks to match this load to a 50 Í2 line. Assuming that the load is matched at 2 GHz, and that the load consists of a resistor and capacitor in series, plot the reflection coefficient magnitude from 1 GHz to 3 GHz for each solution. Solution The first step is to plot the normalized load impedance zl = 1 -2 — jl .6, construct the appropriate SWR circle, and convert to the load admittance, yj,, as shown on the Smith chart in Figure 5.5a. For the remaining steps we consider the Smith chart as an admittance chart. Now notice that the SWR circle intersects the 1 + jb circle at two points, denoted as y\ and yi in Figure 5.5a. Thus the distance d, from the load to the stub, is given by either of these two intersections. Reading the WTG scale, we obtain di = 0.176 - 0.065 = 0.110A, d-i =0.325 - 0.065 = 0.260*.. Actually, there is an infinite number of distances, d, on the SWR circle that intersect the 1 + jb circle. Usually, it is desired to keep the matching stub as close as possible to the load, to improve the bandwidth of the match and to reduce losses caused by a possibly large standing wave ratio on the line between the stub and the load. 230 Chapter 5: Impedance Matching and Tuning At the two intersection points, the normalized admittances arc y, = 1.00 +jl.47, y2 = 1.00-;L47. Thus, the first tuning solution requires a stub with a susceptance of — j 1,47. The length of a short-circuited stub that gives this susceptance can be found on the Smith chart by starting at y = oo (the short circuit) and moving along the outer edge of the chart (g — 0) toward the generator to the -j 1.47 point. The stub length is then lx = 0.095*. Similarly, the required open-circuit stub length for the second solution is £2 = 0.405A. This completes the tuner designs. To analyze the frequency dependence of these two designs, we need to know the load impedance as a function of frequency. The series-RL load impedance is It = 60 - >80 ft at 2 GHz, so R = 60 ft and C = 0.995 pF. The two tuning FIGURE 5.5 Solution to Example 5.2. (a) Smith chart for the shunt-stub tuners. 5.2 Single-Stub Tuning 231 ■0.11QA—*- 60 n -o-AAA-i 50 a 0.O95A Solution ÍH 0.260A-V 60 íi =Z 0.995 pF 50 0.995 pF Solution #2 III FIGURE 5-5 Continued, (b) The two shunt-stub tuning solutions, (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b). circuits are shown in Figure 5.5b. Figure 5.5c shows the calculated reflection coefficient magnitudes for these two solutions. Observe that solution 1 has a significantly better bandwidth than solution 2; this is because both d and t are shorter for solution 1, which reduces the frequency variation of the match. ■ To derive formulas for d and £, let the load impedance be written as ZL = \/¥l = Rl + jXL. Then the impedance Z down a length, d, of line from the load is Z = Z0 (RL+jXL) + jZot Zn + j(RL+jXL)i where t = tan Bd. The admittance at this point is Y = G + JB = L, (5-7) where G = B - Rl(1+i2) R2L-r(XL + ZQt)ir Rjt - (Zg - XLt)(XL + ZaQ ZQ[R2L + (XL + Zot)2] (5.8a) (5.8b) 232 Chapter 5; Impedance Matching and Tuning Now d (which implies r) is chosen so that G — Yq = l/Z(). From (5.8a), this results in a quadra lie equation for t: Zv(RL - Zo)t2 - 2XLZot + (RlZq ~ R\ ~ $1). = °-Solving for t gives Xl±Jrl[{Zí>-RlÝ + X2l\/Zq / =-!-r-=-. for rl f Z0. (5.9) If RL = Z0, then f = ~Xl/2Zq. Thus, the two principal solutions for d are 1 - tan 11, for t > 0 _ __ J 2.T X _ 1 1 i —Or + tan-11), for r < 0. 2tt (5.10) To find the required stub lengths, first use * in (5.8b) to find the stub susceptance, Bs = — B. Then, for an open-circuited stub, while for a short-circuited stub, If the length given by (5.11a) or (5.1 lb) is negative, k/2 can be added to give a positive result. Series Stubs The series stub tuning circuit is shown in Figure 5.4b. We will illustrate the Smith chart solution by an example, and then derive expressions for d and I. EXAMPLE 53 SINGLE-STUB SERIES TUNING Match a load impedance of ZL = 100 + ySO to a 50 Q line using a single series open-circuit stub. Assuming mat the load is matched at 2 GHz, and that the load consists of a resistor and inductor in series, plot the reflection coefficient magnitude from 1 GHz to 3 GHz. Solution The first step is to plot the normalized load impedance, zL = 2 + j 1.6, and draw the SWR circle. For the series-stub design, the chart is an impedance chart. Note that the SWR circle intersects the 1 + jx circle at two points, denoted as z\ and Z2 in Figure 5.6a. The shortest distance, d\, from the load to the stub is, from the WTG scale, £j =0.328 -0.208 =0.120X, while the second distance is d2 = (0.5 - 0.208) + 0.172 - 0.463A. 5.2 Single-Stub Tuning 233 As in the shunt-stub case, additional rotations around the SWR circle lead to additional solutions, but these are usually not of practical interest. The normalized impedances at the two intersection points are zt = 1-/1.33, Z2 = 1 + /1.33. Thus, the first solution requires a stub with a reactance of J1-33. The length of an open-circuited stub that gives this reactance can be found on the Smith chart by starting at z = oo (open circuit), and moving along the outer edge of the chart (r = 0) toward the generator to the jT.33 point. This gives a stub length of 11 = 0.397;,, Similarly, the required open-circuited stub length for the second solution is 12 = 0.103X. This completes the tuner designs (a) FIGURE 5J> Solution to Example 5.3. (a) Smith chart for the series-stub tuners. 234 Chapter 5: Impedance Matching and Tuning 50 ÍÍ Ü.397A ji jf. , 0 j 50 U 0.103A 50 ÍÍ 50 0 100 £1 ) 6.37 nH 50 a :ioon Í 6.37 nH 0.120A- 0.463A- Soiulion 1 Solution 2 30 FIGURE 5.6 Continued, (b) The two series-stub tuning solutions, (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b). If the load is a series resistor and inductor with Zi = 100 + 7 80 £2 at 2 GHz, then R - 100 Q and L = 6.37 nH. The two matching circuits are shown in Figure 5.6b. Figure 5.6c shows the calculated reflection coefficient magnitudes versus frequency for the two solutions, ■ To derive formulas for d and t for the series-stub tuner, let the load admittance be written as Yi = 1 /Zi = Gl+jBl. Then the admittance Y down a length, d, of line from the load is Y= (GL+jBL) + JtYQ YQ + jt(GL + jBLy where r = tan Bd, and Yq — l/Zo- Then the impedance at this point is 2 = /e + ;X=i R = Gi+(BL+m2> (513a) x=Glt-(rr<^ + 0 -^-(;r + tan 1 0 for t < 0. 2jt (5.15) The required stub lengths are determined by first using t in (5.13b) to find the reactance, X. This reactance is the negative of the necessary stub reactance, X*, Thus, for a short-circuited stub, while for an open-circuited stub, 4 -l _j l „i/Zo\ — = — tan I — I = — tan I — I. k lit \XJ In \X} (5.16a) (5.16b) If the length given by (5.16a) or (5.16b) is negative, k/2 can be added to give a positive result. DOUBLE-STUB TUNING The single-stub tuners of the previous section are able to match any load impedance (as long as it has a nonzero real part) to a transmission line, but suffer from the disadvantage of requiring a variable length of line between the load and the smb. This may not be a problem for a fixed matching circuit, but would probably pose some difficulty if an adjustable tuner was desired. In this case, the double-stub tuner, which uses two tuning stubs in fixed positions, can be used. Such tuners are often fabricated in coaxial line, with adjustable siuhs connected in parallel to the main coaxial line. We will see, however, that the double-stub tuner cannot match all load impedances. The double-stub tuner circuit is shown in Figure 5.7 a, where the load may be an arbitrary distance from the first stub. Although this is more representative of a practical situation, the circuit of Figure 5.7b, where the load Y'L has been transformed back to the position of the first stub, is easier to deal with and does not lose any generality. The stubs shown in Figure 5.7 are shunt stubs, which are usually easier to implement in practice than are series stubs; the latter could be used just as well, in principle. In either case, the stubs can be open-circuited or short-circuited. Smith Chart Solution The Smith chart of Figure 5.8 illustrates the basic operation of the double-stub tuner. As in the case of the single-stub tuners, two solutions are possible. The susceptance of 236 Chapter 5: Impedance Matching and Timing PI FIGURE 5,7 Double-stub dining, (a) Original circuit with the load an arbitrary distance from the first stub, (b) Equivalent circuit with load at the first stub. 5.3 Double-Stub Tuning 237 the first stub, b\ (or b\, for the second solution), moves the load admittance to y) (or yj). These points lie on the rotated 1 + j b circle; the amount of rotation is d wavelengths toward the load, where d is the electrical distance between the two stubs. Then trauisfonning y( (or yj) toward the generator through a length, dy of line leaves us at the point y% (or y2), which must be on the 1 + jb circle. The second stub then adds a susceptance b2 (or b2), which brings us to the center of the chart, and completes the match. Notice from Figure 5.8 that if the load admittance, yi, were inside the shaded region of the ga -j- jb circle, no value of stub susceptance b\ could ever bring the load point to intersect the rotated 1 + jb circle. This shaded region thus forms a forbidden range of load admittances, which cannot be matched with this particular double-stub tuner. A simple way of reducing the forbidden range is to reduce the distance, dt between the stubs. This has the effect of swinging the rotated l + jb circle back toward the y = oo point, but d must be kept large enough for the practical purpose of fabricating the two separate stubs. In addition, stub spacings near 0 or A,/2 lead to matching networks that are very frequency sensitive. In practice, stub spacings are usually chosen as a/8 or 3 A/8. If the length of line between the load and the first stub can be adjusted, then the load admittance yi can always be moved out of the forbidden region. EXAMPLE 5.4 DOUBLE-STUB TUNING Design a double-stub shunt tuner to match a load impedance Zt = 60 — j'80 U to a 50 line. The stubs are to be open-circuited stubs, and are spaced kj 8 apart. Assuming that this load consists of a series resistor and capacitor, and that the match frequency is 2 GHz, plot the reflection coefficient magnitude versus frequency from 1 GHz to 3 GHz. Solution The normalized load admittance is = 0,3 4- /0.4, which is plotted on the Smith chart of Figure 5.9a. Next we construct the rotated 1 + jb conductance circle, by moving every point on the g = 1 circle a/8 toward the load. We then find the susceptance of the first stub, which can be one of two possible values: bi = 1.314, or b\ = -0.114. We now transform through the a/8 section of fine by rotating along a constant radius (SWR) circle a/8 toward the generator, litis brings the two solutions to the following points: yi = 1 - J3.38, or y'2 = 1 + 71.38. Then the susceptance of the second stub should be b2 = 3.38, or £4 = -1.38. The lengths of the open-circuited stubs are then found as ft = 0.146a. %:s= 0.482A., or t\ = 0.204a, 1'2 = 0.350a. This completes both solutions for the double-stub tuner design. 238 Chapter 5: impedance Matching and Tuning Now if the resistor-capacitor load Zi = 60 — j 80 £2 at / = 2 GHz, then R = 60 £2 and C = 0,995 pR The two tuning circuits are then shown in Figure 5.9b, and the reflection coefficient magnitudes are plotted versus frequency in Figure 5.9c. Note that the first solution has a much narrower bandwidth than the second (primed) solution, due to the fact that both stubs for the first solution are somewhat longer (and closer to k/2) than the stubs of the second solution. ■ Analytic Solution Just to the left of the first stub in Figure 5,7b, the admittance is y, =GL + j(BL + Bt), (5.17) where YL = GL + jBL is the load admittance and Bt is the susceptance of the first smb. After transforming through a length d of transmission line, the admittance just to the right of the second stub is 0Yc + jt(GL+jBL-rjBi) Rotated FIGURE 5.9 Solution to Example 5.4. (a) Smith chart for the double-stub tuners. A/8-» ft'fl 5.3 Double-Stub Tuning 239 -A/8--- 60« 0.995 pF Solution J lr| Solution 2 / Solution #2 3.0 ZII 0.995 pF FIGURE 5.9 Continued, (b) The two double-stub tuning solutions, (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b). where t = tan fid and Jo = V^o- At this point, the real part of f% must equal Yq, which leads to the equation rl r y 1+t2 . (Y(f — Bjt — B\t) (5.19) Solving for GL gives 1±. 1 - 4tHYQ - BLt - B]()2 F2(l-M2)2 Since Gi is real, the quantity within the square root must be nonnegative, and so 4r2(r„ - BLt - Bxt)2 (5.20) 0 < Y2(l+t2Y < 1. This implies that 0 < GL < y0 l+/2 (5.21) t2 sin2^' which gives the range on Gt that can be matched for a given stub spacing, d. After d 240 Chapter 5: Impedance Matching and Tuning has been fixed, the first stub susceptance can be determined from (5.19) as YQ±J(l+t2)GLYa-G2i2 Bt=-BL +-y--. (5.22) i Then the second stub susceptance can be found from the negative of the imaginary part of (5.18) to be ±Y0JyoGl(1 + t2) - G2Li2 + GLYQ B2----i- -. (5.23) GLt The upper and lower signs in (5.22) and (5.23) correspond to the same solutions. The open-circuited stub length is found as while the short-circuited stub length is found as where B = B\ or Bj. THE QUARTER-WAVE TRANSFORMER As discussed in Section 2.5, the quarter-wave transformer is a simple and useful circuit for matching a real load impedance to a transmission line. An additional feature of the quarter-wave transformer is that it can be extended to multisection designs in a methodical manner, for broader bandwidth. If only a narrow band impedance match is required, a single-section transformer may suffice. But, as we will see in the next few sections, multisection quarter-wave transformer designs can be synthesized to yield optimum matching characteristics over a desired frequency band. We will see in Chapter 8 that such networks are closely related to bandpass filters. One drawback of the quarter-wave transformer is that it can only match a real load impedance. A complex load impedance can always be transformed to a real impedance, however, by using an appropriate length of transmission line between the load and the transformer, or an appropriate series or shunt reactive stub. These techniques will usually alter the frequency dependence of the equivalent load, which often has the effect of reducing the bandwidth of the match. In Section 2.5 we analyzed the operation of the quarter-wave transformer from an impedance viewpoint and a multiple reflection viewpoint. Here we will concentrate on the bandwidth performance of the transformer, as a function of the load mismatch; this discussion will also serve as a prelude to the more general case of multisection transformers in the sections to follow. The single-section quarter wave matching transformer circuit is shown in Figure 5.10. The characteristic impedance of (he matching section is Z, = 4Ítél> (5.25) At the design frequency, /o, the electrical length of the matching section is Arj/4, but at other frequencies the length is different, so a perfect match is no longer obtained. We will now derive an approximate expression for the mismatch versus frequency. 5.4 The Quarter-Wave Transformer 241 FIGURE 5.10 A single-secrion quarter-wave matching transformer. I = k0/4 at the design frequency /0. The input impedance seen looking into the matching section is %^&f£f^ (5.26) where / == tan &t = tan $, and 61 — 6-= tt/2 at the design frequency, fa. The reflection coefficient is then r = h - Z» = ^L-Z0) + jt(Z2-Z0ZL) Zí„ + Zq Zl(Zi + Z0) + jr(zf + Z0ZL)' Since Zf = ZoZt, this reduces to =-ZL - Zfl__ Zt-r-Z0 + j2řVŽ^ŽI The reflection coefficient magnitude is |ZL-Z0| [(ZL + Zo)2+4f3ZoZi]1 [(Zl + Z0)V(Zl - Z^ + ^ZoZlKZl -Za)2]}m 1 {1 + [4Z*ZLf{ZL - Z0)2] + ^ZoZ^VfZi - Z0)2])1/2 1 [\+\4ZQZL/(ZL - Z0)2] sec2*?]' /2' (5.29) since 1 + r2 = 1 + tan2 9 = sec2 0, Now if we assume that the frequency is near the design frequency, fo, then t ~ X0/4 and ^ a* jr/2. Then sec2(9 :§> I, and (5.29) simplifies to |r| - 1 L_=^|cos#1, for B near tt/2. (5.30) 2 V ZqZ/, This result gives the approximate mismatch of the quarter—wave transformer near the design frequency, as sketched in Figure 5.11. If we set a maximum value, rm, of the reflection coefficient magnitude that can be tolerated, then we can define the bandwidth of the matching transformer as A0 = 2(|-0m), (5.31) 242 Chapter 5: Impedance Matching and Tuning in 2 FIGURE 5.11 Approximate behavior of the reflection coefficient magnitude for a single-section quarter-wave transformer operating near its design frequency. since the response of (5.29) is symmetric about 9 = n/2, and T = Tm at 6 = 6m and at e = 7i — 9m. Equating rm to the exact expression for reflection coefficient magnitude in (5.29) allows us to solve for 9m: 1 , Tt \ZL- Z0 Tm 2*JZBZL or cos6i„ = —-—. (5.32) yi -rl \zL - z0| If we assume TEM lines, then ^ vp 4/0 2/o therefore the frequency of the lower band edge at 9 — Bm is , _ 29mh Jm — ( 7T and the fractional bandwidth is, using (5,32), A/ = 2(/0 -/„) _2 2fm =2 40m fa fa h x = 2--cos . —-— . (5.33) The fractional bandwidth is usually expressed as a percentage, 100A///o %. Note that the bandwidth of the transformer increases as Zi becomes closer to Zq (a less mismatched load). The above results are strictly valid only for TEM lines. When non-TEM lines (such as waveguides) are used, the propagation constant is no longer a Linear function of frequency, and the wave impedance will be frequency dependent. These factors serve to complicate the general behavior of quarter-wave transformers for non-TEM lines, but in practice the bandwidth of the transformer is often small enough so that these complications do not substantially affect the result. Another factor ignored in the above analysis is the effect of reactances associated with discontinuities when there is a step change in the dimensions of 5.4 The Quarter-Wave Transformer 243 FIGURE S.12 Reflection coefficient magnitude versus frequency for a single-section quarter-wave matching transformer with various load mismatches. a transmission line. This can often be compensated for by making a small adjustment in the length of the matching section. Figure 5,12 shows a plot of the reflection coefficient magnitude versus normalized frequency for various mismatched loads. Note the trend of increased bandwidth for smaller load mismatches. EXAMPLE 5.5 QUARTER-WAVE TRANSFORMER BANDWIDTH Design a single-section quarter-wave matching transformer to match a 10 Í2 load to a 50 £2 line, at /0 = 3 GHz. Determine the percent bandwidth for which the SWR < 1.5. Solution From (5.25), the characteristic impedance of the matching section is Z, = 7z0Zt = ^(50X10) = 22.36 fl, and the length of the matching section is X/4 at 3 GHz. An SWR of 1.5 corresponds to a reflection coefficient magnitude of SWR+1 1.5 + 1 The fractional bandwidth is computed from (5.33) as A/ /o 244 Chapter 5: Impedance Matching and Tuning THE THEORY OF SMALL REFLECTIONS The quarter-wave transformer provides a simple means of matching any real load impedance to any line impedance. For applications requiring more bandwidth than a single quarter-wave section can provide, multisection transformers can be used. The design of such transformers is the subject of the next two sections, but prior to that material we need to derive some approximate results for the total reflection coefficient caused by the partial reflections from several small discontinuities. This topic is generally referred to as the theory of small reflections [1]. Single-Section Transformer Consider the single-section transformer shown in Figure 5.13; we will derive an approximate expression for the overall reflection coefficient T. The partial reflection and transmission coefficients are P _ Z2 ~~ Z| 1 " z2 + zť ZL-Z2 r3 = Tn = Tl2 = ZL + Z2 i + r, = i + r2 = 2Z2 z, + z2 ■ 2Zj Zi+Z2' (5.34) (5.35) (5.36) (5.37) (5.38) We can compute the total reflection, T, seen by the feed line by the impedance method or by the multiple reflection method, as discussed in Section 2.5. For our present purpose ■pt = 6- FIGURE 5.13 Partial reflections and transmissions on a single-section matching transformer. 5,5 The Theory of Small Reflections 245 the latter technique is preferred, so we can express the total reflection as an infinite sum of partial reflections and transmissions as follows; r = ti + TnTi^e-2^ + w^rlr^-416 + • ■ ■ 00 = Ti 4- 7i2r21iV-2^ £ ^e-2jtt6(5.39) Using the geometric series = -:-, for \x\ < 1, (5.39) can be expressed in closed form as 1 i - r2r3e-2j$ From(5.35),(5.37), and (5.38), we use T2 = -Tlr T2, = 1 + r,,and7'|2 - 1 - T, in(5,40) to give i + r1r3^' ■ } Now if the discontinuities between the impedances Z\, Z2 and Z2, Zi are small, then | Ti F31 < 1, so we can approximate (5.41) as T - Ti + r3e-2je. (5.42) This result states the intuitive idea that the total reflection is dominated by the reflection from the initial discontinuity between Zy and Z2 (Fi), and the first reflection from the discontinuity between Z2 and Zl (Y^e~2^). The e~lje term accounts for the phase delay when the incident wave travels up and down the line. The accuracy of this approximation is illustrated in Problem 5.14. Multisection Transformer Now consider the multisection transformer shown in Figure 5.14. This transformer consists of W equal-length (commensurate) sections of transmission lines. We will derive an approximate expression for the total reflection coefficient r. I'd ri F*t FIGURE 5.14 Partial reflection coefficients for a multisection matching transformer. 246 Chapters: Impedance Matching and Tuning Partial reflection coefficients can be defined at each junction, as follows: Zv + Zo r„ = ln+'~l\ (5.43b) £>n + 1 + *>n T* = (5.43c) We also assume that all Z„ increase or decrease monotonically across the rransformer, and that Zi is real. Tms implies that all Tn will be real, and of the same sign (Fn > OifZj, > Zo; T„ < 0 if ZL < Zo). Then using the results of the previous section, the overall reflection coefficient can be approximated as T(8) = ra + Tie-2^ + r2e~4js +•■■ + TNe^Ne, (5.44) Further assume that the transformer can be made symmetrical, so that To = Ti = Ty-i, T2 = rN_2, etc. (Note that this does not imply that the Z„s are symmetrical.) Then (5.44) can be written as T(9) = e~!m {Ww + e->m] + F, [ejiN~2)$ + e~^'2^ +■■•}. (5.45) If N is odd, the last term is r(#_])/2(e-^ + e~J9\ while if /V is even the last term is F^-. Equation (5.45) is then seen to be of the form of a finite Fourier cosine series in 9, which can be written as T(0) = 2e~jm |Y0cos N9 + V\ cos(N - 2)9 + ■ ■ ■ + r„ cos(W - 2n)9 + •■■+1 ^2] • for N even^ ■ (546a> r(9) = 2e-^9[r0 cos N$ + T, cos(/Y - 2)9 + - ■ ■ + F„ cos(AT - 2n)9 H-----h r(jv_i)/2 cos 9], for N odd. (5.46b) The importance of these results lies in the fact that we can synthesize any desired reflection coefficient response as a function of frequency (#), by properly choosing the r„s and using enough sections (A7). This should be clear from the realization that a Fourier series can approximate an arbitrary smooth function, if enough terms are used. In the next two sections we will show how to use this theory to design multisection transformers for two of the most commonly used passband responses: the binomial (maximally flat) response, and the Chebyshev (equal ripple) response. BINOMIAL MULTISECTION MATCHING TRANSFORMERS The passband response of a binomial matching transformer is optimum in the sense that, for a given number of sections, the response is as flat as possible near the design frequency. Thus, such a response is also known as maximally flat. This type of response is designed, for an A7-section transformer, by setting the first N — 1 derivatives of |r(0)| to zero, at the center frequency /0- Such a response can be obtained if we let F(0) = A(l+e-2Je)N- (5.47) 5.6 Binomial Multisection Matching Transformers 247 Then the magnitude \T(0)\ is \r(d)\ = \A\\e-^\e^+e-^f = 2JV|A||cos0|w (5.48) Note that |r(#)| = Ofortf = 7r/2,andmat(J*|r(0)|)/rf0" = Oat# = jr/2forn = 1, 2_____ N — 1. (f? = jt/2 corresponds to the center frequency fa for which i = 'k/A and 8 = fit = n/2.) We can determine the constant A by letting / —> 0. Then 9 = /J€ = 0, and (5.47) reduces to r(0) = 2»A = f^, since for / = 0 all sections are of zero electrical length. Thus the constant A can be written as A=2~N^—^. (5.49) Now expand V(9) in (5.47) according to the binomial expansion: » r<0) = A<1 +«T^f = Aj^C^e'2-1^ (5.50) where C* = (KJ N\ (5.51) are the binomial coefficients. Note that G* =■ C#_rt! C0* = 1, and C? = N = C$_v The key step is now to equate the desired passband response as given in (5.50), to the actual response as given (approximately) by (5.44): n rtf) = av c*e-lin6 = r0 + f*tr& + r2^-4^9 + ■ ■ ■ + r>- n=0 This shows mat the P„ must be chosen as Tfl = AC?. (5.52) where A is given by (5.49), and C? is a binomial coefficient. At this point, die characteristic impedances Zn can be found via (5.43), but a simpler solution can be obtained using the following approximation [1]. Since we assumed that the r„ are small, we can write Zn +■ i + Z„ 2 Z„ since In* - 2(x - l)/(x + 1). Then, using (5.52) and (5.49) gives in fej a 2rn = 2AC = 2(2^)|^C^ at In §; (5.53) which can be used to find Z„ + i, starting with n = 0, This technique has the advantage of ensuring self-consistency, in that Z#+ ] computed from (5.53) will be equal to ZL, as it should. Exact results, including the effect of multiple reflections in each section, can be found by using the transmission fine equations for each section and numerically solving for the 248 Chapter 5: Impedance Matching and Tuning characteristic impedances [2]. The results of such calculations are listed in Table 5.1, which give the exact line impedances for N = 2, 3,4t 5, and 6 section binomial matching transformers, for various ratios of load impedance, ZL, to feed line impedance, Zo. The table gives results only for Zl/Zq > 1; if Zl/Zq < 1, the results for Zq/Zi should be used, but with Zi starting at the load end. This is because the solution is symmetric about Zl/Zq = 1; the same transformer that matches Zi to Zo can be reversed and used to match Zo to Zi, More extensive tables can be found in reference [2], The bandwidth of the binomial transformer can be evaluated as follows. As in Section 5.4, let Pm be the maximum value of reflection coefficient that can be tolerated over the passband. Then from (5.48), rm=2N|A|cos*0mT where $m ^ 100 zL = In70.7 + 2~3(3)ln = 4,00, Zo 50 100 Z3 = 54.5 a. To use the data in Table 5.1, we reverse the source and load impedances and consider the problem of matching a 100 £2 load to a 50 £2 line. Then Zl/Zq = 2.0, and we obtain the exact characteristic impedances as Zj = 91.7 £2, Z% = 70.7 £2, and Z3 = 54.5 £2, which agree with me approximate results to three significant digits. Figure 5.15 shows the reflection coefficient magnitude versus frequency for exact designs using n = 1,2,3,4, and 5 sections. Observe that greater bandwidth is obtained for transformers using more sections. ■ CHEBYSHEV MULTISECTION MATCHING TRANSFORMERS In contrast with the binomial matching transformer, the Chebyshev transformer optimizes bandwidth at the expense of passband ripple. If such a passband characteristic can be tolerated, the bandwidth of the Chebyshev transformer will be substantially better than that 5.7 Chebyshev Multisection Matching Transformers 251 of the binomial transformer, for a given number of sections. The Chebyshev transformer is designed by equating T0) to a Chebyshev polynomial, which has the optimum characteristics needed for this type of transformer. Thus we will first discuss the properties of the Chebyshev polynomials, and then derive a design procedure for Chebyshev matching transformers using the small reflection theory of Section 5.5. Chebyshev Polynomials The nth order Chebyshev polynomial is a polynomial of degree n, and is denoted by Tn(x). The first four Chebyshev polynomials are 7-,U) = a, (5.56a) T2(x) = 2x2- 1, (5.56b) 7Mjc) = 4x3 - 3x, (5.56c) T4(x) = 8*4 - 8*2 +1. (5.56d) Higher-order polynomials can be found using the following recurrence formula: T0) = teTx-dxZ - T„_2(*). (5.57) The first four Chebyshev polynomials are plotted in Figure 5.16, from which the following very useful properties of Chebyshev polynomials can be noted: • For —1 < x < 1, \TK(x)\ < 1. In this range, the Chebyshev polynomials oscillate between ±1. This is the equal ripple property, and this region will be mapped to the passband of the matching transformer. • For > 1, \Tn(x)\ > 1. This region will map to the frequency range outside the passband. • For j*| > 1, the |r„(je)| increases faster with x as n increases. 252 Chapter 5: Impedance Matching and Tuning Now let* = cos0 for |x| < 1, Then it can be shown that the Chebyshev polynomials can be expressed as rn(cos^) = co$nd, or more generally as T»U) - cos(n cos-1 x), for < L (5.58a) TJx) = cosh(n cosh-1 x), for pr| > 1. (5.58b) We desire equal ripple in the passband of the transformer, so it is necessary to map 6m to x — 1 and jt — 9m to x = —1, where 9m and % - 9m are the lower and upper edges of the passband, as shown in Figure 5.11. This can be accomplished by replacing cos 0 in (5.58a) with cos 9/ cos0m: f cosd \ . / cos# \1 T„ —— = Trt(sec Bm cos 9) = cos« cos"1 -— . (5,59) Vc°s0m/ L Vcos^m/J Then | sec#m cos/2cost? for N odd. As in the binomial transformer case, we can find the constant A by letting 9 = 0, corresponding to zero frequency. Thus, r(0) = ZJ- = A7V(sec6U, so we have ^=Zi-Z£-1 (5 62) 5.7 Chebyshev Multisection Matching Transformers 253 Now if the maximum allowable reflection coefficient magnitude in the passband is fm, then from (5.61) = | A j»since the maximum value of % (sec &m cos &) in the passband is unity. Then, from (5,62) and the approximations introduced in Section 5.6, $m is determined as TN{secSm) = Zl - Z01 . . J__ Zl + Zo 2rm 1 z^ bz7 or, using (5.58b), sec 9m - cosh — cosh rm I zL + z<> In ZL/Zo Once 0m is known, the fractional bandwidth can be calculated from (5.33) as A/ h = 2 4ft, (5.63) (5.64) From (5.61), the TH can be determined using the results of (5.60) to expand Ttffsec 9m cos 8) and equating similar terms of the form cos(iV — 2n)9. The characteristic impedances Z„ can then be found from (5.43); although, as in the case of the binomial transformer, accuracy can be improved and self-consistency can be achieved by using the approximation that - 1 . Z„+] This procedure will be illustrated in Example 57. The above results are approximate because of the reliance on small reflection theory, but are general enough to design transformers with an arbitrary ripple level, Tm. Table 5.2 gives exact results [2] for a few specific values of rm, for N = 2, 3, and 4 sections; more extensive tables can be found in reference 12]. EXAMPLE 5.7 CHEBYSHEV TRANSFORMER DESIGN Design a three-section Chebyshev transformer to match a 100 ft load to a 50 ft line, with rm = 0.05, using the above theory. Plot the reflection coefficient magnitude versus normalized frequency for exact designs using 1, 2, 3, and 4 sections. Solution From (5.61) with N = 3, T(d) = 2e-j™[F0cos3e + T| costf] = Ae-jr"3()7Msectfm cost?). Then, A — Tm= 0.05, and from (5.63), = 1,408, so, ft* = 44.7°. 254 Chapters: impedance Matching and Tuning TABLE 5.2 Chebyshev Transformer Design N = 2 N - = 3 r,„. = 0.05 r,„ = 0.20 rm = 0.05 rm = 0.20 z./z<> z2/z0 Zi/Zo Z2/Z0 z,/z0 Z2/Z0 Z3/Z0 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1,0000 1.0000 1.0000 1.0000 1.0000 1.5 1.1347 1.3219 1.2247 1,2247 1.1029 1.2247 1.3601 1.2247 i.2247 1.2247 2.0 1.2193 1.6402 13161 1.5197 1.1475 1.4142 1.7429 1.2855 1.4142 1.5558 3.0 1.3494 2.2232 1.4565 2.0598 1.217) 1.7321 2.4649 1.3743 1.7321 2.1829 4.0 1.4500 2.7585 1.5651 2.5558 1.2662 2.0000 31591 1.4333 2.0000 2.7908 6.0 1.6047 3.7389 1.7321 3.4641 1.3383 2.4495 4.4833 1.5193 2.4495 3.9492 80 1.7244 4.6393 1.8612 4.2983 1.3944 2.8284 5.7372 1,5766 2.8284 5.0742 10.0 1.8233 5.4845 1.9680 5.0813 1.43S5 3.1623 6.9517 1.6415 3.1623 6.0920 N —4 0.05 = 0.20 Zl/Zq Z2/Zo Z3/Z0 Z4/Z0 z,/z0 z2/z0 Z3/Z0 Z4/Z0 1.0 1.0000 1.0000 1.0000 l.OOOO 1.0000 1.0000 1.0000 1.0000 1.5 1.0892 1.1742 1.2775 1,3772 1.2247 1.2247 1.2247 1.2247 2.0 1.1201 1.2979 1.5409 1.7855 1.2727 1.3634 1.4669 1.5715 3.0 1.1586 1.4876 2.0167 2.5893 1.4879 1.5819 1.8965 2.0163 4.0 1.1906 1.6414 2.4369 3.3597 1.3692 1,7490 2-2870 2.9214 6.0 1.2290 1.8773 3.1961 4.8820 1.4415 2.0231 2.9657 4.1623 8.0 1.2583 2.0657 3.8728 6,3578 1.4914 2.2428 3.5670 5.3641 10.0 1.2832 2.2268 4.4907 7.7930 1.5163 2.4210 4.1305 6.5950 Using (5.60c) for T3 gives 2[r0cos36* + Tj cosfl] = A sec3 6Wcos30 + 3cos9) - 3A sec9m cos8. Equating similar terms in cosn$ gives the following results: cos 39: 2r0 = A sec3#m, To = 0.0698; cos#: 2P| = 3A(sec3ert - sec9m), Fi =0.1037. From symmetry we also have that r3 = r0 = 0.0698, and T2 = T, =0.1037. Then the characteristic impendences are: n = 0 : In Zj = In Zo + 2r0 = In 50 + 2(0.0698) = 4.051 Z| = 57.5 £2 5.8 Tapered Lines 255 0.3 ffi FIGURE 5.17 Reflection coefficient magnitude versus frequency for lie multisection matching transformers of Example 5.7. n~l: lnZ2 = InZi+2Ti = In 57.5 + 2(0.1037) = 4.259 Z2 = 70.7 Q n = 2: In Z3 = In Z2 + 2T2 = In 70,7+ 2(0.1037) = 4.466 Z3 = 87.0 a These values can be compared to the exact values from Table 5.2 of Z\ = 57.37 ft, Z2 = 70.71 ft, and Z3 = 87.15 ft. The bandwidth, from (5.64), is or 101%. This is significantly greater than the bandwidth of the binomial transformer of Example 5.6 (70%), which was for the same type of mismatch. The trade-off, of course, is a nonzero ripple in the passband of the Chebyshev transformer. Figure 5.17 shows reflection coefficient magnitudes versus frequency for the exact designs from Table 5.2 for N = 1. 2,3, and 4 sections. ■ TAPERED LINES In the preceding sections we discussed how an arbitrary real load impedance could be matched to a line over a desired bandwidth by using multisection matching transformers. As the number, N, of discrete sections increases, the step changes in characteristic impedance between the sections become smaller. Thus, in the limit of an infinite number of sections, we approach a continuously tapered line. In practice, of course, a matching transformer must be of finite length, often no more than a few sections long. But instead of discrete sections, the line can be continuously tapered, as suggested in Figure 5.18a, By changing the type of taper, we can obtain different passband characteristics, 256 Chapter 5: Impedance Matching and Tuning Z + AZ -a o __i_|_^ z i + Az z m FIGURE 5.18 A tapered transmission line matching section and the model for an incremeotal length of tapered line, (a) The tapered transmission line matching section, (b) Model for an incremental step change in impedance of the tapered line. In this section we will derive an approximate theory, based on the theory of small reflections, to predict the reflection coefficient response a& a function of the impedance taper, Z(z). We will then apply these results to a few common types of tapers. Consider the continuously tapered line of Figure 5.18a as being made up of a number of incremental sections of length Azt with an impedance change AZ(z) from one section to the next, as shown hi Figure 5.18b. Then the incremental reflection coefficient from the step at z is given by AP = (Z + AZ)-Z * m. (5.65) (Z+AZ) + Z 2Z In the limit as Az -* 0, we have an exact differential: .r dZ l^OnZ/Zo), dv = iž = 2—ír-dz> (5-66) since mfiz)} - 1 dz -J dz • Then, by using the theory of small reflections, the total reflection coefficient at z = 0 can be found by summing all the partial reflections with their appropriate phase shifts: m4i*^Mih <5-67) where $ = 2BÍ. So if Z(z) is known, T(f?) can be found as a function of frequency. Alternatively, if is specified, then in principle Z(z) can be found. This latter procedure is difficult, and is generally avoided in practice; the reader is referred to references [ 1 ], [4] for further discussion of this topic. Here we will consider three special cases of Z(z) impedance tapers, and evaluate the resulting responses. 5.6 Tapered Unes 257 0 ir lit 3ít 4w 5fr gL (b) FIGURE 5.19 A inatching section with an exponential impedance taper, (a) Variation of impedance, (b) Resulting reflection coefficient magnitude response. Exponential Taper Consider first an exponential taper, where Z(z) = Z$eaz, for 0 < z < L, (5.68) as indicated in Figure 5.19a. At z = 0, Z(0) = Zq, as desired. At z = L, we wish to have Z(L) = ZL — Zrje01, which determines the constant a as (5.69) We now find r(9) by using (5.68) and (5.69) in (5.67): 2I -W^dne^ydz dz 2L Ja = Ml/* JfiL*nfiL 2 BL K ' Observe that this derivation assumes that Bt the propagation constant of the tapered line, is not a function of z—an assumption which is generally valid only for TEM lines. The magnitude of the reflection coefficient in (5.70) is sketched in Figure 5.19b; note that the peaks in |T| decrease with increasing length, as one might expect, and mat the length should be greater than k/2 (BL > ji) to minimize the mismatch at low frequencies. 258 Chapter 5: Impedance Matching and Tuning Triangular Taper Next consider a triangular taper for (d In Z/Zq)/^, that is, Then, Z0e2Wln z^z° for 0 < z < L f% Z^^W-mZtm for L/2 < z < L. (5.71) (ln Z/Z0) _ f 4z/L2 ln ZL/Z0 for 0 < z < L/2 dz ~ \ (4/L - 4z/L?) In ZL/Z0 for L/2 < « < L. (5.72) Z(s) is plotted in Figure 5.20a. Evaluating T from (5.67) gives The magnitude of this result is sketched in Figure 5.20b. Note that, for fiL > 27T, the peaks of the triangular taper are lower than the corresponding peaks of the exponential case. But the first null for the triangular taper occurs at pL — 2n, whereas for the exponential taper it occurs at pL a it. Klopfenstein Taper Considering the fact that there is an infinite number of possibilities for choosing an impedance matching taper, it is logical to ask if there is a design which is "best." For a given taper FIGURE 5.20 A matching station with a triangular taper for d(\nZ/Za)/dz. (a) Variation of impedance, (b) Resulting reflection coefficient magnitude response. 5.8 Tapered Lines 259 length (greater than a critical value), the Klopfensteiu impedance taper [4], [5] has been shown to be optimum in the sense that the reflection coefficient is minimum over the pass-band. Alternatively, for a maximum reflection coefficient specification in the passband, the Klopfenstein taper yields the shortest rrwtching section. The Klopfenstein taper is derived from a stepped Chebyshev transformer as the number of sections increases to infinity, and is analogous to the Taylor distribution of antenna array theory. We will not present the details of this derivation, which can be found in references [1], [4]; only the necessary results for the design of Klopfenstein tapers are given below. The logarithm of the characteristic impedance variation for the Klopfenstein taper is given by 2 cosh A A2(x. A) is defined as 4>(x, A) = —4>(~x, A) = f Jo h(Ayľ^ŕ) dy, forU| A. If 8L < A, the cos yf(8L)2 — A2 term becomes cosh -J A2 - (BL)2. In (5.74) and (5.76), To is the reflection coefficient at zero frequency, given a: ZL + Z« 2 \ZJ The passband is defined as BL > A, and so the maximum ripple in the passband i cosh A ^76) V ■8>8) because T(B) oscillates between ±Fo/ cosh A for BE > A. It is interesting to note that the impedance taper of (5.74) has steps at z = 0 and L (the ends of the tapered section), and so does not smoothly join the source and load impedances. A typical Klopfenstein impedance taper and its response are given in the following example. EXAMPLE 5.8 DESIGN OF TAPERED MATCHING SECTIONS Design a triangular taper, an exponential taper, and a Klopfenstein taper (with Tm = 0.02) to match a 50 íi load to a 100 ft line. Plot the impedance variations and resulting reflection coefficient magnitudes versus BL. 260 Chapter 5: Impedance Matching and Tuning Solution Triangular tape r: From (5.71) the impedan ce variation i s Z(z) = Zo e2(z/l}2]nZL/Z A = 3.543 = 1.13ít. Figure 5,21asb shows the impedance variations (versus z/L), and the resulting reflection coefficient magnitude (versus ft L) for the three types of tapers. The Klopfenstein taper is seen to give the desired response of |T| < Tm = 0.02 for ftL > 1,13jt, which is lower than either the triangular or exponential taper responses. Also note that, like the stepped-Chebyshev matching transformer, the response of the Klopfenstein taper has equal-ripple lobes versus frequency in its passband, ■ THE BODE-FANO CRITERION In this chapter we discussed several techniques for matching an arbitrary load at a single frequency, using lumped elements, tuning stubs, and single-section quarter-wave transformers. We then presented multisection matching transformers and tapered lines as a means of obtaining broader bandwidths, with various passband characteristics. We will now close our study of impedance matching with a somewhat qualitative discussion of the theoretical limits that constrain the performance of an impedance matching network. We limit our discussion to the circuit of Figure 5.1, where a lossless network is used to match an arbitrary complex load, generally over a nonzero bandwidth. From a very general perspective, we might raise the following questions in regard to this problem: • Can we achieve a perfect match (zero reflection) over a specified bandwidth? • If not, how well can we do? What is the trade-off between T,„, the maximum allowable reflection in the passband, and the bandwidth? • How complex must the matching network be for a given specification? These questions can be answered by the Bode-Fano criterion [7], [8] which gives, for certain canonical types of load impedances, a theoretical limit on the minimum reflection coefficient magnitude that can be obtained with an arbitrary matching network. The Bode-Fano criterion thus represents the optimum result that can be ideally achieved, even though such a result may only be approximated in practice. Such optimal results are always 262 Chapter 5: Impedance Matching and Tuning important, however, because they give us the upper limit of performance, and provide a benchmark against which a practical design can be compared. Figure 5.22a shows a lossless network used to match a parallel RC load impedance. The Bode-Fano criterion states that where F(a>) is the reflection coefficient seen looking into the arbitrary lossless matching network. The derivation of this result is beyond the scope of this text (the interested reader is referred to references [7] and {8]), but our goal here is to discuss the implications of the above result. Assume that we desire to synthesize a matching network with a reflection coefficient response like that shown in Figure 5.23a. Applying (5.79) to this function gives yet? j C IT / In -doi = / In— In — <—, (5.80) Circuit Bodc-Fano limit 1» Lossless matching network Lossless matching network J ii ■ Jn lrc«)l da* < irRC T(«) Lossless matching network 4:10 1 J VL tc) 3 Lossless matching network 1 In \n»)\ FIGURE 5.22 The Bode-Fano limits for RC and RL loads matched with passive and lossless networks (o>o is the center frequency of the matching bandwidth), (a) Parallel RC. (b) Series RC. (c) Parallel RL. (d) Series RL. References 263 Irl! (*>) FIGURE 5.23 Illustrating the Bode-Fano criterion. (a) A possible reflection coefficient response, (b) Nonrealizable and realizable reflection coefficient responses. which leads to the following conclusions: • For a given load (fixed RC product), a broader bandwidth (Aa>) can be achieved only at the expense of a higher reflection coefficient in the passband (rm). • The passband reflection coefficient F„, cannot be zero unless A is the match frequency, or use CAD to plot the reflection coefficient versus frequency. 5.13 Design a single-section quarter-wave matching transformer to match a 350 load to a 100 £2 line. What is the percent bandwidth of this transformer, for SWR < 2? If the design frequency is 4 GHz, sketch the layout of a microstrip circuit, including dimensions, to implement this matching transformer. Assume the substrate is 0.159 cm thick, with a dielectric constant of 2.2. 5.14 Consider the quarter-wave transformer of Figure 5.13, with Z\ = 100 Q, Z% = 150 £2, and ZL = 225 Q. Evaluate the worst-case percent error in computing |F| from the approximate expression of (5.42), compared to the exact result. 5.15 A waveguide load with an equivalent TE|0 wave impedance of 377 Q must be matched to an air-filled X-band rectangular guide at 10 GHz. A quarter-wave matching transformer is to be used, and is to consist of a section of guide filled with dielectric. Find the required dielectric constant and physical length of the matching section. What restrictions on the load impedance apply to this technique? 5.16 A four-section binomial matching transformer is to be used to match a 12.5 U load to a 50 £2 line at a center frequency of 1 GHz. (a) Design the matching transformer, and compiite the bandwidth for Tffl =0.05. Use CAD to plot the input reflection coefficient versus frequency, (h) Lay out the microstrip implementation of this circuit on an FR4 substrate having e, = 4.2, d = 0.158 cm, tan^ = 0.02, with copper conductors 0.5 mil thick. Use CAD to plot the insertion loss versus frequency. 5.17 Derive the exact characteristic impedance for a two-section binomial matching transformer, for a normalized load impedance Zl/Zq = 1.5. Check your results with Table 5,1. 5.18 Calculate and plot the percent bandwidth for a JV = 1, 2, and 4 section binomial matching transformer, versus ZL/Z0 = t.5 to 6 for Fm = 0.2. 5.19 Using (5.56) and trigonometric identities, verify the results of (5.60). 5.20 Design a four-section Chebyshev matching transformer to match a 40 & line to a 60 £2 load. The maximum permissible SWR over the passband is 1.2. What is the resulting bandwidth? Use the approximate theory developed in the text, as opposed to the tables. Use CAD to plot the input reflection coefficient versus frequency, 5.21 Derive the exact characteristic impedances for a two-section Chebyshev matching transformer, for a normalized load impedance Zt/Zo = 1.5. Check your results with Table 5.2 for T^, = 0.05. 5.22 A load of Zl/Zq = 1.5 is to be matched to a feed line using a multisection transformer, and it is desired to have a passband response with |r(#)| = ,4(0.1 + cos5 9), for 0 < $ < n. Use the approximate theory for multisection transformers to design a two-section transformer. 5.23 A tapered matching section has d(\w ZjZ^ijdz = A sin nz/L. Find the constant A so that Z(0) = Z{) and Z(L) = ZL. Compute T, and plot |T| versus f)L. 5.24 Design an exponentially tapered matching transformer to match a 100 Q load to a 50 Q, line. Plot | T| versus fiL, and find the length of the matching section (at the center frequency) required to obtain |F| < 0.05 over a 100% bandwidth. How many sections would be required if a Chebyshev matching transformer were used to achieve the same specifications? 5.25 An ultra wideband (UWB) radio transmitter, operating from 3.1 to 10.6 GHz, drives a parallel RC load with R = 75 Q and C — 0.6 pF. What is the best return Loss that can be obtained with an optimum matching network? 5.26 Consider a series RL load with R = 80 £2 and L = 5 nH. Design a lumped-element L-section matching network to match this load to a 50 £2 line at 2 GHz. Plot |T| versus frequency for this network to determine the bandwidth for which |T| < rm = 0.1 ■ Compare this with the maximum possible bandwidth for this load, as given by the Bode-Fano criterion. (Assume a square reflection coefficient response like that of Figure 5.23a.) Microwave Resonators Microwave resonators are used in a variety of applications, including filters, oscillators, frequency meters, and tuned amplifiers. Since the operation of microwave resonators is very similar to that of the lumped-element resonators of circuit theory, we will begin by reviewing the basic characteristics of series and parallel RLC resonant circuits. We will then discuss various implementations of resonators at microwave frequencies using distributed elements such as transmission lines, rectangular and circular waveguide, and dielectric cavities. We will also discuss the excitation of resonators using apertures and current sheets. SERIES AND PARALLEL RESONANT CIRCUITS Near resonance, a microwave resonator can usually be modeled by either a series or parallel RLC lumped-element equivalent circuit, and so we will derive some of the basic properties of such circuits below. Series Resonant Circuit A series RLC resonant circuit is shown in Figure 6.1a, The input impedance is 6.1 Zin = R + jioL - ;-—t (6.1) and the complex power delivered to the resonator is (6.2) 266 6-1 Series and Parallel Resonant Circuits 267 0.707 R _l 0 1 (b) FIGURE 6.1 A series RLC resonator and its response, (a) The series RLC circuit, (b) The input impedance magnitude versus frequency. The power dissipated by the resistor, R, is Pious = 2'^'"^' the average magnetic energy stored in the inductor, L, is 4 and the average electric energy stored in the capacitor, C, is (6.3a) (6.3b) (6.3c) where Vc is the voltage across the capacitor. Then the complex power of (6.2) can be rewritten as and the input impedance of (6.1) can be rewritten as 2P,D Ptes + 2j Z - (6.7) Pt Thus Q is a measure of the loss of a resonant circuit—lower loss implies a higher Q. For the series resonant circuit of Figure 6. la, the Q can be evaluated from (6.7) using (6.3), and the fact that Wm — We at resonance, to give Pjoss R (OqRC which shows that Q increases as R decreases. Now consider the behavior of the input impedance of this resonator near its resonant frequency [1]. We let o> = g)q + Ao>, where Aa> is small. The input impedance can then be rewritten from (6.1) as since o>5 = l/LC. Now to2 — — (D^Yoj + coq) = Aa>(2o> — Ato) — livAto for small Aco. Thus, Zin ~ R + j2LA- j = —Q - + j2L(co -«><>) = R + j2LAa>, which is identical to (6.9). This is a useful procedure because for most practical resonators the loss is very small, so the Q can be found using the perturbation method, beginning with the solution for the lossless case. Then the effect of loss can be added to the input impedance by replacing coa with the complex resonant frequency given in (6.10). 6.1 Series and Parallel Resonant Circuits 269 Finally, consider the half-power fractional bandwidth of the resonator. Figure 6.1b shows the variation of the magnitude of the input impedance versus frequency. When the frequency is such that |Zi0|2 = 2R2, then by (6.2) the average (real) power delivered to the circuit is one-half that delivered at resonance. If BW is the fractional bandwidth, then Att)/ít»o = BW/2 at the upper band edge. Then using (6.9) gives Ik + j V?0(BW)|2 = 2R2, or BW = ~. (6.11) Parallel Resonant Circuit The parallel RLC resonant circuit, shown in Figure 6.2a, is the dual of the series RLC circuit. The input impedance is z» = (ž + 7^ + y 2 1 \I\2 ' W\2 which is identical to (6.5). As íd the series case, resonance occurs when Wm ~We. Then from (6.16) and (6.14a) the input impedance at resonance is _ _ P\0%s _ p which is a purely real impedance. From (614b,c), Wm = We implies that the resonant frequency, a>o, should be defined as i*> = ^==( (6.17) which again is identical to die series resonant circuit case. From the definition of (6.7), and the results in (6.14), the Q of the parallel resonant circuit can be expressed as Q^om2^"- == wotfC, (6.18) "loss 6>oL since Wm = We at resonance. This result shows that the Q of the parallel resonant circuit increases as R increases. Near resonance, the input impedance of (6.12) can be simplified using the result that 1 --Sg 1 - x H----. I +x Letting en = to$ + Ao), where Ao) is small, (6.12) can be rewritten as [1] ( 1 Aw \'] R 1 4- 2JAojRC [+2jQ&ü>/gjq (6.19) 6.1 Series and Parallel Resonant Circuits 271 since cojj = I/LC. When R = 00 (6.19) reduces to m j2C((o - o>q) As in the series resonator case, the effect of loss can be accounted for by replacing wo in this expression with a complex effective resonant frequency: o* <— *>o (l + M (6.20) Figure 6.2b shows the behavior of the magnitude of the input impedance versus frequency. The half-power bandwidth edges occur at frequencies (Aco/coo = BW/2), such that which, from (6.19), implies that as in the series resonance case. BW=— (6.21) Q Loaded and Unloaded Q The Q defined in the preceding sections is a characteristic of the resonant circuit itself, in the absence of any loading effects caused by external circuitry, and so is called the unloaded Q. In practice, however, a resonant circuit is invariably coupled to other circuitry, which will always have the effect of lowering the overall, or loaded Q, Qi, of the circuit. Figure 6.3 depicts a resonator coupled to an external load resistor, Ri. If the resonator is a series RLC circuit, the load resistor Ri adds in series with R so that the effective resistance in (6.8) is R + Rl , If the resonator is a parallel RLC circuit, the load resistor R1 combines in parallel with R so that the effective resistance in (6.18) is R Rl/(R + Rl)- If we define an external Q, Qe, as Ö, = Rl for series circuits for parallel circuits. (6.22) then the loaded Q can be expressed as J_ 1 I Ql ~ Qe + Q' Table 6.1 summarizes the above results for series and parallel resonant circuits. (6.23) Resonant circuit Q FIGURE 6.3 A resonant circuit connected to an external load, RL. 272 Chapter 6: Microwave Resonators TABLE 6.1 Summary of Results for Series and Parallel Resonators Quantity Series Resonator Parallel Resonator Input Impedance/admittance 1 Zm = R + jtoL - j— cuC R coL 2RQAa> ^R + j—-- 1 IQAto R J Rojq Power loss PU= \\I\2R 1 |V|2 Stored magnetic energy W,„=l-\I\2L w =Vl2-L Stored electric energy wt = ^\v\2c Resonant frequency 1 Unloaded Q o - _ 1 R wqRC ft Q = a>aRC- —- External Q TRANSMISSION LINE RESONATORS As we have seen, ideal lumped circuit elements are usually unattainable at microwave frequencies, so distributed elements are more commonly used. In this section we will study the use of transmission line sections with various lengths and terminations (usually open or short circuited) to form resonators. Since we will be interested in the Q of these resonators, we must consider lossy transmission lines. Short-Circuited A/2 Line Consider a length of lossy transmission line, short circuited at one end, as shown, in Figure 6.4, The line has a characteristic impedance Zo, propagation constant B, and attenuation constant a. At the frequency to = wq, the length of the line is I = A/2, where K = 2ji/B. From (2.91), the input impedance is Zm = Z(,tetih(a + j8)t. Using an identity for the hyperbolic tangent gives tanhofl + ytan/il ia °1 + j tan^taiuW' Observe that Zia == jZ0 VwB£ if a = 0 (no loss). In practice, most transmission lines have small loss, so we can assume that at \, and so tanh at ~ at. Now lei to = coq + Aoj, where Aw is small. Then, assuming a TEM line, 0)1 Otot Atot vp vp vp 6.2 Transmission Line Resonators 273 ■—^* o- 21,. 0. a -cm FIGURE 6.4 A short — 0(£= X/2), and its input impedance at this frequency is Zin = R = Z^at. Resonance also occurs fori = nX/2,n — 1, 2, 3..... The voltage distributions for the n — 1 and n = 2 resonant modes are shown in Figure 6.4. 274 Chapter 6: Microwave Resonators The Q of this resonator can be found from (6.8) and (6.26) as Q=<*±=* ± (627) * R 2at 2a since fit = it at the first resonance. This result shows that the Q decreases as the attenuation of the line increases, as expected. EXAMPLE 6,1 Q OF HALF-WAVE COAXIAL LINE RESONATORS A k/2 resonator is made from a piece of copper coaxial line, with an inner conductor radius of 1 mm and an outer conductor radius of 4 mm. If the resonant frequency is 5 GHz, compare the Q of an air-filled coaxial line resonator to that of a Teflon-filled coaxial line resonator. Solution Wc must first compute the attenuation of the coaxial line, which can be done using the results of Example 2.6 or 2.7. From Appendix F, the conductivity of copper is a =5.813 x 107 S/m. Then the surface resistivity is Rs = = 1-84 x I0"2 a, and the attenuation due to conductor loss for the air-filled line is & f% i\ ac 2rt]nb/a\a b) (-4r- + 7t-) = 0-022 Np/m. V0.001 0,004/ F 1 .84 x 10 2(377)ln(0.004/0.001) For Teflon, €T — 2.08 and tan S = 0.0004, so the attenuation due to conductor loss for the Teflon-filled line is 1.84 x 10-V2l)8 - ~ 2(377)ln(0.004/0.001) f —J— + ) = 0.032 Np/m. \ 0.001 0.004/ The dielectric loss of the air-filled line is zero, but the dielectric loss of the Teflon-filled line is j—- (104.7)v^08(0.0004) -„-M, =---= 0.030 Np/m. Finally, from (6.27), the Qs can be computed as *™ 2« 2(0.022) _ fi _ 104.7yT08 y-^flon - 2a - 2(0 Q32 - 0 Q30) - um Thus it is seen that the Q of the air-filled line is almost twice that of die Teflon-filled line. The Q can be further increased by using silver-plated conductors. ■ 6,2 Transmission Line Resonators 275 Short-Circuited A/4 Line A parallel type of resonance (antiresonance) can be achieved using a short-circuited transmission line of length A/4. The input impedance of the shorted line of length t is Zm = Z0 tanh(« 4- jB)t tarnW + ; tan Bt = Z0 1 + jim&iisaňioíi = ^-J***"*/*, (6.28) tarnW -jcotBt where the last result was obtained by multiplying both numerator and denominator by —j cot Bt. Now assume that i = a/4 at oj = too, and let m = ojq + Aa). Then, for a TEM line, OJqS. Atot 71 71 Aw m- — + — = t + ^—■ in 7iAo>\ 7i Aw and so cot Bt = cot I — -|--| = - tan- V 2 2a)o } lino Aw —jtAí/j 2o>q 2o>Q Also, as before, tanhůť at for small loss. Using these results in (6.28) gives - = 14-jg/2aip ^ Zp ^ at + j jt Ag>/2u)(í at + jtiAw/Io^' since a in Aůj/2íí>o . Then, tiAco pe = n + and so tan fii — tan Aüj-jT A our and tanhofí ja arí. Using these results in (6.32) gives --—-;—- (6.33) ccl + j(Aom/iúQ) Comparing with the input impedance of a parallel resonant circuit as given by (6,19) suggests that the resistance of the equivalent RLC circuit is R = —£, (6-34a) at o- FIGURE 6-5 An open-circuited length of lossy transmission line, and the voltage distributions for n = l(( = X/2) and n = 2 (f = A) resonators, 6,2 Transmission Line Resonators 277 and the capacitance of the equivalent circuit is C = —^—. (6.34b) 2^o Z"o The inductance of the equivalent circuit is L = —. (6.34c) co2QC From (6.18) and (6.34) the Q is Q = ^RC = ~ = ^~. (6.35) 2nr£ 2a since € = 7r/^ at resonance. EXAMPLE 6.2 A HALF-WAVE M1CROSTRIP RESONATOR Consider a microstrip resonator constructed from a X/2 length of 50 Q open-circuited microstrip line. The substrate is Teflon = 2.08, tan 5 = 0.0004), with a thickness of 0.159 cm. The conductors are copper. Compute the length of the line for resonance at 5 GHz, and the Q of the resonator. Ignore fringing fields al the end of the line. Solution From (3,197), the width of a 50 Q, microstrip line on this substrate is found to be W = 0.508 cm, and the effective permittivity is ft = 1.80. Then the resonant length can be calculated as „ fc % c 3 x 108 * = x = -4 = ^- =-^= = 2.24 cm. 2 2/ 2fJTe 2(5 x 10«)v/L80 The propagation constant is . 2itf 27ifJQ 2jt(5 x 109)VT80 A ,. 8 =-=-— =--— = 151.0 rad/m. vp c 3 x 10s From (3.199), the attenuation due to conductor loss is R, 1.84x10' ctc = —--■ = ——- = 0.0724 Np/m, Z0W 50(0.00508) F where we used Rs from Example 6.1. From (3.198), the attenuation due to dielectric loss is k0€r(€e - 1)tan5 (104.7)(2.08)(0.80)(0.0004) aj =-—-=----= 0,024 Np/m. 2V^(er - 1) 2700(1.08) Then from (6.35) the Q is 278 Chapter 6: Microwave Resonators RECTANGULAR WAVEGUIDE CAVITIES Resonators can also be constructed from closed sections of waveguide, which should not be surprising since waveguides are a type of transmission line. Because of radiation loss from open-ended waveguide, waveguide resonators are usually short circuited at both ends, thus forming a closed box or cavity, Electric and magnetic energy is stored within the cavity, and power can be dissipated in the metallic walls of the cavity as well as in the dielectric filling the cavity. Coupling to the resonator can be by a small aperture or a small probe or loop. We will first derive the resonant frequencies for a general TE or TM resonant mode, and then derive an expression for die Q of the TBm mode. A complete treatment of the Q for arbitrary TE and TM modes can be made using the same procedure, but is not included here because of its length and complexity. Resonant Frequencies The geometry of a rectangular cavity is shown in Figure 6.6. It consists of a length d of rectangular waveguide shorted at both ends (z = 0, d). We first find the resonant frequencies of this cavity under the assumption that the cavity is lossless, then we determine the Q using the perturbation method outlined in Section 2.7, We could begin with the wave equations and use the method of separation of variables to solve for the electric and magnetic fields that satisfy the boundary conditions of the cavity, but it is easier to start with the TE and TM waveguide fields, which already satisfy the necessary boundary conditions on the side walls (x = 0, a and y = 0, b) of the cavity. Then it is only necessary to enforce the boundary conditions that Ex = Ey = 0 on the end walls at z = 0, d. From Table 3.2 the transverse electric fields {Ex, Ey) of the TEmn or TMm)1 rectangular waveguide mode can be written as Et(x, .V, z) = e(x, y)[_A+e-rf">»z + A~ej^"% (6.36) FIGURE 6.6 A rectangular resonant cavity, and the electric field distributions for the TEioi and TE|02 resonant modes. 6.3 Rectangular Waveguide Cavities 279 where e(x, y) is the transverse variation of the mode, and A+, A~ are arbitrary amplitudes of the forward and backward traveling waves. The propagation constant of the m. nth TE or TM mode is where k = oi^/JIi, and ft, € are the permeability of permittivity of the material filling the cavity. Applying the condition that E, = 0 at z = 0 to (6.36) implies that A+ = —A~ (as we should expect for reflection from a perfectly conducting wall), Then the condition that ET = 0 at z — d leads to the equation Et{x, y, d) = -e(x, y)A+2j sin Bmnd = 0. The only nontrivial (A+ 5$ 0) solution thus occurs for &mad = tn, £=1,2,3..... (6.38) which implies that the cavity must be an integer multiple of a half-guide wavelength long at the resonant frequency. No nontrivial solutions are possible for other lengths, or for frequencies other than the resonant frequencies. The rectangular cavity is thus a waveguide version of the short-circuited k/2 transmission line resonator, A resonant wavenumber for the rectangular cavity can be defined as , =0 ' mpl)2 JUr* v a ) + {P>mp) M a ) pdp \pL/ J»(Pnm)> (6.55) where the integral identity of Appendix C. 17 has been used. The power loss in the conducting walls is = T-i I 1" [\H4,(p = a)\2 + \Hz(p=a)\2]addz £ I Jz=0 J=0 + 2T f [\Hp(z = 0)\2 + \Ht))(z = 0)\2]pdpdA Then, from (6.8), the Q of the cavity with imperfectly conducting walls but lossless dielectric is coqW (ka^rjad 1 *(p'nm)2Rs (6.57) From (6.52) and (6.51) we see that B — injd and (ka)2 are constants that do not vary with frequency, for a cavity with fixed dimensions. Thus, the frequency dependence of Qc is given by k/Rs, which varies as i/\fj; this gives the variation in Qc for a given resonant mode and cavity shape (fixed n, m, £, and ajd). Figure 6.10 shows the normalized Q due to conductor loss for various resonant modes of a cylindrical cavity. Observe that the TEon mode has a Q significantly higher than the lower-order TEi11, TMqio. or TMn, modes. To compute the Q due to dielectric loss, we must compute the power dissipated in the dielectric. Thus, 286 Chapter 6: Microwave Resonators Then (6.8) gives the Q as 0^ = — = - = —, (6.59) Pa €" tan 5 where tan 5 is (he loss tangent of the dielectric. This is the same as the result for Qd of (6.48) for the rectangular cavity. When both conductor and dielectric losses are present, the total cavity Q can be found from (6.49). EXAMPLE 6,4 DESIGN OF A CIRCULAR CAVITY RESONATOR A circular cavity resonator with d = 2a is to be designed to resonate at 5.0 GHz in the TE OG. FIGURE 6.12 Magnetic wall boundary condition approximation and distribution of H. versus z for p = 0 of the first mode of the cylindrical dielectric resonator. 6.5 Dielectric Resonators 289 We begin by finding the fields of the TEoi dielectric waveguide mode with a magnetic wall boundary condition at p = a. For TE modes, Ez = 0, and Ht must satisfy the wave equation (V2 + k2)Hz = 0, (6.60) , , Iffik fork|L/2. Since 'd/B

L/2, the propagation constant will be imaginary, so it is convenient to write a = fil - 4 = J^)2-kl (6.66a) and to define a wave impedance in the air region as Zd = fe, (6.66b) a which is seen to be imaginary. From symmetry, the Hz and E^ field distributions for the lowest-order mode will be even functions about z = 0. Thus the transverse fields for the TEq]S mode can be written for |;| < L/2 as E^ — A Jv(kcp) cos fiz. (6.67a) Hp = ~i= JiiKp) sin pz, (6.67b) 290 Chapters: Microwave Resonators and for \z\ > L/2 as £0 = B4(kcp)e-alz\ (6.68a) Hp = —JQ(k L/2 or z < —L/2, respectively. Matching tangential fields at z = L/2 (or z = —L/2) leads to the following two equations: A cos IK« — B which determines the resonant frequency. This solution is relatively crude, since it ignores fringing fields at the sides of the resonator, and yields accuracies only on the order of 10% (not accurate enough for most practical purposes), but it serves to illustrate the basic behavior of dielectric resonators. More accurate solutions are available in the literature [3]. The Q of the resonator can be calculated by determining the stored energy (inside and outside the dielectric cylinder), and the power dissipated in the dielectric and possibly lost to radiation. If the latter is small, the Q can be approximated as 1/ tan 5, as in the case of the metallic cavity resonators. EXAMPLE 6.5 RESONANT FREQUENCY AND Q OF A DIELECTRIC RESONATOR Find the resonant frequency and approximate Q for the TEqk mode of a dielectric resonator made from titania, with er = 95, and tan S = 0.001. The resonator dimensions are a = 0,413 cm, and L = 0.8255 cm. Solution The transcendental equation of (6.70) must be solved for *0, with B and a given by (6.65a) and (6.66a). Thus, BL where cfo c(2.405) ■ „ /2 = — = ~-- = 27.804 GHz. 2jt 2na Using the interval-halving method (see the Point of Interest on root-finding algorithms in Chapter 3) to find die root of the above equation gives a resonant frequency of about 3.152 GHz. This compares with a measured value of about 3.4 GHz from reference [2], indicating a 10% error. The approximate Q, due to dielectric loss, is Qd ~ r~~r = 1000. ■ tan 5 EXCITATION OF RESONATORS We now discuss how the resonators of the previous sections can be coupled to external circuitry. In general, the way in which this is done depends on the type of resonator under consideration; some typical coupling techniques are shown for various resonators in Figure 6,13. In this section we will discuss the operation of some of the more common coupling techniques, notably gap coupling and aperture coupling. First we will illustrate the concept of critical coupling, whereby a resonator can be matched to a feedline, using a lumped-element resonant circuit. Critical Coupling To obtain maximum power transfer between a resonator and a feedline, the resonator must be matched to the feed at the resonant frequency. The resonator is then said to be critically FIG UKE 6.13 Coupling to microwave resonators, (a) A microstrip transmission line resonator gap coupled to a microstrip feedline. (b) A rectangular cavity resonator fed by a coaxial probe, (c) A circular cavity resonator aperture coupled to a rectangular waveguide, (d) A dielectric resonator coupled to a microstrip feedline. 292 Chapter 6: Microwave Besqnators FIGURE 6.14 A series resonant circuit coupled to a feedline. coupled to the feed. We will first illustrate the basic concept of critical coupling by considering the series resonant circuit shown in Figure 6.14. From (6.9), the input impedance near resonance of the series resonant circuit of Figure 6.14 is given by Zit, = R + j2L Aw = R + j —Ž^ŽS (6.71) and the unloaded Q is, from (6.8)T C = ^. (6-72) At resonance, Aw = 0, so from (6.71) the input impedance is Zm = R. In order to match the resonator to the line we must have Then the unloaded Q is From (6.22), the external Q is R = Z0. (6.73) &=M (6.74) ■^0 Qe = = Q, (6.75) which shows that the external and unloaded Qs are equal under the condition of critical coupling. It is useful to define a coefficient of coupling, g, as * = 7P (6.76) which can be applied to both series (_g = Zq/R) and parallel (g = R/Z$) resonant circuits. Then, three cases can be distinguished. 1. g < 1 The resonator is said to be undercoupled to the feedline. 2. g = 1 The resonator is critically coupled to the feedline. 3. g > 1 The resonator is said to be overcoupled to the feedline. Figure 6.15 shows a Smith chart sketch of the impedance loci for the series resonant circuit, as given by (6.71), for various values of R corresponding to the above cases. A Gap-Coupled Microstrip Resonator Next we consider a X/2 open-circuited microstrip resonator coupled to a microstrip feedline, as shown in Figure 6.13a, The gap in the microstrip line can be approximated as a series 6,6 Excitation of Resonators 293 FIGURE 6.15 Smith chart illustrating coupling to a series RLC circuit. capacitor, so the equivalent circuit of this resonator and feed can be constructed as shown in Figure 6.16. The normalized input impedance seen by the feedline is then Z . [1 /coC + Z<, cot pi] Zo Zy ,/tan B£ + be\ where bc = Z0a>C is the normalized susceptance of the coupling capacitor, C. Resonance occurs with z = 0, or when tan pi + bc = 0, (6.78) —II— Feed line Gap capacitance Open-circuit A/2 resonator FIGURE 6.16 Equivalent circuit of the gap-coupled microstrip resonator of Figure 6.13a. 294 Chapter 6: Microwave Resonators The solutions to this transcendental equation are sketched in Figure 6.17. In practice, bc \, and assuming that bc is small. Thus, Z((t>) - Z(ú>i) + íů>- £Ü[) From (6.77) and (6.78), zM = 0. Then, db) + (6.79) dz_ dü) m -jfsgggd(ßt) _ j(\+b2} t ^ bc tan ßi dco ti1 v„ ~ b2 vp since bc <§; 1 and I ~jrvpfo>i, where vp is the phase velocity of the transmission line (assumed TEM). Then the normalized impedance can be written as z(co) = ú>ib2 (6.80) So far we have ignored losses, but for a high-Q cavity loss can be included by replacing the resonant frequency iú\ with the complex resonant frequency given by wi(l + j/2Q), which follows from (6.10). Applying this procedure to (6.80) gives the input impedance of gap-coupled lossy resonator as lQb\ + j 7T(íW -- Of\) o>\b2 (6.81) Note that an uncoupled k/2 open-circuited transmission tine resonator looks like a parallel RLC circuit near resonance, but the present case of a capacitive coupled X/2 resonator looks like a series RLC circuit near resonance. This is because the series coupling capacitor has the effect of inverting the driving point impedance of the resonator (see the discussion of impedance inverters in Section S.5). 6.6 Excitation of Resonators 295 At resonance, then, the input resistance is R = Z^x-/2Q.tr*. For critical coupling we must have R = Zo, or (6,82) The coupling coefficient of (6.76) is o Zo 2Qb-, = _ = _. (6.83) Mbc < V^/2^ttheng < 1 and the resonator is undercoupled; if bc > vW2£?,theng > 1 and the resonator is overcoupled. EXAMPLE 6.6 DESIGN OF A GAP-COUPLED MICROSTRIP RESONATOR A resonator is made from an open-circuited 50 & microstrip line, and is gap-coupled to a 50 £2 feedline, as in Figure 6.13a. The resonator has a length of 2.175 cm, an effective dielectric constant of 1.9, and an attenuation of 0.01 dBA;m near its resonance. Find the value of the coupling capacitor required for critical coupling, and the resulting resonant frequency. Solution The first resonant frequency will occur when the resonator is about t = ksf2 in length. Thus, ignoring fringing fields, the approximate resonant frequency is v„ c 3 x 10s fo = — = :r—= <=-== = 5.00 GHz, ks 2tJTe 2(0.02175 )VT9 which does not include die effect of the coupling capacitor. Then from (6.35) the Q of this resonator is 0=-?- = — = — = n{%J dB/Np) - 628 y 2a ksa 2ta 2(0.02175 m)(ldB/m) From (6.82) the normalized coupling capacitor susceptance is bc = JL = = 0.05, V 2Q V 2(628) so the coupling capacitor has a value of C=^=_°-°5, = 0.032 pp. coZa 2jt(5 x 109)(50) which should result in the critical coupling of the resonator to the 50 Q feedline. Now that C is determined, the exact resonant frequency can be found by solving the transcendental equation of (6.78). Since we know from the graphical solution of Figure 6.17 that the actual resonant frequency is slightly lower than the unloaded resonant frequency of 5.0 GHz, it is an easy matter to calculate (6.78) for several frequencies in this vicinity, which leads to a value of about 4.918 GHz. This is about 1.6% lower than the unloaded resonant frequency. Figure 6.18 shows a Smith chart plot of the input impedance of the gap-coupled resonator for coupling capacitor values that lead to under, critical, and overcoupled resonators. ■ 296 Chapter 6: Microwave Resonators FIGURE 6.18 Smith chart plot of input impedance of the gap-coupled microstrip resonator of Example 6.6 versus frequency for various values of the coupling capacitor. An Aperture-Coupled Cavity As a final example of resonator excitation, we will consider the aperture coupled waveguide cavity shown in Figure 6. J 9. As discussed in Section 4.8, a small aperture in the transverse wall acts as a shunt inductance. If we consider the first resonant mode of the cavity, which occurs when the cavity length i = ks/2, then the cavity can be considered as a transmission line resonator shorted at one end. The aperture-coupled cavity can then be modeled by the equivalent circuit shown in Figure 620. This circuit is basically the dual of the equivalent circuit of Figure 6.16, for the gap-coupled rnicrostrip resonator, so we will approach the solution in the same manner. Aperture - j b f / / / / 0 / 1 circuit ■ / V / 2 ■Waveguide-H-*-Cavity-H FIGURE 6J9 A rectangular waveguide aperture coupled to a rectangular cavity. 6.6 excitation of Resonators 297 FIGURE 6.20 Equivalent circuit of the aperture-coupled cavity. The normalized input admittance seen by the feedline is tan 81 where jcj. = mL/Zq is the normalized reactance of the aperture. An antiresonance occurs when the numerator of (6.84) vanishes, or when tan fit + xL = 0, (6.85) which is similar in form to (6.78), for the case of the gap-coupled microstrip resonator. In practice, xl 1, so the first resonant frequency, co\, will be close to the resonant frequency for which 8t = jz, similar to the solution illustrated in Figure 6.17. Using the same procedure as in the previous section, the input admittance of (6.84) can be expanded in a Taylor series about the resonant frequency, o)^ assuming xL ) = B2cx\ (6.87) In (6.87), ko, B> and xL should be evaluated at the resonant frequency \ in the numerator of (6.87) with co. From (6.84) it is seen that this occurs at a frequency such that tanj3i = 0, or 3t = tz. In this case the cavity is exactly Xs/2 long, so a null in the transverse electric field exists at the aperture plane, and the aperture has no effect. This mode is of Utile practical interest, because of this loose coupling, 298 Chapter 6: Microwave Resonators 6.7 The excitation of a cavity resonator by an electric current probe or loop can be analyzed by the method of modal analysis, similar to that discussed in Sections 4.7 and 4.8. The procedure is complicated, however, by the fact that a complete modal expansion requires fields having irrotational (zero curl) components. The interested reader is referred to references [l]and[4], CAVITY PERTURBATIONS In practical applications cavity resonators are often modified by making small changes in their shape, or by the introduction of small pieces of dielectric or metallic materials. For example, the resonant frequency of a cavity can be easily tuned with a small screw (dielectric or metallic) that enters the cavity volume, or by changing the size of the cavity with a movable wall. Another application involves the determination of dielectric constant by measuring the shift in resonant frequency when a small dielectric sample is introduced into the cavity. In some cases, the effect of such perturbations on the cavity performance can be calculated exactly, but often approximations must be made. One useful technique for doing this is the perturbational method, which assumes that the actual fields of a cavity with a small shape or material perturbation are not greatly different from those of the unperturbed cavity. Thus, this technique is similar in concept to the perturbational method introduced in Section 2.7 for treating loss in good conductors, where it was assumed that there was not a significant difference between the fields of a component with good conductors and one with perfect conductors. In this section we will derive expressions for the approximate change in resonant frequency when a cavity is perturbed by small changes in the material tilling the cavity, or by small changes in its shape. Material Perturbations Figure 6.21 shows a cavity perturbed by a change in the permittivity (Ae), or permeability (Afi), of all or part of the material filling the cavity. If £o, Hq are the fields of the original cavity, and Ě, H are the fields of the perturbed cavity, then Maxwell's curl equations can be written for the two cases as V x Eq — -yoJo/j/Yo. V x Ě — —jw(ß + Aß)H, V x H = jů>(€ + Ae)£, (6.90a) (6.90b) (6.91a) (6.91b) where coq is the resonant frequency of the original cavity and o> is the resonant frequency of the perturbed cavity. (b) FIGURE 6.21 A resonant cavity perturbed by a change in the permittivity or permeability of (he material in the cavity, (a) Original cavity, (b) Perturbed cavity. 6.7 Cavity Perturbations 299 Now multiply the conjugate of (6.90a) by H and multiply (6.91b) by EJ to get H-Vx£0' = jo^iifl >H£t E^ ■ V xH = jit>(€ + A{e + Ae)£^ * E. (6.92a) Similarly, we multiply the conjugate of (6.90b) by £ and multiply (6.91a) by ff0* to get E-Vx ff* = -M)€£*-E, B* - V x E = -jtoQi + An)H£ ■ H. Subtracting these two equations and using vector identity (B.8) gives V ■ (£ x tf0*) = -jto(n + Am)£o ' # + M^o Is (6.92b) Now add (6.92a) and (6,92b), integrate over the volume Vb, and use the divergence theorem to obtain / V * (££ x H + E x H£)dv = & (£j x H + E x /?„*) ■ ds ~ 0 = j I {[woe - + A<0]££ • E + [tj^ii - (w(/i + A/i)]W0* ■ tfjrfv, (6.93) where the surface integral is zero because n x E = 0 on So- Rewriting gives «> - wo - /vi<^< £ ■ £0* + Ajtiff • H£)dv (o j' (e£ ■ + /iff ■ ff0')erturbed cavities, so that the decrease in resonant frequency can be related to the increase in stored energy of the perturbed cavity. EXAMPLE 6.7 MATERIAL PERTURBATION OF A RECTANGULAR CAVITY A rectangular cavity operating in the TEiol mode is perturbed by the insertion of a thin dielectric slab into the bottom of the cavity, as shown in Figure 6.22. Use the perturbational result of (6.95) to derive an expression for the change in resonant frequency. 300 Chapter6: Microwave Resonators FIGURE 6.22 A rectangular cavity perturbed by a thin dielectric slab. Solution From (6.42a-c), the fields for the unperturbed TE101 cavity mode can be written as /:-, = A sin —- sin —. a d —JA ItX 71 z H, = —— sin — cos a d ' JTlA jzx . jiz HT —--cos — sin —, a In the numerator of (6.95), Ae — (er - l)eo for 0 < y < t, and zero elsewhere. The integral can then be evaluated as f (At|£0|2 + An\80\2)dv = & - l)«o [ [ f \Ey\2dzdydx JV Jx=Q Jy=(} Jz=H (*r - l)eoA2ald The denominator of (6.95) is proportional to the total energy in the unperturbed cavity, which was evaluated in (6.43), thus, L (c\E0\£ + fi\HQ\*)dv = Then (6.95) gives the fractional change (decrease) in resonant frequency as (u — a>o —itr — 1)( 2b Shape Perturbations Changing the size of a cavity or inserting a tuning screw can be considered as a change in the shape of the cavity and, for small changes, can also be treated by the perturbation technique. Figure 6.23 shows an arbitrary cavity with a perturbation in its shape; we will derive an expression for the change in resonant frequency. As in the case of material perturbations, let Eq, Ha, too be the fields and resonant frequency of the original cav ity and let E, H, w be the fields and resonant frequency of the perturbed cavity. Then Maxwell's curl equations can be written for the two cases as V x Ea = -jöJoßHa, V x Hq = j(üo€EQ. (6.96a) (6.96b) 6.7 Cavity Perturbations 301 (a) (hj FIGURE 6.23 A resonant cavity perturbed by a change in shape, (a) Original cavity, (b) Perturbed cavity. 7xl = - jtotiH, (6.97a) V x H = jateE. (6.97b) Now multiply the conjugate of (6.96a) by H and multiply (6.97b) by E$ to gel H - V x £* = jaotiff - H£, ■ V xH = jo)€El ■ E. Subtracting these two equations and using vector identity (B.S) then gives V • (El x H) = fttofiH • H$ - jaxEl • E. (6.98a) Similarly, we multiply the conjugate of (6.96b) by E and (6.97a) by Hq to get E ■ V x = -ja>QtE ■ • V x E = -jotfiHS ■ H. Subtracting and applying vector identity (B.S) gives V ■ (E x £0*) = -jwiiH^ ■ H + jo)^E ■ £E^xHds = -j(a> - o>o) y (e£ ■ £J + M?? ■ (6-99) since fix E = 0 on 5. Since the perturbed surface S = So — AS, we can write - coo AWm-AWc (6.103) where AW* and A are the changes in the stored magnetic energy and electric energy, respectively, after the shape perturbation, and Wm -f We is the total stored energy in the cavity. These results show that the resonant frequency may either increase or decrease, depending on where the perturbation is located and whether it increases or decreases the cavity volume. EXAMPLE 6.8 SHAPE PERTURBATION OF A RECTANGULAR CAVITY A thin screw of radius m extends a distance £ through the center of the top wall of a rectangular cavity operating in the TEioi mode, as shown in Figure 6.24. If the cavity is air-rilled, use (6.102) to derive an expression for the change in resonant frequency from the unperturbed cavity. Solution From (6.42a-c), the fields for the unperturbed TEioi cavity can be written as _ . 1ZX . 1ZZ Ev = A sin — sin —, a d -JA , 7tx nz Hx = - — sin — cos Zxe & a jizA jix , nz Hz ---cos — sin —-. kna a d ^ - - - m- ■--------, FIGURE 6.24 A rectangular cavity perturbed by a tuning post in the center of the top wall. Problems 303 Now if the screw is thin, we can assume that the fields are constant over the cross-section of the screw and can be represented by the fields at x = a/2, z = d/2: Hx {x ~\,y,z~ f) = °' Then the numerator of (6.102) can be evaluated as / (M|r?o|2 -€\E0\2)dv^-€0{ A2dv = -e(iA2AV, Jav Jav where A V = itlr\ is the volume of the screw. The denominator of (6.102) is, from (6.43), abdeGA2 Vq€0A2 fv(M\'+^mi)dv= 2 2 where Vb = abd is the volume of the unperturbed cavity. Then (6.102) gives (Q- = it. The phases of the terms having amplitude B are chosen to be 180° apart. Then the scattering matrix has the following form: ro ß L0 a 0 0 -ß ß 0 0 a 0 1 -ß a 0 J (7.18) Note that the two couplers differ only in the choice of reference planes. Also, the amplitudes a and 8 are not independent, as (7.14a) requires that a2 + ß2 = 1. (7.19) Thus, apart from phase references, an ideal directional coupler has only one degree of freedom. Another way for (7.11) and (7.13) to be satisfied is if |5131 = and |%| = |5*34|.If we choose phase references, however, such that S\3 = S24 = and Si 2 = $34 = jB (which satisfies (7.16)), then (7.10a) yields + S^) = 0, and (7.12a) yields 8(S*U -S23) = 0. These two equations have two possible solutions. First, S14 = S23 = 0, which is the same 7.1 Basic Properties of Dividers and Couplers 313 Input CD © Through ! scaled (3) Coupled Input Isolated CD © Through Q Coupled FIGURE 7.4 Two commonly used symbols for directional couplers, and power flow conventions. as the above solution for the directional coupler. The other solution occurs for a = 8 = 0, which implies that Sl2 = = 5M = S34 = 0. This is the case of two decoupled two-port networks (between ports 1 and 4, and ports 2 and 3), which is of trivial interest and will not be considered further. We are thus left with the conclusion that any reciprocal, lossless, matched four-port network is a directional coupler. The basic operation of a directional coupler can be illustrated with the aid of Figure 7.4, which shows two commonly used symbols for a directional coupler and the port definitions. Power supplied to port 1 is coupled to port 3 (the coupled port) with the coupling factor |S)3\2 = J82, while the remainder of the input power is delivered to port 2 (the through port) with the coefficient \Sft\2 = a2 — 1 - 82. In an ideal directional coupler, no power is delivered to port 4 (the isolated port). The following three quantities are generally used to characterize a directional coupler: Coupling = C = 10 log = -201ogj8 dB, Directivity = D = 10 log —^ = 20log ~ - dB, Pi \Su\ -20logj5|4| dB. Isolation = 7 = 10 log — — Pa (7.20a) (7.20b) (7.20c) The coupling factor indicates the fraction of the input power that is coupled to the output port. The directivity is a measure of the coupler's ability to isolate forward and backward waves, as is the isolation. These quantities are then related as I = D + C dB. (7.21) The ideal coupler would have infinite directivity and isolation {S\4 = 0). Then both a and 8 could be determined from the coupling factor, C. Hybrid couplers are special cases of directional couplers, where the coupling factor is 3 dB, which implies that a = 3 = \/V2. There are two types of hybrids. The quadrature hybrid has a 90° phase shift between ports 2 and 3 (# = 4> = when fed at port 1, and is an example of a symmetrical coupler. Its [S\ matrix has the following form: -0 1 j o- 1 1 0 0 j 71 j 0 0 1 -0 j 1 (7.22) 314 Chapter 7: Power Dividers and Directional Couplers The magic-T hybrid or rat-race hybrid has a 180* phase difference between ports 2 and 3 when fed at port 4, and is an example of an ami symmetrical coupler Its [S] matrix has the following form: "0 1 1 o - 1 1 0 0 -1 1 0 0 1 _0 -1 1 0 . POINT OF INTEREST: Measuring Coupler Directivity The directivity of a directional coupler is a measure of the coupler's ability to separate forward and reverse wave components, so applications of directional couplers often require high (35 dB or greater) directivity. Poor directivity will limit the accuracy of a reflectometer, and can cause variations in the coupled power level from a coupler when there is even a small mismatch on the through line. The directivity of a coupler generally cannot be measured directly because it involves a low-level signal that can be masked by coupled power from a reflected wave on the through arm. For example, if a coupler has C = 20 dB and D = 35 dB, with a load having RL = 30 dB, the signal level through the directivity path will beD + C = 55dB below the input power, but the reflected power through the coupled arm will only be RL + C = 50 dB below the input power. One way to measure coupler directivity uses a sliding matched load, as follows. First, the coupler is connected to a source and matched load, as shown in the left-hand figure below, and die coupled output power is measured. If we assume an input power P;, this power will be Pc = C2Pi, where C = 10'"c dBV2° is the numerical voltage coupling factor of the coupler. Now reverse the position of the coupler as shown in the right-hand figure below and terminate the through line with a sliding load. Changing the position of the sliding load introduces a variable phase shift in the signal reflected from the load and coupled to the output port Thus the voltage at the output port can be wriden as where Vj- is the input voltage, D = ]fyDdB)''20 ^ l is the numerical value of the directivity, |F| is the reflection coefficient magnitude of the load, and 0 is the path length difference between die directivity and reflected signals. Moving the sliding load changes 0, so the two signals will combine to trace out a circular locus, as shown in the following figure, 1m Yqf, 7.2 The T-Junction Power Divider 315 The minimum and maximum output powers are given by Now let M and m be defined in terms of these powers as follows: i= Pc { d V = / l + inpy Final These ratios can be accurately measured directly by using a variable attenuator between the source and coupler, The directivity (numerical) can then be found as / 2m \ D = M\ - . This method requires that |T| < 1/d or, in dB, rl > d, Reference- M. Sucher and J. Fox, editors, Handbook of Microwave Measurements, third ediiion, volume II, Polytechnic Press, New York, 1963. THE T-JUNCTION POWER DIVIDER The T-junction power divider is a simple three-port network that can be used for power division or power combining, and can be implemented in virtually any type of transmission line medium. Figure 7.5 shows some commonly used T-junctions in waveguide and micro-strip or stripline form. The junctions shown here are, in the absence of transmission line loss, lossless junctions. Thus, as discussed in the preceding section, such junctions cannot be matched simultaneously at all ports. We will treat such junctions below, followed by a discussion of the resistive divider, which can be matched at all ports but is not lossless. (3 FIGURE 7.5 Various T-junction power dividers, (a) E plane waveguide T. (b) H plane waveguide T. (c) Mkrostrip T-junction. 316 Chapter 7; Power Dividers and Directional Couplers Lossless Divider The lossless T-junctions of Figure 7.5 can all be modeled as a junction of three transmission lines, as shown in Figure 7.6 [3]. In general, there are fringing fields and higher order modes associated with the discontinuity at such a junction, leading to stored energy that can be accounted for by a lumped susceptance, B. In order for the divider to be matched to the input line of characteristic impedance Zq, we must have Yio = jB+^- + ^ = ^r. (7.24) Z] Zi z0 If the transmission lines are assumed to be lossless (or of low loss), then the characteristic impedances are real. If we also assume 5=0, then (7.24) reduces to 1 1 1 --1--= —. (7.25) Z i Z i Zo In practice, if B is not negligible, some type of reactive tuning element can usually be added to the divider to cancel this susceptance, at least over a narrow frequency range. The output line impedances Zt and Z2 can then be selected to provide various power division ratios. Thus, for a 50 U input line, a 3 dB (equal split) power divider can be made by using two 100 Q output lines. If necessary, quarter-wave transformers can be used to bring the output line impedances back to the desired levels. If the output lines are matched, then the input line will be matched. There will be no isolation between the two output ports, and there will be a mismatch looking into the output ports. EXAMPLE 7.1 THE T-JUNCTION POWER DIVIDER A lossless T-junction power divider has a source impedance of 50 £2. Find the output characteristic impedances so that the input power is divided in a 2:1 ratio. Compute the reflection coefficients seen looking into the output ports. Solution If the voltage at the junction is Vo, as shown in Figure 7.6, the input power to the matched divider is 7.2 The T-Junction Power Divider 317 while the output powers arc P - 1 V°2 - 1 P * tg**15* These results yield the characteristic impedances as Z, =3Z0 = 150Í2, Z2 = ^°=75ň. 2 Then the input impedance to the junction is Zta = 75||150 = 5QÍÍ, so that the input is matched to the 50 £2 source. Looking into the 150 £2 output hue, we see an impedance of 50||75 = 30 Í2, while at the 75 Í2 output line we see an impedance of 50| 1150 = 37.5 Í2. Thus, the reflection coefficients seen looking into these ports are 30 - 150 T, =-= -0.666, 30 -(- 150 37.5 -75 ,w f; = ——-= -0.333. 37.5 +75 ■ Resistive Divider If a three-port divider contains lossy components it can be made to be matched at all ports, although the two output ports may not be isolated [3]. The circuit for such a divider is illustrated in Figure 7,7, using lumped-element resistors. An equal-split (-3 dB) divider is shown, but unequal power division ratios are also possible. The resistive divider of Figure 7.7 can easily be analyzed using circuit theory. Assuming that all ports are terminated in the characteristic impedance Zq, the impedance Z, seen FIGURE 7.7 An equal-split three-port resistive power divider. 318 Chapter 7: Power Dividers and Directional Couplers looking into the Zo/3 resistor followed by the output line, is 7.3 Z = ! + 2-0 = 4f>. (7.26) Then the input impedance of the divider is = | + 2~f = Z„, (7.27) which shows that the input is matched to the feed line. Since the network is symmetric from all three ports, the output ports are also matched. Thus, Si t = 522 = £33 = 0. If the voltage at port 1 is Vj, then by voltage division the voltage V at the center of the junction is v = v_22o/3 =2 28 1Zo/3 + 2Z{)/3 3 and the output voltages are, again by voltage division, V2 = V3 = V-—-= - V = - Vi. (7.29) 3 Zo + Z0/3 4 2 Thus, £21 = S31 = S23, = 1/2, which is —6 dB below the input power level. The network is reciprocal, so the scattering matrix is symmetric, and can be written as 0 i[° 1 n 2 1 0 1 Hi 1 0. (7.30) The reader may verify that this is not a unitary matrix. The power delivered to the input of the divider is 1 v? = 53h (7.31) while the output powers are (1/2V!)3 IV.2 1 M>w F2 = P-i = - ——— = ~ -Fmi (7.32 3 2 Z0 8Z0 4 which shows that half of the supplied power is dissipated in the resistors. THE WILKINSON POWER DIVIDER The lossless T-junction divider suffers from the problem of not being matched at ail ports and, in addition, does not have any isolation between output ports. The resistive divider can be matched at all ports, but even though it is not lossless, isolation is still not achieved. From the discussion in Section 7.1, however, we know that a lossy three-port network can be made having all ports matched with isolation between the output ports. The Wilkinson power divider |4] is such a network, with the useful property of being lossless when the output ports are matched; that is, only reflected power is dissipated. The Wilkinson power divider can be made with arbitrary power division, but we will first consider the equal-split (3 dB) case. This divider is often made in microstrip or stripline form, as depicted in Figure 7.8a; the corresponding transmission line circuit is given in Figure 7.8b. We will analyze this circuit by reducing it to two simpler circuits driven by symmetric and antisymmetric sources at the output ports. This "even-odd" mode analysis technique [5] will also be useful for other networks that we will analyze in later sections. 7,3 The Wilkinson Power Divider 319 Even-Odd Mode Analysis For simplicity, we can normalize all impedances to the characteristic impedance Zo, and redraw the circuit of Figure 7.8b with voltage generators at the output ports as shown in Figure 7.9. This network has been drawn in a form that is symmetric across the midplane; the two source resistors of normalized value 2 combine in parallel to give a resistor of normalized value 1, representing the impedance of a matched source. The quarter-wave lines have a normalized characteristic impedance Z, and the shunt resistor has a normalized value of r; we shall show that, for the equal-split power divider, these values should be Z == V2 and r = 2, as given in Figure 7.8. Now we define two separate modes of excitation for the circuit of Figure 7.9: the even mode, where Vg2 = VS3 = 2Vb, and the odd mode, where Vsi — — Vg$ = 2VV Then by superposition of these two modes, we effectively have an excitation of Vs2 = 4 %, Vs3 — 0, from which we can find the S parameters of the network. We now treat these two modes separately. Even mode. For the even-mode excitation, Vg2 = Vgš = 2Vq, and so V% = V{ and there is no current flow through the r/2 resistors or the short circuit between the inputs of the two transmission lines at port L Thus we can bisect the network of Figure 7.9 with open FIGURE 7.9 The Wilkinson power divider circuit in normalized and symmetric form. 320 Chapter 7: Power Dividers and Directional Couplers Port I la) Port 2 I WW-öf FIGURE 7.10 Bisection of the circuit of Figure 7.9. (a) Even-mode excitation, (b) Odd-mode excitation. circuits at these points to obtain the network of Figure 7.10a (the grounded side of the k/4 line is not shown). Then, looking into port 2, we see an impedance 2 (7.33) since the transmission line looks like a quarter-wave transformer. Thus, if Z = -Jl, port 2 will be matched for even mode excitation; men V2 = V0 since Zfn = I. The r/2 resistor is superfluous in this case, since one end is open-circuited. Next, we find Vf from the transmission line equations. If we let x = 0 at port 1 and x = —k/4 at port 2, the voltage on the transmission line section can be written as Then, V(jc) = V+(e-jßx + TeJß*). VÍ = V(~k/4) = jV+(l-r)=V» r + i r -1 (7.34) v? = V(0) = v+u + r) = yVo The reflection coefficient F is that seen at port 1, looking toward the resistor of normalized value 2, so F = and 2 + j§3 w = —jVq\/2. [7.35) Odd mode. For the odd-mode excitation, Vg2 = — Vg^ = 2V0f and so V.2° = — Vj, and there is a voltage null along the middle of the circuit in Figure 7.9. Thus, we can bisect this circuit by grounding it at two points on its midplane to give the network of Figure 7.10b, Looking into port 2, we see an impedance of r/2, since the parallel-connected transmission line is A/4 long and shorted at port 1, and so looks like an open circuit at port 2. Thus, port 2 will 7.3 The Wilkinson Power Divider 321 be matched for odd mode excitation if we select r =2. Then V2° — Vo and Vx° = 0; for this mode of excitation all power is delivered to the r/2 resistors, with none going to port 1. Finally, we must find the input impedance at port 1 of the Winkinson divider when ports 2 and 3 are temtinated in matched loads. The resulting circuit is shown in Figure 7.11a, where it is seen that this is similar to an even mode of excitation, since Vi = Vj. Thus, no current flows through the resistor of normalized value 2, so it can be removed, leaving the circuit of Figure 7-1 lb. We now have the parallel connection of two quarter-wave transformers terminated in loads of unity (normalized). The input impedance is then Zin = ^(V2)2=l. (7.36) In summary, we can establish die following S parameters for the Wilkinson divider: 5,i = 0 (ZLn = 1 at port 1) = 533 = 0 (ports 2 and 3 matched for even and odd modes) 5|2 = s2l (symmetry due to reciprocity) 5,3 = 531 (symmetry of ports 2 and 3) 523 = 532 = 0 (due to short or open at bisection) The preceding formula for 5,2 applies because all ports are matched when terminated with matched loads. Note that when the divider is driven at port 1 and the outputs are matched, no power is dissipated in the resistor. Thus the divider is lossless when the outputs are matched; only reflected power from ports 2 or 3 is dissipated in the resistor. Since S23 = S31 = 0, ports 2 and 3 are isolated. 322 Chapter 7: Power Dividers and Directional Couplers 0*5 /„ /„ U/o FIGURE 7.12 Frequency response of an equal-split Wilkinson power divider. Port 1 is the input port; ports 2 and 3 are the output ports. EXAMPLE 7.2 DESIGN AND PERFORMANCE OF A WILKINSON DIVIDER Design an equal-split Wilkinson power divider for a 50 £2 system impedance at frequency f$, and plot the return loss (Sn )> insertion loss (S21 = S31), and isolation (52J = 533) versus frequency from 0.5/o to 1.5/o- Solution From Figure 7.8 and the above derivation, we have that the quarter-wave transmission lines in the divider should have a characteristic impedance of Z - V2Z0 = 70.7 n, and the shunt resistor a value of r = 2z0 = 100 a. The transmission lines are X/4 long at the frequency /0- Using a computer-aided design program for the analysis of microwave circuits, the S parameter magnitudes were calcul ated and plotted in Figure 7.12. ■ Unequal Power Division and M-Way Wilkinson Dividers Wilkinson-type power dividers can also be made with unequal power splits; a microstrip version is shown in Figure 7.13. If the power ratio between ports 2 and 3 is K2 = P$fPi, FIGURE 7.13 A Wilkinson power divider in microstrip form having unequal power division. 7.4 Waveguide Directional Couplers 323 FIGURE 7.14 An Af-way, equal-split Wilkinson power divider. then the following design equations apply: Zo3 R 1 + K2 tf3 K2Zm « Z^K{\+K2), Zo(* + l). (7.37a) (7.37b) (7.37c) Note that the above results reduce to the equal-split case for K = 1. Also observe that the output lines are matched to the impedances R2 = Z$K and R$ = Zq/K, as opposed to the impedance Z0; matching transformers can be used to transform these output impedances. The Wilkinson divider can also be generalized to an JV-way divider or combiner [4], as shown in Figure 7.14. This circuit can be matched at all ports, with isolation between all ports. A disadvantage, however, is the fact that the divider requires crossovers for the resistors for N > 3. This makes fabrication difficult in planar form. The Wilkinson divider can also be made with stepped multiple sections, for increased bandwidth. A photograph of a 4-way Wilkinson divider network is shown in Figure 7.15. WAVEGUIDE DIRECTIONAL COUPLERS We now turn our attention to directional couplers, which are four-port devices with the characteristics discussed in Section 7,1. To review the basic operation, consider the directional coupler schematic symbols shown in Figure 7.4. Power incident at port 1 will couple to port 2 (the through port) and to port 3 (the coupled port), but not to port 4 (the isolated port). Similarly, power incident in port 2 will couple to ports I and 4, but not 3. Thus, ports I and 4 are decoupled, as are ports 2 and 3. The fraction of power coupled from port 1 to port 3 is given by C, the coupling coefficient, as defined in (7.20a), and the leakage of power from port 1 to port 4 is given by /, the isolation, as defined in (7.20c). Another quantity that can be used to characterize a coupler is the directivity, D = I — C (dB), which is the ratio of the power delivered to the coupled port and the isolated port. The ideal coupler is characterized solely by the coupling factor, as the isolation and directivity are infinite. The ideal coupler is also lossless and matched at all ports. Directional couplers can be made in many different forms. We will first discuss waveguide couplers, followed by hybrid junctions. A hybrid junction is a special case of a directional coupler, where the coupling factor is 3 dB (equal split), and the phase relation between the output ports is either 90° (quadrature hybrid), or 18011 (magic-Tor rat- race hybrid). Then we will discuss the implementation of directional couplers in coupled transmission line form. 324 Chapter 7: Power Dividers and Directional Couplers FIGURE 7.15 Photograph of a four-way corporate power divider network using three microstrip Wilkinson power dividers. Note the isolation chip resistors. Courtesy of M. D. Abouzahra, MIT Lincoln Laboratory, Lexington, Mass. Bethe Hole Coupler The directional property of all directional couplers is produced through the use of two separate waves or wave components, which add in phase at the coupled port and are canceled at the isolated port. One of the simplest ways of doing this is to couple one waveguide to another through a single small hole in the common broad wall between the two guides. Such a coupler is known as a Bethe hole coupler, two versions of which are shown in Figure 7.16. From the small-aperture coupling theory of Section 4.8, we know that an aperture can be replaced with equivalent sources consisting of electric and magnetic dipole moments [6]. The normal electric dipole moment and the axial magnetic dipole moment radiate with even symmetry in the coupled guide, while the transverse magnetic dipole moment radiates with odd symmetry, Thus, by adjusting the relative amplitudes of these two equivalent sources, we can cancel the radiation in the direction of the isolated port, while enhancing the radiation in the direction of the coupled port. Figure 7.16 shows two ways in which these wave amplitudes can be controlled; in the coupler shown in Figure 7.16a, the two guides are parallel and the coupling is controlled by s, the aperture offset from the sidewall of the guide. For the coupler of Figure 7.16b, the wave amplitudes are controlled by the angle, $, between the two guides. First consider the configuration of Figure 7.16a, with an incident TEio mode into port 1. These fields can be written as £v = Asin— e~JP\ (7.38a) a — A 7T r H, = — sin— e-#K (7.3Sb) Zio a jizA nx 7.4 Waveguide Directional Couplers 325 >'4 (CoupJed) © (Input) (T) y ;--------s '---- —7— ® {Isolated) (2) (Through) la) (Coupled) (3) (Input) (T) (Through) FIGURE 7.16 Two versions of the Bethe hole directional coupler, (a) Parallel guides, (b) Skewed guides. where Z\o = ^oWČ is the wave impedance of the TE|0 mode. Then, from (4.] 24) and (4,125), this incident wave generates the following equivalent polarization currents at the aperture at x = s, y = b, z = 0: 7tS Pe = €qvceyA sin —5{x — s)6(y — b)S(z), a r — X 7TS jit 71S~\ Pm = -a„A — sin — + t~~ cos — Six - s)8(y - b)S(z), IZm a paZw a J (7.39a) (7.39b) Using (4.128a,b) to relate Pe and Pm to the currents 7 and M, and then using (4.118), (4.120), (4.122), and (4.123) gives the amplitudes of the forward and reverse traveling waves in the top guide as A+ to — ■ Mdv A\o — -T f ^-Jdv + ^j h; •no Jy r\a jv —j(oA f 7 7TS iiO«jH f . -> xs it2 7 7rs\~\ —- (fyty Sill'----r- \m--h "TT 2 COS _ Í ' /"ě+ - Jdv + -±- { H+.Mdv Jv -no Ja We sinJ — + —j- suť--cos^ — , (7.40a) -1 -Ja>A fid (7.40b) where i^o = ab(Z\§ is the power normalization constant. Note from (7,40a,b) that the amplitude of the wave excited toward port 4 (A +0) is generally different from that excited toward port 3 (A j"0) (because i? + = - iJ/) so we can cancel the power delivered to port 4 by setting A^ = 0. If we assume that the aperture is round, then Table 4.3 gives the polarizabilities Chapter 7: Power Dividers and Directional Couplers as cce = 2/g/3 and am = 4^/3, where ro is the radius of the aperture. Then from (7.40a) we obtain the following condition: L 4mo\ . 2ns 4?r 24 + dB. (7.42a) (7.42b) Thus, a Bethe hole coupler of the type shown in Figure 7.16a can be designed by first using (7.41) to find s, the position of the aperture, and then using (7.42a) to determine the aperture size, ro, to give the required coupling factor. For the skewed geometry of Figure 7.16b, the aperture may be centered at s = a/2, and the skew angle 0 adjusted for cancellation at port 4. In this case, the normal electric field does not change with 0, but the transverse magnetic field components are reduced by cos 0. We can thus account for the skew by replacing ctm in the previous derivation by am cos 0. The wave amplitudes of (7.40a,b) then become, for s =-a/2. A+ -^10 — Aio — - jdiA ( €(>ae Z2 €ßfl* + -^2~ cost? (7.43a) (7.43b) Setting A*q = 0 results in the following condition for the angle 9: 2^0--tt cos# = 0, 7- or cos-0 = 2ß2 The coupling factor then simplifies to C = 20 log = -20 log 4Afo 2„3 dB. (7.44) (7.45) The geometry of the skewed Bethe hole coupler is often a disadvantage in terms of fabrication and application. Also, both coupler designs operate properly only at the design frequency; deviation from this frequency will alter the coupling level and the directivity, as shown in the following example. 7.4 Waveguide Directions Couplers 327 EXAMPLE 7.3 BETI1E HOLE COUPLER DESIGN AND PERFORMANCE Design a Bethe hole coupler of the type shown in Figure 7.16a for X-band waveguide operating at 9 GHz, with a coupling of 20 dB. Calculate and plot the coupling and directivity from 7 to 11 GHz. Assume a round aperture. Solution For X-band waveguide at 9 GHz, we have the following constants: a = 0.02286 m, b = 0.01016 m, kG = 0.0333 m, k0 = 188.5 nT1, B = 129.0 m-1, Zio = 550.9 />,,) = 4.22 x 10-7m2/a, Then (7,41) can be used to find the aperture position s; sin — = = 0.972, 0.972 = 0.424« = 9.69 mm. a s = — sin 71 The coupling is 20 dB, so or C = 20dB = 201og A '10 Mo = 1020'20 = 10. thus, |Af0/A| = 1/10. We now use (7.40b) to find r0: Mo 1 Qi \{ fW*m\ Since ct< = 2rl /3 and am = 4r$ /3, we obtain — . or 0.1 = 1.44 x 106^, ro — 4.15 mm. This completes the design of the Bethe hole coupler. To compute the coupling and directivity versus frequency, we evaluate (7.42a) and (7.42b), using the expressions for Aj"0 and given in (7.40a) and (7.40b). In these expressions, the aperture position and size are fixed at \ = 9.69 mm and ro — 4.15 mm, and the frequency is varied. A short computer program was used to calculate the data shown in Figure 7.17. Observe that the coupling varies by less than 1 dB over the band. The directivity is very large (>60 dB) at the design frequency, but decreases to 15-20 dB at the band edges. The directivity is a more sensitive function of frequency because it depends on the cancellation of two wave components. H Design of Multihole Couplers As seen from Example 73, a single-hole coupler has a relatively narrow bandwidth, at least in terms of its directivity. But if the coupler is designed with a series of coupling holes, the 328 Chapter 7: Power Dividers and Directional Couplers 7.Ü 7.5 8.0 S.5 9.0 9.5 10.0 10.5 11.0 Frequency GHz FIGURE 7,17 Coupling and directivity versus frequency for the Bethe hole coupler of Example 7.3. extra degrees of freedom can be used to increase this bandwidth. The principle of operation and design of such a multihole waveguide coupler is very similar to that of the multisection matching transformer. First let us consider the operation of the two-hole coupler shown in Figure 7,18. Two parallel waveguides sharing a common broad wall are shown, although the same type of structure could be made in microstrip or stripline form. Two small apertures are spaced Ag /4 apart, and couple the two guides. A wave entering at port 1 is mostly transmitted through to port 2, but some power is coupled through the two apertures. If a phase reference is taken at the first aperture, then the phase of the wave incident at the second aperture will be —90°. Each aperture will radiate a forward wave component and a backward wave component into the upper guide; in general, the forward and backward amplitudes are different. In the direction of port 3, both components are in phase, since both have traveled kg/4 to the second aperture. But we obtain a cancellation in the direction of port 4, since the wave coming through the second aperture travels Xg/2 further than the wave component coming through the first aperture. Clearly, this cancellation is frequency sensitive, making the directivity a sensitive function of frequency. The coupling is less frequency dependent, since the path lengths from port 1 to port 3 are always the same. Thus, in the multihole coupler design, we synthesize the directivity response, as opposed to the coupling response, as a function of frequency. We now consider the general case of the multihole coupler shown in Figure 7.19, where N + 1 equally spaced apertures couple two parallel waveguides. The amplitude of the incident wave in the lower left guide is A and, for small coupling, is essentially the same as the amplitude of the through wave. For instance, a 20 dB coupler has a power coupling factor (4) (Isolated) (Coupled) (5) = rN, r\ = rN-\, etc. Then from (7.53) we can write S as S = N E h=0 3e-2j*e n/2 = 2^r„3cos(iV-2n)ö! n=0 7.4 Waveguide Directional Couplers 331 where 9 = 3d. To achieve a Chebyshev response we equate this to the Chebyshev polynomial of degree N: Nil S = 2^rlcos(iV -2n)9-k\TN(sec9m cos»)\, (7.56) where k and 0m are constants to be determined. From (7.53) and (7.56), we see that for 0 = 0, S = rn = £|Tjv(sec0m)|. Using this result in (7.51) gives the coupling as C = -20log |Ks\ -20logS IT/vfsec 9m cos0)|. So if Dmin is the specified minimum value of directivity in the passband, then 9m can be found from the relation Daia = 2Olog7/1v(sec0ffl) dB. (7.59) Alternatively, we could specify the bandwidth, which then dictates Sm and i?mjn. In either case, (7.57) can then be used to find k, and then (7.56) solved for the radii, rn. If N is odd (an even number of holes), the results for C, D, and £>m;rt in (7.57), (7.58), and (7.59) still apply, but instead of (7.56), the following relation is used to find the aperture radii: S = 2 r*cos(N -2nW =k\TN(secemcose)\. (7.60) «=o EXAMPLE 7A MULTIHOLE WAVEGUIDE COUPLER DESIGN Design a four-hole Chebyshev coupler in X-band waveguide using round apertures located at s = a jA. The center frequency is 9 GHz, the coupling is 20 dB, and the minimum directivity is 40 dB. Plot the coupling and directivity response from 7 to 11 GHz. Solution For X-band waveguide at 9 GHz, we have the following constants: a = 0.02286 m, b = 0.01016 m, X0 = 0.0333 m, kn = 188.5 m"1, 8 = 129.0 m-', Zio = 550.9 Q, P\o = 4,22 x 10"7 m2m. 332 Chapter 7: Power Dividers and Directional Couplers From (7.40a) and (7.40b), we obtain for an aperture at s = a/4: tir. i 2k0 r . j JTii 2£2 { . jits it2 , Jr^M » „_„ = ^—sin---sin — + T^TS cos- — = 3.953 x 10s, \Kh\ = —— sin2 — + 73- sm2--cos2 — = 3.454 x 105. For a four-hole coupler, N = 3, so (7.59) gives 40 = 20 log 73(secem) dB. 100 = 73(sec^) = cosh^cosbrVsec^)), sec 6m = 3.01, where (5.58b) was used. Thus Qm = 70.6° and 109.4° at the band edges. Then from (7.57) we can solve for k: C = 20 = -201og(3.953 x 105) - 20log* - 40 dB, 20 log it = -171.94, it = 2,53 x 10-!>. Finally, (7.60) and the expansion from (5.60c) for T3 allow us to solve for the radii as follows: S = 2[V^cos30 + rj* cosfl] a= *[W 0m(co$ 36" + 3cos#) - 3 sec0m cos0], 2r^ = k sec3 9m => **o = n = 3.26 mm, 2r] = 3A(sec3 8m - stc8m) =4 n = r2 = 4.51 mm. The resulting coupling and directivity are plotted in Figure 7.20; note the increased directivity bandwidth compared to that of the Bethe hole coupler of Example 7.3. Frequency GHz FIGURE 7.20 Coupling and directivity versus frequency for the four-hole coupler of Example 7.4. (Input)© (Isolated) © 3 FIGURE 7.21 Geometry of □ branch-li ne coupler. 7.5 The Quadrature (90°) Hybrid 333 (2) (Output) © (Output) THE QUADRATURE (90°) HYBRID Quadrature hybrids are 3 dB directional couplers with a 90° phase difference in the outputs of the through and coupled arms. This type of hybrid is often made in microstrip or stripline form as shown in Figure 7.21, and is also known as a branch-line hybrid. Other 3 dB couplers, such as coupled line couplers or Lange couplers, can also be used as quadrature couplers; these components will be discussed in later sections. Here we will analyze the operation of the quadrature hybrid using an even-odd mode decomposition technique similar to that used for the Wilkinson power divider. With reference to Figure 7.21 the basic operation of the branch-line coupler is as follows. With all ports matched, power entering port 1 is evenly divided between ports 2 and 3, with a 90° phase shift between these outputs. No power is coupled to port 4 (the isolated port). Thus, the [S] matrix will have the following form: [5] = — ro j 1 o- -1 j 0 0 1 n l 0 0 i _o 1 j 0_ (7.61) Observe that the branch -line hybrid has a high degree of symmetry, as any port can be used as the input port. The output ports will always be on the opposite side of the junction from the input port, and the isolated port will be the remaining port on the same side as the input port. This symmetry is reflected in the scattering matrix, as each row can be obtained as a transposition of the first row. Even-Odd Mode Analysis We first draw the schematic circuit of the branch-line coupler in normalized form, as in Figure 7.22, where it is understood that each line represents a transmission line with indicated characteristic impedance normalized to Zo- The common ground return for each transmission line is not shown. We assume that a wave of unit amplitude Ai = 1 is incident at port 1. Now the circuit of Figure 7.22 can be decomposed into the superposition of an even-mode excitation and an odd-mode excitation [5], as shown in Figure 7.23. Note that superimposing the two sets of excitations produces the original excitation of Figure 7,22, and since the circuit is linear, the actual response (the scattered waves) can be obtained from the sum of the responses to the even and odd excitations. 334 Chapter 7: Power Dividers and Directional Couplers A, = l FIGURE 7,22 Circuit of the branch-line hybrid coupler in normalized form. Because of the symmetry or antisymmetry of the excitation, the four-port network can be decomposed into a set of two decoupled two-port networks, as shown in Figure 7.23. Since the amplitudes of the incident waves for these two-ports are ±1 /2, the amplitudes of the emerging wave at each port of the branch-line hybrid can be expressed as B?> = - Te - -T„, B4 — -re — -r0, (7.62a) (7.62b) (7.62c) (7.62d) Q) i 1/V2 i © -o-o--- + 1/2 + 1/2 —a o (4) 1 / 1/V2 Line of symmetry / = 0 V = max © I 1/Vä 1 1 + 1/2 -1/2 ->- © 1 / «NS Line of antisymmetry / = max 1 1/V2 1 1 r. S iE fl UJ2 1 Open-circuited stubs (2 separate 2-ports) 1 (a) +1/2 -- I 1/V2 1 1 i/ulis 1 Short-circuited stubs (2 separate 2-pQTts) 1« (b) FIGURE 7.23 Decomposition of the branch-line coupler into even- and odd-mode excitations, (a) Even mode (e). (b) Odd mode (0). 7.5 The Quadrature (90°) Hybrid 335 where rec and te(l are the even- and odd-mode reflection and transmission coefficients for the two-port networks of Figure 7.23. First consider the calculation of Te and te, for the even-mode two-port circuit. This can best be done by multiplying the abcd matrices of each cascade component in that circuit, to give \a Biri oir o umn °i_jr-i n Shunt Jl/4 Shunt Y = j Transmission F = j line (7.63) where the individual matrices can be found from Table 4.1, and the admittance of the shunt open-circuited X/8 stubs is y = j tan/Jt! = j. Then Table 4.2 can be used to convert from abcd parameters (defined here with z0 — 1) to 5 parameters, which are equivalent to the reflection and transmission coefficients. Thus, a+b-c-d (-1 +j-J + D/V5 1% —---———- =-= = 0, (7.64a) a + b + c+d (-] +y +j - l)/V2 te =----— =---- = ^-L(l + j). (7.64b) a + b + c + d (-1 + y- + ;_i)A/2 72 Similarly, for the odd mode we obtain j i (7.65) V2 which gives the reflection and transmission coefficients as rp = 0, (7.66a) T0 = ^=(l-7). (7.66b) V2 Then using (7.64) and (7.66) in (7.62) gives the following results: 5, = 0 (port 1 is matched), (7.67a) B2 = —~ (half-power, -90° phase shift from port 1 to 2), (7.67b) V2 b$ = —1_ (half-power, -180° phase shift from port 1 to 3), (7.67c) v 2 B4 = 0 (no power to port 4). (7.67d) These results agree with the first row and column of the [SJ matrix given in (7,61); the remaining elements can easily be found by transposition. In practice, due to the quarter-wave length requirement, the bandwidth of a branch-line hybrid is limited to 10-20%. But as with multisection matching transformers and muttihole directional couplers, the bandwidth of a branch-line hybrid can be increased to a decade or more by using multiple sections in cascade. In addition, the basic design can be modified for unequal power division and/or different characteristic impedances at the output ports. Another practical point to be aware of is the fact that discontinuity effects at the junctions of the branch-line coupler may require that the shunt arms be lengthened by 10°-20°. Figure 7.24 shows a photograph of a quadrature hybrid. 336 Chapter 7: Power Dividers and Directional Couplers FIGURE 724 Photograph of a microstrip quadrature hybrid prototype. Courtesy of M. D. Abouzahra, MIT Lincoln Laboratory, Lexington, Mass. EXAMPLE 7.5 DESIGN AND PERFORMANCE OF A QUADRATURE HYBRID Design a 50 £2 branch-line quadrature hybrid junction, and plot the 5 parameter magnitudes from 0.5/o to 1.5fo, where /o is the design frequency. Solution After the preceding analysis, the design of a quadrature hybrid is trivial. The lines are A/4 at the design frequency /q, and the branch-line impedances are The calculated frequency response is plotted in Figure 7.25. Note that we obtain perfect 3 dB power division in ports 2 and 3, and perfect isolation and return loss at ports 4 and 1, respectively, at the design frequency /<>. All of these quantities, however, degrade quickly as the frequency departs from /0. ■ [%] -20 0.5/o fa Wo FIGURE 7.25 5 parameter magnitudes versus frequency for the branch-line coupler of Exam-" pie 7.5. 7.6 Coupled Line Directional Couplers 337 NX\\V\\NXX>^V.y\V,\\\\\v\\sXXX\-..\\\\\\\^ I tw> s > J£» ^VvN\\\Vs^\'y\x^:v\\\\\\\V.\\-..--.\\v\v\\-,\ W; (a) (b) w — ....., it; FIGURE 7.26 Various coupled transmission line geometries, (a) Coupled stripline (planar, or edge-coupled), (b) Coupled stripline (stacked, or broadside-coupled), (c) Coupled microstrip. COUPLED LINE DIRECTIONAL COUPLERS When two unshielded transmission lines are close together, power can be coupled between the lines due to the interaction of the electromagnetic fields of each line. Such lines are referred to as coupled transmission fines, and usually consist of three conductors in close proximity, although more conductors can be used. Figure 7.26 shows several examples of coupled transmission lines. Coupled transmission lines are usually assumed to operate in the TEM mode, which is rigorously valid for stripline structures and approximately valid for microstrip structures. In general, a three-wire line, like those of Figure 7.26, can support two distinct propagating modes. This feature can be used to implement directional couplers, hybrids, and filters. We will first discuss the theory of coupled lines and present some design data for coupled stripline and coupled microstrip. Then we wdl analyze the operation of a single-section directional coupler, and extend these results to multisection coupler design. Coupled Line Theory The coupled fines of Figure 7.26, or any other three-wire line, can be represented by the structure shown in Figure 7.27. If we assume TEM propagation, then the electrical characteristics of die coupled lines can be completely determined from the effective capacitances between the lines and the velocity of propagation on the line. As depicted in Figure 7.27, C\2 represents the capacitance between the two strip conductors, while and C22 represent the capacitance between one strip conductor and ground. If the strip conductors are identical in size and location relative to the ground conductor, then Cn = C22. Note that the designation of "ground" for the third conductor has no special relevance beyond the fact that it is convenient, since in many applications this conductor is the ground plane of a stripline or microstrip circuit. w2 fI-W-lc ,\VVVV-,v.xV\^.^xVVl\Vx\lvl\lAV^ 77777777777777777777777777 FIGURE 7.27 A three-wire coupled transmission line and its equivalent capacitance network. Chapter 7: Power Dividers and Directional Couplers + v T77777777777777777777h S ^777, I ! E-wall rHr-^Hh-x ^77h7777777777777r77777Z FIGURE 7.28 Even- and odd-mode excitations for a coupled line, and the resulting equivalent capacitance networks, (a) Even-mode excitation, (fj) Odd-mode excitation. Now consider two special types of excitations for the coupled line: the even mode, where the currents in the strip conductors are equal in amplitude and in the same direction, and the odd mode, where the currents in the strip conductors are equal in amplitude but in opposite directions. The electric field lines for these two cases are sketched in Figure 7.28. For the even mode, the electric field has even symmetry about the center line, and no current flows between the two strip conductors. This leads to the equivalent circuit shown, where C\% is effectively open-circuited. Then the resulting capacitance of either line to ground for the even mode is C„_, = £.',,= C72, (7.68) assuming that the two strip conductors are identical in size and location. Then the characteristic impedance for the even mode is j~L JhCe 1 where vp is the phase velocity of propagation on the line. For the odd mode, the electric field lines have an odd symmetry about the center line, and a voltage null exists between the two strip conductors. We can imagine this as a ground plane through the middle of C]2} which leads to the equivalent circuit as shown. In this case, the effective capacitance between either strip conductor and ground is Co = C\i 4r 2C\2 = C22 + 2C]2, (7.70) 7.6 Coupted Line Directional Couplers 339 and the characteristic impedance for the odd mode is (7.71) In words, Zq^Zoo) is the characteristic impedance of one of the strip conductors relative to ground when the coupled line is operated in the even (odd) mode. An arbitrary excitation of a coupled line can always be treated as a superposition of appropriate amplitudes of even and odd modes. This analysis assumes the lines are symmetric, and that fringing capacitances are identical for even and odd modes. If the coupled line is purely TEM, such as coaxial, parallel plate, or stripline, analytical techniques such as conformal mapping [7} can be used to evaluate the capacitances per unit length of line, and the even- and odd-mode characteristic impedances can then be deter-mined. For quasi-TEM lines, such as microstrip, these results can be obtained numerically or by approximate quasi-static techniques [8J. In either case, such calculations are generally too involved for our consideration, so we will present only two examples of design data for coupled lines. For a symmetric coupled stripline of the type shown in Figure 7.26a, the design graph in Figure 7.29 can be used to determine the necessary strip widths and spacing for a given set of characteristic impedances, Zae and Z^, and the dielectric constant. This graph should cover ranges of parameters for most practical applications, and can be used for any dielectric constant, since stripline supports a purely TEM mode. For microstrip, the results do not scale with dielectric constant, so design graphs must be made for specific values of dielectric constant. Figure 7.30 shows such a design graph for coupled microstrip tines on a substrate with er = 10. Another difficulty with microstrip coupled lines is the fact that the phase velocity is usually different for the two modes of 0.01 1.1) 0.2 2.0 2!J 20 i i 1 i ' i_I_!_I_I_I_i i I 40 60 30 100 120 140 160 FIGURE 7.29 Normalized even- and odd-mode characteristic impedance design data for edge-coupled striplines. 340 Chapter 7: Power Dividers and Directional Couplers FIGURE 7.30 Even- and odd-mode characteristic impedance design data for coupled microstrip lines on a substrate with €r = 10. propagation, since the two modes operate with different field configurations in the vicinity of the air-dielectric interface. This can have a degrading effect on coupler directivity. EXAMPLE 7.6 IMPEDANCE OF A SIMPLE COUPLED LINE For the broadside coupled stripline geometry of Figure 7.26b, assume W > S and W ^> so that fringing fields can be ignored, and detennine the even- and odd-mode characteristic impedances. Solution We first find the equivalent network capacitances, Cn and C\i (since the line is symmetric, C22 = Ci 1) The capacitance per unit length of broadside parallel lines with width, W, and separation, d, is C = — Fd/m. d with e being the substrate permittivity. This formula ignores fringing fields. Cn is fonned by the capacitance of one strip to the ground planes. Thus the capacitance per unit length is - 2€,€QW C\ 1 = —-Fd/m. b — s The capacitance per unit length between the strips is Cn = —-— Fd/m. 7.6 Coupled Line Directional Couplers 341 Then from (7.68) and (7.70), the even- and odd-mode capacitances are /? - S Cff = Cn +2Cl2 = 2^e0W^^^-r^ Fd/m. The phase velocity on the line is vp = 1 /^/erfo^o = cj*Jk~r, so the characteristic impedances are ^o«- = —s = «o; 2op = I = m vpC0 ,v2W^[l/(b-S)+l/S] Design of Coupled Line Couplers With the preceding definitions of the even- and odd-mode characteristic impedances, we can apply an even-odd mode analysis to a length of coupled line to arrive at the design equations for a single-section coupled line coupler. Such a line is shown in Figure 7.31. This four-port network is terminated in the impedance Zo at three of its ports, and driven with a voltage generator of 2V0 and internal impedance Zo at port 1. We will show that a coupler can be designed with arbitrary coupling such that the input (port 1) is matched, while port 4 is isolated. Port 2 is the through port, and port 3 is the coupled port. In Figure 7.31, a ground conductor is understood to be common to both strip conductors. Isolated jAVvV^—7#T ÜF—^vVW]_ fb> FIGURE 7.31 A single-section coupled line coupler, (a) Geometry and port designations, (b) The schematic circuit. 342 Chapter 7: Power Dividers and Directional Couplers (a) (b) j7"fr~JWvVl_ Zu aaaag_ FIGURE 7.32 Decomposition of the coupled line coupler circuit of Figure 7.31 into even- and odd-mode excitations, (a) Even mode, (b) Odd mode. For this problem we will apply the even-odd mode analysis technique in conjunction with the input impedances of the line, as opposed to the reflection and transmission coefficients of the line. So by superposition, the excitation at port 1 in Figure 7.31 can be treated as the sum of the even- and odd-mode excitations shown in Figure 7.32. From symmetry, we can see that If = /|, f| = Jjf, V{ = V/, and V4* = V{ for the even modes, while If = 1% = w = -Vf, and V4° = -V2° for the odd mode. The input impedance at port 1 of the coupler of Figure 7.31 can thus be expressed as Now if we let Zfn be the input impedance at port 1 for the even mode, and Z? be the input impedance for the odd mode, then we have Zb + jZto tan fl Zm - Z0e--, 7 ,-771 (7.73a) Zoe + ;Zotan0 Zto + ;Zotan0 since, for each mode, the line looks like a transmission line of characteristic impedance Z&. or Zoo, terminated in a load impedance, Z0. Then by voltage division Z'n (7.74b) = "»«AB- 7.6 Coupled Line Directional Couplers 343 Vb If = -TT^-—, (7-75a) Zin + z0 Vb /f = ——. (7.75b) ' Z*4-Z0 Using these results in (7.72) yields Zrn(Z?n+Zo) + ZfD(Z«+Z0) 2(ZPZfn - Zg) r? Ziu + Zifl + 2Z0 Zin + Zin + 2Z0 Now if we let Zq = v/ZfeZoo, (7.77) then (7.73aTb) reduce to zf = z VZq7 + 7^/2^7tane 2o _ z VZqT-r- j VZp7 tanc? \/Zo7 + j'v/Zfetanf)' so that Z*Z£, = Zo Z0(? = Z£\ and (7.76) reduces to Zin * Z0. (7.78) Thus, as long as (7.77) is satisfied, port 1 (and, by symmetry, all other ports) will be matched. Now if (7.77) is satisfied, so that Zi„ = Zo, we have that V\ = V0, by voltage division. The voltage at port 3 is ft = n + % = n - W = v-c - j^rj-l ®m LZiB+2o Z^ + ZoJ where (7.74) has been used. From (7.73) and (7.77), we can show that ZfD Zrj + jZte tanfJ Zg + Za 2ZQ + j(Z(* + Z0(>) tan $' Z? + Zo 2Zo + j(Zoe + Zoo) tan 0' so that (7.79) reduces to jiZ^ - Zoo)tan# Vi - Vb-—-—^-—-r. 7.80) Now define C as c Zo.-Zfe, (?81) Zfe + Zop which we will soon see is actually the midband voltage coupling coefficient, V3/ Vb- Then, 2Z0 Zq^ 4- Z^ so that V3 = V0 , k_-- (7.82) Vl - C2 + / tan# 344 Chapter 7: Power Dividers and Directional Couplers Similarly, we can show that V4 = V% + VX = V{ - V2° = 0, (7,83) and V2 = V{ + W = V 10 dB), and that each section is A/4 long (6 = jt/2) at the center frequency. For a single coupled line section, with C 1, (7.82) and (7.84) simplify to Yi JCtanS Vl -C2 4- jim9 l + /ian0 J cos & + j sin 9 — e (7.88a) (7,88b) Then for 9 = n/2, we have that V^/V\ = Cand Vi/V\ — — j. This approximation is equivalent to assuming that no power is lost on the through path from one section to the next, and is similar to the multisection waveguide coupler analysis. It is a good assumption for small CT even though power conservation is violated. Using these results, the total voltage at the coupled port (port 3) of the cascaded coupler in Figure 7.35 can be expressed as Vj = (JCi $in0e_J'*)Vi + (yC2sin0 100 dB 2 3 4 Frequency (GHz) FIGURE 7.37 Coupling versus frequency for the three-section binomial coupler of Example 7.1 7.7 The Lange Coupler 349 21 = 50^^1=56.69^, , /0875 The couphng and directivity for this coupler are plotted in Figure 7,37. ■ THE LANGE COUPLER Generally the coupling in a coupled line coupler is too loose to achieve coupling factors of 3 dB or 6 dB. One way to increase the couphng between edge-coupled lines is to use several lines parallel to each other, so that the fringing fields at both edges of a line contribute to the coupling. Probably the most practical implementation of this idea is the Lange coupler [10], shown in Figure 7.38a. Here, four coupled lines are used with interconnections to provide tight coupling, This coupler can easily achieve 3 dB coupling ratios, with an octave or more bandwidth. The design tends to compensate for unequal even- and odd-mode phase velocities, which also improves the bandwidth. There is a 90* phase difference between the output lines (ports 2 and 3), so the Lange coupler is a type of quadrature hybrid. The main FIGURE 738 The Lange coupler, (a) Layout in microstrip form, (b) The unfolded Lange coupler. 350 Chapter 7: Power Dividers and Directional Couplers Coupled Input (a) Isolated Through Coupled Input -*- --90"-- CD CD ö - Isolated -»- © Through FIGURE 739 Equivalent circuits for the unfolded Lange coupler, (a) Four-wire coupled line model, (b) Approximate two-wire coupled line model. disadvantage of the Lange coupler is probably practical, as the lines are very narrow, close together, and it is difficult to fabricate the necessary bonding wires across the lines. This type of coupled line geometry is also referred to as interdigitated; such structures can also be used for filter circuits. The unfolded Lange coupler [11], shown in Figure 7.38b, operates essentially the same as the original Lange coupler, but is easier to model with an equivalent circuit. Such an equivalent circuit consists of a four-wire coupled line structure, as shown in Figure 7.39a. AH the lines have the same width and spacing. If we make the reasonable assumption that each line couples only to its nearest neighbor, and ignore more distant couplings, men we effectively have a two-wire coupled line circuit, as shown in Figure 7.39b. Then, if we can derive the even- and odd-mode characteristic impedances, Ze\ and Z^, of the four-wire circuit of Figure 7.39a in terms of and Z$Q, the even- and odd-mode characteristic impedances of any adjacent pair of lines, we can apply the coupled line coupler results of Section 7.6 to analyze the Lange coupler. Figure 7.40a shows the effective capacitances between the conductors of the four-wire coupled line of Figure 7.39a. Unlike the two-line case of Section 7.6, the capacitances of the FIGURE 7.40 Effective capacitance networks for the unfolded Lange coupler equivalent circuits of Figure 7.39. (a) Effective capacitance for the four-wire model, (b) Effective capacitance for the two-wire model. 7.7 The Lange Coupler 351 four lines 10 ground are different depending on whether the line is on the outside (1 and 4), or on the inside (2 and 3). An approximate relation between these capacitances is given as [12J Cin = Ces--^-^. (7.92) For an even-mode excitation, all four conductors in Figure 7.40a are at die same potential, so Cm has no effect and the total capacitance of any line to ground is Cm = C» + Citl. (7.93a) For an odd-mode excitation, electric walls effectively exist through the middle of each Cmt so the capacitance of any line to ground is Crt = C^ + Cin + 6Cm. (7.93b) The even- and odd-mode characteristic impedances are then 2,4 = —n O -94a) VpCt4 = ^— (7.94b) VpCtf where vp is the phase velocity of propagation on the line. Now consider any isolated pair of adjacent conductors in the four-line model; the effective capacitances are as shown in Figure 7,40b. The even- and odd-mode capacitances are Ce = C,s, (7.95a) C„ = Cex + 2COT. (7.95b) Solving (7.95) for Cex and Cmt and substituting into (7.93) with the aid of (7.92) gives the even-odd mode capacitances of the four-wire line in terms of a two-wire coupled line: CeQCe + CfJ) C= -. (7.96a) Since characteristic impedances are related to capacitance as Zr> = l/vpC, we can rewrite (7.96) to give the even/odd mode characteristic impedances of the Lange coupler in terms of the characteristic impedances of a two-conductor line which is identical to any pair of adjacent lines in the coupler: Zm = T~,—r^Zoi, (7.97a) % = Zto + Z°* z0(J, (7.97b) where Z^, Z&, are the even- and odd-mode characteristic impedances of the two-conductor pair. 352 Chapter 7: Power Dividers and Directional Couplers 7.8 Now we can apply the results of Section 7.6 to the coupler of Figure 7.39b. From (7.77) the characteristic impedance is ZqěZooÍZqo + Zoe)2 (3Zoc + Z^OZfie + Zqo) ' while the voltage coupling coefficient is, from (7.81 )T ZeU — Zo4 . ^{Z^g ~ Zq^) Ze4 + Z^ ^{Zq6 + Zfo (7.98) C = (7.99) where (7.97) was used. For design purposes, it is useful to invert these results to give the necessary even- and odd-mode impedances for a desired characteristic impedance and coupling coefficient: Ztie = Zqo = 4C - 3 + y/'9~- 8C~2 2CV(1 -O/O + C)' 4C + 3- V9-8C2 2C^(1+C)/(1 - O" (7.100a) (7.100b) These results are approximate because of the simplifications involved with the application of two-line characteristic impedances to the four-line circuit and because of the assumption of equal even- and odd-mode phase velocities. In practice, however, these results generally give sufficient acuracy. If necessary, a more complete analysis can be made to directly determine Z^ and Z^ for the four-line circuit, as in reference [13]. THE 180° HYBRID The 180° hybrid junction is a four-port network with a 180° phase shift between the two output ports. It can also be operated so that the outputs are in phase. With reference to the 180° hybrid symbol shown in Figure 7.41, a signal applied to port 1 will be evenly split into two in-phase components at ports 2 and 3, and port 4 will be isolated. If the input is applied to port 4, it will be equally split into two components with a 180° phase difference at ports 2 and 3, and port 1 will be isolated. When operated as a combiner, with input signals applied at ports 2 and 3, the sum of the inputs will be formed at port 1, while the difference will be formed at port 4. Hence, ports 1 and 4 are referred to as the sum and difference ports, respectively. The scattering matrix for the ideal 3 dB 180° hybrid thus has the following form: -o 1 1 o - 1 l 0 0 -1 ■Jl i 0 0 1 .0 -1 1 0 _ (7.101) The reader may verify that this matrix is unitary and symmetric. The 180° hybrid can be fabricated in several forms. The ring hybrid, or rat-race, shown in Figures 7.42 and 7.43a, can easily be constructed in planar (microstrip or stripline) form, FIGURE 7.41 Symbol for a 180° hybrid junction. 7.8 The 180° Hybrid 353 FIGURE 7.43 Hybrid junctions, (a) A ring hybrid, or rat-race, in microstrip or stripline form, (b) A tapered coupJed line hybrid, (c) A waveguide hybrid junction, ormagic-T. 354 Chapter 7: Power Dividers and Directional Couplers although waveguide versions are aJso possible. Another type of planar 180° hybrid uses tapered matching lines and coupled lines, as shown in Figure 7.43b. Yet another type of hybrid is the hybrid waveguide junction, or magic-T, shown in Figure 7.43c. We will first analyze the ring hybrid, using an even-odd mode analysis similar to that used for the branch-line hybrid, and use a similar technique for the analysis of the tapered line hybrid. Then we will qualitatively discuss the operation of the waveguide magic-T. Evan-Odd Mode Analysis of the Ring Hybrid First consider a unit amplitude wave incident at port 1 (the sum port) of the ring hybrid of Figure 7.43a. At the ring junction this wave will divide into two components, which both arrive in phase at ports 2 and 3, and 180° out of phase at pott 4. Using the even-odd mode analysis technique [5], we can decompose this case into a superposition of the two simpler circuits and excitations shown in Figure 7.44. Then the amplitudes of the scattered waves from the ring hybrid will be Bi = Bi = B2 = B, = Ir-lr 2Ie r°- (7.102a) (7.102b) (7.102c) (7,102d) We can evaluate the required reflection and transmission coefficients defined in Figure 7.44 using the ABCD matrix for the even- and odd-mode two-port circuits in Figure 7.44. O.e. => (a) --S.C. =£> + 1/2 I CD o.c. +1/2 (D -c— V2 A/8 SC. A/4 V2 AM — A/4 © + 1/2 T ■—— y- e vT A/8 3A/S o.c. i O.C, (a) s.c. ==> CD A/4 -1/2 + t/2 T ; c j- A/8 V2 3A/S S.C. 1 S.C. FIGURE 7.45 Even- and odd-mode decomposition of the ring hybrid when port 4 is excited with a unit amplitude incident wave, (a) Even mode, (b) Odd mode. The abcd matrices for the even- and odd-mode circuits of Figure 7.45 are \A Bl _T -I yV2-| [c olrhvi 1 J' rA m r l ;V2-i [c dI Ijji -i J; Then from Table 4.2, the necessary reflection and transmission coefficients are j V21 Using these results in (7.106) gives To % Bi -J Vi' -J = o, vT -J ß4 =0, (7.107a) (7.107b) (7,108a) (7.108b) (7.108c) (7.108d) (7.109a) (7.109b) (7.109c) (7.109d) 7.8 The 18CP Hybrid 357 which shows that the input port is matched, port 1 is isolated, and the input power is evenly divided into ports 2 and 3 with a 180° phase difference. These results form the fourth row and column of the scattering matrix of (7.101), The remaining elements in this matrix can be found from symmetry considerations. The bandwidth of the ring hybrid is limited by the frequency dependence of the ring lengths, but is generally on the order of 20-30%. Increased bandwidth can be obtained by using additional sections, or a symmetric ring circuit as suggested in reference [14]. EXAMPLE 73 DESIGN AND PERFORMANCE OF A RING HYBRID Design a ISO" ring hybrid for a 50 Q. system impedance, and plot the magnitude of the S parameters (Si j) from 0.5 /o to 1.5 /o, where /o is the design frequency. Solution With reference to Figure 7.43a, the characteristic impedance of the ring transmission line is V2Z0 = 70.7fl, while the feedline impedances are 50ft, The S parameter magnitudes are plotted versus frequency in Figure 7.46. ■ Even-Odd Mode Analysis of the Tapered Coupled Line Hybrid The tapered coupled line 180° hybrid [15], shown in Figure 7.43b, can provide any power division ratio with a bandwidth of a decade or more. This hybrid is also referred to as an asymmetric tapered coupled line coupler. The schematic circuit of this coupler is shown in Figure 7.47; the ports have been numbered to correspond functionally to the ports of the 180° hybrids in Figures 7.41 and 7.43. The coupler consists of two coupled lines with tapering characteristic impedances over the length 0 < z < L. At z = 0 the lines are very weakly coupled so that Zoe(0) = Zoo(0) = Z0, while at z = L the coupling is such that Z^{L) = Z0/k and Z^iL) = kZ0, where 0 < k < 1 is a coupling factor which we will relate to the voltage coupling factor. The even mode of the coupled line thus matches a load impedance of Z(,/k (at z = L) to Z0t while the odd mode matches a load of kZo to Zq; note that Z^i,z)Z^0{z) = Z\ for all z. The Klopfenstein taper is generally used for these tapered matching lines. For L < % < 2L, the lines are uncoupled, and both have a characteristic impedance Zq\ these lines are required FIGURE 7.46 S parameter magnitudes versus frequency for the ring hybrid of Example 7.9. Chapter 7: Power Dividers and Directional Couplers Output Difference input ^ 1 2o fa) (b) CD (5 Sum input A/Wv-. Output 21 21 FIGURE 7.47 (a) Schematic diagram of the tapered coupled line hybrid, (b) The variation of characteristic impedances, for phase compensation of the coupled line section. The length of each section, & at BL, must be the same, and should be electrically long to provide a good impedance match over the desired bandwidth. First consider an incident voltage wave of amplitude V0 applied to port 4, the difference input. This excitation can be reduced to the superposition of an even-mode excitation and an odd-mode excitation, as shown in Figure 7.48a.b. At the junctions of the coupled and V0/2 -V0/2 Zo 3fi 3> "^AAA^ FIGURE 7.48 Excitation of die tapered coupled line hybrid, (a) Even-mode excitation, (b) Odd-mode excitation. 7.8 The 180' Hybrid 359 uncoupled lines (z = L), the reflection coefficients seen by the even or odd modes of the tapered lines are Zq - Zo/* k-l S-S+S^RT (7110a) r-f^ = I^. (7..iob) Zo+AZ0 1+k Then at z = 0 these coefficients are transformed to * +1 r„ = ~e-2^. (7.iiib) l + & Then by superposition the scattering parameters of ports 2 and 4 are as follows: 544 = ^(^ + ^ = 0, (7.112a) 524 = kvt - rd) = fc4r**i <7-112b> 2 k + 1 By symmetry, we also have that S22 = 0 and 542 — $24- To evaluate the transmission coefficients into ports 1 and 3, we will use the ABCD parameters for the equivalent circuits shown in Figure 7.49, where the tapered matching sections have been assumed to be ideal, and replaced with transformers. The ABCD matrix of the transmission line-transformer-transmission line cascade can be found by multiplying the three individual ABCD matrices for these components, but it is easier to use the fact that the transmission fine sections affect only the phase of the transmission coefficients. The ABCD matrix of the transformer is, for the even mode, 'VI 0 1 , 0 l/Vk}' Vit: 1 "0-L J-O- (a) 1 : VF (b) T° FIGURE 7.49 Equivalent circuits for the tapered coupled line hybrid, for transmission from port 4 to port 3. (a) Even-mode case, (b) Odd-mode case. 360 Chapter 7: Power Dividers and Directional Couplers and for the odd mode is n/Vk o i L o Vk, Then the even- and odd-mode transmission coefficients are Te = T0 = k+ 1 (7.113) since T = 2/(A -V B/Z0 + CZ0 + D) = 2«Jkf(k + 1) for both modes; the e~lj9 factor accounts for the phase delay of the two transmission line sections. We can then evaluate the following S parameters: 2«fk k + i -2ß The voltage coupling factor from port 4 to port 3 is then 2jk ß = |534| = k + r while the voltage coupling factor from port 4 to port 2 is k-l 0 < ß < 1, a = I Sy I = - 0 (b) FIGURE 7.50 Electric field lines for a waveguide hybrid junction, (a) Incident wave at port 1. (b) Incident wave at port 4. Waveguide Magic-T The waveguide magic-T hybrid junction in Figure 7.43c has terminal properties similar to those of the ring hybrid, and a scattering matrix similar in form to (7.101). A rigorous analysis of this junction is too complicated to present here, but we can explain its operation in a qualitative sense by considering the field lines for excitations at the sum and difference pons. First consider a TEl0 mode incident at port 1. The resulting £¥ field lines are illustrated in Figure 7.50a, where it is seen that there is an odd symmetry about guide 4. Since the field lines of a TEio mode in guide 4 would have even symmetry, there is no coupling between ports 1 and 4. There is identical coupling to ports 2 and 3, however, resulting in an in-phase, equal-split power division. For a TEjo mode incident at port 4, the field lines are as shown in Figure 7.50b. Again ports 1 and 4 are decoupled, due to symmetry (or reciprocity). Ports 2 and 3 are excited equally by the incident wave, but with a 180° phase difference. In practice, mning posts or irises are often used for matching; such components must be placed symmetrically to maintain proper operation of the hybrid. OTHER COUPLERS While we have discussed the general properties of couplers, and have analyzed and derived design data for several of the most frequently used couplers, there are many other types of couplers that we have not treated in detail. In this section, we will briefly describe some of these, Moreno crossed-guide coupler. This is a waveguide directional coupler, consisting of two waveguides at right angles, with coupling provided by two apertures in the common broad 362 Chapter 7: Power Dividers and Directional Couplers Through isolated Coupled P Inpul FIGURE 7.51 The Moreno crossed-guide coupler, waU of the guides. See Figure 7.51. By proper design [16], the two wave components excited by these apertirres can be made to cancel in the back direction. The apertures usually consist of crossed slots, in order to couple tightly to the fields of both guides. Schwinger reversed-phase coupler. This waveguide coupler is designed so that the path lengths for the two coupling apertures are the same for the uncoupled port, so that the directivity is essentially independent of frequency. Cancellation in the isolated port is accomplished by placing the slots on opposite sides of the centerline of the waveguide walls, as shown in Figure 752, which couple to magnetic dipoles with a 180& phase difference. Then, the kg/4 slot spacing leads to in-phase combining at the coupled (backward) port, but this coupling is very frequency sensitive. This is the opposite situation from that of the multihole waveguide coupler discussed in Section 7,4. Riblet short-slot coupler. Figure 7.53 shows a Riblet short-slot coupler, consisting of two waveguides with a common sidewalk Coupling takes place in the region where part of the common wall has been removed. In this region, both the TEio (even) and the TE20 (odd) mode are excited, and by proper design can be made to cause cancellation at the isolated port and addition at the coupled port. The width of the interaction region must generally be reduced to prevent propagation of the undesired TE/jo mode. This coupler can usually be made smaller than other waveguide couplers. Through Coupled Input FIGURE 7.52 The Schwinger reversed-phase coupler. 7.9 Oliver Couplers 363 Symmetric tapered coupled line coupler. We saw that a continuously tapered transmission line matching transformer was the logical extension of the multisection matching transformer. Similarly, the multisection coupled line coupler can be extended to a continuous taper, yielding a coupled line coupler with good bandwidth characteristics. Such a coupler is shown in Figure 7.54. Generally, both the conductor width and separation can be adjusted to provide a synthesized coupling or directivity response. One way to do mis involves the computer optimization of a stepped-section approximation to the continuous taper [17]. This coupler provides a 90° phase shift between the outputs. Couplers with apertures in planar lines. Many of the above-mentioned waveguide couplers can also be fabricated with planar lines such as microstrip, stripline, dielectric image lines, or various combinations of these. Some possibilities are illustrated in Figure 7.55, In principle, the design of such couplers can be carried out using the small-hole coupling theory and analysis techniques used in this chapter. The evaluation of the fields of planar lines, however, is usually much more complicated than for rectangular waveguides. POINT OF INTEREST: The Reflectometer A reflectometer is a circuit thai uses a directional coupler to isolate and sample the incident and reflected powers from a mismatched load, It forms the heart of a scalar or vector network analyzer, as it can be used to measure the reflection coefficient of a one-port network and, in a more general configuration, the S parameters of a two-port network. It can also be used as an SWR meter, or as a power monitor in systems applications. The basic reflectometer circuit shown on the next page can be used to measure the reflection coefficient magnitude of an unknown load. If we assume a reasonably matched coupler with loose FIGURE 7-54 A symmetric tapered coupled line coupler. Chapter 7: Power Dividers end Directional Couplers Coupling aperture Microstrip lines (a) Coupling aperture Coupling aperture (b) Microstrip line g Dielectric image guide Microstrip line Waveguide FIGURE 7.55 Various aperture coupled planar line couplers, (a) Mcrosuip-to-microsuip coupler, (b) Microstrip-to-waveguide coupler, (c) Microstrip-to-dieleciric image line coupler. coupling (C 1), so that y/l—C1 — 1, then the circuit can be represented by the signal flow graph shown below. In operation, the directional coupler provides a sample, V;, of the incident wave, and a sample, Vr, of the reflected wave. A ratio meter with an appropriately calibrated scale can then measure these voltages and provide a reading in terms of reflection coefficient magnitude, or SWR. C D ® CD r Load Realistic directional couplers, however, have finite directivity, which means thai both the incident and reflected powers will contribute to both and Vr, leading to an error. If we assume a unit incident wave from the source, inspection of the signal flow graph leads to the following expressions for Vt and Vr: v, = ~ + crv*, where T is the reflection coefficient of the load, D = 10*D dB'a)1 is the numerical directivity of the coupler, and $ are unknown phase delay differences through the circuit, Then the maximum and minimum values of the magnitude of Vr/ Vj can be written as 4 Problems 365 For a coupler with infinite directivity this reduces to the desired result of ir|. Otherwise a measurement uncertainty of approximately ±1/D is introduced. Good accuracy thus requires a coupler with high directivity, preferably greater than 40 dB. REFERENCES 11 ] A, E. Bailey, Ed., Microwave Measurement, Peter Peregrmus, London, 1985. [2] R. E. Collin, Foundations for Microwave Engineering, Second Edition, McGraw-Hill, N.Y., 1992, [3] F. E. Gardiol, Introduction to Microwaves, Artech House, Dedham, Mass., 1984. [4] E. Wilkinson, "An Af-Way Hybrid Power Divider," IRE Trans, on Microwave Theory and Techniques, -vol. MTT-8, pp. 116-118, January 1960. [5] J. Reed and G. J. Wheeler, "A Method of Analysis of Symmetrical Four-Port Networks," IRE Trans, on Microwave Theory and Techniques, vol. MTT-4, pp. 246-252, October 1956. [6] C, G, Montgomery, R. H, Dicke, and E. M. Purcell, Principles of Microwave Circuits, MIT Radiation Laboratory Series, vol. 8, McGraw-Hill, N.Y., 1948, [71 H- Howe, Stripiine Circuit Design, Artech House, Dedham, Mass., 1974. [8] K. C, Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slot Lines, Artech House, Dedham, Mass., 1979. [9] L. Young, "The Analytical Equivalence of the TEM-Mode Directional Couplers and Transmission-Line Stepped Impedance Filters," Proc. IEEE, vol. 110, pp. 275-281, February 1963. [10] J. Lange, "lnterdigitated Stripiine Quadrature Hybrid,"' IEEE Trans. Microwave Theory and Techniques, vol. MTT-17, pp. 1150-1151, December 1969. [11] R. Waugh and D. LaCombe, "Unfolding (he Lange Coupler," IEEE Trans. Microwave Theory and Techniques, vol. MTT-20, pp. 777-779, November 1972. [12] W. P. Ou, "Design Equations for an lnterdigitated Directional Coupler," IEEE Trans. Microwave Theory and Techniques, vol. MTT-23, pp. 253-255, February 1973. [13] D. Paolino, "Design More Accurate lnterdigitated Couplers," Microwaves, vol. 15, pp. 34-38, May 1976. [14] J. Hughes and K. Wdson, "High Power Multiple IMPATT Amplifiers," Proc. European Microwave Conference, pp. 118-122, 1974. [15] R. H. DuHamel and M. E. Armstrong, "The Tapered-Line Magic-T," Abstracts of 15th Annual Symposium of the USAF Antenna Research and Development Program, Monticello, 111.. October 12-14, 1965. [16] T. N. Anderson, "Directional Coupler Design Nomograms," Microwave Journal, vol. 2, pp. 34—38, May 1959. [17] D. W, Kammler, 'The Design of Discrete W-Section and Continuously Tapered Symmetrical Microwave TEM Directional Couplers," IEEE Trans, on Microwave Theory and Techniques, vol. MTT-17, pp. 577-590, August 1969. PROBLEMS 7.1 Consider the T-junction of three tines with characteristic impedances Z\, Z2, and Zj, as shown below. Demonstrate that it is impossible for all three lines to be matched, when looking toward the junction. 366 Chapter 7: Power Dividers and Directional Couplers 7,2 A directional coupler has die scattering matrix given below. Find the directivity, coupling, isolation, and return loss at the input port when the other ports are terminated in matched loads. [S] = ■0.05 iM 0.96^0 0.1 m 0,05Z°0-| 0.%Z0 0.05^30 0.05 Z5Q 0.1 Z50 0.1 m 0.05Z9J 0.04/30 0.961Ü .0.05ZÍS o.l O.96Z0 0.05^30 7.3 Two identical 90" couplers with C = 8.34 dB are connected as shown below. Find the resulting phase and amplitudes at ports 1' and 3', relative to port 1. X ® CD © © CD 7.4 A 2 W power source is connected to the input of a directional coupler with C = 20 dB, D = 25 dB, and an insertion loss of 0.7 dB. Find the output powers (in dBm) at the through, coupled, and isolated ports. Assume all ports to be matched. 7.5 Design a lossless T-junction divider with a 30 source impedance to give a 3:1 power split. Design quarter-wave matching transformers to convert the impedances of the output lines to 30 £1. Determine the magnitude of the S parameters for this circuit, using a 30 Q. characteristic impedance. 7.6 Consider the T and % resistive attenuator circuits shown below. If the input and oumut are matched to Zo, and the ratio of output voltage to input voltage is a, derive the design equations for R{ and &> for each circuit. If Z0 = 50 Í2, compute f?i and Třj for 3 dB, 10 dB, and 20 dB attenuators of each type. *i + o- V0 ^vWV ■A o - 7.7 Design a three-port resistive divider for an equal power split and a 100 £2 system impedance. If port 3 is matched, calculate the change in output power at port 3 (in dB) when port 2 is connected first to a matched load, and then to a load having a mismatch of P = 0.3. Problems 367 7.8 Consider the general resistive divider shown below. For an arbitrary power division ratio,« = derive expressions for the resistors RitR->, and í3, and the output characteristic impedances Za> Zaj so that all ports are matched, assuming the source impedance is Z0. 7.9 Design a Wil kinson power divider w itb a power division ratio of P^ / ?t = 1/3, and a source i mpedance of50S2. 7.10 Derive the design equations in (7.37a,b,c) for the unequal-split Wilkinson divider. 7.11 For the Bcthc hole coupler of the type shown in Figure 7.16a, derive a design for $ so that port 3 is the isolated port. 7.12 Design a Bethe hole coupler of the type shown in Figure 7,16a for Ku-band waveguide operating at 11 GHz. The required coupling is 20 dB. 7.13 Design a Bethe hole coupler of the type shown in Figure 7.16b for Ku-band waveguide operating at 17 GHz. The required coupling is 30 dB. 7.14 Design a five-hole directional coupler in Ku-band waveguide with a binomial directivity response. The center frequency is 17.5 GHz, and the required coupling is 20 dB. Use round apertures centered across the broad wall of die waveguides. 7.15 Repeat Problem 7.14 for a design with a Chebyshev response, having a minimum directivity of 30 dB. 7.16 Develop die necessary equations required to design a two-hole directional coupler using two waveguides with apertures in a common sidewall, as shown below. 7.17 Consider the general branch-line coupler shown on the next page having shunt arm characteristic impedances Za, and series arm characteristic impedances Zb. Using an even-odd mode analysis, derive design equations for a quadrature hybrid coupler with an arbitrary power division ratio of a = Pil P%y and with the input port (port 1) matched. Assume all arms are X/4 long. Is port 4 isolated, in general? Pi-* So 388 Chapter 7: Power Dividers and Directional Couplers 7.18 An edge-coupled stripline with a ground plane spacing of 0.32 cm and a dielectric constant of 2.2 is required to have even- and odd-mode characteristic impedances of = 70 £2 and Zq,, = 40 Q. Find the necessary strip widths and spacing. 7.19 A coupled microstrip line on an FR-4 substrate with tr = 4.2 and d = 0.158 cm has strip widths of 0.30 cm and a strip spacing of 0.1173 cm. Find the even- and odd-mode characteristic impedances, 7.20 Repeat the derivation in Section 7.6 for the design equations of a single-section coupled line coupler using reflection and transmission coefficients, instead of voltages and currents. 7.21 Design a single-section coupled line coupler with a coupling of 19.1 dB, a system impedance of 60 £2, and a center frequency of 8 GHz. If the coupler is to be made in stripline (edge-coupled), with er =2.2 and b = 0.32 cm, find the necessary strip widths and separation. 7.22 Repeat Problem 7.2 J for a coupling factor of 5 dB. Is this a practical design? 7.23 Derive Equations (7-83) and (7.84). 7.24 A 20 dB three-section coupled line coupler is required to have a maximally flat coupling response, with a center frequency of 3 GHz, and Z0 = 50 £2. (a) Design the coupler and find Z& and Z^ for each section. Use CAD to plot the resulting coupling (in dB) from 1 to 5 GHz. (b) Lay out the microstrip implementation of the coupler on an FR4 substrate having tr = 4.2, d = 0.158 cm, tan & = 0.02, with copper conductors 0.5 mil thick. Use CAD to plot the insertion Joss versus frequency. 7.25 Repeat Problem 7.24 for a coupler with an equal-ripple coupling response, where the ripple in the coupling is 1 dB over the passband. 7.26 For the Lange coupler, derive the design equations (7-100) for Zp* and Zoo from (7,98) and (7.99). 7.27 Design a 3 dB Lange coupler for operation at 5 GHz. If the coupler is to be fabricated in miciostrip on an alumina substrate with = 10 and d = 1.0 mm, compute Zfe and Zoo for the two adjacent lines, and find the necessary spacing and widths of the lines. 7.28 Consider the four-port hybrid transformer shown below. Determine the scattering matrix for this device, and show that it is similar in form to the scattering matrix for the 180° hybrid. Let the port characteristic impedances be Zm = Z(m = Z0; Z02 = Z<% — 2Zq, (This type of transformer is often used in telephone circuits.) 7.29 An input signal V, is applied to the sum port of a 180f hybrid, and another signal V4 is applied to the difference port. What are the output signals? Problems 369 7.30 Calculate the even- and odd-mode characteristic impedances for a tapered coupled line 180" hybrid coupler with a 3 dB coupling ratio and a 50 Q characteristic impedance, 7.31 Find the S parameters for the four-port Bag ley polygon power divider shown below. 732 For the symmetric hybrid shown below, calculate the output voltages if port 1 is fed with an incident wave of 1 zfi V. Assume the outputs are matched. 7.33 The Bailey unequal-split power divider uses a 90° hybrid coupler and a T-jimction, as shown below. The power division ratio is controlled by adjusting the feed position,«, along the transmission line of length b that connects ports 1 and 4 of the hybrid. A quartet-wave transformer of impedance Z0/\/2 is used to match the input of the divider, (a) For b = X/4, show that the output power division ratio is given by Pi/Pt = tanI<^a/2&). (b) Using a branch-line hybrid with Z0 = 50 Ci, design a power divider with a division ratio of P2./P2 — 0-5, and plot the resulung input return loss and transmission coefficients versus frequency. Chapter Eight Microwave Filters A microwave filter is a two-port network used to control the frequency response at a certain point in a microwave system by providing transmission at frequencies within the passband of the filter and attenuation in the stopband of the filter. Typical frequency responses include low-pass, high-pass, bandpass, and band-reject characteristics. Applications can be found in virtually any type of microwave communication, radar, or test and measurement system. Microwave filter theory and practice began in the years preceding World War II, by pioneers such as Mason, Sykes, Darlington, Fano, Lawson, and Richards. The image parameter method of filter design was developed in the late 1930s and was useful for low-frequency filters in radio and telephony. In the early 1950s a group at Stanford Research Institute, consisting of G. Matlhaei, L. Young, E. Jones, S. Cohn, and others, became very active in microwave filter and coupler development. A voluminous handbook on filters and couplers resulted from this work and remains a valuable reference [1], Today, most microwave filter design is done with sophisticated computer-aided design (CAD) packages based on the insertion loss method. Because of continuing advancements in network synthesis with distributed elements, the use of low-temperature superconductors, and the incorporation of active devices in filter circuits, microwave filter design remains an active research area. We begin our discussion of filter theory and design with the frequency characteristics of periodic structures, which consist of a transmission line or waveguide periodically loaded with reactive elements. These structures are of interest in themselves, because of the application to slow-wave components and traveling-wave amplifier design, and also because they exhibit basic passband-stopband responses that lead to the image parameter method of filter design. Filters designed using the image parameter method consist of a cascade of simpler two-port filter sections to provide the desired cutoff frequencies and attenuation characteristics, but do not allow the specification of a frequency response over the complete operating range, Thus, although the procedure is relatively simple, the design of filters by the image parameter method often must be iterated many times to achieve the desired results, A more modern procedure, called the insertion loss method, uses network synthesis techniques to design filters with a completely specified frequency response. The design is simplified by beginning with low-pass filter prototypes that are normalized in terms of impedance and 370 8.1 Periodic Structures 371 frequency. Transformations are then applied to convert the prototype designs to the desired frequency range and impedance level. Both the image parameter and insertion loss method of filter design provide lumped-element circuits. For microwave applications such designs usually must be modified to use distributed elements consisting of transmission line sections. The Richard's transformation and the Kuroda identities provide this step. We will also discuss transmission line filters using stepped impedances and coupled lines; filters using coupled resonators will also be briefly described. The subject of microwave filters is quite extensive, due lo the importance of these components in practical systems and the wide variety of possible implementations. We give here a treatment of only the basic principles and some of the more common filter designs, and refer the reader to references such as [1], [2], [3], and [4] for farther discussion. PERIODIC STRUCTURES An infinite transmission line or waveguide periodically loaded with reactive elements is referred to as a periodic structure. As shown in Figure 8.1, periodic structures can take various forms, depending on the transmission line media being used. Often the loading elements are formed as discontinuities in the line, but in any case they can be modeled as lumped reactances across a transmission fine as shown in Figure S.2. Periodic structures support slow-wave propagation (slower than the phase velocity of the unloaded line), and (b) FIGURE 8.1 Examples of periodic structures, (a) Periodic srubs on a microstrip line, (b) Periodic diaphragms in a waveguide. Chapter 8: Microwave Fitters -o- —o— ft Vfí Unit cell FIGURE 8.2 Equivalent circuit, of a periodically loaded transmission line. The unloaded line has characteristic impedance Z0 and propagation constant k. have passband and stopband characteristics similar to those of filters; they find application in traveling-wave tubes, masers, phase shifters, and antennas. Analysis of Infinite Periodic Structures We begin by studying the propagation characteristics of the infinite loaded line shown in Figure 8.2. Each unit cell of this line consists of a length d of transmission line with a shunt susceptance across the midpoint of the line; the susceptance b is normalized to the characteristic impedance, Z0. If we consider the infinite line as being composed of a cascade of identical two-port networks, we can relate the voltages and currents on either side of the ňth unit cell using the ABCD matrix: m-[c B»ivc\> (8.1) where A, B, C, and D are the matrix parameters for a cascade of a transmission fine section of length df% a shunt susceptance b> and another transmission line section of length d/2. From Table 4.1 we then have, in normalized form, cos e 2 , e t sin - ■ 2 . 8-i 0 cos -2 J 1 Jb e cos -2 . $ jsin- 9 n JSm2 e cos -2 / b \ ( b b\ I cos 8--sin $ \ j [ sin B + - cos B--) V 2 } \ 2 2) /, b b\ { n b . \ j I sin# + - cos£ + - J I cos# - - sm# 1 (8.2) where 9 —kd, and £ is the propagation constant of the unloaded line, The reader can verify that A D — BC = 1, as required for reciprocal networks. For a wave propagating in the +z direction, we must have V(z)= V(0)e-^, 7(^ = 7(0)^% (8.3a) (83b) for a phase reference at z = 0. Since the structure is infinitely long, the voltage and current at the nth terminals can differ from the voltage and current at the n + 1 terminals only by 8.1 Periodic Structures 373 the propagation factor, e rd. Thus, &H = Vne-*d, (8.4a) = 1^. (S.4b) Using this result in (8.1) gives the following: ri_ra BirK+ii.r^+i^ For a nonuivial solution, the determinant of the above matrix must vanish: AD + -(A + D)eYd -BC = 0, (8.6) or. since AD — BC = 1, 1 + &t* - (a + D)eyd s= 0. e~yd+eyd = A + D, A + D b cosh yd1 — -= cos 61 — - sinf9, (8.7) 2 2 where (8.2) was used for the values of A and D. Now if y = a + we have that cosh yd = cosh ad cos Bd + j sinh ad sin fid = cos # - | sin 0. (8.8) Since the right-hand side of (8.8) is purely real, we must have either w = 0 or B — 0. G*k i: a = 0, /0 ^ 0. This case corresponds to a nonattenuating, propagating wave on the periodic structure, and defines the passband of the structure. Then (8.8) reduces to b cos Bd = cos f? - - sin B. (8.9a) which can be solved for 8 if the magnitude of the right-hand side is less than or equal to unity. Note that there are an infinite number of values of B that can satisfy (8.9a), Case 2-. a / 0, 8 = 0, jr. In this case the wave does not propagate, but is attenuated along the fine; this defines the stopband of the structure. Because the line is lossless, power is not dissipated, but is reflected back to the input of the line. The magnitude of (8.8) reduces to cosh ad = b cos 8--sin# 2 > 1, (8,9b) which has only one solution (a > 0) for positively traveling waves; a < 0 applies for negatively traveling waves. If cos 6 - (b/2) sinf? < — 1, (8.9b) is obtained from (8.8) by letting B = 7i; then all the lumped loads on the line are X/2 apart, yielding an input impedance the same as if p = 0. Thus, depending on the frequency and normalized susceptance values, the periodically loaded line will exhibit either passbands or stopbands, and so can be considered as a type of filter. It is important to note that the voltage and current waves defined in (8.3) and (8.4) are meaningful only when measured at the terminals of the unit cells, and do not apply to voltages and currents that may exist at points within a unit cell. These waves are similar to the elastic waves {Bloch waves) that propagate through periodic crystal lattices. Chapter 8: Microwave Filters Besides the propagation constant of the waves on the periodically loaded line, we will also be interested in the characteristic impedance for these waves. We can define a characteristic impedance at the unit cell terminals as ZB = Z0^±1, (8.10) in+l since V*fl+i and In+\ in the above derivation were normalized quantities. This impedance is also referred to as the Btoch impedance. From (8.5) we have that (A-e^)Vn+i+BIn+i =0. so (8.10) yields -BZo From (8.6) we can solve for eyd in terms of A and D as follows: yd_ (A + D)±J(A + D)2-4 € * 2 Then the Bloch impedance has two solutions given by ,±__— 2ffZp 2A - A - D =p J {A + Df - 4 = ^7—^-, (8.11) For symmetrical unit cells (as assumed in Figure 8.2) we will always have A = D. In this case (8,11) reduces to 2±=±BZ»= The ± solutions correspond to the characteristic impedance for positively and negatively traveling waves, respectively. For symmetrical networks these impedances are the same except for the sign; the characteristic impedance for a negatively traveling wave turns out to be negative because we have defined /„ in Figure 8,2 as always being in the positive direction. From (8.2) we see that B is always purely imaginary. If a = 0, & ^ 0 (passband), then (8.7) shows that cosh yd = A < 1 (for symmetrical networks) and (8.12) shows that ZB will be real. If o* ^ 0, B = 0 (stopband), then (8.7) shows that cosh yd = A > 1, and (8.12) shows that Zg is imaginary. This situation is similar to that for the wave impedance of a waveguide, which is real for propagating modes and imaginary for cutoff, or evanescent, modes. Terminated Periodic Structures Next consider a truncated periodic structure, terminated in a load impedance ZL, as shown in Figure 8.3. At the terminals of an arbitrary unit cell, the incident and reflected voltages FIGURE 8.3 A periodic structure terminated in a normalized load impedance ZL. 6.1 Periodic Structures 375 and currents can be written as (assuming operation in the passband) % = V+e"^nd + V^hd, (8.13a) f„ = I^e-^d 4- «J= + ^e^nd, (8.13b) where we have replaced yz in (8.3) with j&nd, since we are interested only in terminal quantities. Now define the following incident and reflected voltages at the nth unit cell; V* = V+e-rf"d, (8.14a) V- = V~e^nd. (8.14b) Then (8.13) can be written as At the load, where n - N, we have VN V« = V;+V-, (8.15a) y+ y- h=^: + ~- (8.15b) •B so the reflection coefficient at the load can be found as r = ^ = -Zl/z*~l. (8.17) v+ zjz- - 1 If the unit cell network is symmetrical (A = D), then Z£ = — Zg = Zj), which reduces (8.17) to the familiar result that r = |^ (s.18) i£ + Lb So to avoid reflections on the terminated periodic structure, we must have ZL = Zfl, which is real for a lossless structure operating in a passband. If necessary, a quarter-wave transformer can be used between the periodically loaded line and the load. k-fi Diagrams and Wave Velocities When studying the passband and stopband characteristics of a periodic structure, it is useful to plot the propagation constant, 6, versus the propagation constant of the unloaded line, k (or in). Such a graph is called a k-3 diagram, or Briilouin diagram (after L. Brillouin, a physicist who studied wave propagation in periodic crystal structures). The k-B diagram can be plotted from (8.9a), which is the dispersion relation for a general periodic structure. In fact, a k-B diagram can be used to study the dispersion characteristics of many types of microwave components and transmission lines. For instance, consider the dispersion relation for a waveguide mode: (8.19) 376 Chapter8: Microwave Fitters Slope = v&lc Slope = vp/c FIGURE 8.4 k-fi diagram for a waveguide mode. where kc is the cutoff wavenumber of the mode, k is the free-space wavenumber, and 8 is the propagation constant of the mode. Relation (S. 19) is plotted in the k-8 diagram of Figure 8.4. For values of k < kc, there is no real solution for 8, so the mode is nonpropagating. For k > the mode propagates, and k approaches 8 for large values of 8 (TEM propagation). The k~& diagram is also useful for interpreting the various wave velocities associated with a dispersive structure. The phase velocity is (8.20) which is seen to be equal to c (speed of light) times the slope of the line from the origin to the operating point on the k-f) diagram. The group velocity is iho dk dp=Cdp' (8.21) which is the slope of the k-8 curve at the operating point. Thus, referring to Figure 8.4, we see that the phase velocity for a propagating waveguide mode is infinite at cutoff and approaches c (from above) as k increases. The group velocity, however, is zero at cutoff and approaches c (from below) as k increases. We finish our discussion of periodic structures with a practical example of a capacitively loaded line. EXAMPLE 8.1 ANALYSIS OF A PERIODIC STRUCTURE Consider a periodic capacitively loaded line, as shown in Figure 8.5 (such a line may be implemented as in Figure 8.1 with short capacitive stubs). If Zq = 50 £2. d 1.0 cm, and Co = 2,666 pF, sketch the k~B diagram and compute the propagation constant, phase velocity, and Bloch impedance at / = 3.0 GHz. Assume k = fca- FIGURE 8.5 A capacitively loaded line. 8.1 Periodic Structures 377 Solution We can rewrite the dispersion relation of (8.9a) as cos fid = cos kcjd — I —-— I k$d sin&nrf. Then CqZqc _ (2.666 x 1Q-12)(5Q)(3 x 10s) 2d ~ ~ 2(0.01) = 2.0, so we have cos ßd = cos k&i - 2kod sin kGd. The most straightforward way to proceed at this point is to numerically evaluate the right-hand side of the above equation for a set of values of k&i starting at zero. When the magnitude of the right-hand side is unity or less, we have a passband and can solve for fid. Otherwise we have a stopband. Calculation shows that the first passband exists for 0 < k{,d < 0.96. The second passband does not begin until the sin£oJ term changes sign at kod = x. As kod increases, an infinite number of passbands are possible, but they become narrower. Figure 8.6 shows the k-fi diagram for the first two passbands. At 3.0 GHz, we have so fid — 1.5 and the propagation constant is 8 = 150 rad/m. The phase velocity is which is much less than the speed of light, indicating that this is a slow-wave 2ji0 x 109) 3 x 10s (0.01) = 0.6283 = 36% 4 i Passband 3 2 Passband FIGURE 8.6 k-ß diagram for Example 8.1. 378 Chapterd: Microwave Filters 8.2 structure. To evaluate the Bloch impedance, we use (8.2) and (8.12): ft = ojCoZo = tm 2 2 9 = kQd = 36°, b A — cos 0 — - sinů = 0.0707, / b b\ B — j I sin0 + -cos#- - J = jO.3479. Then, Z„ = ^== OU347JX50) =n4n VA2 - 1 jVl - (0.0707)2 FILTER DESIGN BY THE IMAGE PARAMETER METHOD The image parameter method of filter design involves the specification of passband and stopband characteristics for a cascade of two-port networks, and so is similar in concept to the periodic structures that were studied in Section 8.1. The method is relatively simple but has the disadvantage that an arbitrary frequency response cannot be incorporated into the design. This is in contrast to the insertion loss method, which is the subject of the following section. Nevertheless, the image parameter method is useful for simple filters and provides a link between infinite periodic structures and practical filter design. The image parameter method also finds application in solid-state traveling-wave amplifier design. Image Impedances and Transfer Functions for Two-Port Networks We begin with definitions of the image impedances and voltage transfer function for an arbitrary reciprocal two-port network; these results are required for the analysis and design of filters by the image parameter method. Consider the arbitrary two-port network shown in Figure 8.7, where the network is specified by its ABCD parameters. Note that the reference direction for the current at port 2 has been chosen according to the convention for ABCD parameters. The image impedances, Zf( and Z,2, are defined for this network as follows; Z/i = input impedance at port 1 when port 2 is tenninated with Zn, Zn = input impedance at port 2 when port 1 is terminated with 3si? Thus both ports are matched when terminated in their image impedances. We will now derive expressions for the image impedances in terms of the ABCD parameters of a network. 'In i in 2 FIGURE 8.7 A two-port network terminated in its image impedances. 8.2 Filter Design by the Image Parameter Method 379 The port voltages and currents are related as V]=AV2 + B12, (8.22a) h = CV2 + DI2. (8.22b) The input impedance at port 1, with port 2 terminated in Zi2> is V, AV2 + BI2 AZl2 + B t\ CV2 + DI2 CZi2 + D since Vj = Zi2i2. Now solve (8.22) for V2r h by inverting the ABCD matrix. Since A D - BC = 1 for a reciprocal network, we obtain V2 = DV]-BI]. (8.24a) /2 = -CVi + Ah. (8-24b) Then the input impedance at port 2, with port 1 cerminated in Z,|, can be found as -V2 DVt - BIy DZn + B ,n2 '-CVx+Ali - CZa + A' ^m since V] = — Zi\I\ (circuit of Figure 8.7). We desire that Zin1 = ZiX and Zm2 = Zl2, so (8.23) and (8.25) give two equations for the image impedances: Z,, (CZl2 + D) = AZi2 + B. (8.26a) ZjiD — B = Z;2(A — CZi\). (8.26b) Solving for Z; i and Zi2 gives z'2 = V Äc' (S'27b) with Z,2 = DZfi/A. If the network is symmetric, then A = D and Z,i = Zl2 as expected. Now consider the voltage transfer function for a two-port network terminated in its image impedances. With reference to Figure 8.8 and (8,24a), the output voltage at port 2 can be expressed as V2 = DVy - g/, = £)- — ) V, (8.28) FIGURE 8.8 A two-port network terminated in its image impedances and driven with a voltage generator. 380 Chapter 8: Microwave Filters (since we now have Vi = /| Z; i) so the voltage ratio is ^ = d-^- = D- BJ^ ~ I~{VaB - vbc). (8,29a) v\ Zn V ab V a Similarly, the current ratio is § = + A = -czi} +a = J~(VaD - vsc). (8.29b) The factor *JDJA occurs in reciprocal positions in (8.29a) and (8,29b), and so can be interpreted as a transformer turns ratio. Apart from this factor, we can define a propagation factor for the network'as = JAD - 4b~C, (8.30) withy = = 0. The propagation factor, also from Table 8.1, is 2ft) 2w / to2 e* = \-—- + — ./^-l. (8.36) ft>f V Now consider two frequency regions: 1. For oj < tt>c\ This is the passband of the filter section. Equation (8.35) shows that Z,T is real, and (8.36) shows that y is imaginary, since to2/to2 — 1 is negative and kn-i: 2. For ft) > ft>c: This is the stopband of the filter section. Equation (8.35) shows that Z,t is imaginary, and (8.36) shows that eY is real and —1 < ey < 0 (as seen from the limits as to (i>c and to -> 00). The attenuation rate for to > toe is 40 dB/decade. Typical phase and attenuation constants are sketched in Figure 8.10. Observe that the attenuation, a, is zero or relatively small near the cutoff frequency, although a —> 00 as to —*■ 00. This type of filter is known as a consmnt-k low-pass prototype. There are only 382 ChapterG: Microwave Filters FIGURE 8.10 Typical passband and stopband characteristics of the low-pass constant-^ sections of Figure 8.9. two parameters to choose (L and C), which are determined by L and Z2 = 1 /joiC, so the propagation factor is the same as that for the low-pass T network. The cutoff frequency, o>Cf and nominal characteristic impedance, Rq, are the same as the corresponding quantities for the t network as given in (S.33) and (8.34). At Final m-derived section, m-Derived Filter Sections We have seen that the constant-^ filter section suffers from the disadvantages of a relatively slow attenuation rate past cutoff, and a nonconstant image impedance. The m-derived filter section is a modification of the constant-fe section designed to overcome these problems. As shown in Figure 8,l2a,b the impedances Z\ and Z2 in a constant-fc T-section are replaced with ZJ and Z'2, and we let Z; = mZ,. (8.39) Then we choose Z2 to obtain the same value of Zu as for the constant-^ section. Thus, from Table 8.1, Z,t = J Z, Z2 + S = Jz'i Z'2 + S = JmZ, Z^ + ^2%. (8.40) Solving for Z'2 gives Z2 Zi mZj Z2 (1 — m2) -j z; = — + -1 - —- = — + ^—---Zj, (8.41) m 4m 4 tn 4m Because the impedances Z\ and Z2 represent reactive elements, Z2 represents two elements in series, as indicated in Figure 8.12c. Note that tn = 1 reduces to the original constant-fc section. For a low-pass filter, we have Z\ = jcoL and Zi — XfjioC. Then (8.39) and (8.41) give the m-derived components as ZJ = jcoLm, (8.42a) 1 , (l-m2). Z2 = — r + " ,-JojL, )b)Cm 4m which results in the circuit of Figure 8.13. Now consider the propagation factor for the 384 Chapter 8: Microwave Filters mU2 mU2 Am (a) Cm) + jcoLQ -m2)/4m ~ 1 - (I - hi2XM)2 If we restrict 0 < m < 1, then these results show that eY is real and \ey\ > 1 for m > a>c. Thus hie stopband begins at w = M- Note that (8.44) indicates that co^ > (t>c, so infinite attenuation occurs after the cutoff frequency, coc, as illustrated in Figure 8.14. The position of the pole at can be controlled with the value of m. FIGURE 8.14 Typical attenuation responses for constant-fc, m-derived, and composite filters. 385 mZj/2 I mZxtl mZx!2 \ mZ^il FIGURE 8.15 Development of an m-derived jr-section. (a) Infinite cascade of m-derived T-sections, (b) A de-embedded ;r-equivalent, We now have a very sharp cutoff response, but one problem with the m-derived section is that its attenuation decreases for oj > fc^. Since it is often desirable to have infinite attenuation as co -> oo, the m-derived section can be cascaded with a constant-/: section to give the composite attenuation response shown in Figure 8.14. The m-derived T-section was designed so that its image impedance was identical to that of the constant-t section (independent of m), so we still have the problem of a nonconstant image impedance. But the image impedance of the jt-equivalent will depend on m, and this extra degree of freedom can be used to design an optimum matching section. The easiest way to obtain the corresponding ;r-section is to consider it as a piece of an infinite cascade of m-derived T -sections, as shown in Figure 8.l5a,b, Then the image impedance of this network is, using the results of Table S.l and (8.35), Now ZLZ2 - L/C = R^ and Z\ = -oP-L2 = -4/^(w/&^)2, so (8.45) reduces to _ l-(l-m2)(a>M-)2a {9A,, Zin =-. —/&> (8-46) $ - (o/«c)2 Since this impedance is a function of m, we can choose m to minimize the variation of Zin over the passband of the filter. Figure 8.16 shows this variation with frequency for several values of m; a value of m = 0.6 generally gives the best results. This type of m-derived section can then be used at the input and output of the filter to provide a nearly constant impedance match to and from Rq. But the image impedance of the constant-/: and m-derived T-sections, Z;r> does not match Z-in% this problem can be surmounted by bisecting the ^-sections, as shown in Figure 8.17. The image impedances 386 ChapterS: Microwave Filters Of FIGURE 8.16 Variation of ZJJF in the passband of a low-pass m -derived section for various values of m. of this circuit are Z(l = Z,t and Zj2 = Z\n, which can be shown by finding its ABCD parameters: Z\ 4Z2' B C 7' Zl 2 ' 1 2~T2' D = l, and then using (8,27) for Z,\ and Z!2: Z[\ = Jz\z'2 + -± — Z,T, Za = l + z;/4Z2 Ztr 7' 7' _ fl^2 _ g where (8.40) has been used for Z/7. (8.47a) (8.47b) (8.47c) (S.47d) (8.48a) (8.48b) Composite Filters By combining in cascade the constant-fc, m-derived sharp cutoff, and the m-derived matching sections we can realize a filter with the desired attenuation and matching properties. This c-WW FIGURE 8.17 A bisected ^-section used to match Z,* to ZiT. 8.2 Filter Design by the Image Parameter Method 387 o- Matching section «i = 0.6 1 High-/ cutoff Constant k T FIGURE 8.18 The final four-stage composite filter. Sharp cutoff nkO.6 T Matching section -iT m = 06 1 -a type of design is called a composite filter, and is shown in Figure 8.18. The sharp-cutoff section, with m < 0.6, places an attenuation pole near the cutoff frequency to provide a sharp attenuation response; the constant-Zc section provides high attenuation further into the stopband. The bisected-^ sections at the ends of the filter match the nominal source and load impedance, Rq, to the internal image impedances, Z,t. of the constant-k and řw-derived sections. Table 8.2 summarizes the design equations for low- and high-pass TABLE 8.2 Summary of Composite Filter Design Low-Pass High-Pass Constants T section La in «0 = JUČ L = 2/řůfcj[. m-derived T section mL!2 mUl L, C Same as constant-* section *J 1 - (ptc/„)2 for sharp-cutoff 0.6 for rfl&tchiug Bisected-rr matching section mlj'l mLn. Constant-* T section 1C 2C —\\-?-IK ao = JUČ L = R„/2ii>r m-de rived 7 section ICIw lOm -w- Um) 4m <] - Jfl-I -c L, C Same as constant-/; section J1 - (íú„Attt.)2 for sharp-cutoff 0.6 for matching Bisected-jt matching section ICIm ICim -o-c-II- 2L/m U-m2) "J" 388 Chapter 8: Microwave Filters composite filters; notice that once the cutoff frequency and impedance are specified, there is only one degree of freedom (the value of m for the sharp-cutoff section) left to control the filter response. The following example illustrates the design procedure. EXAMPLE 8.2 LOW-PASS COMPOSITE FILTER DESIGN Design a low-pass composite filter with a cutoff frequency of 2 MHz and impedance of 75 Q. Place the infinite attenuation pole at 2.05 MHz, and plot the frequency response from 0 to 4 MHz. Solution All the component values can be found from Table 8.2. For the constant-^ section: 2R 2 L = — = 11.94 /xH, C = :-= 2.122 nF For the m-derived sharp-cutoff section: / / f m -Hi in L = 1,310 mH, mC = 465.8 pF, —L = 12.94/iH. 4m For the m = 0.6 matching sections: ~ = 3.582 mH, — 636.5 pF, 1 - m: 2m L = 6.36S iM. The completed filter circuit is shown in Figure 8.19; the series pairs of inductors between the sections can be combined. Figure 8.20 shows the resulting frequency response for \S\2\- Note the sharp dip at / = 2.05 MHz due to the m = 0.2195 section, and the pole at 2.50 MHz, which is due to the m = 0.6 matching sections. ■ 3,582 fiM 5.97/aH 5,97 yH 1,310 1,310 ííH 3.5Ě2 jiH d= 2122 pF 12.94 fiH 465.8 pF I o- 6.368 fxH 636.5 pF -o Matching Constam-jfc m-derived FIGURE 8.19 Low-pass composite niter for Example 8.2. Matching 8.3 Fitter Design by the Insertion Loss Method 389 FILTER DESIGN BY THE INSERTION LOSS METHOD The perfect filter would have zero insertion loss in the passband, infinite attenuation in the stopband, and a linear phase response (to avoid signal distortion) in the passband, Of course, such filters do not exist in practice, so compromises must be made; herein lies the art of filter design. The image parameter method of the previous section may yield a usable filter response, but if not there is no clear-cut way to improve the design. The insertion loss method, however, allows a high degree of control over the passband and stopband amplitude and phase characteristics, with a systematic way to synthesize a desired response. The necessary design trade-offs can be evaluated to best meet the application requirements. If, for example, a minimum insertion loss is most important, a binomial response could be used; a Chebyshev response would satisfy a requirement for the sharpest cutoff. If it is possible to sacrifice the attenuation rate, a better phase response can be obtained by using a linear phase filter design. And in all cases, the insertion loss method allows filter performance to be improved in a straightforward manner, at the expense of a higher order filter. For the filter prototypes to be discussed below, the order of the filter is equal to the number of reactive elements. Characterization by Power Loss Ratio In the insertion loss method a filter response is defined by its insertion loss, or power loss radoy Plr: Power available from source P^c 1 LR ~ Power delivered to load ~ P^ ~ 1 - \r{a>)\2' Observe that this quantity is the reciprocal of \S\2\2 if both load and source are matched. The insertion loss (IL) in dB is IL= 101ogPLft. (8.50) 390 Chapter 8: Microwave Filters From Section 4.1 we know that |P(tó)l2 is an even function of or, therefore it can be expressed as a polynomial in o>2. Thus we can write M{ar) M{ü>2} + Nico2)' (8.51) where M and N are real polynomials in to2. Substituting this form in (8.49) gives the following: M{o)2) N(co2)' (8.52) Thus, for a filter to be physically realizable its power loss ratio must be of the form in (8.52). Notice that specifying the power loss ratio simultaneously constrains the reflection coefficient, T(o>). We now discuss some practical filter responses. Maximally fiat. This characteristic is also called the binomial or Butterworth response, and is optimum in the sense that it provides the flattest possible passband response for a given filter complexity, or order. For a low-pass filter, it is specified by Pis = l+k' / to \ 2N \toc) (S.53) where N is the order of the filter, and u>t is the cutoff frequency. The passband extends from to = 0 to co = toc; at the band edge the power loss rafio is 1 + kl. If we choose this as the —3 dB point, as is common, we have k — 1, which we will assume from now on. For 0) > o>c, the attenuation increases monotonically with frequency, as shown in Figure 8.21. For co > coCt PLR ~ k2(o)/ojc)2n, which shows that the insertion loss increases at the rate of 20JV dB/decade. Like the binomial response for multisection quarter-wave matching transformers, the first (2rV — 1) derivatives of (8.53) are zero at to = 0. Equal ripple. If a Chebyshev polynomial is used to specify the insertion loss of an N -order low-pass filter as »CÖ- (8.54) then a sharper cutoff will result, although the passband response will have ripples of amplitude I + k2, as shown in Figure 8,21, since Tn(x) oscillates between ±1 for I*| < 1. Thus, 'lr Equal ripple Maximally flat oic the insertion loss becomes which also increases at the rate of 20 N dB/decade. But the insertion loss for the Chebyshev case is (22/v)/4 greater than the binomial response, at any given frequency where o> 3> (oc. Elliptic function. The maximally flat and equal-ripple responses both have monotonically increasing attenuation in the stopband. In many applications it is adequate to specify a minimum stopband attenuation, in which case a better cutoff rate can be obtained. Such filters are called elliptic function filters [3], and have equal-ripple responses in the passband as well as the stopband, as shown in Figure 8.22. The maximum attenuation in the passband, AmM, can be specified, as well as the minimum attenuation in the stopband, A^. Elliptic function filters are difficult to synthesize, so we will not consider them further; the interested reader is referred to reference [3]. Linear phase. The above filters specify the amplitude response, but in some applications (such as multiplexing filters for communication systems) it is important to have a linear phase response in the passband to avoid signal distortion. Since a sharp-cutoff response is generally incompatible with a good phase response, the phase response of a filter must be deliberately synthesized, usually resulting in an inferior attenuation characteristic. A linear phase characteristic can be achieved with the following phase response: where 0(w) is the phase of the voltage transfer function of the filter, and p is a constant. A related quantity is the group delay, defined as which shows that the group delay for a linear phase filter is a maximally flat function. More general filter specifications can be obtained, but the above cases are the most common. We will next discuss the design of low-pass filter prototypes which are normalized in terms of impedance and frequency; this normalization simplifies the design of filters for (8.56) 392 ChapterS: Microwave Filters Filter specifications Low-pass prototype design Scaling and conversion Imple mentation FIGURE 8.23 The process of filter design by the insertion loss method. arbitrary frequency, impedance, and type (low-pass, high-pass, bandpass, or bandstop). The low-pass prototypes are then scaled to the desired frequency and impedance, and the lumped-element components replaced with distributed circuit elements for implementation at microwave frequencies. This design process is illustrated in Figure 8.23. Maximally Flat Low-Pass Filter Prototype Consider the two-element low-pass filter prototype shown in Figure 8.24; we will derive the normalized element values, L and C, for a maximally flat response. We assume a source impedance of 1 Q,, and a cutoff frequency wť = 1, From (8.53), the desired power loss ratio will be, for N = 2, PLR = 1 + oj\ (8.57) The input impedance of this filter is ZiD=j2L2 F to4L2C2R2 - la?LCR2) 4R = 1 + -^[(1 - R)2 + (R2C2 + L2- 2LCR2)a>2 + L2C2R2úí4]. (8.60) 4R Notice that this expression is a polynomial in co2. Comparing to the desired response of (8.57) shows that R = 1, since Plr = 1 for a> = 0. In addition, the coefficient of c = 1, however, the element values for the ladder-type circuits of Figure 8.25 can be tabulated [1]. Table 8.3 gives such element values for maximally flat low-pass filter prototypes for N = 1 to 10. (Notice that the values forN = 2 agree with the above analytical solution.) This data is used with either of the ladder circuits of Figure 8,25 in the following way. The element values are numbered from g0 at the generator impedance to at the load impedance, for a filter having N reactive elements. The elements alternate between FIGURE 8.25 Ladder circuits for low-pass filter prototypes and their element definitions, (a) Prototype beginning with a shunt element, (b) Prototype beginning with a series element. 394 Chapter 8: Microwave Filters TABLE 8 J Element Values for Maximally Flat Low-Pass Filter Prototypes (g« = 1, uc = 1, N = 1 to 10) ft Si Si £5 gl 1 2.0000 1.0000 2 1,4142 1.4142 1.0000 3 1.0000 2.0000 1.0000 1.0000 4 0.7654 1,8478 1.8478 0.7654 1.0000 5 0.6180 1.6180 2.0000 1.6180 0.6180 1.0000 6 0.5176 1.4142 1,9318 1.9318 1.4142 0.5176 1.0000 7 0.4450 1.2470 1.8019 2.0000 1.8019 1.2470 0.4450 1.0000 S 0.3902 1.1111 1.6629 1.9615 1.9615 1.6629 l.nii 0.3902 1.0000 9 0.3473 1.0000 1.5321 1.8794 2.0000 1.8794 1.5321 1.0000 0.3473 i.0000 10 0.3129 0.9080 1.4142 1.7820 1.9754 1.9754 1.7820 1.4142 0.9080 0,3129 Source: Reprinted from G. L. Matthaei, L. Young, and E. M. X Jones, Microwave Fitters, Impedance-Matching Networks, and Coupling Structures (Dedham, Mass.: Ailcch House, 19S0) with permission. series and shunt connections, and gk has the following definition: {generator resistance (network of Figure 8.25a) generator conductance (network of Figure 8.25b) {inductance for series inductors capacitance for shunt capacitors J load resistance if gN is a shunt capacitor ght+\ — J conductance if gN is a series inductor Then the circuits of Figure 8.25 can be considered as the dual of each other, and both will give the same response. Finally, as a matter of practical design procedure, it will be necessary to determine the size, or order, of the filter. This is usually dictated by a specification on the insertion loss at some frequency in the stopband of the filter. Figure 8.26 shows the attenuation characteristics for various N, versus normalized frequency. If a filter with N > 10 is required, a good result can usually be obtained by cascading two designs of lower order, Equal-Ripple Low-Pass Filter Prototype For an equal-ripple low-pass filter with a cutoff frequency ojc = 1, the power loss ratio from (8.54) is PLR = l+k2T2(a>), (8.63) where 1 + k2 is the ripple level in the passband. Since the Chebyshev polynomials have the property mat f 0 WO) = i for /V odd, for N even, equation (8.61) shows that the filter will have a unity power loss ratio at o> = 0 for N odd, but a power loss ratio of 1 + k2 at ca — 0 for N even. Thus, there are two cases to consider, depending on N. 8.3 Fitter Design by the Insertion Loss Method 395 FIGURE 8.2ti Attenuation versus normalized frequency for maximally fiat filter prototypes. Adapted from G- L. Matttiaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures (Dedham, Mass.: Artech House, 1980) with permission. For the two-element filter of Figure 8.24, the power loss ratio is given in terms of the component values in (8.60). From (5.56b), we see mat Ti(x) = 2xz — 1, so equating (8.61) to (8.60) gives 1 + k2(4a>4 - 4co2 + 1) = 1 + — [(1 - R)2 + (R2C2 + L2 - 2LCR2W + L2C2R2o>% 4R (8.62) which can be solved for /?, L, and C if the ripple level (as determined by k2) is known. Thus, at a) = 0 we have that a _ g ~ V1 K " 4R ' or R = 1 + 2k2 ± 2ky/\ + k1 (for N even). (8.63) Equating coefficients of co2 and oř yields the additional relations 4k2 = —L2C2R2, 4R -4k2 = + L2- 1LCR \ which can be used to find L and C. Note that (8.63) gives a value for R that is not unity, so there will be an impedance mismatch if the load actually has a unity (normalized) 396 Chapter 8: Microwave Filters TABLE 8.4 Element Values for Equal-Ripple Low-Pass Filter Prototypes <#> = 1, wc = 1, N = 1 to 10, 0,5 dB and 3.0 dB ripple) 0.5 dB Ripple N gl gi g* gs £3 gw an 1 0.6986 1.0000 2 1.4029 0.7071 1.9841 3 1.5963 1.0967 1.5963 1.0000 4 1.6703 1.1926 2.3661 0.8419 1.9841 5 1.7058 1.2296 2.5408 1.2296 1.7058 1.0000 6 1.7254 1.2479 2.6064 1.3137 2.4758 0.8696 1.9841 7 1.7372 1.2583 2.6381 1.3444 2.6381 1.2583 1.7372 1,0000 8 1.7451 1.2647 2.6564 1.3590 2.6964 1.3389 2.5093 0.8796 1.9841 9 1.7504 1.2690 2.6678 1.3673 2.7239 1.3673 2.6678 1.2690 1.7504 1.0000 10 1.7543 1,2721 2.6754 1.3725 2.7392 1.3806 2.7231 1.3485 25239 0.8842 1.9841 3.0 dB Ripple AT g] $i g<> gl & gift in 1 1.9953 1.0000 2 3,1013 0.5339 5.8095 3 3.3487 0.7117 3.3487 1.0000 4 3.4389 0,7483 4.3471 0.5920 5.8095 5 3.4817 0.7618 4.5381 0.7618 3.4817 1.0000 6 3.5045 0.7685 4.6061 0.7929 4.4641 0.6033 5.8095 7 3.5182 0.7723 4.6386 0.8039 4,6386 0.7723 3.5182 1.0000 8 3.5277 0.7745 4.6575 O.S089 4.6990 0.8018 4.4990 0.6073 5.8095 9 3.5340 0,7760 4.6692 0.8118 4.7272 0.8118 4,6692 0.7760 3.5340 1.0000 10 3.5384 0.7771 4,6768 0.8136 4.7425 0.8164 4.7260 0.8051 4.5142 0.6091 5,8095 Source: Reprinted from O. L. Maithaei, L. Young, and E, M. T- Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures (Dedham, Mass.: Artech House. 1980) with permission. impedance; this can be corrected with a quarter-wave transformer, or by using an additional filter element to make jV odd. For odd N, it can be shown that R = 1. (This is because there is a unity power loss ratio at to = 0 for N odd.) Tables exist for designing equal-ripple low-pass filters with a normalized source impedance and cutoff frequency (u>'£ — 1) [I], and can be applied to either of the ladder circuits of Figure 8.25. This design data depends on the specified passband ripple level; Table 8,4 lists element values for normalized low-pass filter prototypes having 0.5 dB or 3.0 dB ripple, for N = 1 to 10. Notice that the load impedance gN+l ^ 1 for even N. If the stopband attenuation is specified, the curves in Figures 8.27a,b can be used to detennine the necessary value of N for these ripple values. Linear Phase Low-Pass Filter Prototypes Filters having a maximally flat time delay, or a linear phase response, can be designed in the same way, but things are somewhat more complicated because the phase of the voltage transfer function is not as simply expressed as is its amplitude. Design values have been 8.3 Filter Design by the Insertion Loss Method 397 70 60 50 I d o 40 1 30 20 10 j i 1, / >:< * / * / / ^ y 0.01 0.02 0,03 0.05 0.07 ft 10 0.20 0.30 0,50 0.70 1,0 2 0 30 5.0 7.0 10.0 # -I (a) 0.01 0.02 0.03 0,05 0.07 0.10 0.20 0,30 0,50 0.70 l.O 2.0 3.0 5,0 7,0 100 l^-l-l (b) FIGURE 8.27 Attenuation versus normalized frequency for equal-ripple filter prototypes, (a) 0.5 dB ripple level, (b) 3.0 dB ripple level. Adapted from G. L. Matthaei, L. Young, and E M. T. Jones, Microwave Filters, Itnpedance-Matchmg Networks, and Coupling Structures (Dedham, Mass.: Artech House, 1980) with permission. 398 Chapters: Microwave Filters TABLE 83 Element Values for Maximally-Flat Time Delay Low-Pass Filter Prototypes (gn = 1, u>c = 1, A' = 1 to 10) N Si Si gj S* gs £]i> 1 2,0000 1.0000 2 1.5774 0.4226 1.0000 3 1.2550 0.5528 0.1922 1.0000 4 1.0598 0.5116 0.3181 0.1104 1.0000 5 0.9303 0.4577 0.3312 0.2090 0.0718 1.0000 6 0.8377 0.4116 0.3158 0.2364 0.1480 0.0505 1.0000 7 0.7677 0.3744 0.2944 0.2378 0.1778 0.1104 0.0375 1.0000 8 0.7125 0.3446 0.2735 0.2297 0.1867 0.1387 0.0855 0.0289 1.0000 9 0,6678 0.3203 0.2547 0.2184 0.1859 0.1506 0.1111 0.0682 0.0230 1.0000 10 0.6305 0.3002 0.2384 0.2066 0.1808 0.1539 0.1240 0.0911 0,0557 0,0187 Source- Reprinted from G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters Impedance-Matching Networks, and Coupling Structures (Dedhain, Mass.: Artech House, 1980) with permission. derived for such filters [1], however, again for the ladder circuits of Figure 8.25, and are given in Table 8,5 for a normalized source impedance and cutoff frequency £ft£ = 1). The resulting group delay in the passband will be xd = l/(o'c = 1. FILTER TRANSFORMATIONS The low-pass filter prototypes of the previous section were normalized designs having a source impedance of Rs = 1 £2 and a cutoff frequency of a>c = 1. Here we show how these designs can be scaled in terms of impedance and frequency, and converted to give high-pass, bandpass, or bandstop characteristics. Several examples will be presented to illustrate the design procedure. Impedance and Frequency Scaling impedance scaling. In the prototype design, the source and load resistances are unity (except for equal-ripple fillers with even /V, which have nonunity load resistance). A source resistance of Ra can be obtained by multiplying the impedances of the prototype design by R0. Then, if we let primes denote impedance scaled quantities, we have the new filter component values given by L' — Ri)L, (8.64a) C' = ^-, (S,64b) m Rrs=Ro, (8.64c) R'l = RqRl. (8.64d) where L, C. and Ri are the component values for the original prototype. 8.4 Filter Transformations 399 Frequency scaling for low-pass filters. To change the cutoff frequency of a low-pass prototype from unity to a>c requires that we scale the frequency dependence of the filter by the factor 1 /(oc, which is accomplished by replacing a> by to/coc: to (8.65) Then the new power loss ratio will be where w€ is the new cutoff frequency; cutoff occurs when a>/ft>e = 1, or o> = (oc. This transformation can be viewed as a stretching, or expansion, of the original passband, as illustrated in Figure 8,28a,b. The new element values are determined by applying the substitution of (8.65) to the series reactances, jcuLk, and shunt susceptances, ja>Ck, of the prototype filter. Thus, jXt = j—Lk = jwLk, o>c ft) jBk = j—Ck =jcoCfc, which shows that the new element values are given by — r> - Ck — — (8.66a) (8.66b) When both impedance and frequency scaling are required, the results of (8.64) can be combined with (8.66) to give C'k = Raa>c (8.67a) (8.67b) -Cúr 0 (c) FIGURE 8.28 Frequency scaling for low-pass filters and transformation to a high-pass response. (a) Low-pass filler prototype response for toir = 1. (b) Frequency scaling for low-pass response, (c) Transformation to high-pass response. 400 Chapters: Microwave Fitters Low-pass to high-pass transformation. The frequency substitution where to *----. (8.68) oj can be used to convert a low-pass response to a high-pass response, as shown in Figure 8.28c. This substitution maps a) = 0 to to = ±oo, and vice versa; cutoff occurs when o> = ±Ck, of the prototype filter gives jXk = -J — Lk = to jaiC't (1>C _ 1 JBk = —J — = -——, to jo)Lk which shows that series inductors niust be replaced with capacitors C£, and shunt capacitors Ck must be replaced with inductors L'k. The new component values are given by 2 I Impedance scaling can be included by using (8.64) to give 1 Ck = -^, (8.69a) Vk = —(8.69b) C; = n T , (8.70a) L'k = ^r. (8.70b) EXAMPLE 8 J LOW-PASS FILTER DESIGN COMPARISON Design a maximally flat low-pass filter with a cutoff frequency of 2 GHz, impedance of 50 Q, and at least 15 dB insertion loss at 3 GHz, Compute and plot the amplitude response and group delay for / = 0 to 4 GHz, and compare with an equal-ripple (3.0 dB ripple) and linear phase filter having the same order. Solution First find the required order of the maximally flat filter to satisfy the insertion loss specification at 3 GHz. We have that \( \ — 1 = 0.5; from Figure 8.26 we see that N — 5 will be sufficient. Then Table 8.3 gives the prototype element values as g\ = 0.618, gi = 1-618, g3 = 2.000, 84 = 1.618, g5 = 0.618. 8.4 Filter Transformations 401 FIGURE 8.29 Low-pass maximally flat filter circuit for Example 8.3. Then (8.67) can be used to obtain the scaled element values: C\ = 0.984 pF, L2 = 6.438 nH, C'3 =3.183 pF, L; = 6.438 nH, C'5 =0.984 pF. The final filter circuit is shown in Figure 8.29; the ladder circuit of Figure 8.25a was used, but that of Figure 8.25b could have been used just as well. The component values for the equal-ripple filter and the linear phase filter, for N = 5, can be determined from Tables 8.4 and 8.5. The amplitude and group delay results for these three filters are shown in Figure 8.30. These results clearly show the trade-offs involved with the three types of filters. The equal-ripple response has the sharpest cutoff, but the worst group delay characteristics. The maximally flat response has a flatter attenuation characteristic in the passband, but a slightly lower cutoff rate. The linear phase filter has the worst cutoff rate, but a very good group delay characteristic. ■ Bandpass and Bandstop Transformations Low-pass prototype filter designs can also be transformed to have the bandpass or bandstop responses illustrated in Figure 8.31. If a>i and coi denote the edges of the passband, then a bandpass response can be obtained using the following frequency substitution: OX) ~4-- (02 ^/^_^ = i(^_f*y (8.71) — tti\ \COq (.0 I A \COq co / where A = ——— (8.72) Wo is the fractional bandwidth of the passband. The center frequency, coo, could be chosen as the arithmetic mean of a>\ and a>2, but the equations are simpler if it is chosen as the geometric mean: i -til] VD] ti>2 tii —U>2 2 *J (a) (b> (c) FIGURE 8.31 Bandpass and bandstop frequency transformations, (a) Low-pass filter prototype response for co< = 1. (b) Transformation to bandpass response, (c) Transformation to bandstop response. B.4 Filter Transformations 403 low-pass response of Figure 8.31a as follows: When co — ojq, i When co = ai, i When o) = toi, CO COo (Oq CO ů) loo (Of) CO CO con coo o* (8.74c) (8.74d) The low-pass filter elements are thus converted to series resonant circuits (low impedance at resonance) in the series arms, and to parallel resonant circuits (high impedance at resonance) in the shunt arms. Notice that both series and parallel npKAtnr ^ments haw a resnnant frequency of ojtj. resonator elements have a resonant [uency of ojtj. The inverse transformation can be used to obtain a bandstop response. Thus, co <— A (~— ftlf, (8.75) where A and coq have die same definitions as in (8.72) and (8.73). Then series inductors of the low-pass prototype are convened to parallel LC circuits having element values given by AL, C — cl — cooALk' (8.76a) (8.76b) 404 Chapter 8r Microwave Fitters TABLE 8.6 Summary of Prototype Filter Transformations ( A = u}2 — u>\ w0 Low-pass High-pass Bandpass Rand stop ] 1 LA «0 T (y.-C A T ___C_ The shunt capacitor of the low-pass prototype is converted to series LC circuits having element values given by 1 C -H — Aft (8J6c) (8.76d) The element transformations from a low-pass prototype to a highpass, bandpass, or bandstop filter are summarized in Table 8.6. These results do not include impedance scaling, which can be made using (8.64). EXAMPLE 8.4 BANDPASS FILTER DESIGN Design a bandpass filter having a 0.5 dB equal-ripple response, with N = 3. The center frequency is 1 GHz, the bandwidth is 10%, and the impedance is 50 Q, c; 50 Q FIGURE 832 Bandpass filter circuit for Example 84. 8.5 Filter Implementation 405 0.75 1,0 Frequency (GHz) ] 5 FIGURE 8.33 Amplitude response for the bandpass filter of Example 8 4. Solution From Table 8.4 the element values for the low-pass prototype circuit of Figure 8.25b are given as $ = 1.5963 = L,( gi = 1.0967 = C2, g3 = 1.5963 = L3, ,?4 = 1-000 = RL. Then (8.64) and (8.74) give the impedance-scaled and frequency-transformed element values for the circuit of Figure 8.32 as L[Z0 L\ = C\ = L'2 = 1% a>0A A AZ = 127.0 nH, = 0199pF, plane to the Q. plane, which repeats with a period of t»e/vP = 2ir. This transformation was introduced by P. Richard [6] to synthesize an LC network using open- and short-circuited transmission lines. Thus, if we replace the frequency variable oj with £1, the reactance of an inductor can be written as jXL = j S2L = jL tan Bt, (8.78a) and the susceptance of a capacitor can be written as jBc ~ j&C = jC tan Bt. (S.78b) These results indicate that an inductor can be replaced with a short-circuited stub of length Bt and characteristic impedance L, while a capacitor can be replaced with an open-circuited stub of length ft I and characteristic impedance 1/C, A unity filter impedance is assumed. Cutoff occurs at unity frequency for a low-pass filter prototype; to obtain the same cutoff frequency for the Richard's-transformed filter, (8.77) shows that Q, = 1 = tan/31, which gives a stab length of i = X/8, where X is the wavelength of the line at the cutoff frequency, a>c. At the frequency ^ = 2&jc, the lines will be a/4 long, and an attenuation pole will occur. At frequencies away from a>c, the impedances of the stubs will no longer match the original lumped-element impedances, and the filter response will differ from the desired prototype response. Also, the response will be periodic in frequency, repeating every 4ioc. In principle, then, the inductors and capacitors of a lumped-element filter design can be replaced with short-circuited and open-circuited stubs, as illustrated in Figure 8.34. Since the lengths of all the stubs are the same (X /8 at tac), these lines are cail&dcommensurate lines. Kuroda's Identities The four Kuroda identities use redundant transmission line sections to achieve a more practical microwave filter implementation by performing any of the following operations; • Physically separate transmission line stabs • Transform series stubs into shunt stubs, or vice versa • Change impractical characteristic impedances into more realizable ones 8.5 Filter Implementation 407 A/8 at a)„ o— s,c. (a) A/8 at =}=C j£c^> O.C o D FIGURE 8.34 Richard's transformation, (a) For an inductor to a short-circuited stub, (b) For a capacitor to an open-circuited stub. The additional transmission line sections are called unit elements and are A/8 long at a>c; the unit elements are thus commensurate with the stubs used to implement the inductors and capacitors of the prototype design. The four identities are illustrated in Table 8,7, where each box represents a unit element, or transmission line, of the indicated characteristic impedance and length (A/8 at a>c). The inductors and capacitors represent short-circuit and open-circuit stubs, respectively. We TABLE 8.7 The Four Kuroda Identities (n2 = 1 + Z2/ Zt) 408 Chapter 8: Microwave Filters senes stub Z,lnl Unit element rr = l+Z2tZl FIGURE 8,35 Equivalent circuits illustrating Kuroda identity (a) in Table 8.7. will prove the equivalence of the first case, and then show how to use these identities in Example 8,5. The two circuits of identity (a) in Table 8.7 can be redrawn as shown in Figure 8,35; we will show that these two networks are equivalent by showing that their ABCD matrices are identical. From Table 4.1, the ABCD matrix of a length t of transmission line with characteristic impedance Z\ is A C — sin 31 L Zi co$8t 1 1 jQZi /ft L Z, l (8.79) where Í2 = lanBL Now the open-circuited shunt stub in the first circuit in Figure 8.35 has an impedance of — jZi cot 81 = —jZ2( £2, so the ABCD matrix of the entire circuit is A B C D " 1 0" " 1 jilZ\" - z2 _ L77 1 J I Vi + si2 i VT + £22 jQ, jQZi " -i Z j 1 - Or— ZZJ (8.80a) The short-circuited series stub in the second circuit in Figure 8.35 has an impedance OÍ j(Z]/n2) tan fit = /S2Zi/n2, so ůi&ABCD matrix of the entire circuit is i L Z2 1 Vl + Í22 .flZ2 -| J n2 1 1 0 1 ] n/1 + Í22 1 4r(Zj + Z2) L Z2 1 -Í2 Z2 -I (8.80b) 8.5 fitter Implementation 409 The results in (8.80a) and (8.80b) are identical if we choose n1 = 1 + Z2/Zi ■ The other identities in Table 8.7 can be proved in the same way. EXAMPLE 8.5 LOW-PASS FILTER DESIGN USING STUBS Design a low-pass filter for fabrication using microstrip lines. The specifications are: cutoff frequency of 4 GHz, third order, impedance of 50 fi, and a 3 dB equal-ripple characteristic. Solution From Table 8.4, the normalized low-pass prototype element values are g, = 3.3487 = L|. £3 = 0.7117 = 02, g3 = 3.3487 = L3, g4 = 1.0000 = RL, with the lumped-element circuit shown in Figure 8.36a. 1 L, = 3.3487 L, = 3.3487 m FIGURE 8,36 Filter design procedure for Example 8.5. (a) Lumped-element low-pass filter prototype, (b) Using Richard's trans formations to convert inductors and capacitors to series and shunt stubs, (c) Adding unit elements at ends of filter. Chapters: Microwave Filters so a Z,, = 4.350 Zn = 4,350 m Zi, = 217,511 Zn = 217.5 Í1 / = A/8 at w = 1 50 ň 64.9 n i = m& ai4GHi 50 Q 217.5 a 217.5 a so a FIGURE 8.36 Continued, (d) Applying the second Kuroda identity, (e) After impedance and frequency scaling. <0 Microstrip fabrication of final filter. The next step is to use Richard's transformations to convert series inductors to series stubs, and shunt capacitors to shunt stubs, as shown in Figure 8.36b. According to (8,78), the characteristic impedance of a series stub (inductor) is L, and the characteristic impedance of a shunt stub (capacitor) is 1/C. For commensurate line synthesis, all stubs are k/H long at a> = toc. (It is usually most convenient to work with normalized quantities until the last step in the design.) The series stubs of Figure 8.36b would be very difficult to implement in microstrip form, so we will use one of the Kuroda identities to convert these to shunt stubs. First we add unit elements at either end of the filter, as shown in Figure 8.36c. These redundant elements do not affect filter performance since they are matched to the source and load (Zo = 1). Then we can apply Kuroda identity (b) from Table 8.7 to both ends of the filter. In both cases we have that „2 = 1 + ^ = ] + 3.3487 = 1.299. The result is shown in Figure 8.36d. Finally, we impedance and frequency scale the circuit, which simply involves multiplying the normalized characteristic impedances by 50 £2 and choosing the line and stub lengths to be A./8 at 4 GHz. The final circuit is shown in Figure 8.36e, with a microstrip layout in Figure 8.36f. 8.5 Filter Implementation 411 u) Distributed elements 40 0 Frequency {GHz) FIGURE 8.37 Amplitude responses of lumped-element and distributed-element low-pass filter of The calculated amplitude response of this design is plotted in Figure 8.37, along with the response of the lumped-element version. Note that the passband characteristics are very similar up to 4 GHz, but the distributed-elernent filter has a sharper cutoff. Also notice that the distributed-elernent filter has a response which repeats every 16 GHz, as a result of the periodic nature of Richard's trans-break formation. ■ Similar procedures can be used for bandstop filters, but the Kuroda identities are not useful for high-pass or bandpass filters. Impedance and Admittance Inverters As we have seen, it is often desirable to use only series, or only shunt, elements when implementing a filter with a particular type of transmission line. The Kuroda identities can be used for conversions of this form, but another possibility is to use impedance (AT) or admittance (J) inverters [1], [4], [7J. Such inverters are especially useful for bandpass or bandstop filters with narrow (<10%) bandwidths. The conceptual operation of impedance and admittance inverters is illustrated in Figure 8.38; since these inverters essentially form the inverse of the load impedance or admittance, they can be used to transform series-connected elements to shunt-connected elements, or vice versa. This procedure will be illustrated in later sections for bandpass and bandstop filters. hi its simplest form, a J or K inverter can be constructed using a quarter-wave transformer of the appropriate characteristic impedance, as shown in Figure 8.38b. This implementation also allows the ABCD matrix of the inverter to be easily found from the ABCD parameters for a length of transmission line given in Table 4.1. Many other types of circuits can also be used as J or K inverters, with one such alternative being shown in Figure 8.38c. Inverters of this form turn out to be useful for modeling the coupled resonator filters of Section 8.8. The lengths, 8/2, of the transmission line sections are generally required to be negative for this type of inverter, but this poses no problem if these lines can be absorbed into connecting transmission lines on either side. Example 8.5. 412 Chapters: Microwave Filters Impedance inverters Admittance inverters ■A/4- Za = K -A/4- /: = Zo tan 10/2| x--L e=-tanl P -C (b) (c) (d) -«—9/2 —^ „-*^6f2—> Kb ■o— y=y0taule/2| J_ 1 - TO2 1 2B 0 = -tan -c HO 8.6 FIGURE 838 Impedance and admittance inverters, (a) Operation of impedance and admittance inverters, (b) Implementation as quarter-wave transformers, (c) Implementation using transmission lines and reactive elements, (d) Implementation using capacitor networks. STEPPED-IMPEDANCE LOW-PASS FILTERS A relatively easy way to implement low-pass filters in microstrip or stripline is to use alternating sections of very high and very low characteristic impedance lines. Such filters are usually referred to as stepped-impedance, or hi-2, low-Z filters, and are popular because they are easier to design and take up less space than a similar low-pass filter using stubs. Because of the approximations involved, however, their electrical performance is not as good, so the use of such filters is usually limited to applications where a sharp cutoff is not required (for instance, in rejecting out-of-band mixer products). Approximate Equivalent Circuits for Short Transmission Line Sections We begin by finding the approximate equivalent circuits for a short length of transmission line having either a very large or very small characteristic impedance. The ABCD 8.6 Stepped-lmpedance Low-Pass Filters 413 (a) X=Z$l (b) m FIGURE 8,39 Approximate equivalent circuits for short sections of transmission lines. (a) 7-equivalent circuit for a b^srnissionline section having fit < tt/2. (b) Equivalent circuit for small pi and large Z0. (c) Equivalent circuit for small pi and small Zo. parameters of a length, ť, of line having characteristic impedance Zo are given in Table 4, J j the conversion in Table 4.2 can then be used to find the Z-parameters as Zii = Z22 = — = -;Zc,cot^€, Z12 = Zn = ^ = -jZQcscpe. (8.81a) (8.81b) The series elements of the T-equivalent circuit are 'cosjffť - Zii - Z]2 = -j'Zo sin££ -]= jZ0 tan (^), (8.82) while the shunt element of the 7"-equivalent is Z^. So if pi < 7r/2, the series elements have a positive reactance (inductors), while the shunt element has a negative reactance (capacitor). We thus have the equivalent circuit shown in Figure 8.39a, where X — = Zy tan B = I sin pt. Zo (8.83a) (8.83b) Now assume a short length of line (say pi < x/4) and a large characteristic impedance. Then (8.83) approximately reduces to x - zm, B ~ 0. (8.84a) (8.84b) which implies the equivalent circuit of Figure 8.39b (a series inductor). For a short length of line and a small characteristic impedance, (8.83) approximately reduces to X-0, B ~ Yopt, (8.85a) (8.85b) which implies the equivalent circuit of Figure 8.39c (a shunt capacitor). So the series inductors of a low-pass prototype can be replaced with high-impedance line sections (Zo = Z/,), and the shunt capacitors can be replaced with low-impedance line sections (Zo = Zt). The 414 Chapter 8: Microwave Filters ratio Z,,/Zť should be as high as possible, so the actual values of Z& and Zt are usually set to the highest and lowest characteristic impedance that can be practically fabricated. The lengths of the lines can then be determined from (8.84) and (8.85); to get the best response near cutoff, these lengths should be evaluated at o> — toc. Combining the results of (8,84) and (8.85) with the scaling equations of (8.67) allows the electrical lengths of the inductor sections to be calculated as zh (inductor), and the electrical length of the capacitor sections as CZt Bt = (capacitor), (8.86a) (8.86b) where Ro is the filter impedance and L and C are the normalized element values (the g^s) of the low-pass prototype. EXAMPLE 8.6 STEPPED-EVfPEDANCE FILTER DESIGN Design a stepped-impedance low-pass filter having a maximally flat response and a cutoff frequency of 2.5 GHz. It is necessary to have more than 20 dB insertion loss at 4 GHz. The filter impedance is 50 £2; the highest practical line impedance is 120 Q, and the lowest is 20 £1 Consider the effect of losses when this filter is implemented with a microstrip substrate having d — 0.158 cm, €r = 4.2, tan 5 = 0,02, and copper conductors of 0.5 mil thickness. Solution To use Figure 8,26 we calculate to --1 4.0 --1 = 0.6; 2.5 =F 3 (a) i, l2 % U k O O -O O Z, &8 H fí -O——c- -ř Zj, Zt Zh — ' o o o- (b) FIGURE 8.40 Filter design for Example 8.6. (a) Low-pass filter prototype circuit, (b) Stepped-impedance implementation, (c) Microstrip layout of final filter. 8.6 Stepped-lmpedance Low-Pass Filters 415 then the figure indicates N = 6 should give the required attenuation at 4.0 GHz. Table 8.3 gives the low-pass prototype values as Si = 0.517 = Cu §2 = 1414 = 83 = 1.932 = g* = 1.932 = £4, £5 = 1.414 = S6 = 0.517 = u. The low-pass prototype filter is shown in Figure 8.40a. Next, (8.86a,b) are used to replace the series inductors and shunt capacitors with sections of low-impedance and high-impedance lines. The required electrical line lengths, along with the physical mkroslrip line widths, Wr, and lengths, ti, are given in the table below. Section Zi = Z(. or Zh % (mm) 1 20 n UJf6 11,3 2.05 2 33.S-5 0.428 6.63 3 200 44.3° 11.3 7.69 4 120 £2 46.P 0.428 9.04 5 20 S2 32.4° 11.3 5.63 6 120 Q 123° 0.428 2.41 The final filter circuit is shown in Figure 8,40b, with Zt = 20 ft and Zh = 120 ft. Note that &l < 45° for all but one section. The microstrip layout of the filter is shown in Figure 8.40c. Figure 8.41 shows the calculated amplitude response of the filter, with and without losses. The effect of loss is to increase the passband attenuation to about \ \ - \ >v Lumped element \ X* \ ^ 1 1 ( \ . 0 1.0 2.0 3.0 4.0 5.0 Frequency (GHz) FIGURE 841 Amplitude response of the stepped-impedance low-pass filter of Example 8.6, with (dotted line) and without (solid line) losses. The response of the corresponding lumped-element filter is also shown. 416 Chapters: Microwave Fitters 1 dB at 2 GHz. The response of the corresponding lumped-element filter is also shown in Figure 8.41. The passband characteristic is similar to that of the stepped impedance filter, but the lumped-element filter gives more attenuation at higher frequencies. This is because the stepped-impedance filter elements depart significantly from the lumped-element values at higher frequencies. The stepped-impedance filter may have other pass bands at higher frequencies, but the response will not be perfectly periodic because the lines are not commensurate. ■ COUPLED LINE FILTERS The parallel coupled transmission lines discussed in Section 7.6 (for directional couplers) can also be used to construct many types of filters. Fabrication of multisection bandpass or bandstop coupled line filters is particularly easy in microstrip or stripline form, for band widths less than about 20%. Wider bandwidth filters generally require very tightly coupled lines, which are difficult to fabricate. We will first study the filter characteristics of a single quarter-wave coupled line section, and then show how these sections can be used to design a bandpass filter [7]. Other filter designs using coupled lines can be found in reference [1]. Filter Properties of a Coupled Line Section A parallel coupled line section is shown in Figure 8.42a, with port voltage and current definitions. We will derive the open-circuit impedance matrix for this four-port network by considering the superposition of even- and odd-mode excitations [8], which are shown in Figure 8.42b. Thus, the current sources i\ and £3 drive the line in the even mode, while i 2 and i*4 drive the line in the odd mode. By superposition, we see that the total port currents, I i, can be expressed in terms of the even- and odd-mode currents as /. = h + hr (8.87a) h = ii-i2, (8.87b) l3=h-i4, (8.87c) h = h + (8.87d) First consider the line as being driven in the even mode by the i i current sources. If the other ports are open-circuited, the impedance seen at port 1 or 2 is Zein^-jZ0e cot 8t. (8.88) The voltage on either conductor can be expressed as «J(S) = vl(z) = V+le-™-* + ^-<>] = 2Ve+cos 8(1 -z). (8.89) so the voltage at port 1 or 2 is i>i<0) = 1.1(0) = 2Ve+ cos 81 = hZl. This result and (8.88) can be used to rewrite (8.89) in terms of i \ as ^ = ^z) = -jZ0g^-Z\. (8.90) 8.7 Coupled Line Filters 417 -..... 0» (c) FIGURE 8.42 Definitions pertaining to a coupled line filter section, (a) A parallel coupled line section with port voltage and current definitions, (b) A parallel coupled line section with even- and odd-mode current sources, (c) A two-port coupled line section having a bandpass response, Similarly, the voltages due to current sources t3 driving the line in the even mode are if , im .7 cosßz. va(z) = vb(z) = ~jZ()e —-i3. (8-91) Now consider the line as being driven in the odd mode by current i2. If the other ports are open-circuited, the impedance seen at port 1 or 2 is 2? =-;Z0ocot^. (8.92) The voltage on either conductor can be expressed as vliz) - -v2b(z) = Vgl0)2 esc2 $ - (Zot + Ze*,)2 c°t2 í?- (8.101) When the coupled line section is X/4 long (9 = n/2), the image impedance reduces to Zi — 2^0e — Z|)(,), (8.102) which is real and positive, since Zoe > Zqo- But when 0 —» 0 or jt, Z,- *-* ±joo, indicating a stopband. The real part of the image impedance is sketched in Figure 8.43, where the cutoff frequencies can be found from (8.101) as cos 6*i = —cos 82 = Zfe — Z On Zfe -\- Zoo The propagation constant can also be calculated from the results of Table 8,1 as Zfe + Z00 cos ß — Zu Z33 Zf* hi Zii cos#, (8.103) Z\ie — Zoo which shows 6 is real for $1 < 0 < 02 — n — &\> where cos $1 = (Z^ — Zoo)/(Zfre + Zoo). Design of Coupled Line Bandpass Filters Narrowband bandpass filters can be made with cascaded coupled line sections of the form shown in Figure 8.42c. To derive the design equations for filters of this type, we first show that a single coupled line section can be approximately modeled by the equivalent circuit shown in Figure 8.44. We will do this by calculating the image impedance and propagation constant of the equivalent circuit and showing that they are approximately equal to those 8.7 Coupled Line Filters 421 —o FIGURE 8.44 Equivalent circuit of the coupled line section of Figure 8.42c of the coupled line section for 0 = tt/27 which will correspond to the center frequency of the bandpass response. The ABCD parameters of the equivalent circuit can be computed using the ABCD matrices for transmission lines from Table 4, l: [i s]-[ cosö j'Zosinö = I j sin 6 —- COSČ Zo 0 -J/J 0 cosO ,/Zosinr? j sin# JL Zo 2sin20 - COSÖ COS2f?\ -| j (r—z sin2 $ — J cos2 Ö ) ( 7 Zo + -p— J sin 8 cos 0 (8.104) The ABCD parameters of the admittance inverter were obtained by considering it as a quarter-wave length of transmission of characteristic impedance, 1/7. From (8.27) the image impedance of the equivalent circuit is t/ZSsfaTO -(1/7) cos2 # (l/7Z2)sin:0 - 7 cos2 ft' which reduces to the following value at the center frequency, 6 = jt/2: Zf = 7 Z\. From (8,31) the propagation constant is cosB = A — \JZ(, + -y=-\ sin0cos6. (8.105) (8.106) (8.107) Equaling the image impedances in (8.102) and (8.106), and the propagation constants of (8.103) and (8.107), yields the following equations: —(Zo,? — Zoo) — J Zq, Zde + Z (to Z(je — Z = 7Z0-f where we have assumed sin # ~ 1 for 8 near jt/2, These equations can be solved for the even- and odd-mode line impedances to give Z0e = Z0[1 + 7Z0 + (7Z0)2]. Zto = Z0[l - 7Z0 + (7Z0)2], (8.108a) (8,108b) Now consider a bandpass filter composed of a cascade of /V + 1 coupled line sections, as shown in Figure 8.45a. The sections are numbered from left to right, with the load on the 422 Chapter 8: Microwave Filters Z(v» Zoo -1 N (a) N+ 1 to -90u * —>- Z() Zq -90* Zf> Z0 20 (b) -/Z0 cot B -jZ0 cot 0 1 : - 1 °—WW—|-WW-1 i— => sin IS -o -o (c) (d) 1 : /Z0 *-A/4 • o-i i-0 d Z0/, : 1 1 : Z„J, y2 z0 (JV = 2) (e) L'2 C'j -j|-< Z0 tt/2 the series arm impedances of (8.111) are near zero, and can also be ignored, The shunt impedance Zn, however, looks like the impedance of a parallel resonant circuit for $ ^ jt/2. If we let to = too + Aw, where $ = x/2 at the center frequency wq, then we have 26 = 01 = a>£/vp = (too + Ato)Tr/tOQ = 7TÍ1 + Ato/woX so (8.110a) can be written for small Ato as Zi2 = yZo -jZfjtOQ sinjr(l + Aúí/ťtírj) it (to — too) From Section 6.1 the impedance near resonance of a parallel LC circuit is -jLtol Z = 2((o - ú>o) ' (8.112) (8,113) with to2, = 1 / LC. Equating this to (8.112) gives the equivalent inductor and capacitor values as L = C = 2Z0 ■If <4L 2Zq(ůq (8.114a) (8.114b) The end sections of the circuit of Figure 8.45b require a different treatment. The lines of length B on either end of the filter are matched to Zo, and so can be ignored. The end inverters, J\ and /jv+i, can each be represented as a transformer followed by a X/4 section 424 Chapter 8: Microwave Fitters of line, as shown in Figure 8.45d. The ABCD matrix of a transformer with a turns ratio N in cascade with a quarter-wave line is -JZo 1 o L Z0 N 0 (8,115) Comparing this to the ABCD matrix of an admittance inverter (part of (8.104)) shows that the necessary turns ratio is N = JZo. The A./4 line merely produces a phase shift, and so can be ignored. Using these results for the interior and end sections allows the circuit of Figure 8.45b to be transformed into the circuit of Figure 8.45e, which is specialized to the N = 2 case. We see that each pair of coupled line sections leads to an equivalent shunt LC resonator, and an admittance inverter occurs between each pair of LC resonators. Next, we show that the admittance inverters have the effect of transforming a shunt LC resonator into a series LC resonator, leading to the final equivalent circuit of Figure 8.45f (shown for N — 2). This will then allow the admittance inverter constants, Jn, to be determined from the element values of a low-pass prototype. We will demonstrate this for the N = 2 case. With reference to Figure 8.45e, the admittance just to the right of the J% inverter is j<0c2 + J- + z*H = jM + ZoJl jtt>L2 y L2 \tOo to / since the transformer scales the load admittance by the square of the rums ratio. Then the admittance seen at the input of the filter is 4 jVCyL^ [ico/co0) - (m/co)] + ZoJf 3} f272 J*JC2/L2 [(co/coo) - (ojo/co)] + Zott (8.116) These results also use the fact, from (8.114), that L„C„ = 1 /a>\ for all LC resonators. Now the admittance seen looking into the circuit of Figure 8.45f is Y = jcoC[ + = J jcoL\ j(oL'2 + \/}0)C2 + Zo 1 {(co/m) - (too/to)] + Zo (8.117) which is identical in form to (8.116). Thus, the two circuits will be equivalent if the following conditions are met: (8.118a) J2Z312 Ji ^0 J3 J2 = Zo. (8.118b) (8.118c) 8.7 Coupled Line Filters 425 We know l„ and C„ from (8.114); l'n and C'n are determined from the element values of a lumped-element low-pass prototype which has been impedance scaled and frequency transformed to a bandpass filter. Using the results in Table 8.6 and the impedance scaling formulas of (8.64) allows the l'n and C'n values to be written as L; = ^, (8.119a) C\ = , (8.119b) IÍ = SSí, (8.119c) A coo CI--~, (8.119d) coagiZo where A = {coi — co\)/coo is the fractional bandwidth of the filter. Then (8.118) can be solved for the inverter constants with the following results (for N = 2): 7/c2c;\l/4 tta hz^hzl l-r-rr) =^1=' (8-120b) J\ v 2#2 After the i„s are found, Zo> and Z&, for each coupled line section can be calculated from (8.108). The above results were derived for the special case of A' = 2 (three coupled fine sections), but more general results can be derived for any number of sections, and for the case where Zi ^ Zo (or gN+\ 1, as in the case of an equal-ripple response with N even). Thus, the design equations for a bandpass filter with N + 1 coupled line sections are ■A Zq = (8.121a) for «=2,3.....(8.121b) (8.121c) The even and odd mode characteristic impedances for each section are then found from (8.108). EXAMPLE 8.7 COUPLED LINE BANDPASS FILTER DESIGN Design a coupled line bandpass filter with N = 3 and a 0.5 dB equal-ripple response. The center frequency is 2,0 GHz, the bandwidth is 10%, and Zo = 50 Q. What is the attenuation at 1.8 GHz? 426 Chapter 8: Microwave Filters Solution The fractional bandwidth is A = 0.1. We can use Figure 8,27a to obtain the attenuation at 1.8 GHz, but first we must use (8.71) to convert this frequency to the normalized low-pass form (a>c = 1): £ff «|\ _L (11 = _211 W *~ A \o>q to) 0.1 1,2.0 1.8) Then the value on the horizontal scale of Figure 8.27a is - 1 = 1-2.111 - J = 1.11, which indicates an attenuation of about 20 dB for N = 3. The low-pass prototype values, gn, are given in Table 8.4; then (8.121) can be used to calculate the admittance inverter constants, /„. Finally, the even- and odd-mode characteristic impedances can be found from (8.108). These results are summarized in the following table: " S; Z()7n Zfe^) Ztieitt) 1 1.5963 0.3137 70.61 39.24 2 1.0967 0.1187 56.64 44 77 3 1.5963 0.1187 56.64 44,77 4 1.0000 0.3137 70.61 39.24 Note that the filter sections are symmetric about the midpoint. The calculated response of this filler is shown in Figure 8.46; passbands also occur at 6 GHz, 10 GHz, etc. Many other types of filters can be constructed using coupled line sections; most of these are of the bandpass or bandstop variety. One particularly compact design is the interdigitated filter, which can be obtained from a coupled line filter by folding the lines at their midpoints; see [1] and [3] for details. ■ 10 1.5 2.0 2.5 3.0 Frequency (GHz) FIGURE 8.46" Amplitude response of the coupled line bandpass filter of Example 8.7. 8.8 Filters Using Coupled Resonators 427 FILTERS USING COUPLED RESONATORS We have seen that bandpass and bandstop filters require elements that behave as series or parallel resonant circuits; the coupled line bandpass filters of the previous section were of this type. Here we will consider several other types of microwave filters that use transmission line or cavity resonators. Bandstop and Bandpass Filters Using Quarter-Wave Resonators From Chapter 6 we know that quarter-wave open-circuited or short-circuited transmission line stubs look like series or parallel resonant circuits, respectively. Thus we can use such stubs in shunt along a transmission line to implement bandpass or bandstop filters, as shown in Figure 8.47. Quarter-wavelength sections of line between the stubs act as admittance inverters to effectively convert alternate shunt resonators to series resonators. The stubs and the transmission line sections are X/4 long at the center frequency, ojq. For narrow bandwidths the response of such a filter using N stubs is essentially the same as that of a coupled line filter using N + 1 sections. The internal impedance of the stub filter is Zq, while in the case of the coupled line filter end sections are required to transform the impedance level. This makes the stub filter more compact and easier to design. A disadvantage, however, is that a filter using stub resonators often requires characteristic impedances that are difficult to realize in practice. We first consider a bandstop filter using N open-circuited stubs, as shown in Figure 8.47a. The design equations for the required stub characteristic impedances, Zq„, will be derived in terms of the element values of a low-pass prototype through the use of an equivalent circuit. The analysis of the bandpass version, using short-circuited stubs, follows the same procedure so the design equations for this case are presented without detailed derivation. As indicated in Figure 8.48a, an open-circuited stub can be approximated as a series LC resonator when its length is near 90c. The input impedance of an open-circuited transmission FIGURE 8.47 Bandstop and bandpass filters using shunt iransmission line resonators (# = jt/2 at the center frequency), (a) Bandstop filter, (b) Bandpass filter. 428 Chapters: Microwave Filters b - irf2 at iu = u>a (a) (b> 1} G FIGURE 8.48 Equivalent circuit for the bandstop filter of Figure 8.47a. (a) Equivalent circuit of open-circuited stub for# near jt/2. = = << o>o, then 0 = tt/2(1 + Atfj/ťUo), and this impedance can be approximated as Z = ;Z0fr tan ti A o) j Za,, n (o> — iOii) 2(tiQ 2(ůq (8.122) for frequencies in the vicinity of the center frequency, ťWo. The impedance of a series LC circuit is Z = jcoLn + ——- = ;, —---os 2y /---- 3?Zj*(# - ttíg), (8.123) where L„Ctl = 1/itfjj. Equating (8.122) and (8.123) gives the characteristic impedance of the stub in terms of the resonator parameters: Zon — 4&?qL,i (8.124) 8.8 Filters Using Coupled Resonators 429 Then, if we consider the quarter-wave sections of line between the stubs as ideal admittance inverters, the band stop filter of Figure 8.47a can be represented by the equivalent circuit of Figure 8.48b. Next, the circuit elements of this equivalent circuit can be related to those of the lumped-element bandstop filter prototype of Figure 8.48c. With reference to Figure 8.48b, the admittance, Y, seen looking toward the L2C2 resonator is Y = 1 ML2 + a/jcoC2) 1 + -I— z\ + 1 i + l/;'ü>Ci Zn 1 1 + — Zq \ j^/Li/c] [(a}/(Oo) - (wo/a})] The admittance at the corresponding point in the circuit of Figure 8.48c is -i-i (8.125) Y = 1 1 ffrLi + 1 /ja>C2 ljidC[ +1 iJioL\ 1 T+Zo jy/L'2/c2 limfw) - froM) i 4- —j These two results will be equivalent if the following conditions are satisfied: (8.126) (8.127a) (8.127b) Since LnCn = VnC'n = l/a>o, these results can be solved for L„: Z2 1,- ° w2L\ L2 = L'2. (8.128a) (8.128b) Then using (8.124) and the impedance-scaled bandstop filter elements from Table 8.6 gives the stub characteristic impedances as Zoi = Z{i2 = 4Zg 4Z0 4(tíoL2 4Zo (8.129a) (8,129b) 7t ng2A where a = (a^ — a>} )/(Oo is the fractional bandwidth of the filter. It is easy to show that the general result for the characteristic impedances of a bandstop filter is 4Z0 Zon = JTSnA' (8.130) 430 ChapterS; Microwave Fitters For a bandpass filter using short-circuited stub resonators the corresponding result is Zo* = ^. (8.131) These results only apply to fillers having input and output impedances of Z<>, and so cannot be used for equal-ripple designs with N even. EXAMPLE 8.8 BANDSTOP FILTER DESIGN Design a bandstop filter using three quarter-wave open-circuit stubs. The center frequency is 2.0 GHz, the bandwidth is 15%, and the impedance is 50 Q. Use an equal-ripple response, with a 0.5 dB ripple level. Solution The fractional bandwidth is A = 0.15. Table 8.4 gives the low-pass prototype values, gn, for N — 3. Then the characteristic impedances of the stubs can be found from (8.130). The results are listed in the following table: a l 1.5963 265.9 £2 2 1.0967 387.0 a 3 1.5963 265.9 £2 The filter circuit is shown in Figure 8.47a, with all stubs and transmission line sections k/4 long at 2.0 GHz. The calculated attenuation for this filter is shown in Figure 8.49; the ripple in the passbands is somewhat greater than 0.5 dB, as a result of the approximations involved in the development of the design equations. ■ The performance of quarter-wave resonator filters can be improved by allowing the characteristic impedances of the interconnecting lines to be variable; then an exact correspondence with coupled line bandpass or bandstop filters can be demonstrated. Design details for this case can be found in reference [1]. Frequency (GHz) FIGURE 8.49 Amplitude response of the bandstop filter of Example 8,8, i jj / * *5 (a) 8,8 Filters Using Coupled Resonators 431 o\\o ---——&-| (-0 ßti a ßn+l z% At -0-c-o—<3 0-0- Z[) Zq -o - o—o ff) c—o-||-o—o ßv*l 1-1 * S /w/f t 1 y £ —o—T—ojjo 6i o— 2o 2a Zfl Zy --o o—h3 0 ■ FIGURE 8.50 Development of the equivalence of a capacitive-gap coupled resonator bandpass filter to the coupled line bandpass filter of Figure 8.45, (a) The capacitive-gap coupled resonator bandpass filter, (b) Transmission line model, (c) Transmission line model with negative-length sections forming admittance inverters ( = n at cuo). This circuit is now identical in form with the coupled line bandpass filter equivalent circuit in Figure 8.45b. Bandpass Filters Using Capacitively Coupled Series Resonators Another type of bandpass filter that can be conveniently fabricated in microstrip or stripline form is the capacitive-gap coupled resonator filter shown in Figure 8.50, An jVth order filter of this form will use N resonant series sections of transmission line with N 4- 1 capacitive gaps between them. These gaps can be approximated as series capacitors; design data relating the capacitance to the gap size and transmission line parameters is given in graphical form in reference [1]. The filter can then be modeled as shown in Figure 8.50b. The resonators are approximately A/2 long at the center frequency, too. Next, we redraw the equivalent circuit of Figure 8.50b with negative-length transmission line sections on either side of the series capacitors. The lines of length 0 will be X/2 long at too, so the electrical length, 6it of the *th section in Figures 8.50a,b is 0, = jt + ^0, + , for i = 1.2,.... N. (8.132) with tj>i < 0. The reason for doing this is that the combination of series capacitor and negative-length transmission lines forms the equivalent circuit of an admittance inverter, as seen from Figure 8.38c. In order for this equivalence to be valid, the following relationship 432 Chapter 8: Microwave Filters must hold between the electrical length of the lines and the capacitive susceptance: & = -tan-1(2Z()B;). (8.133) Then the resulting inverter constant can be related to the capacitive susceptance as (These results are given in Figure 8.38, and their derivation is requested in Problem 8.15.) The capacitive-gap coupled filter can then be modeled as shown in Figure 8.50d. Now consider the equivalent circuit shown in Figure 8.45b for a coupled line bandpass filter Since these two circuits are identical (as tj> = 26 = n at the center frequency), we can use the results from the coupled line filter analysis to complete the present problem. Thus, we can use (8.121) to find the admittance inverter constants, J*, from the low-pass prototype values (gi) and the fractional bandwidth, A. As in the case of the coupled line filter, there will be N 4- 1 inverter constants for an /Vth order filter. Then (8.134) can be used to find the susceptance, B,, for the ith coupling gap. Finally, the electrical length of the resonator sections can be found from (8.132) and (8.133): d,:= 7T - i[tan-1(2Z0S,) + tan-l(2Z0SJ+1)]. (8.135) 2 EXAMPLE 8.9 CAPACITTVELY COUPLED SERIES RESONATOR BANDPASS FILTER DESIGN Design a bandpass filter using capacitive coupled series resonators, with a 0.5 dB equal-ripple passband characteristic. The center frequency is 2.0 GHz, the bandwidth is 10%, and the impedance is 50 Í2, At least 20 dB of attenuation is required at 2.2 GHz. Solution We first determine the order of the filter to satisfy the attenuation specification at 2.2 GHz. Using (8.71) to convert to normalized frequency gives 1_/^_^\=J_/212_210\ ]9] A \(Uo to j 0.Í \2.0 2.2/ Then. -1 = 1.91-1.0 = 0.91. From Figure 8.27a, we see that N = 3 should satisfy the attenuation specification at 2.2 GHz. The low-pass prototype values are given in Table 8.4, from which the inverter constants can be calculated using (8.121). Then the coupling susceptances can be found from (8.134), and the coupling capacitor values as C = ^ Finally, the resonator lengths can be calculated from (8.135). The following table summarizes these results. 8.8 Fitters Using Coupled Resonators 433 o 10 1 20 S •a 1 | 30 40 "1.0 1.5 2.0 2.5 3.0 Frequency (GHz) FIGURE 8.51 Amplitude response for the capachive-gap coupled series resonator bandpass filter of Example 8.10. n gn Zo J„ cn 0» 1 1.5963 0.3137 6.96 x 10~3 0.554 pF 155.8" 2 1.0967 0.1187 2.41 x 10~;t 0.192 pF 166.5" 3 1.5963 0.1187 2.41 x 10j3 0,192 pF 155.8-= 4 1.0000 0,3137 6.96 x 10"3 0.554 pF — The calculated amplitude response is plotted in Figure 8,51. The specifications of this filter are the same as the coupled line bandpass filter of Example 8.8, and comparison of the results in Figures 8.51 and 8.46 shows that the responses are identical near the passband region. ■ Bandpass Filters Using Capacitively Coupled Shunt Resonators A related type of bandpass filter is shown in Figure 8.52, where short-circuited shunt resonators are capacitively coupled with series capacitors. An Mh order filter will use N stubs, which are slightly shorter than a /4 at the filter center frequency. The short-circuited stub resonators can be made from sections of coaxial line using ceramic materials having very high dielectric constant and low loss, resulting in a very compact design even at UHF frequencies [9]. Such filters are often referred to as ceramic resonator filters, and are FIGURE 8.52 A bandpass tiller using capacitively coupled shunt stub resonators. 434 Chapters: Microwave Filters presently the most common type of RF bandpass filter used in portable wireless systems. Virtually every modern cellular/PCS telephone, wireless LAN, and GPS receiver employs between two and four of these filters. Operation and design of this filter can be understood by beginning with the general bandpass filter circuit of Figure 8.53a, where shunt LC resonators alternate with admittance inverters. As in the case of previous coupled resonator bandpass and bandstop filters, the function of the admittance inverters is to convert alternate shunt resonators to series resonators; the extra inverters at the ends serve to scale the impedance level of the filter to a realistic level. Using an analysis similar to that used for the bandstop filter, the admittance inverter constants can be derived as Z0jol- —. (8.136a) til Z0yh.rt+, = , (8.136b) 4V£o£n+l 71A Z0./*f,;v+i = ,/--. (8.136c) 4g/v£*+i Similarly, the coupling capacitor values can be found as Cm :-- 701 (8.137a) CWi = &Af (S.137b) -n.n+l — Jn.n+\ oJo\/l — (Zn./jv,/v+i)2 (8.137c) Note that the end capacitors are treated differently than the internal elements. Now replace the admittance inverters of Figure 8.53a with the equivalent iz-network of Figure 8.38d, to produce the equivalent Jumped-element circuit shown in Figure 8.53b. Note that the shunt capacitors of the admittance inverter circuits are negative, but these elements combine in parallel with the larger capacitor of the LC resonator to yield a positive capacitance value. The resulting circuit is shown in Figure 8.53c, where the effective resonator capacitor values are given by C'n=C„ + AC, = - C„-,.„ - C,„+l, (8.138) where AC„ = — Ch-\t„ — C,iiA+\ represents the change in the resonator capacitance caused by the parallel addition of the inverter elements. Finally, the shunt LC resonators of Figure 8.53c are replaced with short-circuited transmission stubs, as in the circuit of Figure 8.52. Note that the resonant frequency of the stub resonators is no longer oio, since the resonator capacitor values have been modified •iV,JV+l 1! u ---o oo bo FIGURE 8.53 Equivalent circuit for the bandpass filter of Figure 8.52. (a) A general bandpass filter circuit using shunt resonators with admittance inverters, (b) Replacement of admittance inverters with the circuit implementation of Figure 8.38d. (c) After combining shunt capacitor elements, (d) Change in resonant stub length caused by a shunt capacitor. C w 5" to 9 c -o o. a I 3 Chapter 8: Microwave Filters by the ACns. This implies that the length of the resonator is less than k/4 at ton, the filter center frequency. The transformation of the stub length to account for the change in capacitance is illustrated in Figure 8,53d. A short-circuited length of line with a shunt capacitor at its input has an input admittance of y=.yl + jtooC, (8.139a) where YL = -^cot 6t. If die capacitor is replaced with a short length, At, of transmission line, the input admittance would be yl + j~lanBAt y = ~ n—--= n + J ?F ■ <»-i39b) Z°^ + jTitan^A€ Z° The last approximation follows for BAl c - 1 = 1-3.6671 - 1 = 2.667, from which we find the attenuation as 35 dB. Next we calculate the admittance inverter constants and coupling capacitor values using (8.136) and (8.137): ii C„-i.„ (pF) 1 1.5963 Zo/oi = 0.2218 C0, = 0.2896 2 1.0967 Zo/,2 = 0.0594 C,2 = 0.0756 3 1.5963 Z()J2i = 0.0594 Ca = 0.0756 4 1.0000 ZoJiA = 0.2218 Cu = 0.2896 Then we use (8.138), (8.140), and (8.141) to find the required resonator lengths: AC„ (pF) 1 2 3 -0.3652 -0.1512 -0.3652 -0.04565 -0.0189 -0.04565 73.6° S3.2e 73.6U Note that the resonator lengths are slightly less than 90° (X/4). The calculated amplitude response of this design is shown in Figure 8.54. The stopband rolloff at high frequencies is less than at lower frequencies, and the attenuation at 3 GHz is seen to be about 30 dB, while our calculated value for a canonical lumped-element bandpass filter was 35 dB. ■ L00 1.50 2 00 2.50 3.00 3.50 4.00 Frequency (GHz) FIGURE 8.54 Amplitude response of the capacitively coupled shunt resonator bandpass filter of Example 8.10. 438 Chapters: Microwave Filters REFERENCES [1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Artech House, Dedham, Mass., 1980. [21 R. E, Collin, Foundations for Microwave Engineering, Second Edition, McGraw-Hill, N.Y., 1992, [31 J. A. G. Malherbe, Microwave Transmission Line Filters, Artech House, Dedham, Mass., 1979. [4J W. A. Davis, Microwave Semiconductor Circuit Design, Van Nostrand Remhold, N.Y., 1984. [5] R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, N.Y., 1961. [6] P. I. Richard, "Resistor-Transmission Line Circuits," Proc, of the IRE, vol. 36, pp. 217-220, February 1948. [7] S. B. Coho, "Parallel-Coupled Transmission-Line-Resonator Filters " IRE Trans, Microwave Theory and Techniques, vol. MTT-6, pp. 223-231, April 1958. [8] E. M. T. Jones and J. T. Bolljahn, "Coupled-Strip-Transmission Line Filters and Directional Couplers," IRE Trans. Microwave Theory and Techniques, vol. M'lT-4, pp. 78-81, April 1956. [9] M. Sagawa, M. Makimoto, and S. Yamashita, "A Design Method of Bandpass Filters Using Dielectric-Filled Coaxial Resonators," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-33, pp. 152-157, February 1985. PROBLEMS 8.1 Consider the finite periodic structure shown below, consisting of eight 80 Q. resistors spaced at intervals of A/2 along a transmission line with Z0 = 50 £2. Find the voltage V(z) along the line, and plot | V(z)\ versus z- ion n 0.5 1.0 1.5 2,0 2.5 3,0 Xi ilk 8.2 Sketch the k-fi diagram for the infinite periodic structure shown below. Assume Z0 = 100 fi, d = 1.0 cm, k = fco, and i0 = 3.0 nH. 8-3 Verify the expression for the image impedance of a it -network given in Table 8.1, 8.4 Compute the image impedances and propagation factor for the network shown below. 8.5 Design a composite low-pass filter by the image parameter method with the following specifications: Ra — 50 Q, fc = 50 MHz, and ws 52 MHz. Use CAD to plot the insertion loss versus frequency. Problems 439 8.6 Design a composite high-pass filter by the image parameter method with die following specifications: Rq = 75 £2, fc = 50 MHz. and /M =■ 48 MHz. Use CAD to plot the insertion Joss versus frequency. 8.7 Solve the design equations in Section 8.3 for the elements of an N = 2 equaJ-ripple filter if the ripple specification is 1.0 dB. 8.8 Design a low-pass maximally fiat lumped-element filter having a passband of 0 to 3 GHz, and an attenuation of 20 dB at 5 GHz. The characteristic impedance is 75 Q. Use CAD to plot the insertion loss versus frequency. 8.9 Design a five-section high-pass lumped-element filter with a 3 dB equal-ripple response, a cutoff frequency of 1 GHz, and an impedance of 50 Q. What is the resulting attenuation at 06 GHz? Use CAD to plot the insertion loss versus frequency. 8.10 Design a four-section bandpass lumped-element filter having a maximally flat group delay response. The bandwidth should be 5% with a center frequency of 2 GHz. The impedance is 50 £2. Use CAD to plot the insertion loss versus frequency. 8.11 Design a three-section bandstop lumped-element filter with a 0.5 dB equal-ripple response, a bandwidth of 10% centered at 3 GHz, and an impedance of 75 £2. What is the resulting attenuation at 3.1 GHz'? Use CAD to plot the insertion loss versus frequency. 8.12 Verify the second Kuroda identity in Table 8.7 by calculating the ABCD matrices for both circuits. 8.13 Design a low-pass third-order maximally flat filter using only series stubs. The cutoff frequency is 6 GHz and the impedance is 50 £2. Use CAD to plot the insertion loss versus frequency. 8.14 Design a low-pass fourth-order maximally flat filter using only shunt stubs. The cutoff frequency is 8 GHz and the impedance is 50 £2. Use CAD to plot die insertion loss versus frequency. 8.15 Verify the operation of the admittance inverter of Figure S.38c by calculating its ABCD matrix and comparing it to the ABCD matrix of the admittance inverter made from a quarter-wave line. 8.16 Show that the n equivalent circuit for a short length of transmission line leads to equivalent circuits identical to those in Figure 8.39b and c, for large and small characteristic impedance, respectively. 8.17 Design a stepped-impedance low-pass filter having a cutoff frequency of 3 GHz, and a filth-order 0,5 dB equal-ripple response. Assume Rr, = 50 £2, Zt — 1 5 £2, and Zh = 120 £2. (a) Find the required electrical lengths of the five sections, and use CAD to plot the insertion loss from 0 to 6 GHz. (b) Lay out the microstrip implementation of die filter on an FR4 substrate having (r = 4.2, d = 0079 cm, tan $ = 0.02, with copper conductors 0.5 mil thick. Use CAD to plot the insertion loss versus frequency in the passband of the filter, and compare with the lossless case. 8.18 Design a stepped-impedance low-pass filter with fc = 2.0 GHz and R$ — 50£2, using the exact transmission line equivalent circuit of Figure 8,39a. Assume a maximally flat jV = 5 response, and solve for the necessary line lengths and impedances if Z( = 10 £2 and Zft s= 150 £2. Use CAD to plot the insertion loss versus frequency. 8.19 Design a four-section coupled line bandpass filter with a 0.5 dB equal ripple response. The center frequency is 2.45 GHz, the bandwidth is 10%, and the impedance is 50 £3. (a) Find the required even and odd mode impedances of the coupled line sections, and calculate the expected attenuation at 2.1 GHz, Use CAD to plot the insertion loss from 1,55 to 3.35 GHz. (b) Lay out the microstrip implementation of the filter on an FR4 substrate having i, = 4.2, d = 0.158 cm, tan S = 0.01, with copper conductors 0.5 mil thick. Use CAD to plot the insertion loss versus frequency in the passband of the filter, and compare with the lossless case. 8.20 Design a maximally flat bandstop filter using four open-circuited quarter-wave stub resonators. The center frequency is 3 GHz, the bandwidth is 15%, and the impedance is 40 £2. Use CAD to plot the insertion loss versus frequency. 8.21 Design a bandpass filter using three quarter-wave short-circuited stub resonators. The filter should have a 0.5 dB equal-ripple response, a center-frequency of 3 GHz, a 20% bandwidth, and an impedance of 100 £2. (a) Find the required characteristic impedances of the resonators, and use CAD to plot the insertion loss from 1 to 5 GHz. (b) Lay out the microstrip implementation of the filter on an FR4 substrate having er — 4.2, d = 0.079 cm, tan 5 = 0.02, with copper conductors 0.5 mil thick. Use 440 Chapter 8: Microwave Filters CAD to plot the insertion loss versus frequency in the passband of the filter, and compare with the lossless case. 8.22 Derive the design equation of (8-131) for bandpass filters using quarter-wave shorted stub resonators. 8.23 Design a bandpass filter using capacitive-gap coupled resonators. The response should be maximally flat, with a center frequency of 4 GHz, a bandwidth of 12%, and at least 12 dB attenuation at 3.6 GHz. The characteristic impedance is 50 £2. Find the electrical line lengths and the coupling capacitor values. Use CAD to plot the insertion loss versus frequency. 8.24 A bandpass filter is to be used in a PCS receiver operating in the 824-849 MHz band, and must provide at least 30 dB isolation at the lowest end of the transmit frequency band (869-894 MHz). Design a 1 dB equal-ripple bandpass filter meeting these specifications using capacitively coupled short-circuited shunt stub resonators. Assume an impedance of 50 £2. 8.25 Derive the design equations of (8.136} and (8.137) for the capacitively coupled shunt stub resonator bandpass filter. Chapter Nine Theory and Design of Ferrimagnetic Components The components and networks discussed up to this point have all been reciprocal. That is, the response between any two ports, i and j, of a component did not depend on the direction of signal flow (thus, Sjj = Sji). This will always be the case when the component consists of passive and isotropic material, but if anisotropic (different properties in different directions) materials are used, nonreciprocal behavior can be obtained. This allows the implementation of a wide variety of devices having directional properties. In Chapter 1 we discussed materials with electric anisotropy (tensor permittivity), and magnetic anisotropy (tensor permeability). The most practical anisotropic materials for microwave applications are ferrimagnetic compounds such as YIG (yttrium iron garnet), and ferrites composed of iron oxides and various other elements such as alurninum, cobalt, manganese, and nickel. In contrast to ferromagnetic materials (e.g., iron, steel), ferrimagnetic compounds have high resistivity and a significant amount of anisotropy at microwave frequencies. As we will see, the magnetic anisotropy of a ferrimagnetic materia] is actually induced by applying a DC magnetic bias field. This field aligns the magnetic dipoles in the ferrite material to produce a net (nonzero) magnetic dipole moment, and causes the magnetic dipoles to process at a frequency controlled by the strength of the bias field. A microwave signal circularly polarized in the same direction as this precession will interact strongly with the dipole moments, while an oppositely polarized field will interact less strongly. Since, for a given direction of rotation, the sense of polarization changes with the direction of propagation, a microwave signal will propagate through a ferrite differently in different directions. This effect can be utilized to fabricate directional devices such as isolators, circulators, and gyrators. Another useful characteristic of ferrimagnetic materials is that the interaction with an applied microwave signal can be controlled by adjusting the strength of the bias field. This effect leads to a variety of control devices such as phase shifters, switches, and tunable resonators and filters. We will begin by considering the microscopic behavior of a ferrimagnetic material and its interaction with a microwave signal to derive the permeability tensor. This macroscopic description of the material can then be used with Maxwell's equations to analyze wave propagation in an infinite ferrite medium, and in a ferrite-loaded waveguide. These canonical problems will illustrate the nonreciprocal propagation properties of ferrimagnetic materials, including 441 442 Chapter 9: Theory and Design of Ferrimagnetic Components Faraday rotation and birefringence effects, and will be used in later sections when discussing the operation and design of waveguide phase shifters and isolators. 9.1 BASIC PROPERTIES OF FERRIMAGNETIC MATERIALS In this section we will show how the permeability tensor for a ferrimagnetic material can be deduced from a relatively simple microscopic view of the atom. We will also discuss how loss affects the permeability tensor, and the demagnetization field inside a finite-sized piece of ferrite. The Permeability Tensor The magnetic properties of a material are due to the existence of magnetic dipole moments, which arise primarily from electron spin. From quantum mechanical considerations [1], the magnetic dipole moment of an electron due to its spin is given by m = ^-= 9.27 x 10"24 A-nv\ (9.1) 2me where H is Planck's constant divided by 2tt, q is the electron charge, and me is the mass of the electron. An electron in orbit around a nucleus gives rise to an effective current loop, and thus an additional magnetic moment, but this effect is generally insignificant compared to the magnetic moment due to spin. TheLand^ g factor is a measure of the relative contributions of the orbital moment and the spin moment to the total magnetic moment; $ = 1 when the moment is due only to orbital motion, and g = 2 when the moment is due only to spin. For most microwave ferrite materials, g is in the range of 1.98 to 2.01, so g = 2 is a good approximation. In most solids, electron spins occur in pairs with opposite signs so the overall magnetic moment is negligible. In a magnetic material, however, a large fraction of the electron spins are unpaired (more left-hand spins than right-hand spins, or vice versa), but are generally oriented in random directions so that the net magnetic moment is still small. An external magnetic field, however, can cause the dipole moments to align in the same direction to produce a large overall magnetic moment. The existence of exchange forces can keep adjacent electron spins aligned after the external field is removed; the material is then said to be permanently magnetized. An electron has a spin angular momentum given in terms of Planck's constant as [1], [2] *•-£ <9-2> The vector direction of this momentum is opposite the direction of the spin magnetic dipole moment, as indicated in Figure 9.1. The ratio of the spin magnetic moment to the spin angular momentum is a constant called the gyromagnetic ratio: >'=- = — = 1.759 x 1011 C/Kg, (9.3) 9.1 Basic Properties of Ferrimagnetic Materials 443 Spinning electron FIGURE 9.1 Spin magnetic dipole moment and angular momentum vectors for a spinning electron. where (9.1) and (9.2) have been used. Then we can write the following vector relation between the magnetic moment and the angular momentum; m = —ys, (9.4) where the negative sign is due to the fact that these vectors are oppositely directed. When a magnetic bias field //n = zH$ is present, a torque will be exerted on the magnetic dipole: 7" = m x B{\ = fiQm x Hq = — ixoys x Ha. Since torque is equal to the time rate of change of angular momentum, we have ds —1 dm — =--— = T = Horn x tf0, dt y di (9.5) or dm -a dt (9.6) This is the equation of motion for the magnetic dipole moment, m. We will solve this equation to show that the magnetic dipole precesses around the rVo'ficld vector, as a spinning top precesses around a vertical axis. Wridng (9.6) in terms of its three vector components gives dmx dt dm dt dm. dt = 0. (9,7a) (9.7b) (9.7c) 444 Chapter 9: Theory and Design of Ferrimagoetic Components Now use (9.7a,b) to obtain two equations for mx and my: ~- -f afyo* = 0, (9.8a) (9.8b) where cuo = poyHa (9.9) is called the Larmor, or precession, frequency. One solution to (9.8) that is compatible with (9.7a,b) is given by mx —■ A cos (Dor, (9.10a) my — A sin ojo?, (9.10b) Equation (9.7c) shows that mz is a constant, and (9.1) shows that the magnitude of m is also a constant, so we have the relation that '"Z|2= (It) = ml + ml+ml = 42 + mJ. (9.11) Thus the precession angle, 6, between m and Ho (the z-axis) is given by Jml + ml A sinfJ = -^-—-= —. (9J2) |m| M The projection of m on the xy plane is given by (9.10), which shows that m traces a circular path in this plane. The position of this projection at time / is given by $ = toot, so the angular rate of rotation is dd>/dt = too, the precession frequency. In the absence of any damping forces, the actual precession angle will be determined by the initial position of the magnetic dipole, and the dipole will precess about Ho at this angle indefinitely (free precession). In reality, however, the existence of damping forces will cause the magnetic dipole moment to spiral in from its initial angle until m is aligned with Ho (0 = 0). Now assume that there are N unbalanced electron spins (magnetic dipoles) per unit volume, so that the total magnetization is M — Nm, (9.13) and the equation of motion in (9.6) becomes dM di = -pLoyM*H. (9.14) where H is the internal applied field. (Note: In Chapter 1 we used Pm for magnetization and M for magnetic currents; here we use M for magnetization, as this is common practice in ferrimagnetics work. Since we will not be using magnetic currents in this chapter, there should be no confusion.) As the strength of the bias field Ho is increased, more magnetic dipole moments will align with Wo until all are aligned, and M reaches an upper limit. See Figure 9.2. The material is then said to be magnetically saturated, and Ms is denoted as the saturation magnetization. Ms is thus a physical property of the ferrite material, and typically ranges from4^M, — 300 to 5000 gauss. (Appendix H lists the saturation magnetization and 9.1 Basic Properties of Femmagnettc Materials 445 o Applied bias field Ht FIGURE 92 Magnetic moment of a ferrimagnetic material versus bias field, Hu. other physical properties of several types of microwave fertile materials.) Below saturation, ferrite materials can be very lossy at microwave frequencies, and the RF interaction is reduced. Thus ferrites are usually operated in the saturated state, and this assumption is made for the remainder of this chapter. The saturation magnetization of a material is a strong function of temperature, decreasing as temperature increases. This effect can be understood by noting that the vibrational energy of an atom increases with temperature, making it more difficult to align all the magnetic dipoles. At a high enough temperature the thermal energy is greater than the energy supplied by the internal magnetic field, and a zero net magnetization results. This temperature is called the Curie temperature, 7c- We now consider the interaction of a small AC (microwave) magnetic field with a magnetically saturated ferrite material. Such a field will cause a forced precession of the dipole moments around the H^iz) axis at the frequency of the applied AC field, much like die operation of an AC synchronous motor. The small-signal approximation will apply to all the ferrite components of interest to us, but there are applications where high-power signals can be used to obtain useful nonlinear effects. If /? is the applied AC field, the total magnetic field is where we assume that \H\ <§C Ho. This field produces a total magnetization in the ferrite material given by where M(is the (DC) saturation magnetization and M is the additional (AC) magnetization (in the xy plane) caused by H. Substituting (9,16) and (9.15) into (9.14) gives the following component equations of motion: (9.15) (9.16) dMx (9.17a) dt dMy = ixayMJHo + Hz) - Poy(M, + M,)H (9.17b) (9,17c) 446 Chapter 9: Theory and Design of Ferrimagnetic Components since dMs/dt = 0. Since \H\ «C H0, we have \M\\H\ « \M\H0 and \M\\H\ « M,{ff\, so we can ignore MH products. Then (9.17) reduces to dt dMj_ dt dMz dt = -aX)My +a}mHy, — ojqMx - (omHx, = 0, (9.18a) (9.18b) (9.18c) where wo = iitzyHo mdtom = (AoyMs. Solving (9.18a,b) for A/^ andMy gives the following equations: d2M (9.19a) (9.19b) These are the equations of motion for the forced precession of the magnetic dipoles, assuming small-signal conditions. It is now an easy step to arrive at the permeability tensor for ferrites; after doing this, we will try to gain some physical insight into the magnetic interaction process by considering circularly polarized AC fields. If the AC H field has an e'*0A time-harmonic dependence, the AC steady-state form of (9.19) reduces to the following phasor equations: {u>\ - ar)Mx = (dQojm Hx + jtowmHy, (a?2, - co2)My = -jcoiomHx + coQO)„Hy, (9.20a) (9,20b) which shows the linear relationship between H and M. As in (1.24), (9.20) can be written with a tensor susceptibility, [x], to relate H and M: M = [X)H = Xxx Xxy 0 T Xyx Xyy 0\H, . o o oj (9.21) where the elements of [x ] are given by Xxx ~ Xyy -Xxy = ~ X$i COn — CO1 (9.22a) (9.22b) The z component of f? does not affect the magnetic moment of the material, under the above assumptions. To relate B and H, we have from (1.23) that where the tensor permeability [p.] is given by [ix] = Mo([t/] + [*]) = H jk 0 _ -jk fX 0 L 0 0 Mo- (z bias). (9.23) (9.24) 9.1 Basic Properties of Ferrimagnetic Materials 447 The elements of the permeability tensor are then 1 + ~~2-7 )' (Of} — to / (oa>„ Í = -JßüXxy = JßdXyx = ß0~ iajj - to2 (9.25a) (9.25b) A material having a permeability tensor of this form is called gyrotropic; note that an x (or y) component of H gives rise to both x and y components of B, with a 90° phase shift between them. If the direction of bias is reversed, both H$ and Ms will change signs, so oj$ and oim will change signs. Equation (9.25) then shows that p will be unchanged, but k will change sign. If the bias field is suddenly removed {Ha — 0), the ferrite will generally remain magnetized (0 < \M\ < Afj); only by demagnetizing the ferrite (with a decreasing AC bias field, for example) can M = 0 be obtained. Since the results of (9.22) and (9.25) assume a saturated ferrite sample, both Ms and H$ should be set to zero for the unbiased, demagnetized case. Then to$ = tom = 0, and (9.25) show that p, = pa and k = 0, as expected for a nonmagnetic material. The tensor results of (9.24) assume bias in the z direction, [f the ferrite is biased in a different direction the permeability tensor will be transformed according to the change in coordinates. Thus, if H0 = xHq, the permeability tensor will be [ß] = ßö 0 0 0 ß JK L 0 -jk [a (x bias), (9.26) while if H0 = yH0 the permeability tensor will be 0 -jtc ßo o L jk 0 ß ß 0 {y bias). (9.27) A comment must be made about units. By tradition most practical work in magnetics is done with CGS units, with magnetization measured in gauss (1 gauss = 10-4 weber/m2), and field strength measured in oersteds {4it x 1CTJ oersted = I A/m), Thus, p-o = 1 gauss/ oersted in CGS units, implying that B and H have the same numerical values in a nonmagnetic material. Saturation magnetization is usually expressed as 4jtM^ gauss; the corresponding MKS value is then paMs weber/m2 = 10~4 (4ttMs gauss). In CGS units, the Larmor frequency can be expressed as fa = to^/ln = p^yHajliz — (2.8 MHz/oersted) {Hq oersted), and fr, = com/2x = payMx/2n = (2.8 MHz/oersted) ■ (4xMj gauss). In practice, these units are convenient and easy to use. Circularly Polarized Fields To get a better physical understanding of the interaction of an AC signal with a saturated ferrimagnetic material we will consider circularly polarized fields. As discussed in Section 1.5, a right-hand circularly polarized field can be expressed in phasor form as H+ = H+(x-jy), (9.28a) and in time-domain form as H+ 7= Re{/?+^ft"} = H^ix cosojr + ysinarf), (9.28b) 446 Chapter 9: Theory and Design of Ferrimagnetic Components where we have assumed the amplitude H+ as real. This latter form shows that H+ is a vector which rotates with time, such that at time t it is oriented at the angle cot from the *-axis; thus its angular velocity is co. (Also note that \7i+\ = H+ £ |/?+|.) Applying the RHCP field of (9.28a) to (9.20) gives the magnetization components as too — to * OJq — co so the magnetization vector resulting from can be written as M+ = M+i + -^~H+(x -jy\ (9.29) y cot) — iff which shows that the magnetization is also RHCP, and so rotates with angular velocity to in synchronism with the driving field, H+. Since M+ and H+ are vectors in the same direction, we can write = #o(M+ + H+) = ti+H+, where n+ is the effective permeability for an RHCP wave given by &^h±J&A (9-30) \ coo - 0)J The angle, 8m, between M+ and the ^-axis is given by \M+\ tomH+ coqH+ tan% = —- =--— =--—, (9.31) Ms [co(j - to)Ms (gjq - to)H0 while the angle, 8H, between H+ and the z-axis is given by ITJ+I ff + tan^ = L-i = —. (9,32) m no For frequencies such that to < Itoo, (9.31) and (9.32) show that 8M > QH, as illustrated in Figure 9.3a. In this case the magnetic dipole is precessing in the same direction as it would freely precess in the absence of H+. Now consider a left-hand circularly polarized field, expressed in phasor form as H~ = H-(x + jy), (9.33a) and in time-domain form as ft = Rt{H-ejM} = H(x cos tot - y sin tot). (9.33b) Equation (9.33b) shows that H" is a vector rotating in the -to (left-hand) direction. Applying the LHCP field of (9.33a) to (9.20) gives the magnetization components as AC = — H . too + to My = -^-H~, so the vector magnetization can be written as to, M~ = M;x + M~y =-^—H-(x + jy). (9.34) * too + to 9.1 Basic Properties of Ferrimagnetic Materials 449 which shows that the magnetization is LHCP, rotating in synchronism with H . Writing B~ =\ }i$(M~ + H~) = jx~H~ gives the effective permeability for an LHCP wave as ^-=Mo(l + -^Y (9.35) \ tijQ +(0/ The angle, 8Mi between M~ and the ^-axis is given by \M~\ tomR- (oqH- tan $m =--=-=-, (9.36) Ms (m + o))Ms (o. This effect is known as gyromagnetie resonance^ and occurs when the forced precession frequency is equal to the free precession frequency. In the absence of loss the response may be unbounded, in the same way that the response of an LC resonant circuit will be unbounded when driven with an AC signal having a frequency equal to the resonant frequency of the LC circuit. All real ferrite materials, however, have various magnetic loss mechanisms that damp out such singularities. 450 Chapter 9: Theory and Design of Fenrimagnetic Components As with other resonant systems, loss can be accounted for by making the resonant frequency complex,: coo <— too + jccoj, (9,37) where a is a damping factor, Substituting (9.37) into (9.22) makes the susceptibilities complex: Xxy^Xty+JXxy (9.38b) where the real and imaginary parts are given by ^-fi»2(l+«2)]2 + 4^V [4 - (o2(l + a2)f + 4col2^ Equation (9.37) can also be applied to (9,25) to give a complex p — p' — and k = k' - }k"; this is why (9.38b) appears to define x'xy and %L backward, as %xy = yfc/>o. For most ferrite materials the loss is small, so a , resonance occurs when Hq = Hr, such that to$ = ^nyHr. The linewidth, AH, is defined as the width of the curve of xxx versus Hq where x"x has decreased to half its peak value. If we assume (1 + a2) ~ 1, (9.39b) shows that the maximum value of Xxx is (om/2aoj, and occurs when co = coq. Now let u;02 be the Larmor frequency for which Ho — H2. where xxx has decreased to half its maximum value. Then we can solve (9.39b) for « in terms of &>02: axoo)m {o)22 + (jo2) ojm Hi - a1? + 4^2o>2a2 ^to' 4aV = {a?m -co2)2, o)02 = wVl + 2a S co(\ + a). Then Am - 2(om>2 — wo) ~ 2[w( I -\-ct) — (o] = 2ccco, and using (9.9) gives the line-width as AH=^ = ^. (9.40) Typical linewidths range from less than 100 Oe (for yttrium iron garnet) to 100-500 Oe 9.1 Basic Properties of Ferrimagnetic Materials 451 (hi FIGURE 9.4 Complex susceptibilities for a typical ferrite. (a) Real and imaginary parts of ^rjt, (b) Real and imaginary parts of x*>■■ (for ferrites); single-crystal YIG can have a linewidth as low as 0.3 Oe. Also note that this loss is separate from the dielectric loss that a ferrimagnetic material may have. Demagnetization Factors The DC bias field, Ho, internal to a ferrite sample is generally different from the externally applied field, Hit, because of the boundary conditions at the surface of the ferrite. To illustrate this effect, consider a thin ferrite plate, as shown in Figure 9.6. When the applied field is normal to the plate, continuity of Bn at the surface of the plate gives Bn = MoHa - itxNy ^ ^ff_ »^ D D My = -^"xe H--—-riye, (y.4JC) where D = (l + X,, 1 + Xyyty£ - Xy*X*yNxNy. (9.44) This result is of the form M = \xAH< where the coefficients of Hxe and Hye in (9.43) can be defined as "external" susceptibilities since they relate magnetization to the external RF fields. For an infinite ferrite medium gyromagnetic resonance occurs when the denominator of the susceptibilities of (9.22) vanishes, at the frequency o>r = a> = ^ But for a finite-sized ferrite sample the gyromagnetic resonance frequency is altered by the demagnetization factors, and given by the condition that D = 0 in (9.43). Using the expressions in (9.22) for the susceptibilities in (9.44), and setting the result equal to zero gives ( (OQ(OmNx\ f awomNy\ oP-orm K+14^) V + ^)' mm?** = After some algebraic manipulations this result can be reduced to give the resonance frequency, o)r, as toy = co- yfitm + Nx)&m + o)mNy). (9.45) Since ojo = tkxyHq — &oy(H of the material. This quantity characterizes the strength of the magnet, so generally a magnet material is chosen to have a large remanence. Another important parameter is the coercivity, H[7 which is the value of the negative H field required to reduce the magnetization to zero. A good permanent magnet should have a high coercivity to reduce the effects of vibration, temperature changes, and external fields, which can lead to a toss of magnetization. An overall figure of merit for a permanent magnet is sometimes given as the maximum value of the BH product, (flifW, on the demagnetization curve. This quantity is essentially die maximum magnetic energy density that can be stored by the magnet, and can be useful in electromechanical applications. The following table lists the remanence, coercivity, and (it H)mM for some of the most common permanent magnet materials. Material Composition (txHx +JKHy), (9.49a) -j&Ex = -JQ)(-jlCHX + jlHy), (9.49b) 0 = (9.49c) (9.49d) -m* = jOitEy, (9.49e) 0 = jai€Ez. (9.49f) Equations (9.49c) and (9.490 show that Ez^ Hz= 0, as expected for TEM plane waves. We also have V • D = V • B = 0, since 9/3x = 9/9y = 0. Equations (9.49d,e) give relations between the transverse field components as Y=HL = Zlk = ^ (9.50) EX Ey, B where Y is the wave admittance. Using (9.50) in (9.49a) and (9.49b) to eliminate Hx and Hy gives the following results: j(o2€KEx + (p2 - o?iit)Ey = 0, (9.51a) {p2 - o>2fie)Ex - jto2€KEy = 0. (9.51b) For a nontrivial solution for Ex and Ey the determinant of this set of equations must vanish: &>4* V - (p2 - a>v)2 = 0, or p± = a>Jc(fir±ic). (9.52) So mere are two possible propagation constants, P+ and j6_. First consider the fields associated with fi+, which can be found by substituting fi+ into (9.51a), or (9.5lb); ja>2(KEx + <02£tcEy = 0, or Ey = -jEx. 456 Chapter 9: Theory and Design of Femmagnetic Components Then the electric field of (9.48a) must have the following form: E+ = Ea(x - jy)e^\ (9.53a) which is seen to be a right-hand circularly polarized plane wave. Using (9.50) gives the associated magnetic field as H+ « EQY+(jx + W^H (9-53b) where Y+ is the wave admittance for this wave: Y+ = * = L* (9.53c) 0+ V*+* Similarly, the fields associated with B- are left-hand circularly polarized: E. = E0(x + jy)e-^-\ (9.54a) /?_ = EoY^-jx + y)e~ilf-\ (9.54b) where K_ is the wave admittance for this wave: r.-»- J-^U (9.54c) £- V ft - 8 Thus we see that RHCP and LHCP plane waves are the source-free modes of the 1 -biased ferrite medium, and these waves propagate through the ferrite medium with different propagation constants. As discussed in the previous section, the physical explanation for this effect is that the magnetic bias field creates a preferred direction for magnetic dipole precession, and one sense of circular polarization causes precession in this preferred direction while the other sense of polarization causes precession in the opposite direction. Also note that for an RHCP wave, the ferrite material can be represented with an effective permeability of it 4- a:, while for an LHCP wave the effective permeability is p, — k. In mathematical terms, we can state that (ii + k) and {p. — k), or and B-, are the eigenvalues of the system of equations in (9.51), and that E+ and £_ are the associated eigenvectors. When losses are present, the attenuation constants for RHCP and LHCP waves will also be different. Now consider a linearly polarized electric field at z = 0, represented as the sum of an RHCP and an LHCP wave: E|z=0 = xE0 = - jy) + + jy). (9.55) The RHCP component will propagate in the z direction as e~J^z, and the LHCP component will propagate as e~^-z, so the total field of (9.55) will propagate as E = ^C* - jy)e->^ + ^(x + J9)e~^ = Eo [l cos f * - 7 sin (^T^) »1 e-j(^-™2. (9.56) This is still a linearly polarized wave, but one whose polarization rotates as the wave propagates along the r-axis. At a given point along the z-axis the polarization direction 9,2 Plane Wave Propagation in a Ferrite Medium 457 measured from the .r-axis is given by

at. Then B+ > B-, and (9.57) shows that $ becomes more negative as z increases, meaning that the polarization (direction of E) rotates counterclockwise as we look in the +z direction. Reversing the bias direction (sign of Ho and Ms) changes the sign of k, which changes the direction of rotation to clockwise. Similarly, for +z bias, a wave traveling in the — z direction will rotate its polarization clockwise as we look in the direction of propagation (—z)\ if we were looking in the -\-z direction, however, the direction of rotation would be counterclockwise (same as a wave propagating in the +z direction). Thus, a wave that travels from z = 0 to z = L and back again to z = 0 undergoes a total polarization rotation of 20, where (f> is given in (9.57) with z = L. So, unlike the situation of a screw being driven into a block of wood and then backed out, the polarization does not "unwind" when the direction of propagation is reversed. Faraday rotation is thus seen to be a nonreciprocal effect. EXAMPLE 9.1 PLANE WAVE PROPAGATION IN A FERRITE MEDIUM Consider an infinite ferrite medium with 4n Ms = 1800 gauss, AH =75 oersted, €r = 14, and taná = 0.001. If die bias field strength is H0 = 3570 oersted, calculate and plot the phase and attenuation constants for RHCP and LHCP plane waves versus frequency, for / = 0 to 20 GHz. Solution The Larmor precession frequency is (Oft fo = ;p = (2.8 MHz/oersted)(3570 oersted) - 10.0 GHz, 2jt and fm = ^ = (2.8 MHz/oersted) (1800 gauss) = 5.04 GHz. 2jt At each frequency we can compute the complex propagation constant as V± = «i + jB± = jto-Jt(}X ± k), where e = e<>er(l - j tan5) is die complex permittivity, and fx, k are given by (9.25). The following substitution for &>o is used to account for ferrimagnetic loss: .MY AH coq <— ^o-f J---, (2.8 MHz/Oe)(75 Oe) or f0 <— /o + j---- - (10.4 ,0.105) GHz, Chapter 9: Theory and Design of Ferrimagnetic Components which is derived from (9.37) and (9.40). The quantities (}x ± k) can be simplified to the following, by using (9.25): li + k = fi0 ( 1 h--}, V ojq - co/ /. o)m \ fi -k =mo 1 + -—.— )■ The phase and attenuation constants are plotted in Figure 9.8, normalized to the free-space wavenumber, to- Observe that 6+ and ct+ (for an RHCP wave) show a resonance near / = /0 = 10 GHz; 8- and t*_ (for an LHCP wave) do not, however, because the singularities in fx and tc cancel in the (fx — k) term contained in y_. Also note from Figure 9.7 that a stopband (8+ near zero, large a+) exists for RHCP waves for frequencies between /$ and /0 + fm (between co® and coq 4- co„,). For frequencies in this range, the above expression for (p. + it) shows that this quantity is negative, and 8+ = 0 (in the absence of loss), so an RHCP wave incident on such a ferrite medium would be totally reflected. ■ Propagation Transverse to Bias (Birefringence) Now consider the case where an infinite ferrite region is biased in the x direction, transverse to the direction of propagation; the permeability tensor is given in (9.26). For plane wave fields of the form in (9.48), Maxwell's curl equations reduce to = -jcofioHx, (9.58a) -}8EX = -jtoiiiHy + jkHz), (9.58b) 0 = -jco{-JKHy -r-ixHz\ (9.58c) jSHy = jCO€Ex, (9.58d) — jtOtEy, (9.58c) 0 = jcoie E,. (9.58f) FIGURE 9,7 Normalized phase and attenuation constants for circularly polarized plane waves in the ferrite medium of Example 9.1. 9.2 Plane Wave Propagation in a Ferrite Medium 459 Then Ez = 0, and V ■ D - 0 since d/dx = d/dy = 0. Equations (9.58d,e) give an admittance relation between the transverse field components: f-gt.i&.Jffi, (9.59) EX Ey 8 Using (9.59) in (9.58a,h) to eliminate Hx and Hy, and using (9.58c) in (9.58b) to eliminate Hz, gives the following results: 82Ey=a>2iiQ€Ey, (9.60a) H(62 - co2fi€)Ex = -co2ttc2Ex. (9.60b) One solution to (9.60) occurs for p\ = «V7^F, (9.61) with Ex = 0. Then the complete fields are E0 = yEQe-^\ (9.62a) Hn = ~xE0Yoe-JI*°z, (9.62b) since (9.59) shows that Hy = 0 when Ex = 0, and (9.58c) shows that Hz = 0 when Hy = 0. The admittance is Y° = ir = xh- (9-63) Po V AO This wave is called the ordinary wave, because it is unaffected by the magnetization of the ferrite. This happens whenever the magnetic field components transverse to the bias direction are zero (Hy = Hl=0). The wave propagates in either the +z or — z direction with the same propagation constant, which is independent of H{\. Another solution to (9.60) occurs for pe = odJpZ^, (9.64) with Ey = 0, where p,e is an effective permeability given by This wave is called the extraordinary wave, and is affected by the ferrite magnetization. Note that the effective permeability may be negative for certain values of m, The electric field is Ef = xEoe-i^. (9.66a) Since Ey = 0, (9.58e) shows that Hx = 0. Hy can be found from (9.58d), and Hz from (9.58c), giving the complete magnetic field as He = EqY£ + £^ J (9.66b) where Ye = ^ = ff. (9.67) These fields constitute a linearly polarized wave, but note that the magnetic field has a component in the direction of propagation. Except for the existence of Hz, the extraordinary 460 Chapter 9: Theory and Design of Ferrimagnetic Components -AirM, = 3000G 6GHz^£N -----4-n-M, = T700G 1 11 GHz \ > 7^*- 6 GHz \ I i TV. i i \ 11 GHz I 1 1 0 200 400 600 800 1000 1200 I4O0 1600 1800 2000 HQ (OERSTEDS) FIGURE 9.8 Effective permeability, ft?, versus bias field, H^, for various saturation magnetizations and frequencies. wave has electric and magnetic fields that are perpendicular to the corresponding fields of the ordinary wave. Thus, a wave polarized in the y direction will have a propagation constant p\ (ordinary wave), but a wave polarized in the x direction will have a propagation constant fte (extraordinary wave). This effect, where the propagation constant depends on the polarization direction, is called birefringence [2]. Birefringence often occurs in optics work, where the index of refraction can have different values depending on the polarization. The double image seen through a calcite crystal is an example of this effect. From (9.65) we can see that the effective permeability for the extraordinary wave, can be negative if k2 > ii2. This condition depends on the values of m, or /, Ho, and Ms, but for a fixed frequency and saturation magnetization there will always be some range of bias field for which }i£ < 0 (ignoring loss). When this occurs fie will become imaginary, as seen from (9.64), which implies that the wave will be cutoff, or evanescent. An x polarized plane wave incident at the interface of such a ferrite region would be totally reflected. The effective permeability is plotted versus bias field strength in Figure 9.8, for several values of frequency and saturation magnetization. PROPAGATION IN A FERRITE-LOADED RECTANGULAR WAVEGUIDE In the previous section we introduced the effects of a ferrite material on electromagnetic waves by considering the propagation of plane waves in an infinite ferrite medium. In practice, however, most ferrite components use waveguide or other types of transmission lines loaded with ferrite material. Most of these geometries are very difficult to analyze. Nevertheless, it is worth the effort to treat some of the easier cases, involving ferrite-loaded rectangular waveguides, in order to quantitatively demonstrate the operation and design of several types of practical ferrite components. TEm0 Modes of Waveguide with a Single Ferrite Slab We first consider the geometry shown in Figure 9.9, where a rectangular waveguide is loaded with a vertical slab of ferrite material, biased in the y direction. This geometry and 9.3 Propagation in a Ferrite-Loaded Rectangular Waveguide 461 FIGURE 93 Geometry of a rectangular waveguide loaded with a transversely biased ferrite slab. its analysis will be used in later sections to treat the operation and design of resonance isolators, field-displacement isolators, and remanent (nonreciprocal) phase shifters. In the ferrite slab, Maxwell's equations can be written as V x E = -jai[fi]H, V x H = jateE, (9.68a) (9.68b) where \fx\ is the permeability tensor for y bias, as given in (9.27). Then if we let E(x ,y,z} = [e(*, y) + y)\e->fr and H(x, y, z) = [h(x, y) + zhz(x, y)]e~^z, (9.68) reduces to -I- jBey = -jai({ihx - JKhz), By dey Sex . ■ . ----r— = -JCO(jfchx + Ltk,), dx by dhz — +jBhy = jaxex, dy dhy dhx ~dx _~3y (9.69a) (9.69b) (9.69c) (9.69d) (9.69e) (9.690 For TEm0 modes, we know that Ez = 0 and d/dy = 0, Then (9.69b) and (9.69d) imply that ex = hy = 0 (since B2 ^ to2fiQe for a waveguide mode) and so (9.69) reduces to three equations: jBey ~- -ja>(ixhx - JKhz), dey —- = -jo>(JKhx + fihz), ox ju)€ey = -jBhx - —. (9.70a) (9.70b) (9.70c) 462 Chapter 9: Theory and Design of Ferrimagnetic Components We can solve (970a,b) for hx and h, as follows. Multiply (9.70a) by p and (9.70b) by jk, then add to obtain hx = —L_ (~»Bey - AX (9.71a) Now multiply (9.70a) by jk and (9.71a) by /i, then add to obtain itiilile \ OX ) where fie = (ji2 — K2)/fi,. Substituting (9.71) into (9.70c) gives a wave equation for ey: \ y dx J (D^Lf/.,. \ dx 8x2 y or where k f is defined as a cutoff wavenumber for the ferrite: kj = o?\ttz - a2. (9.72) (9.73) We can obtain the corresponding results for the air regions by letting = yxo. % = 0, and ir — 1, to obtain where ka is the cutoff wavenumber for the air regions: kl=k2-B2, The magnetic field in the air region is given by ■1 j dey The solutions for ey in the air-ferrite-air regions of the waveguide are then A sin kax, for 0 < x < c, B sin£/(T —c) + C sinkf(c -f- ( — x), for c < * < c + r, /> sin — x)t (otc + th = (jka A /(o/a0) cos kax, forO < x < c. {j/oiiJ.iie){KB[B sinJt^(x — c) + CsinA-/(c + / - x)] + fxkf[B coskf(x — c) — Ccos kf(c +1 — x)]}, for c < x < c +1, (—jkaDfo>ii0)Qoska{a - x), forc + r„ to be complex, as in (9.37). In later sections we will also need to evaluate the electric field in the guide, as given in (9.77a). If we choose the arbitrary amplitude constant as A, then B, C, and D can be found in terms of A by using (9.78a), (9.78b), and (9.78c). Note from (9.75) that if B > k0i then k„ will be imaginary. In this case, the sin kax function of (9.77a) becomes j sinh \ka \x, indicating an almost exponential variation in the field distribution. A useful approximate result can be obtained for the differential phase shift, B+ — /L, by expanding B in (9.79) in a Taylor series about / = 0. This can be accomplished with implicit differentiation after using (9.73) and (9.75) to express k/ and ka in terms of 8 [4 J. The result is B± - 8. * sin2kcc = 2kc-—r sinz^c, (9.80) afx {i S where kc = nja is the cutoff frequency of the empty guide, and AS/S = t/ais the filling factor, or ratio of slab cross-sectional area to waveguide cross-sectional area. Thus, this formula can be applied to other geometries such as waveguides loaded with small ferrite strips or rods, although the appropriate demagnetization factors may be required for some ferrite shapes. The result in (9.80) is accurate, however, only for very small ferrite cross sections, typically for AS/S < 0.01. This same technique can be used to obtain an approximate expression for the forward and reverse attenuation constants, in terms of the imaginary parts of die susceptibilities defined in (9.39): AS {Phs* sin2 kcx + $*jfj cos2kcx T sin2kcx), (9.81) 464 Chapter 9: Theory and Design of Ferrimagnetic Components Magnetic wall FIGURE 9.10 Geometry of a rectangular waveguide loaded with two symmetrical ferrite slabs. where B0 = ^fk2. — k} is the propagation constant of the empty guide. This result will be useful in the design of resonance isolators. Both (9.80) and (9.81) can also be derived using a perturbation method with the empty waveguide fields [4], and so are usually referred to as the perturbation theory results. TEm0 Modes of Waveguide with Two Symmetrical Ferrite Slabs A related geometry is the rectangular waveguide loaded with two symmetrically placed ferrite slabs, as shown in Figure 9.10. With equal but opposite y-directed bias fields on the ferrite slabs, this configuration provides a useful model for the nonreciprocal remanent phase shifter, which will be discussed in Section 9.5. Its analysis is very similar to that of the single-slab geometry. Since the hy and hz fields (including the bias fields) are antisymmetric about the midplane of the waveguide at jc = a/2, a magnetic wall can be placed at this point. Then we only need to consider the region for 0 < x < a/2. The electric field in this region can be written as A sinA^*, 0 < x < c, ev =■ Bsink/ix-c) +Csinkf(c + t—x), c'A yi (a) (b) FIGURE 9.11 Two resonance isolator geometries, (a) f-plane, full-height slab, (b) //-plane slab. ferrite slab or strip mounted at a certain point in a waveguide. We will discuss the two isolator geometries shown in Figure 9.11. Ideally, the RF fields inside the ferrite material should be circularly polarized, In an empty rectangular waveguide the magnetic fields of the TEio mode can be written as where kc = itja is the cutoff wavenumber and &0 — ^jk2 — k2 is the propagation constant of the empty guide. Since a circularly polarized wave must satisfy the condition that Hx/Hz = ±j, the location, of the CP point of the empty guide is given by tankcx = ±^. (9.86) Po Ferrite loading, however, may perturb the fields so that (9,86) may not give the actual optimum position, or it may prevent the internal fields from being circularly polarized for any position. First consider the full-height £-plane slab geometry of Figure 9.1 la; we can analyze this case using the exact results from the previous section. Alternatively, we could use the perturbation result of (9.81), but this would require the use of a demagnetization factor for A*, and would be less accurate than the exact results. Thus, for a given set of parameters, (9.79) can be solved numerically for the complex propagation constants of the forward and reverse waves of the ferrite-loaded guide. It is necessary to include the effect of magnetic loss, which can be done by using (9.37) for the complex resonant frequency, coq, in the expressions for Lt and tc. The imaginary part of too can be related to the linewidth, AH, of the ferrite through (9.40). Usually the waveguide width, a, frequency, co, and ferrite parameters 4nMSi and er will be fixed, and the bias field and slab position and thickness will be determined to give the optimum design. Ideally, the forward attenuation constant (a+) would be zero, with a nonzero attenuation constant (a_) in the reverse direction. But for the E-plane ferrite slab there is no position x = c where the fields are perfecdy CP in the ferrite (this is because the demagnetization factor Nx — 1 [4]). Hence the forward and reverse waves both contain an RHCP component and an LHCP component, so ideal attenuation characteristics cannot be obtained. The optimum design, then, generally minimizes the forward attenuation, which determines the slab position. Alternatively, it may be desired to maximize the ratio of the reverse to forward 9.4 Ferrite Isolators 467 attenuations. Since the maximum reverse attenuation generally does not occur at the same slab position as the minimum forward attenuation, such a design will involve a trade-off of the forward loss. For a long, thin slab, the demagnetization factors are approximately those of a thin disk: Nx ^* 1, Ny = Nz — 0. It can then be shown via the Kittel equation of (9.45) that the gyromagnetic resonance frequency of the slab is given by a> - vW«o + ov), (9.87) which determines Hq, given the operating frequency and saturation magnetization. This is an approximate result; the transcendental equation of (9.79) accounts for demagnetization exactly, so the actual internal bias field, H0, can be found by numerically solving (9.79) for the attenuation constants for values of Ho near the approximate value given by (9.87). Once the slab position, c, and bias field, H0, have been found the slab length, L, can be chosen to give the desired total reverse attenuation (or isolation) as (a_) L. The slab thickness can also be used to adjust this value. Typical numerical results are given in Example 9.2. One advantage of this geometry is that the full-height slab is easy to bias with an external C-shaped permanent magnet, with no demagnetization factor. But it suffers from several disadvantages: • Zero forward attenuation cannot be obtained because the internal magnetic field is not truly circularly polarized. • The bandwidth of the isolator is relatively narrow, dictated essentially by the line-width, AH, of the ferrite. • The geometry is not well suited for high-power applications because of poor heat transfer from the middle of the slab, and an increase in temperature will cause a change in M,, which will degrade performance. The first two problems noted above can be remedied to a significant degree by adding a dielectric loading slab; see reference [5] for details. EXAMPLE 9.2 FERRITE RESONANCE ISOLATOR DESIGN Design an £-plane resonance isolator in X-band waveguide to operate at 10 GHz with a minimum forward insertion loss and 30 dB reverse attenuation. Use a 0.5 mm thick ferrite slab with AnMs = 1700G, AH = 200 Oe, and e: = 13. l>termine the bandwidth for which the reverse attenuation is at least 27 dB. Solution The complex roots of (9.79) were found numerically using an interval-halving routine followed by a Newton-Raphson iteration. The approximate bias field, Ho, given by (9.87) is 2820 Oe, but numerical results indicate the actual field to be closer to 2840 Oe for resonance at 10 GHz. Figure 9.12a shows the calculated forward (a+) and reverse (a J) attenuation constants at 10 GHz versus slab position, and it can be seen that the minimum forward attenuation occurs for c/a = 0.125; the reverse attenuation at this point is «_ = 12,4 dB/cm. Figure 9.12b shows the attenuation constants versus frequency for this slab position. For a total reverse attenuation of 20 dB, the length of the slab must be 463 Chapter 9: Theory and Design of Ferrimagnetic Components da /(GHz) (a) (b) 47rWr=1700G fl-0 = 2840Oe ArY = 200Oe a = 2.286cm f=0.05cm er = 13 FIGURE 9.12 Forward and reverse attenuation constants for the resonance isolator of Example 9.2. (a) Versus slab position, (b) Versus frequency. For the total reverse attenuation to be at least 27 dB, we must have 27 dB a_ >-- 11.3 dB/cm. 2.4 cm So the bandwidth according to the above definition is, from the data of Figure 9.12b, less than 2%. This result could be improved by using a ferrite with a larger linewidth, at the expense of a longer or thicker slab and a higher forward attenuation, ■ Next we consider a resonance isolator using the //-plane slab geometry of Figure 9.1 lb. If the slab is much thinner than it is wide, the demagnetization factors will approximately be Nx = Nt = 0, Ny = 1. This means that a stronger applied bias field will be required to produce the internal field, Ho, in the y direction. But the RF magnetic field components, hx and kz, will not be affected by the air-ferrite boundary since Nx = N, = 0, and perfect circular polarized fields will exist in the ferrite when it is positioned at the CP point of the empty guide, as given by (9.86). Another advantage of this geometry is that it has better thermal properties than the E-plane version, since the ferrite slab has a large surface area in contact with a waveguide wall for heat dissipation. Unlike the full-height E-plane slab case, the H -plane geometry of Figure 9.1 lb cannot be analyzed exactly. But if the slab occupies only a very small fraction of the total guide cross section (AS/S < 1, where AS and S are the cross sectional areas of the slab and waveguide, respectively), the perturbational result forar+ in (9.81) can be used with reasonable results. This expression is given in terms of the susceptibilities %xx = x'xx ~ Jx"x> Xzz ~ X!zz -jx"z, and Xxy = Xxy + J Xxy>as defined for a y-biased ferrite in a manner similar to (9.22). For ferrite shapes other than a thin H-plane slab, these susceptibilities would have to be modified with the appropriate demagnetization factors, as in (9.43) [4]. 9.4 Ferrite Isolators 469 As seen from the susceptibility expressions of (9.22), gyromagnelic resonance for this geometry will occur when to = too, which determines the internal bias field, //[>. The center of the slab is positioned at the circular polarization point of the empty guide, as given by (9.86), This should result in a near-zero forward attenuation constant. The total reverse attenuation, or isolation, can be controlled with either the length, L, of the ferrite slab or its cross section AS, since (9.8J) shows a± is proportional to AS/S. If AS/S is too large, however, the purity of circular polarization over the slab cross section will be degraded, and forward loss will increase. One practical alternative is to use a second identical ferrite slab on the top wall of the guide, to double AS/S without significantly degrading polarization purity. The Field Displacement Isolator Another type of isolator uses the fact that the electric field distributions of the forward and reverse waves in a ferrite slab-loaded waveguide can be quite different. As illustrated in Figure 9.13, the electric field for the forward wave can be made to vanish at the side of the ferrite slab at x — c -f- while the electric field of the reverse wave can be quite large at this same point. Then if a thin resistive sheet is placed in this position, the forward wave will be essentially unaffected while die reverse wave will be attenuated. Such an isolator is called a field displacement isolator, high values of isolation with a relatively compact device can be obtained with bandwidths on the order of 10%. Another advantage of the field displacement isolator over the resonance isolator is that a much smaller bias field is required, since it operates well below resonance. The main problem in designing a field displacement isolator is to determine the design parameters that produce field distributions like those shown in Figure 9.13. The general form of the electric field is given in (9.77a), from the analysis of the ferrite slab-loaded waveguide. This shows that for the electric field of the forward wave to have a sinusoidal dependence for c +1 < x < a, and to vanish at x = c + i, the cutoff wavenumber k+ must be real and satisfy the condition that where d = a —c — i. In addition, the electric field of the reverse wave should have a hyperbolic dependence for c +1 < x < a, which implies that k~ must be imaginary. Since (9.88) Resistive sheet FIGURE 9.13 Geometry and electric fields of a field displacement isolator. 470 Chapter 9: Theory and Design of Fenrimagnetic Components from (9.75), k2 = fc2 - 82, the above conditions imply that 8+ < k0 and 8" > kQ, where fco = coJJU^t. These conditions on B± depend critically on the slab position, which must be determined by numerically solving (9.79) for the propagation constants. The slab thickness also affects this result, but less critically; a typical value is t — a/10. It also turns out that in order to satisfy (9.88), to force Ey — 0 at x = c +1, pie = (jLt2 - k2)/ijl must be negative. This requirement can be intuitively understood by thinking of the waveguide mode for c +1 < x < a as a superposition of two obliquely traveling plane waves. The magnetic field components Hx and Hz of these waves are both perpendicular to the bias field, a situation which is similar to the extraordinary plane waves discussed in Section 9.2, where it was seen that propagation would not occur for \xe < 0. Applying this cutoff condition to the ferrite-loaded waveguide will allow a null in Ey for the forward wave to be formed at x = c +1. The condition that ii? be negative depends on the frequency, saturation magnetization, and bias field. Figure 9.8 shows the dependence of ja? versus bias field for several 2.0 1.8 1.6 i2 1.0 o.s 0.6 \_' l_J_!_I_I_I_I_I_I 0 0.02 0.04 0.06 0.08 0 10 da (a) 1.0 *, 0.9 I 0.8 | 0.7 I 0.6 I 0.3 \ 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0-9 L0 xla (b) FIGURE 9.14 Propagation constants and electric field distribution for the field displacement isolator of Example 9.3. (a) Forward and reverse propagation constants versus slab position, (b) Electric field amplitudes for the forward and reverse waves. 9.5 Ferrite Phase Shifters 471 frequencies and saturation magnetization. This type of data can be used to select the saturation magnetization and bias field to give (Ji€ < 0 at the design frequency. Observe that higher frequencies will require a ferrite with higher saturation magnetization, and a higher bias field, but pe < 0 always occurs before the resonance in ne at V The electric fields are plotted in Figure 9.14b. Note that the forward wave has a null at the face of the ferrite slab, while the reverse wave has a peak (the relative amplitudes of these fields are arbitrary). Then a resistive sheet can be placed at this point to attenuate the reverse wave. The actual isolation will depend on the resistivity of this sheet; a value of 75 £2 per square is typical, ■ FERRITE PHASE SHIFTERS Another important application of ferrite materials is in phase shifters, which are two-port components that provide variable phase shift by changing the bias field of the ferrite. (Microwave diodes and FETs can also be used to implement phase shifters; see Section 10.3.) Phase shifters find application in test and measurements systems, but the most significant use is in phased array antennas where the antenna beam can be steered in space by electronically controlled phase shifters. Because of this demand, many different types of phase shifters have been developed, both reciprocal (same phase shift in either direction) and nonreciprocal [2], [6]. One of the most useful designs is the latching (or remanent) nonreciprocal phase shifter using a ferrite toroid in a rectangular waveguide; we can analyze this geometry with a reasonable degree of approximation using the double ferrite slab geometry discussed in Section 9.3. Then we will qualitatively discuss the operation of a few other types of phase shifters. Nonreciprocal Latching Phase Shifter The geometry of a latching phase shifter is shown in Figure 9.15; it consists of a toroidal ferrite core symmetrically located in the waveguide with a bias wire passing through its center. When the ferrite is magnetized, the magnetization of the sidewalls of the toroid will 472 Chapter 9: Theory and Design of Ferrimagnetic Components / Bias / line Toroidal ferrite FIGURE 9.15 Geometry of a nonreciprocaj latching phase shifter using a ferrite toroid. be oppositely directed and perpendicular to the plane of circular polarization of the RF fields. Since the sense of circular polarization is also opposite on opposite sides of the waveguide, a strong interaction between the RF fields and the ferrite can be obtained. Of course, the presence of the ferrite perturbs the waveguide fields (the fields tend to concentrate in the ferrite), so the circular polarization point does not occur at tan kcx — kc/8o, as it does for an empty guide. In principle, such a geometry can be used to provide a continuously variable (analog) phase shift by varying the bias current. But a more useful technique employs the magnetic hysteresis of the ferrite to provide a phase shift that can be switched between two values (digital). A typical hysteresis curve is shown in Figure 9.16, showing the variation in magnetization, M, with bias field, Hq. When the ferrite is initially demagnetized and the bias field is off, both M and Ho are zero. As the bias field is increased, the magnetization increases along the dashed line path until the ferrite is magnetically saturated, and M — Ms. If the bias field is now reduced to zero, the magnetization will decrease to a remanent condition (like a permanent magnet), where M = Mr ■ A bias field in the opposite direction will saturate the ferrite with M = — Ms, whereupon the removal of the bias field will leave the ferrite in a remanent state with M = —Mr. Thus we can "latch" the ferrite magnetization in one of two states, where M = ±Mr, giving a digital phase shift. The amount of differential phase shift between these two states is controlled by the length of the ferrite toroid. In practice, several sections having individual bias lines and decreasing lengths are used in series to give binary differential phase shifts of 180s, 90°, 45°, etc. to as fine a resolution as desired (or can be afforded). An important advantage of the latching mode of operation is that the bias current does not have to be continuously applied, but only pulsed with one polarity or the other to change the polarity of the remanent magnetization; switching speeds can be on the order of a few microseconds. The bias wire can be oriented perpendicular to the electric field in 9.5 Fenrite Phase Shifters 473 the guide, with a negligible perturbing effect. The top and bottom walls of the fertile toroid have very little magnetic interaction with the RF fields because the magnetization is not perpendicular to the plane of circular polarization, and the top and bottom magnetizations are oppositely directed. So these walls provide mainly a dielectric loading effect, and the essential operating features of the remanent phase shifters can be obtained by considering the simpler dual ferrite slab geometry of Section 9.3, For a given operating frequency and waveguide size, the design of a remanent dual slab phase shifter mainly involves the determination of the slab thickness, l, the spacing between the slabs, s = 2d — a — 2c — 2t (see Figure 9.10), and the length of the slabs for the desired phase shift. This requires the propagation constants, B±, for the dual slab geometry, which can be numerically evaluated from the transcendental equation of (9.84). This equation requires values for fi and which can be determined from (9.25) for the remanent state by setting H$ = Q (a>G = 0) and Ms = Mr (ojm = fx0yMr): tt = Mo, (9.89a) K = -/to—. (9.89b) The differentia! phase shift, 8+ — ft-, is linearly proportional to for k/mo up to about 0.5. Then, since at is proportional to Mr> as seen by (9.89b), it follows that a shorter ferrite can be used to provide a given phase shift if a ferrite with a higher remanent magnetization is selected. The insertion loss of the phase shifter decreases with length, but is a function of the ferrite linewidth, AH. A figure of merit commonly used to characterize phase shifters is the ratio of phase shift to insertion loss, measured in degrees/dB. EXAMPLE 9.4 REMANENT PHASE SHIFTER DESIGN Design a two-slab remanent phase snifter at 10 GHz using X-band waveguide with ferrite having AirMt = 1786G and €r = 13. Assume that the ferrite slabs are spaced 1 mm apart. Determine the slab thicknesses for maximum differential phase shift, and the lengths of the slabs for 180° and 9(f phase shifter sections. Solution From (9.89) we have M_ _ Mo K Mo " Using a numerical root-finding technique, such as interval halving, we can solve (9.84) for the propagation constants j}+ and P- by using positive and negative values of k. Figure 9.17 shows the resulting differential phase shift, (B+ — ft-)Jh), versus slab thickness, f, for several slab spacings. Observe that the phase shift increases as the spacing, s, between the slabs decreases, and as the slab thickness increases, for t(a up to about 0.12. From the curve in Figure 9.17 for s = 1 mm, we see that the optimum slab thickness for maximum phase shift is tja =0.12, or t = 2.74 mm, since a — 2.286 cm for X-band guide. The corresponding normalized differential phase shift is 0.40, so ^^_=0.^ = 0.4(M^) \ cm ) = 0.836 rad/cm = 48' / cm. that = 1, - 4J£ = ±(2-8MHz/0e><]786G> & ±() 5 co 10,000 MHz 474 Chapter 9: Theory and Design of Ferrimagnetic Components 0.« i Ho \ 1 s 0,04 0.08 0.12 0.16 Slab thickness tia FIGURE 9.17 Differential phase shift for the two-slab remanent phase shifter of Example 94. The ferrite length required for the 180e phase shift section is then 180° L = ——■— = 3.75cm, 48°/cm while the length required for a 90° section is 90" L = 48c/cm = 1.88 cm. Other Types of Ferrite Phase Shifters Many other types of ferrite phase shifters have been developed, with various combinations of rectangular or circular waveguide, transverse or longitudinal biasing, latching or continuous phase variation, and reciprocal or nonreciprocal operation. Phase shifters using printed transmission lines have also been proposed. Even though PIN diode and FET circuits offer a less bulky and more integratable alternative to ferrite components, ferrite phase shifters often have advantages in terms of cost, power handling capacity, and power requirements. But there is still a great need for a low-cost, compact phase shifter. Several waveguide phase shifter designs are derived from the nonreciprocal Faraday rotation phase shifter shown in Figure 9.18. In operation, a rectangular waveguide TEui mode entering at the left is converted to a TEn circular waveguide mode with a short transition section. Then a quarter-wave dielectric plate, oriented 45° from the electric field vector, converts the wave to an RHCP wave by providing a 90° phase difference between the field components that are parallel and perpendicular to the plate. In the ferrite-loaded region the phase delay is jS+z, which can be controlled with the bias field strength. The second quarter-wave plate converts the wave back to a linearly polarized field. The operation is similar for a wave entering at the right, except now the phase delay is 3-z; the phase shift is thus nonreciprocal. The ferrite rod is biased longitudinally, in the direction of propagation, with a solenoid coil. This type of phase shifter can be made reciprocal by using nonreciprocal quarter-wave plates to convert a linearly polarized wave to the same sense of circular polarization for either propagation direction. 9.5 Ferrfte Phase Shifters 475 The Reggia-Spencer phase shifter, shown in Figure 9.19, is a popular reciprocal phase shifter. In either rectangular or circular waveguide form, a longitudinally biased ferrite rod is centered in the guide. When the diameter of the rod is greater than a certain critical size, the fields become tightly bound to the ferrite and are circularly polarized. A large reciprocal phase shift can be obtained over relatively short lengths, although the phase shift is rather frequency sensitive. The Gyrator An important canonical nonreciprocal component is the gyrator, which is a two-port device having a ISO* differential phase shift. The schematic symbol for a gyrator is shown in Figure 9.20, and the scattering matrix for an ideal gyrator is [S] - ~\ oj' (9,90) which shows that it is lossless, matched, and nonreciprocal. Using the gyrator as a bask nonreciprocal building block in combination with reciprocal dividers and couplers can lead to useful equivalent circuits for nonreciprocal components such as isolators and circulators. FIGURE 9.19 Reggia-Spencer reciprocal phase shifter. 476 Chapter 9: Theory and Design of Ferr(magnetic Components I? FIGURE 9.20 Symbol for a gyrator, which has a differential phase shift of 18(F. 9.6 Figure 9.21, for example, shows an equivalent circuit for an isolator using a gyrator and two quadrature hybrids. The gyrator can be implemented as a phase shifter with a 180" differential phase shift; bias can be provided with a permanent magnet, making the gyrator a passive device. FERRITE CIRCULATORS As we discussed in Section 7.1, a circulator is a three-port device that can be lossless and matched at all ports; by using the unitary properties of the scattering matrix we were able to show how such a device must be nonreciprocal. The scattering matrix for an ideal circulator thus has the following form: 0 1 L0 0 0 1 11 0 0 (9.91) which shows that power flow can occur from pons 1 to 2,2 to 3, and 3 to 1, but not in the reverse direction. By transposing the port indices, the opposite circularity can be obtained. In practice, this result can be produced by changing the polarity of the ferrite bias field. Most circulators use permanent magnets for the bias field, but if an electromagnet is used the circulator can operate in a latching (remanent) mode as a single-pole double-throw (SPOT) switch. A circulator can also be used as an isolator by terminating one of the ports with a matched load. A junction circulator is shown in Figure 9.22. We will first discuss the properties of an imperfectly matched circulator in terms of its scattering matrix. Then we will analyze the operation of the stripline junction circulator. The operation of waveguide circulators is similar in principle. Properties of a Mismatched Circulator If we assume that a circulator has circular symmetry around its three ports and is lossless, but not perfectly matched, its scattering matrix can be written as rr d ) is passed, while the reverse wave («-) is absorbed in the matched load of the first hybrid. 9.6 Ferrite Circulators 477 FIGURE 9.22 Photograph of a disassembled ferrite junction circulator, showing the strtpline conductor, the ferrite disks, and the bias magnet. The middle port of the circulator is terminated with a marched load, so this circulator is actually configured as an isolator. Note the change in the width of the stripline conductors, due to die different dielectric constants of the ferrite and the surrounding plastic material. Since the circulator is assumed lossless, [S] must be unitary, which implies the following two conditions: ir|2 + l^l2 + l«l2 = i, rr + «r* + pa* = o. (9.93a) (9.93b) If the circulator were matched (T = 0), then (9.93) shows that either « = 0 and \B\ = 1, or & = 0 and |«| = 1; this describes the ideal circulator with its two possible circularity states. Observe that this condition depends only on a lossless and matched device. Now assume small imperfections, such that |T| o (not to be confused with 0)0 = FMo#o): *n 1841 , ,„R 0^0 = —^= = —=. (9.101) When the ferrite is magnetized there are two possible resonant modes for each value of n, as associated with either a e**^ variation or e~!n4> variation. The resonance condition for the two n — 1 modes is — Ji(x)±J{(x) = 0. (9.102) fix 460 Chapter 9: Theory and Design of Ferrimagnetic Components where x = ka. This result shows the nonreciprocal property of the circulator, since changing the sign of k (the polarity of the bias field) in (9.102) leads to the other root and propagation in the opposite direction in 4>- If we let x+ and jc„ be the two roots of (9.102), then the resonant frequencies for these two n = I modes can be expressed as o>± = Xi (9.103) We can develop an approximate result for to± if we assume that k/(x is small, so that o>± will be close to (oq of (9.101). Using a Taylor series about jc0 for the two terms in (9.102) gives the following results, since Jfoo) = 0; J\(x) ~ y,(^0) + (x - -vo)j|'(*o) = Mxri), Ji(x) - J{(x0) + (x- *b)/ft*tf = -(x - xn) (\ - -pj Jl&&% Then (9.102) becomes — =F(jf± -*o>( 1 - \ J =0, (i ±0.418^, or x± ~ *0 ^1 ± 0418^J i 104^ since xn = 1.841. This result gives the resonant frequencies as w± ^cod^I ±0.418^. (9.105) Note that co± approaches a* as k -» 0, and that Now we can use these two modes to design a circulator. The amplitudes of these modes provide two degrees of freedom that can be used to provide coupling from the input to the output port, and to provide cancellation at the isolated port. It will turn out that &>q will be the operating frequency, between the resonances of the co± modes. Thus, ^ 0 over the periphery of the ferrite disks, since o> ^ co±. If we select port 1 as the input, port 2 as the output, and port 3 as the isolated port, as in Figure 9.23, we can assume the following Ez field at the ports at p = a: Ez{p = a, = 120° (Port 2), (9.106a) 0, for = 240° (Port 3). If the feedlines are narrow, the Ez field will be relatively constant across their width. The corresponding field should be H^p =at<}>) = Hq, for —if < < Hq, for 120° -if < $ < 120° (9.106b) 0, elsewhere. 9.6 Ferrite Circulators 481 Equating (9.106a) to Ez of (9.100a) gives the mode amplitude constants as 2Ji(ka) ' Eo(l-j/^3) Uiikd) Then (9.100a,b) can be reduced to give the electric and magnetic fields as sin0\ (9.107a) (9.107b) EoMkp) / = —, „, ( COS Ó--=- i, (9.108a) 4-m hp fi (9.108b) To approximately equate to in (9.106b) requires that H$ be expanded in a Fourier series: 2H^ h=—co 7t + 5 ]T p + e^2™'V* + (1 + e^™V)e-M\ sin/itfr x-. n (9.109) The n = 1 term of this result is - j n/Š/ío sin f H\iP = a,(j>) = 2n [('^'-('-éH which can now be equated to (9,108b) for p — a. Equivalence can be obtained if two conditions are met: J[{ka) = 0, YEqk v^/Vosin^ and kap, 7T The first condition is identical to the condition for resonance in the absence of bias, which implies that the operating frequency is ojo, as given by (9.101). For a given operating frequency, (9.101) can then be used to find the disk radius, a. The second condition can be related to the wave impedance at port 1 or 2: Eq VŽkap, sin yV ^ psinÝ izYk kY (9.110) 462 Chapter 9: Theory and Design of FerrimagnetJc Components l£,l t t t Input Output isolated FIGURE 9.24 Magnitude of the electric field around the periphery of the junction circulator. since \flka}ii = V3( 1 -841 )/ir ~ 1.0. Thus, Zw can be controlled for impedance matching by adjusting k via the bias field. We can compute the power flows at the three ports as follows: P\n = Pi = —p ■ Ě X H* = PM = P2 = p.Ěx H* = ~ELH EqHqsin^ _ EIkY <í=íl 71 Hp Eotfflsin^r EqkY Plso = P3 = p ■ Ě x H* = -EZH4 ^=120' 0=240'- 7t TtfX - 0. (9.111a) (9.111b) (9.111c) This shows that power flow occurs from port 1 to 2, but not from 1 to 3, By the azimuthal symmetry of the circulator, this also implies that power can be coupled from port 2 to 3, or from port 3 to 1, but not in the reverse directions. The electric field of (9.108a) is sketched in Figure 9.24 along the periphery of the circulator, showing that the amplitudes and phases of the e±J modes are such that their superposition gives a null at the isolated port, with equal voltages at the input and output ports This result ignores the loading effect of the input and output lines, which will distort the field from that shown in Figure 9.24. This design is narrowband, but bandwidth can be improved using dielectric loading; the analysis then requires consideration of higher order modes. REFERENCES [ 1] R. F. Soohoo, Microwave Magnetics, Harper and Row, N.Y., 1985. [2] A. J. Baden Fuller, Ferriies at Microwave Frequencies, Peter Peregrinus, London, 1987. [3] R. E. Collin, Field Theory of Guided Waves, McGraw-Hill, N.Y.. 1960. [4] B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnencs^ McGraw-Hill, N.Y., 1962. [5] F. E. Gardiol and A. S. Vander Vorst, "Computer Analysis of E-plane Resonance Isolators," IEEE Trans. Microwave Theory and Techniques, vol. MTT-19, pp. 315-322, March 1971. [6] G. P. Rodrigue, "A Generation of Microwave Ferrite Devices," Proc. IEEE, vol. 76, pp. 121-137, February 19S8. [7] C. E. Fay and R. L. Comstock, "'Operation of the Ferrite Junction Circulator," IEEE Trans. Microwave Theory and Techniques, vol. MTT-13, pp. 15-27. January 1965. Problems 483 PROBLEMS 9.1 A certain ferrite material has a saturation magnetization of 4jr Ms = 1780G. Ignoring loss, calculate the elements of the permeability tensor at / = 10 GHz for two cases: (a) no bias field and ferrite demagnetized (Ms = Hi, = 0) and (b) a z-directed bias field of 1000 oersted. 9.2 Consider the following field transformations from rectangular to circular polarized components: «* = + jBy)/2, H+ = (Hx +jHy)/2, B~ = (& - j By)/2, H~ = (H, - jfiy)/2, For a s-biased ferrite medium, show that the relation between B and H can be expressed in terms of a diagonal tensor permeability as follows: 0 0 * -H+- B~ 0 (il-K) 0 H- B; . 0 0 ■ m i 9.3 A Y1G sphere with AnMs = 1780G lies in a uniform magnetic field having a strength of 1200 Oe. What is the magnetic field strength inside the YIG sphere? 9.4 A thin rod is biased along its axis with an external applied field of Ha = 1000 Oe. If 4tcMs = 600 G, calculate the gyromagnetic resonance frequency for the rod. 9.5 An infinite lossless ferrite medium with a saturation magnetization of 4n Ms = 1200 G and a dielectric constant of 10 is biased to a field strength of 500 oersted. At 8 GHz, calculate the differential phase shift per meter between an RHCP and an LHCP plane wave propagating in the direction of bias. If a linearly polarized wave is propagating in this material, what is die distance it must travel in order that its polarization is rotated 90"? 9.6 An infinite lossless ferrite medium with a saturation magnetization of 4:r Afs = 1780 G and a dielectric constant of 13 is biased in the Jt direction with a field strength of 2000 oersted, At 5 GHz, two plane waves propagate in the +z direction, one linearly polarized in x and the other linearly polarized in v. What is the distance these two waves must travel so that the differential phase shift between them is 180"? 9.7 Consider a circularly polarized plane wave normally incident on an infinite ferrite medium, as shown in the following figure. Calculate the reflection and transmission coefficients for an RHCP (T+. T+) and an LHCP T~) incident wave. HINT: The transmitted wave will be polarized in the same sense as the incident wave, but the reflected wave will be oppositely polarized. 484 Chapter 9: Theory and Design of Ferrimagnetic Components 9.8 An infinite lossless ferrite material with An Ms — 1200G is biased in the x direction with H$ = Hgx. Determine the range of Ha, in oersteds, where an extraordinary wave (polarized in propagating in Z) will be cutoff. The frequency is 4 GHz. 9.9 Find the forward and reverse propagation constants for a waveguide half-filled with a transversely biased ferrite. (The geometry of Figure 9.9 with c — 0 and r = a/2,) Assume a = 1,0 cm, / = 10 GHz, AnM, = 1700 G, and e, = 13. Plot versus #„ = 0 to 1500 Oe. Ignore loss and the fact that the ferrite may not be saturated for small H0, 9.10 Find the forward and reverse propagation constatits for a waveguide filled with two pieces of oppositely biased ferrite. (The geometry of Figure 9.10 wirh c = 0 and t — a/2.) Assume a — 1.0 cm, / — 10 GHz, An Mt = t 700 G, and €, = 13. Plot versus H0 = 0 to 1500 Oe. Ignore loss and the fact that the ferrite may not be saturated for small Hg. 9.11 Consider a wide, thin ferrite slab in a rectangular X-band waveguide, as shown in Figure 9.1 lb. If / = 10 GHz, Ax Ms = 1700 G, c = a/4, and AS = 2 mm2, use the perturbation formula of (9.80) to plot the differential phase shirt, tp* - £_)/£&, versus the bias field for Ha = 0 to 1200 Oe. Ignore loss, 9.12 An E-plane resonance isolator with the geomeuy of Figure 9.1J a is to be designed to operate at 8 GHz, with a ferrite having a saturation magnetization of 4ttMs — \ 500 G. (a) What is the approximate bias field, H0, required for resonance? (b) What is the required bias field if the //-plane geometry of Figure 9.11 b is used? 9.13 Design a resonance isolator using the //-plane ferrite slab geometry of Figure 9.11b in an X-band waveguide. The isolator should have minimum forward insertion loss, and a reverse attenuation of 30 dB at 10 GHz. Use a ferrite slab having AS/S = 0.01, AnMs = 1700 G, and AH = 200 Oe. 9.14 Calculate and plot the two normalized positions, x/a, where the magnetic fields of the TEl0 mode of an empty rectangular waveguide are circularly polarized, for k$ = i& to 2k?, 9.15 The latching ferrite phase shifter shown in the figure below uses the birefringence effect. In state 1, the ferrite is magnetized so that H0 — 0 and M = Mrx. In state 2, the ferrite is magnetized so thai H0 = 0 and M = Mry. If / = 5 GHz, €r = 10, AnMr = 1200 G, and L = 3.65 cm, calculate the differentia] phase shift between the two states. Assume the incident plane wave is x polarized for both states, and ignore reflections. 9.16 Rework Example 9.4 with a slab spacing of s = 2 nun, and a remanent magnetization of 1000G. (Assume all other parameters as unchanged, and that the differential phase shift is linearly proportional to a:.) 9.17 Consider a latching phase shifter constructed with a wide, tbin if-plane ferrite slab in an X-band waveguide, as shown in Figure 9.11b. If / = 9 GHz, AnMr = 1200G, c = a/4, and AS = 2mm2, use the perturbation formula of (9.80) to calculate the required length for a differential phase shift of 22.5". 9.18 Design a gyrator using the twin //-plane ferrite slab geometry shown on the next page. The frequency is 9.0 GHz, and the saturation magnetization is 4wMs = 1700 G, The cross-sectional area of each slab is 3,0 mm2, and the guide is X-band waveguide. The permanent magnet has a field strength of H9 = 4000 Oe. I>eterrnine the internal field in the ferrite, Ha, and use the perturbation formula of (9.80) to Problems 485 determine the optimum location of the slabs and the length, L, to give the necessary 180° differential phase shift. 9.19 Draw an equivalent circuit for a circulator using a gyrator and two couplers. 920 A certain lossless circulator has a return loss of 10 dB, What is the isolation? What is the isolation if the return loss is 20 dB? Noise and Active RF Components The effect of noise is critical to the performance of most RF and microwave communications, radar, and remote sensing systems because noise ultimately determines the threshold for the minimum signal that can be reliably detected by a receiver. Noise power in a receiver will be introduced from the external environment through the receiving antenna, as well as generated internally by the receiver circuitry. Here we will study the sources of noise in microwave systems, and the characterization of microwave components in terms of noise temperature and noise figure, including the effect of impedance mismatch. We will also discuss the related topics of dynamic range and intermodulation distortion, which are important considerations when large signal levels are present in nonlinear components such as diodes and transistors. The additional noise-related topics of oscillator phase noise and antenna noise temperature will be discussed in Chapters 12 and 13. The components and circuits that we have discussed so far have been linear and passive, but practical microwave systems invariably require the use of some nonlinear and active components. Such devices, which include diodes, transistors, and electron tubes, can be used for signal detection, mixing, amplification, frequency multiplication, switching, and as sources of microwave and RF signals. We will discuss some of the basic characteristics of microwave diodes and transistors in this chapter, and present equivalent circuits for these devices. We will avoid any discussion of the physics of diodes or transistors (see references [l]-[6] tor such material), since for our purposes it will be adequate to characterize these devices in terms of their circuit properties. These results will be used to study some basic diode detector and control circuits, and in later chapters for the design of amplifier, mixer, and oscillator circuits using diodes and transistors. We conclude this chapter with an overview of microwave integrated circuits. The earliest detector diode was probably the "cat-whisker" crystal detector used in early radio work. The advent of tubes used as detectors and amplifiers eliminated this component in most radio systems, but the crystal diode was later used by Southworth in his 1930s experiments with waveguides, since the tube detectors of that era could not operate at such high frequencies. Frequency conversion and heterodyning were also first developed for radio applications, in the 1920s. These same techniques were later applied to microwave radar receiver design at the MIT Radiation Laboratory during World War II using crystal diodes as mixers [1], but it was 486 487 not until the 1960s that microwave solid-state devices saw significant development. PIN diodes were invented, and used as microwave switches and phase shifters. The basic theory of the field effect transistor (FET) was developed by Shockley in 1952, and the first FETs were fabricated on silicon. The first microwave gallium arsenide FETs were developed in the late 1960s [2], The logical trend for microwave circuits has since been to integrate transmission lines, active devices, and other components on a single semiconductor substrate to form a monolithic microwave integrated circuit (MMIC). The first single-function MMICs were developed in the late 1960s, but more sophisticated circuits such as multistage FET amplifiers, 3- or 4-bit phase shifters, complete uransrnit/receive radar modules, and other circuits are now being fabricated as MMICs [2], The present trend is toward MMICs with higher performance, tower cost, and greater complexity. NOISE IN MICROWAVE CIRCUITS Noise power is a result of random processes such as the flow of charges or holes in an electron tube or solid-state device, propagation through the ionosphere or other ionized gas, or, most basic of all, the thermal vibrations in any component at a temperature above absolute zero. Noise can be passed into a microwave system from external sources, or generated within the system itself. In either case the noise level of a system sets the lower limit on the strength of a signal that can be detected in the presence of the noise. Thus, it is generally desired to minimize the residual noise level of a radar or communications receiver, to achieve the best performance. In some cases, such as radiometers or radio astronomy systems, the desired signal is actually the noise power received by an antenna, and it is necessary to distinguish between the received noise power and the undesired noise generated by the receiver system itself. Dynamic Range and Sources of Noise In previous chapters we have implicitly assumed that all components were linear, meaning that the output is directly proportional to the input, and deterministic, meaning that the output is predictable from the input. In reality no component can perform in this way over an unlimited range of input/output signal levels. In practice, however, there is a range of signal levels over which such assumptions are valid; this range is called the dynamic range of the component. As an example, consider a realistic microwave transistor amplifier having a gain of 10 dB, as shown in Figure 10.1. If the amplifier were ideal, the output power would be related to the input power as foui = 10fm: and this relation would hold true for any value of PiD. Thus if Pia = 0, we would have Font = 0, and if Fin = 106 W, we would have Pout = 107 W. Obviously neither of these 488 Chapter 10; Noise and Active RF Components P«i jf. -70 -60 -50 -40 -30 -20 -10 0 10 20 30 Pin (dBmj FIGURE 10.1 Illustrating the dynamic range of a realistic amplifier. conditions will be true in practice. Because of noise generated by the amplifier itself, a certain nonzero noise power will be delivered by the amplifier even when the input power is zero. For very high input powers, the amplifier will be destroyed. Thus, the actual relation between the input and output power will be as shown in Figure 10.1. At very low input power levels, the output will be dominated by the noise of the amplifier. This level is often called the noise jioor of the component or system; typical values may range from —60 dBm to -100 dBm over the bandwidth of the system, with lower values being obtainable with cooled components. Above the noise floor, the amplifier has a range of input power for which Pom = \0Pm is closely approximated. This is the usable dynamic range of the component. At the upper end of the dynamic range, the output begins to saturate, meaning that the output power no longer increases linearly as the input power increases. A quantitative measure of the onset of saturation is given by the 1 dB compression point, which is defined as the input power for which the output is 1 dB below that of the ideal amplifier (the corresponding ounput power level can also be used to specify this point). If the input power is excessive, the amplifier can be destroyed. Noise is usually generated by me random motions of charges or charge carriers in devices and materials. Such motions can be caused by any of several mechanisms, leading to various sources of noise: • Thermal noise is the most basic type of noise, being caused by thermal vibration of bound charges. Also known as Johnson or Nyquist noise. • Shot noise is due to random fluctuations of charge carriers in an electron tube or solid-state device. • Flicker noise occurs in solid-state components and vacuum tubes. Flicker noise power varies inversely with frequency, and so is often called 1 //-noise, • Plasma noise is caused by random motion of charges in an ionized gas, such as a plasma, the ionosphere, or sparking electrical contacts. • Quantum noise results from the quantized nature of charge carriers and photons; often insignificant relative to other noise sources. 10.1 Noise in Microwave Circuits 489 FIGURE 10.2 A random voltage generated by a noisy resistor. It is sometimes necessary for measurement purposes to have a calibrated noise source. Passive noise generators consist of a resistor held at a constant temperature, either in a temperature-controlled oven or a cryogenic flask. Active noise sources can be made using gas-discharge tubes or avalanche diodes; such sources generally give much higher noise power than passive sources. Noise Power and Equivalent Noise Temperature Consider a resistor at a temperature of T degrees kelvin (K), as depicted in Figure 10.2. The electrons in this resistor are in random motion, with a kinetic energy that is proportional to the temperature, T. These random motions produce small, random voltage fluctuations at the resistor terminals, as illustrated in Figure 10.2. This voltage has a zero average value, but a nonzero tms value given by Planck's black body radiation law, h = 6.626 x 10"34 J-sec is Planck's constant. k = 1.380 x 10"23J/°K is Boltzmann's constant. T is the temperature in degrees kelvin (K). B is the bandwidth of the system in Hz. / is the center frequency of the bandwidth in Hz. R is the resistance in £2. This result comes from quantum mechanical considerations, and is valid for any frequency, f. At microwave frequencies the above result can be simplified by making use of the fact that hf -o-■• FIGURE 10,3 Equivalent circuit of a noisy resistor delivering maximum power to a load resistor through an ideal bandpass filter. approximation may be invalid, in which case (10.1) should be used. Note that this noise power is independent of frequency; such a noise source has a power spectral density that is constant with frequency, and is referred to as a white noise source. The noise power is direcUy proportional to the bandwidth, which in practice is usually limited by the passband of the microwave system. Since independent white noise sources can be treated as Gaussian distributed random variables, the noise powers (variances) are additive. The noisy resistor of Figure 10.2 can be replaced with a Thevenin equivalent circuit consisting of a noiseless resistor and a generator with a voltage given by (10,2), as shown in Figure 10.3. Connecting a load resistor R results in maximum power transfer from the noisy resistor, with the result that power delivered to the load in a bandwidth B, is p^(mf*^m since V„ is an rms voltage. This important result gives the maximum available noise power from the noisy resistor at temperature T. Observe the following trends: • As B -¥ 0, P„ 0, This means that systems with smaller bandwidths collect less noise power. • As T —► 0, P„ —* 0. This means that cooler devices and components generate less noise power. • As B —>■ co, P„ —* oo. This is the so-called ultraviolet catastrophe, which does not occur in reality because (10.2-10.3) are not valid as / (or B) oo; (10.1) must be used in this case. If an arbitrary source of noise (thermal or nonthermal) is "white," so uiat the noise power is not a strong function of frequency, it can be modeled as an equivalent thermal noise source, and characterized with an equivalent noise temperature. Thus, consider the arbitrary white noise source of Figure 10.4, which has a driving-point impedance of R and delivers a noise power Nc to a load resistor R. This noise source can be replaced by a noisy resistor of value Rt at temperature TV, where Te is an equivalent temperature selected so Arbitrary white noise source FIGURE 10.4 The equivalent noise temperature, Tei of an arbitrary white noise source. 10.1 Noise in Microwave Circuits 491 that the same noise power is delivered to the load. That is, Components and systems can then be characterized by saying that they have an equivalent noise temperature, Te; this implies some fixed bandwidth, b, which is generally the bandwidth of the component or system For example, consider a noisy amplifier with a bandwidth fl and gain G. Let the amplifier be matched to noiseless source and load resistors, as shown in Figure 10.5. If the source resistor is at a (hypothetical) temperature of Ts = 0 A\ then the input power to the amplifier will be = 0, and the output noise power N0 will be due only to the noise generated by the amplifier itself. We can obtain the same load noise power by driving an ideal noiseless amplifier with a resistor at a temperature, so that the output power in both cases is N0 = GkT£B. Then Te is the equivalent noise temperature of the amplifier. Active noise sources use a diode or tube to provide a calibrated noise power output, and are useful for test and measurement applications. Active noise generators can be characterized by an equivalent noise temperature, but a more common measure of noise power for such components is the excess noise ratio (ENR), defined as ENR(dB) = 10 log —^77—^ = 10 log fcS, (10.6) where Ns and Tg are the noise power and equivalent temperature of the generator, and N0 and Tb are the noise power and temperature associated with a room-temperature passive source (a matched load). Solid-state noise generators typically have ENRs ranging from 20 to40dB. figure 10.5 Defining the equivalent noise temperature of a noisy amplifier, (a) Noisy amplifier, (b) Noiseless amplifier. 492 Chapter 10: Noise and Active RF Components ration I—WVv—3 FIGURE 10.6 The T*-factor method for measuring the equivalent noise temperature of an amplifier. Measurement of Noise Temperature In principle, the equivalent noise temperature of a component can be detennined by measuring the output power when a matched load at 0 K. is connected at the input of die component. In practice, of course, the 0 K source temperature cannot be achieved, so a different method must be used. If two loads at significantiy different temperatures are available, then the Y-factor method can be applied. This technique is illustrated in Figure 10.6, where the amplifier (or other component) under test is connected to one of two matched loads at different temperatures, and the output power is measured for each case. Let Ti be the temperature of the hot load, and T2 the temperature of the cold load (T\ > T±\ and let P\ and Pi be the respective powers measured at the amplifier output. The output power consists of noise power generated by the amplifier as well as noise power from the source resistor. Thus we have Ni = GkT\ B + GkTeB, (10,7a) N2 = GkT2B + GkTeB, (10.7b) which are two equations for the two unknowns, Te and GB (the gain-bandwidth product of the amplifier). Define the Y-factor as X2 T2 + Te which is determined via the power measurements. Then (10.7) can be solved for the equivalent noise temperature, Te=TA^- H0.9) in terms of the load temperatures and the Y-factor. Observe that to obtain accurate results from this method, the two source temperatures must not be too close together. If they are, N\ will be close to N2, Y will be close to unity, and the evaluation of (10.9) will involve die subtractions of numbers close to each other, resulting in a loss of accuracy, In practice, one noise source is usually a load resistor at room temperature (Z0), while the other noise source is either "hotter" or "colder," depending on whether Te is greater or lesser than Tq. An active noise generator can be used as a "hotter" source, while a "colder" source caD be obtained by immersing a load resistor in liquid nitrogen (T = 11K), or liquid helium (T = 4 K). EXAMPLE 10.1 NOISE TEMPERATURE MEASUREMENT An X-band amplifier has a gain of 20 dB and a 1 GHz bandwidth. Its equivalent noise temperature is to be measured via the T-factor method. The following data is obtained: for ti = 290 K, TV] = -62.0 dBm. for T2 = 77 K, N2 = -64.7 dBm. 10.1 Noise in Microwave Circuits 493 Determine the equivalent noise temperature of the amplifier. If the amplifier is used with a source having an equivalent noise temperature of Ts = 450 K, what is the output noise power in dBm? Solution From (10.8), the K-factor in dB is Y = (N[ - N2) dB = (-62.0) - (-64.7) = 2.7 dB. which is a numeric value of Y = 1.86. Then using (10.9) gives the equivalent noise temperature as = Tj - YT2 _ 290-(1.86X77) _ y-1 1,86-1 If a source with an equivalent noise temperature of 7V = 450 K drives the amplifier, the noise power into the amplifier will be kT^B, The total noise power out of the amplifier will be N0 = GkTsB 4- GkTeB = 100(1.38 x 10-23)(109)(450+ 170) = 8.56 x 10^10 W = -60.7dBm. ■ Noise Figure We have seen that a noisy microwave component can be characterized by an equivalent noise temperature. An alternative characterization is the noise figure of the component, which is a measure of the degradation in the signal-to-noise ratio between the input and output of the component. The signal-to-noise ratio is the ratio of desired signal power to undesired noise power, and so is dependent on the signal power. When noise and a desired signal are applied to the input of a noiseless network, both noise and signal will be attenuated or amplified by the same factor, so that the signal-to-noise ratio will be unchanged. But if the network is noisy, the output noise power will be increased more than the output signal power, so that the output signal-to-noise ratio will be reduced. The noise figure, F, is a measure of this reduction in signal-to-noise ratio, and is defined as F = > 1, (10.10) where S,, $j are the input signal and noise powers, and S0, Mb are the output signal and noise powers. By definition, the input noise power is assumed to be the noise power resulting from a matched resistor at r0 = 290 K; that is, Nt = kTGB. Consider Figure 10.7, which shows noise power Nt and signal power 5, being fed into a noisy two-port network. The network is characterized by a gain G, a bandwidth B, and an equivalent noise temperature, Te. The input noise power is rV,- = kToB, and the FIGURE 10.7 Deterniining the noise figure of a noisy network. 494 Chapter 10: Noise and Active RF Components \ \ N;=kTB I -—»- £.. T, Z„ = R = kTB -3 FIGURE 10.8 Determining the noise figure of a lossy line or attenuator with loss L and temperature T. output noise power is a sum of the amplified input noise and the internally generated noise: N0 = kGB(T$ + Te). The output signal power is S„ = GSj. Using these results in (10.10) gives the noise figure as kTaB GSj T0 In dB, F — 101og(l + Tp/To) dB > 0. If the network were noiseless, Tc would be zero, giving F - 1, or 0 dB. Solving (10.11) for Te gives Te = (F-m- (10.12) It is important to keep in mind two things concerning the definition of noise figure: noise figure is defined for a matched input source, and for a noise source that consists of a resistor at temperature T0 = 290 K. Noise figure and equivalent noise temperatures are interchangeable characterizations of the noise properties of a component. An important special case occurs in practice when the two-port network is a passive, lossy component, such as an attenuator or lossy transmission fine, held at a temperature, T. Consider such a network with a matched source resistor, which is also at temperature T, as shown in Figure 10.8. The gain, G, of a lossy network is less than unity; the loss factor, L, can be defined as L = 1 / G > 1. Because the entire system is in thermal equilibrium at the temperature T, and has a driving point impedance of R, the output noise power must be N0 = kTB. But we can also think of this power as coming from the source resistor (through the lossy line), and from the noise generated by the line itself. Thus we also have that N0 = kTB = GkTB + GN^aai- (10.13) where /Va, and the reflection coefficient looking toward the 10.1 Noise in Microwave Circuits 499 © L T, Za. 3 FIGURE 10.12 A lossy transmission line at temperature T with an impedance mismarch at its input port, generator can be written as Z„ — Zn rs -- J—t~ ž 0- Zg + Za The scattering matrix of the lossy line of characteristic impedance Zq can be written as rsK> i] (10.30) where /3 is the propagation constant of the line. Using (10.26) gives the reflection coefficient looking into port 2 of the line as raut = S22 +--rr-p- = —e JF ■ 1 — Aljl j i (10.311 Then the available gain, from (10.25), is I G21 = r(i-ir,|-) L(1_,r^) ouU (10.32) We can verify two limiting cases of (10.32): when L = 1 we have Gu = 1, and when r, = 0 we have G2\ = \jL. Using (10.32) in (10.28) gives the equivalent noise temperature of the mismatched lossy line as _ 1 - G21 ^ _ jL- lKL + mp), (10.33) The corresponding noise figure can then be evaluated using (10.11). Observe that when the line is matched, Vs = 0 and (10.33) reduces to Te = (L - 1)7, in agreement with the result for the matched lossy line given by (10.15). If the line is lossless, then L — 1 and (10.33) reduces to 7^ = 0 regardless of mismatch, as expected, But when the line is lossy and mismatched, so that L > 1 and \TS\ > 0, then the noise temperature given by (10.33) is greater than Te = (L - 1 )T, the noise temperature of the matched lossy line. The reason for this increase is that the lossy line actually delivers noise power out of both its ports, but when the input port is mismatched some of the available noise power at port 1 is reflected from the source back into port 1, and appears at port 2. When the generator is matched to port 1, none of the available power from port 1 is reflected back into the line, so the noise power available at port 2 is a minimum. EXAMPLE 10 J APPLICATION TO A WILKINSON POWER DIVIDER Find the noise figure of a Wilkinson power divider when one of the output ports is terminated in a matched load. Assume an insertion loss factor of L from the input to either output port. 500 Chapter 10: Noise and Active RF Components Solution From Chapter 7 the scattering matrix of a Wilkinson divider is given as: [S] = V2L "0 1 f I 0 0 1 0 0 where the factor L > 1 accounts for the dissipative loss from port 1 to port 2 or 3. To evaluate the noise figure of the Wilkinson divider, we first terminate port 3 with a matched load; this converts the 3-port device to a 2-port device. If we assume a matched source at port 1, we have Fs = 0. Equation (10.26) then gives roul = S22 = 0, and so the available gain can be calculated from (10,25) as 021 — \S2]\2 = ^TJ-Then the equivalent noise temperature of the Wilkinson divider is, from (10.28), T, = 2iT = (21 - 1)7, G21 where T is the physical temperature of the divider. Using (10.11) gives the noise figure as F = l + £ = l + (2L-l)£. Observe that if the divider is at room temperature, then T = To and the above reduces to F = 2L. If the divider is at room temperature and lossless, this reduces to F = 2 = 3 dB. In this case the source of the noise power is the isolation resistor contained in the Wilkinson divider circuit. Because the network is matched at its input and output, it is easy to obtain these same results using Uie mermodynarnic argument directiy. Thus, if we apply an input noise power of kTB to port 1 of the matched divider at temperature T, the system will be in thermal equilibrium and the output noise power must therefore be kTB. We can also express the output noise power as the sum of the input power times the gain of the divider, and A^dd^, the noise power added by the divider itself (referenced to the input to the divider): kTB = ——h --—. 2L 2L Solving for /V^ded gives A^ded = kTB(2L — 1), so the equivalent noise temperature is kB in agreement with the above, ■ DYNAMIC RANGE AND INTERMODULATION DISTORTION Since thermal noise is generated by any lossy component, and all realistic components have at least a small loss, the ideal linear component or network does not exist in the sense that its output response is always exactly proportional to its input excitation. Thus, all realistic devices are nonlinear at very low power levels due to noise effects. In addition, all practical 10.2 Dynamic Range and Intermodulation Distortion 501 Nonlinear v0 device or network FIGURE 10.13 A general nonlinear device or network. components also become nonlinear at high power levels. This may ultimately be the result of catastrophic destruction of the device at very high powers or, in the case of active devices such as diodes and transistors, due to effects such as gain compression or the generation of spurious frequency components due to device nonlinearities. In either case these effects set a minimum and maximum realistic power range, or dynamic range, over which a given component or network will operate as desired. In this section we will study dynamic range, and the response of nonlinear devices in general. These results will be useful for our later discussions of amplifiers (Chapter 11), mixers (Chapter 12), and wireless receiver design (Chapter 13). Devices such as diodes and transistors are nonlinear components, and it is this nonlin-earity that is of great utility for functions such as amplification, detection, and frequency conversion [1J. Nonlinear device characteristics, however, can also lead to undesired responses such as gain compression and the generation of spurious frequency components. These effects may produce increased losses, signal distortion, and possible interference with other radio channels or services. Figure 10.13 shows a general nonlinear network, having an input voltage V[ and an output voltage tv In the most general sense, the output response of a nonlinear circuit can be modeled as a Taylor series in terms of the input signal voltage; v0 — «o + a) Vi + a2v] + a-$vf j----. (10.34) where the Taylor coefficients are defined as ao = va(0) (DC output) (10.35a) (linear output) (10.35b) (squared output) (10.35c) and higher order terms. Thus, different functions can be obtained from the nonlinear network depending on the dominance of particular terms in the expansion. If ao is the only nonzero coefficient in (10.34), the network functions as a rectifier, converting an AC signal to DC. If a i is the only nonzero coefficient, we have a linear attenuator (a \ < 1) or amplifier (a i > 1). If «2 is the only nonzero coefficient, we can achieve mixing and other frequency conversion functions. Usually, however, practical devices have a series expansion containing many nonzero terms, and a combination of several of these effects will occur. We consider some important special cases below. Gain Compression First consider the case where a single frequency sinusoid is applied to the input of a general nonlinear network, such as an amplifier: a-> — dvo dVj d2v, tivl Bj=0 Vi = VoCOSftV. (10.36) 502 Chapter 10: Noise and Active RF Components Then (10.34) gives the output voltage as V0 = ffo + fll Vq COS COQt + £72 V02 cos2 0>ol + #3 Vo3 cos3 COqZ H---- = (a0 + \a2 V02) + Vo + | «3 Vq3) cos + ^2 V02 cos 2o' + ■■■■ (10.37) This result leads to the voltage gain of the signal component at frequency coq: _^_a,y0 + j^o3 3 2 _ -~|7--a' + 4<*3 V0 ' (10. JS) where we have retained only terms through the third order. The result of (10.38) shows that the voltage gain is equal to the «j coefficient, as expected, but with an additional term proportional to the square of the input voltage amplitude. In most practical amplifiers a% is typically negative, so that the gain of the amplifier tends to decrease for large values of Vq. This effect is called gain compression, or saturation. Physically, this is usually due to the fact that the instantaneous output voltage of an amplifier is limited by the power supply voltage used to bias the active device. Smaller values of «3 will lead to higher output voltages. A typical amplifier response is shown in Figure 10.14, For an ideal linear amplifier a plot of the output power versus input power is a straight line with a slope of unity, and the gain of the amplifier is given by the ratio of the output power to the input power. The amplifier response of Figure 10.14 tracks the ideal response over a limited range, then begins to saturate, resulting in reduced gain. To quantify the linear operating range of the amplifier, we define the I dB compression point as the power level for which the output power has decreased by 1 dB from the ideal characteristic. This power level is usually denoted by Pj, and can be stated in terms of either input power or output power, For amplifiers Pi is usually specified as an output power, while for mixers Pj is usually specified in terms of input power. Inlermodulation Distortion Observe from the expansion of (10.37) that a portion of the input signal at frequency too is converted to other frequency components. For example, the first term of (10.37) represents 10.2 Dynamic Range and Intermodulation Distortion 503 a DC voltage, which would be a useful response in a rectifier application. The voltage components at frequencies 2coq or 3o>o can be useful for frequency multiplier circuits. In amplifiers, however, the presence of other frequency components will lead to signal distortion if those components are in the passband of the amplifier. For a single input frequency, or tone, (Oq, the output will in general consist of harmonics of the input frequency of the form ncoa> for n = 0,1,2, — Usually these harmonics lie outside the passband of the amplifier, and so do not interfere wim the desired signal at frequency a>o. The situation is different, however, when the input signal consists of two closely spaced frequencies. Consider a two-tone input voltage, consisting of two closely spaced frequencies, a>i and ioj: vt = V*o(cosa>i? + costly/). (10.39) From (10.34) the output is v6 =gq + a\ V0(cos io\ t + cos co^t) + ai V02(cos im\ t + cos ant)1 + 03 (cos a>i t + cos o>itf H---- = ao + a{ Vocostoif -Mi Vocostt^r -\-^atV^{\ + cos2*^0 +^2V02(1 + cosloy^t) + a2V02 cos(<0] - ff>i)t + 02V02 cos(wi + tt -1- | cos 32t) + + a-% V03 [| cosa^r + I cos(2a*i - a>2)t + | cos(2a>j + on)t\ + + a3 V§. [f cosftj,f + I cos(2a»2 - <£>\)t + | cosQxoi + ] + rto*2, (10.41) withm, n — 0, ±1, ±2, ±3.....These combinations of the two input frequencies are called intermodulation products, and the order of a given product is defined as \m\ + \n |. For example, the squared term of (10.40) gives rise to the following four intermodulation products of second order: 2(oy (second harmonic ofwi) m = 2 n = Q order = 2, 2ol>2 (second harmonic of a>i) m = 0 n — 2 order — 2, $1 — a>2 (difference frequency) m = 1 n = -1 order = 2, a>i + CO2 (sum frequency) m = 1 n = 1 order = 2. All of these second-order products are undesired in an amplifier, but in a mixer the sum or difference frequencies form the desired outputs. In either case, if i, 3^, 2w\ + a>i, 2o>2 + o>i, 2oi\ — i, and 2oh — • The first tour of these will again be located far from u\ or (02, and will typically be outside the passband of the component. But the 604 Chapter 10: Noise and Active RF Components 0 i-j2-„>, ft>2 +«t>i 3"i / \ 3«S FIGURE 10.15 Output spectrum of second- and third-order two-tone intermodulation products, assuming tu, < a>2, two difference terms produce products located near the original input signals at oji and a>2, and so cannot be easily filtered from the passband of an amplifier. Figure 10.15 shows a typical spectrum of the second- and third-order two-tone intermodulation products. For an arbitrary input signal consisting of many frequencies of varying amplitude and phase, the resulting in-band intermodulation products will cause distortion of the output signal. This effect is called third-order intermodulation distortion. Third-Order Intercept Point Equation (10.40) shows that as the input voltage Vb increases, the voltage associated with the third-order products increases as V03. Since power is proportional to the square of voltage, we can also say that the output power of third-order products must increase as the cube of the input power. So for small input powers the third-order intermodulation products must be very small, but will increase quickly as input power increases. We can view this effect graphically by plotting the output power for the first- and third-order products versus input power on log-log scales (or in dB), as shown in Figure 10.16. The output power of the first-order, or linear, product is proportional to the input power, and so the line describing this response has a slope of unity (before the onset of compression). The line describing the response of the third-order products has a slope of 3. (The second-order products would have a slope of 2, but since these products are generally not in the passband of the component, we have not plotted their response in Figure 10.16.) Both the linear and third-order responses will exhibit compression at high input powers, so FIGURE 10.16 Third-order intercept diagram for a nonlinear component. 10.2 Dynamic Range and Intermediation Distortion 505 we show the extension of their idealized responses with dotted lines. Since these two lines have different slopes, they will intersect, typically at a point above the onset of compression, as shown in the figure. This hypothetical intersection point, where the first-order and third-order powers are equal, is called the third-order intercept point, denoted PjT and specified as either an input or an output power. Usually P3 is referenced at the output for amplifiers, and at the input for mixers. As depicted in Figure 10.16, P3 generally occurs at a higher power level than P%, the 1 dB compression point. Many practical components follow the approximate rule that Pj is 12 to 15 dB greater than Pi, assuming these powers are referenced at the same point. We can express P3 in terms of the Taylor coefficients of the expansion of (10.40) as follows. Define P^ as the output power of the desired signal at frequency a>\. Then from (10.40) we have %y = M vl 00-42) Similarly, define P^-wi as the output power of the intermodulation product of frequency 2co\ - a>2. Then from (10.40) we have P^-^ = | H^Vif = ^a|V06. (10.43) By definition, these two powers are equal at the third-order intercept point. If we define the input signal voltage at the intercept point as V;f, then equating (10.42) and (10,43) gives 2al VIP — 32"3 Vtf Solving for V!F yields (10.44) Since P3 is equal to the linear response of P^, at the intercept point, we have from (10.42) and (10.44) that ^=^lv0=v,P = H^ = ^ 00.45) where P3 in this case is referred to the output port. This expression will be useful in the following section. Dynamic Range We can define dynamic range in a general sense as die operating range for which a component or system has desirable characteristics. For a power amplifier this may be the power range that is limited at the low end by noise and at the high end by the compression point. This is essentially the linear operating range for the amplifier, and is called the linear dynamic range (DR(). For low-noise amplifiers or mixers, operation may be limited by noise at the low end and the maximum power level for which intermodulation distortion becomes unacceptable. This is effectively the operating range for which spurious responses are minimal, and is called the spurious-free dynamic range (DRy). We thus compute the linear dynamic range DRi as the ratio of Pi, the 1 dB compression point, to the noise level of the component, as shown in Figure 10.17. These powers can be referenced at either the input or the output of die device. Note that some authors prefer 506 Chapter 10: Noise and Active Ft F Components -60 -30 P, (] —il>2 — 32 4o[ 9al fiy1 (Py)2 (10.47) where (10.42) and (10.45) have been used. Observe that this result clearly shows that the third-order intermodulation power increases as the cube of the input signal power. Solving (10.47) for , and applying the result to (10.46) gives the spurious free dynamic range in terms of P$ and N0, the output noise power of the component: DRf = Pliat — -m1 (10.48) This result can be written in terms of dB as DRf(dB)=UP3-N0) (10.49) for P3 and expressed in dB or dBm. If the output SNR is specified, this can be added to N0 to give the spurious free dynamic range in terms of the minimum detectable signal level. Finally, although we derived this result for the 2j product, the same result applies for the 2oj2 — toj product, 10.2 Dynamic Range and Intermodulation Distortion 507 EXAMPLE 10.4 DYNAMIC RANGES A receiver has a noise figure of 7 dB, a I dB compression point of 25 dBm (referenced to output), a gain of 40 dB, and a third-order intercept point of 35 dBm (referenced to output). If the receiver is fed with an antenna having a noise temperature of TA — 150 K, and the desired output SNR is 10 dB, find the linear and spurious free dynamic ranges. Assume a receiver bandwidth of 100 MHz. Solution The noise power at the receiver output can be calculated as N0 =* GkB[TA + (F - 1)7;] = 104(1.38 x 10-")(108)[150+ (4,01)(290)] = 1.8 x 10 8 W = -47.4 dBm. Then the linear dynamic range is, in dB, DRt = P\. - N0 = 25 dBm + 47.4 dBm = 72 dB. Equation (10.49) gives the spurious free dynamic range as DRf = =(/>3 - N0 - SNR) = §(35 + 47.4 - 10) = 48.3 dB. Observe that DRf <£DRt. ■ Intercept Point of a Cascaded System As in the case of noise figure, the cascade connection of components has the effect of degrading (lowering) the third-order intercept point. Unlike the case of a cascade of noisy components, however, the intermodulation products in a cascaded system are deterministic (coherent), so we cannot simply add powers, but must deal with voltages. With reference to Figure 10.18, let G \ and be the power gain and third-order intercept point for the first stage, and G2 and P% be the corresponding values for the second stage. From (10.47) the third-order distortion power at the output of the first stage is fp' Ý rkt-es = ^gU (10-50> where 3fc is the desired signal power at frequency o>i at the output of the first stage. The voltage associated with this power is fig = fit Zn = P' where Z0 is the system impedance. (10.51) ft \k\ik lák G2 GlG2 Pi V, P' P? V", P" Pi (a) (b) FIGURE 10.18 Third-order intercept point for a cascaded system, (a) Two cascaded networks, (b) Equivalent network. 508 Chapter 10: Noise and Active RF Components The total third-order distortion voltage at the output of the second stage is the sum of this voltage times the voltage gain of the second stage, and the distortion voltage generated by the second stage. This is because these voltages are deterministic and phase-related, unlike the uncorrelated noise powers that occur in cascaded components, Adding these voltages gives the worst-case result for the distortion level, because there may be phase delays within the stages that could cause partial cancellation. Thus we can write the worst-case total distortion voltage at the output of the second stage as Pj Pj Since = G2P;(, we have Then the output distortion power is P" _0^-aJ__/_|__I 1 v (p»\3 - (P<"i) GO 53) Thus the third-order intercept point of the cascaded system is ^(m*iT- (1054) Note that P3 = G2Pj for P§ -v 00, which is the limiting case when the second stage has no third-order distortion. This result is also useful for transferring P3 between input and output reference points. EXAMPLE 10.5 CALCULATION OF CASCADE INTERCEPT POINT A low-noise amplifier and mixer are shown in Figure 10.19. The amplifier has a gain of 20 dB and a third-order intercept point of 22 dBm (referenced at output), and the mixer has a conversion loss of 6 dB and a third-order intercept point of 13 dBm (referenced at input). Find the intercept point of the cascade network. Solution First we transfer die reference of P3 for the mixer from its input to its output: P3" = 13 dBm - 6 dB = 7 dBm (referenced at output). Converting the necessary dB values to numerical values yields: p3 = 22 dBm = 158 mW (for amplifier), P3" - 7 dBm = 5 mW (for mixer), G2 = -6 dB = 0.25 (for mixer). FIGURE 10.19 System for Example 10.5. 10.3 RF Diode Characteristics 509 Then using (10.54) gives the intercept point of the cascade as 1 1 • I G2P> Pi <0.25)(158) 5 = 4.4 mW = 6.4dBm, which is seen to be much lower than the P% of the individual components. ■ Passive Intermodulation The above discussion of intermodulatlon distortion was in the context of active circuits involving diodes and transistors, hut it is also possible for intermodulation products to be generated by passive nonlinear effects in connectors, cables, antennas, or almost any component where there is a metal-to-metal contact. This effect is called passive intermodulation (P1M) and, as in the case of intermodulation in amplifiers and mixers, occurs when signals at two or more closely spaced frequencies mix to produce spurious products. Passive intermodulation can be caused by a number of factors, such as poor mechanical contact, oxidation of junctions between ferrous-based metals, contamination of conducting surfaces at RF junctions, or the use of nonlinear materials such as carbon fiber composites or ferromagnetic materials. In addition, when high powers are involved, thermal effects may contribute to the overall nonlinearity of a junction. It is very difficult to predict PfM levels from first principles, so measurement techniques must usually be used. Because of the third-power dependence of the third-order intermodulation products with input power, passive intermodulation is usually only significant when input signal powers are relatively large, This is frequendy the case in cellular telephone base station transmitters, which may operate with powers of 30-40 dBm, with many closely spaced RF channels. It is often desired to maintain the PIM level below —125 dBm, with two 40 dBm transmit signals. This is a very wide dynamic range, and requires careful selection of components used in the high-power portions of the transmitter, including cables, connectors, and antenna components. Because these components are often exposed to the weather, deterioration due to oxidation, vibration, and sunlight must be offset by a careful maintenance program. Passive intermodulation is generally not a problem in receiver systems, due to the much lower power levels. In this section we will discuss characteristics of the major types of diodes used in RF and microwave circuits. A diode is a two-terminal semiconductor device having a nonlinear V-I relationship. This nonlinearity can be exploited for the useful functions of signal detection, demodulation, switching, frequency multiplication, and oscillation [1]. Diodes can be packaged as axial lead components, surface mountable chips, or monolithically integrated with other components on a single semiconductor substrate, Schottky Diodes and Detectors The classical pn junction diode commonly used at low frequencies has a relatively large junction capacitance that makes it unsuitable for high frequency application. The Schottky barrier diode, however, relies on a semiconductor-metal junction that results in a much lower junction capacitance. The primary application of diodes of this type is in frequency conversion of an input signal. Figure 10.20 illustrates the three basic frequency conversion operations of rectification (conversion to DC), detection (demodulation of an amplitude modulated signal), and mixing (frequency shifting). 10.3 RF DIODE CHARACTERISTICS 510 Chapter 10: Noise and Active RF Components WW- RF f DC . i Modulated RF Hi m Modulation , i ■j /Rf Ti n /rf~Ao /rf+/lo l .0 FIGURE 10.20 Basic frequency conversion operations of rectification, detection, and mixing, (a) Diode rectifier, (b) Diode detector, (c) Mixer. The Schottky diode can be modeled as a nonlinear resistor, with a small-signal V-/ relationship expressed as I(V) = Is{eaV - %% (10.55) where or = q/nkT, and q is the charge of an electron, k is Boltzmann's constant, T is temperature, n is the idealty factor, and Is is the saturation current [4]-[6]. Typically, ls is between 10-6 and 10-15 A, and a = q/nkT is approximately 1/(25 mV) for T = 290K. The ideally factor, n, depends on the structure of the diode itself, and can vary from 1.2 for Schottky barrier diodes to about 2.0 for point-contact silicon diodes. Figure 10.21 shows a typical diode VW characteristic. Small-signal approximation. Now let the diode voltage be V — Vo + v, (10,56) where V0 is a DC bias voltage and v is a small AC signal voltage. Then (10.55) can be expanded in a Taylor series about Vo as follows: dV 1 2d2I (10.57) 10.3 RF Diode Characteristics 511 FIGURE 10.21 V-I characteristics of a Schottky diode. where 1« = f{Vc,) is the DC bias current. The first derivative can be evaluated as dl dV 1 = uIseaV* = «(/„ + /,) = Gd = —, (10.58) which defines Rj, the junction resistance of the diode, and Gj = 1//?;, which is called the dynamic conductance of the diode. The second derivative is d2I dGd dV = c*2lfeaV« = ct\k + Is) = aGd = G'd. (10.59) Then (10.57) can be rewritten as the sum of the DC bias current, In, and an AC current, i: f&) = Io + i = h + vGd + —G'd + - - . (10.60) The three-term approximation for the diode current in (10.60) is called the small-signal approximation, and will be adequate for most of our purposes. The small-signal approximation is based on the DC voltage-current relationship of (10.55), and shows that the equivalent circuit of a diode will involve a nonlinear resistance. In practice, however, the AC characteristics of a diode also involve reactive effects due to the structure and packaging of the diode. A typical equivalent circuit for a diode is shown in Figure 10.22. The leads and contacts of the diode package lead to a series inductance, Lp, and shunt capacitance, Cp. The series resistor, %, accounts for contact and current-spreading resistance. C; and Rj are the junction capacitance and resistance, and are bias-dependent. Diode rectifiers and detectors. In a rectifier application, a diode is used to convert a fraction of an RF input signal to DC power. Rectification is a very common function, and is used for power monitors, automatic gain control circuits, and signal strength indicators. If the diode voltage consists of a DC bias voltage and a small-signal RF voltage, V = Vq + vo cos coqI, (10.61) + o FIGURE 10.22 EqLiValent AC circuit model for a Schottky diode. 512 Chapter 10: Noise and Active RF Components then (10.60) shows that the diode current will be I = + vqGj cosiOat + -^G'd cos2toot = Iq + -£Gd + VoGd cos mot + —Gd co$2wot. (10.62) /o is the bias current and v\GdjA is the DC rectified current. The output also contains AC signals of frequency o (and higher-order harmonics), which are usually filtered out with a simple low-pass filter. A current sensitivity, fr, can be defined as a measure of the change in DC output current for a given input RF power. From (10.60) the RF input power is v2Gj/2 (using only the first term), while (10.62) shows the change in DC current is vlG'd/4. The current sensitivity is then Ř = A/dc G> A/W. (10.63) An open-circuit voltage sensitivity, 6V, can be defined in terms of the voltage drop across the junction resistance when the diode is open-circuited. Thus, By, = &iRy (10.64) Typical values for the voltage sensitivity of a diode range from 400 to 1500 mV/mW. In a detector application the nonlinearity of a diode is used to demodulate an amplitude modulated RF carrier. For this case, the diode voltage can be expressed as v(t) = v<,{\ + nt cos ft>mí)cosa)t)í, (10.65) where com is the modulation frequency, con is the RF carrier frequency (too ^> d + m2 ml ——h 2m cos comt + — cos 2comr + cos 2ú\-,f in + m cos(2újo + tom)t + m cos(2tuo — ^- t" "^D^ Y1G oscillator PIN Diodes and Control Circuits Switches are used extensively in microwave systems, for directing signal or power flow between other components. Switches can also be used to construct other types of control circuits, such as phase shifters and attenuators. Mechanical switches can be made in waveguide or coaxial form, and can handle high powers, but are bulky and slow. PIN diodes, however, can be used to construct an electronic switching element easily integrated with planar circuitry and capable of high-speed operation. (Switching speeds of 10 nanoseconds or less are typical.) FETs can also be used as switching elements. PIN diode characteristics. The PIN diode has V-I characteristics that make it a good RF switching element. When reverse biased, a small series junction capacitance leads to a relatively high diode impedance, while a forward bias current removes the junction capacitance and leaves the diode in a low impedance state [5], Equivalent circuits for these two states are shown íd Figure 10.25. Typical values for the parameters are: C-; — 1 pE or less; L, = 0.5 nH, or less; RT = 5 Q, or less; R/ = 1 fi, or less. The equivalent circuits do not include parasitic effects due to packaging, which may be important. The forward bias 10,3 RF Diode Characteristics 515 FIG UHF 10.25 AWv-K (a) (b) Equivalent circuits for the ON and OFF states of a PIN diode, (a) Reverse bias (OFF) state, (b) Forward bias (ON) state. current is typically 10-30 mA, and the reverse bias voltage is typically 40^60 V. The bias signal must be applied to the diode with RF chokes and DC blocks to isolate it from the RF signal. Single-pole PIN diode switches. A PIN diode can be used in either a series or a shunt configuration to form a single-pole, single-throw RF switch. These circuits are shown in Figure 10.26, with bias networks. In the series configuration of Figure 10.26a, the switch is on when the diode is forward biased, while in the shunt configuration the switch is on when the diode is reversed biased. In both cases, input power is reflected when the switch is in the OFF state. The DC blocks should have a very low impedance at the RF operating frequency, while the RF choke inductors should have a very high RF impedance. In some designs, high impedance quarter-wavelength lines can be used in place of the chokes, to provide RF blocking. Ideally, a switch would have zero insertion loss in the ON state, and infinite attenuation in the OFF state. Realistic switching elements, of course, result in some insertion loss for the ON state, and finite attenuation for the OFF state. Knowing the diode parameters for the equivalent circuits of Figure 10.25 allows the insertion loss for the ON and OFF states to be calculated for the series and shunt switches. With reference to die simplified switch circuits of Figure 10.27, we can define the insertion loss in terms of the actual load voltage, V*t, and Vb, which is the load voltage which would appear if the switch (Zd) were absent: VL IL = -20 log -f . (10.67) Vb o Bias q Bias RF choke DC Z0 hlpck I —M— Diode DC block RF choke fa) Chnke( X DC block DC block Diode (a) FIGURE 10,26 Singte-pole PIN diode switches, (a) Series configuration, (b) Shunt configuration. 516 Chapter 10: Noise and Active RF Components 2j - ft) FIGURE 10 27 Simplified equivalent circuits for the scries and shunt single-pole PIN diode switches, (a) Series switch, (b) Shunt switch. Simple circuit analysis applied to the two cases of Figure 10.27 gives the following results: 2Z0 IL = -20 log IL = -20 log 2Z0 + Zrf 2Zd (series switch), (shunt switch). (10.68a) (10.68b) In both cases, Zj is the diode impedance for either the reverse or forward bias state. Thus, Zr — Rr + j(o>Ls — 1 /wCj) for reverse bias for forward bias. (10.69) The ON state or OFF stale insertion loss of a switch can usually be improved by adding an external reactance in series or in parallel with the diode, to compensate for the reactance of the diode. This technique usually reduces the bandwidth, however, Several single-throw switches can be combined to form a variety of multiple-pole and/or multiple-throw configurations [6J. Figure 10,28 shows series and shunt circuits for a single-pole, double-throw switch; such a switch requires at least two switching elements. In operation, one diode is biased in the low impedance state, with the other diode biased in the high impedance state. The input signal is switched from one output to the other by reversing the diode states. The quarter-wave lines of the shunt circuit limit the bandwidth of this configuration. EXAMPLE 10.6 SINGLE-POLE PIN DIODE SWITCH A single-pole switch is to be constructed using a PIN diode with the following parameters: Cj = 0.1 pF, Rr = 1 £1, Rf = 5.0 £2, L,- = 0.4 nH. If the operating frequency is 5 GHz, and Z0 = 50 Í2, what circuit (series or shunt) should be used to obtain the greatest ratio of off-to-on attenuation? Solution We first compute the diode impedance for the reverse and forward bias states, using (10.69): [ Zr = Rr + j((oLi - l/ft>C,) = 1.0 - /305.7 Si Zd = ( Zf = Rf + ja>Li = 0.5 + ,/12.6 Q. 10.3 RF Diode Characteristics 517 Output 1 Output 1 F1GVJRE 10.28 Circuits for single-pole double-throw PIN diode switches, (a) Series, (b) Shunt. Then using (10.68) gives the insertion losses for the ON and OFF states of the series and shunt switches as follows: For the series circuit. 1^ = -20 log /L«ff = -20 log For the shunt circuit, /Lon =s -20 log lLtf = -20 log 2Z0 2Zo + Zf 2Z0 + Zr 2Zr = 0.11 dB. = 10.16dB. 2Zr + Z0 2Zf + Z0 = 0.03 dB, = 7.07 dB. So the series configuration has the greatest difference in attenuation between the ON and OFF states, but the shunt circuit has the lowest ON insertion loss. ■ PIN diode phase shifters. Several types of microwave phase shifters can be constructed with PIN diode switching elements. Compared with ferrite phase shifters, diode phase shifters have the advantages of small size, integrability with planar circuitry, and high speed. The power requirements for diode phase shifters, however, are generally greater than those for a latching ferrite phase shifter, because diodes require continuous bias current 518 Chapter 10: Noise and Active RF Components FIGURE 10.29 A switched-line phase shifter. while the latching ferrite device requires only a pulsed current to change its state. There are basically three types of PIN diode phase shifters: switched line, haded line,, and reflection. The switched-line phase shifter is the most straightforward type, using two single-pole double-throw switches to route the signal flow between one of two transmission lines of different length. See Figure 10.29. The differential phase shift between the two paths is given by A = 8{t2-t,\ (10,70) where B is the propagation constant of the line. If the transmission lines are TEM (or quasi-TEM, like miorostrip), this phase shift is a linear function of frequency, which implies a true time delay between the input and output ports. This is a useful feature in broadband systems, because distortion is minimized. This type of phase shifter is also inherently reciprocal, and can be used for both receive and transmit functions. The insertion loss of the switched line phase shifter is equal to the loss of the SPDT switches plus line losses. Like many other types of phase shifters, the switched-line phase shifter is usually designed for binary phase shifts of A0 = 180°, 90°, 45°, etc. One potential problem with this type of phase shifter is that resonances can occur in the OFF line, if its length is near a multiple of k/2. The resonant frequency will be slightly shifted due to the series junction capacitances of the reversed biased diodes, so the lengths l\ and t2 should be chosen with this effect taken into account. A design that is useful for small amounts of phase shift (generally 45°, or less) is the loaded-line phase shifter. The basic principle of this type of phase shifter can be illustrated with the circuit of Figure 10.30a, which shows a transmission line loaded with a shunt susceptance, jB. The reflection and transmission coefficients can be written as i-n+m = mm i+d+m i+jb' T = 1 4- r = —^—-, (10.71b) 2 + jb where b = BZq is the normalized susceptance. Thus the phase shift in the transmitted wave introduced by the load is i b &4> = tan-1 -, (10.72) 2 which can be made positive or negative, depending on the sign of b. A disadvantage is the insertion loss that is inherently present, due to the reflection from the shunt load. And increasing b to obtain a larger A

at the output. Ideally, the diodes would look like short circuits in their on state, and open circuits in their off state, so that the reflection coefficients at the right side of the hybrid can be written as r = e~for the diodes in their ON state, and T = e~M+Ěí^ for the diodes in their OFF state. There are infinite numbers of choices of line lengths that give the desired A0 (that is, the value of tp is a degree of freedom), but it can be shown that bandwidth is optimized if the reflection coefficients for the two states are phase conjugates. Thus, if &4> - 90°, the best bandwidth will be obtained for 4> = 45°. A good input match for the reflection-type phase shifter requires that the diodes be well-matched. The insertion loss is limited by the loss of the hybrid, as well as the forward and reverse resistances of the diodes. Impedance transformation sections can be used to improve performance in this regard. Varactor Diodes A varactor diode provides a junction capacitance that varies with bias voltage, thus providing an electrically adjustable reactive circuit element. The most common application of varactor diodes is to provide electronic frequency tuning of the local oscillator in a multichannel receiver, such as those used in cellular telephones, wireless local area network radios, and television receivers. This is accomplished by using a varactor diode in the resonant circuit of a transistor oscillator, and changing the DC bias voltage applied to the diode, The nonlinearity of varactors also makes them very useful for frequency multipliers (to be discussed further in Chapter 12). Varactor diodes are usually made from silicon or gallium arsenide semiconductors, A simplified equivalent circuit for a reverse-biased varactor diode is shown in Figure 10.32. The junction capacitance is dependent on the junction bias voltage, V, o—WW FIGURE 1032 Equivalent circuit of a reverse biased varactor diode. 10.3 RF Diode Characteristics 521 according to C;(V) =-—-j (10.76) J (1 - VjVaY where Co and Vo are constants, and y is an exponent that varies between 1/3 and % depending on the semiconductor doping profile used in the diode. A constant doping profile results in y = 1/2. Rj is the junction resistance, which for reverse bias is typically greater than I06 £2, and can usually be ignored. Rs is the series junction and contact resistance, typically on the order of a few ohms. A typical varactor diode might have Co = 0.2 pF and Vo = 0.5 V, resulting in a junction capacitance that varies from about 0.1 pF to 0.2 pF as the bias voltage ranges from 2.0 to 0 volts. Other Diodes Here we summarize the characteristics of several other diode devices that are commonly used in microwave circuits. These devices are most useful at millimeter wave frequencies since three-terminal devices (junction and field effect transistors) usually offer better performance at lower frequencies. More details on these and related diode devices can be found in the references. IMPATT diodes. An IMPATT (Impact Avalanche and Transit Time) diode has a physical structure similar to a PIN diode, but relies on an avalanche effect for its unique properties. It exhibits negative resistance over a broad frequency band that extends into the submillimeter range, and can therefore be used to directly convert DC to RF power. Typical IMPATTs operate at frequencies from 10 to 300 GHz, at relatively high powers, with efficiencies ranging up to 15%. IMPATT diodes are the only practical solid-state device that can provide fundamental frequency power above 100 GHz. IMPATT devices can also be used for frequency multiplication and amplification. Silicon IMPATT diodes can provide CW power ranging from 10 W at 10 GHz, to 1 W at 94 GHz, with efficiencies typically below 10%. GaAs IMPATTs can provide CW power ranging from 20 W at 10 GHz, to 5 mW at 130 GHz. Pulsed operation generally results in higher powers and higher efficiencies. Because of the low efficiency of these devices, thermal considerations are the limiting factor for both CW and pulsed operation. IMPATT oscillators can be both mechanically and electrically tuned. A disadvantage of IMPATT oscillators is that their AM noise level is generally higher than that of other sources. Gunn diodes. The Gunn diode has an l-V characteristic that exhibits a negative differential resistance (negative slope) that can be used to generate RF power from DC. Its operation is based on the transferred electron effect (also known as the Gunn effect), which was discovered by J. B. Gunn in 1963. Practical Gunn diodes typically use either GaAs or InP materials. Gunn diodes can produce continuous power of up to several hundred milliwatts, at frequencies from 1 to 100 GHz, with efficiencies ranging from 5% to 15%. Oscillator circuits using Gunn diodes require a high-Q resonant circuit or cavity, which is often tuned mechanically. Electronic tuning by bias adjustment is limited to 1% or less, but varactor diodes are sometimes included in the resonant circuit to provide a greater range of electronic tuning. Gunn diode sources are used extensively in low-cost applications such as traffic radars, motion detectors for door openers and security alarms, and test and measurement systems. BARTTT diodes. A BAR1TT (Barrier Injection Transit Time) diode has a structure similar to a junction transistor without a base contact. Like the IMPATT diode, it is a transit-time 522 Chapter 10: Noise and Active RF Components device. It generally has a lower power capability than the IMPATT diode, but the advantage of lower AM noise. This makes it useful for local oscillator applications, at frequencies up to 94 GHz. BARITT diodes are also useful for detector and mixer applications. RF TRANSISTOR CHARACTERISTICS Transistors are critical components of modern RF and microwave systems, finding application as amplifiers, oscillators, switches, phase shifters, mixers, and active filters. Transistor devices can be categorized as either junction transistors, or field effect transistors [5]-[9]. Junction transistors include bipolar junction transistors (BJTs) and heterojunction bipolar transistors (HBTs), in either npn or pnp configurations. Modern junction transistors are made using silicon, silicon-germanium, gallium arsenide, and indium phosphide materials. The silicon junction transistor is one of the oldest and most popular active RF devices because of its low cost and good operating performance in terms of frequency range, power capacity, and noise characteristics. Silicon junction transistors are useful for amplifiers up to the range of 2-10 GHz, and in oscillators up to about 20 GHz. Bipolar transistors typically have very low l/f noise characteristics, making them well suited for oscillators with low phase noise. Recent developments with junction transistors using SiGe have demonstrated much higher cutoff frequencies, making these devices useful in low-cost circuits operating at frequencies of 20 GHz or higher. Heterojunction bipolar transistors may use GaAs or InP materials, and can operate at frequencies exceeding 100 GHz. Field effect transistors (FETs) can take many forms, including the MESFET (metal semiconductor FET), the HEMT (high electron mobility transistor), the PHEMT (pseudo-morphic HEMT), the MOSFET (metal oxide semiconductor FET), and the MISFET (metal insulator semiconductor FET). FET transistor technology has been under continuous development for more than 50 years—the first junction field effect transistors were developed in the 1950s, while the HEMT was proposed in the early 1980s. Unlike junction transistors, which are current-controlled, FETs are voltage-controlled devices, and can be made with either a p-channel or an n-channel. GaAs MESFETs are one of the most commonly used transistors for microwave and millimeter wave applications, being usable at frequencies up to about 40 GHz. Even higher operating frequencies can be obtained with GaAs HEMTs, GaAs FETs and HEMTs are especially useful for low-noise amplifiers, since these transistors have lower noise figures than any other active devices. Table 10.2 summarizes the performance characteristics of some of the most popular microwave transistors. In this section we will give a brief discussion of the basic construction of microwave FETs and bipolar transistors, along with small-signal equivalent circuit models for these TABLE 10.2 Performance Characteristics of Micrwave Transistors Si Si SiGe GaAs GaAs GaAs Device BJT CMOS HBT MESFET HEMT HBT Useful frequency range (GHz) 10 20 30 40 100 60 Typical gain (dB) 10-15 10-20 10-15 5-20 10-20 10-20 Noise figure (dB) 2.0 1.0 0.6 1.0 0.5 4.0 (frequency) (2 GHz) (4 GHz) (8 GHz) (10 GHz) (12 GHz) (12 GHz) Power capacity High Low Medium Medium Medium High Cost Low Low Medium Medium High High Single polarity supply? Yes Yes Yes No Yes 10.4 10.4 RF Transistor Characteristics 523 Source Gale Drain N* Epitaxial layer j-0.3flm Buffer layer I~3jUm Drain „ Source High-resistivity substrate ^ 00 um Ga*e FIGURE 10.33 (a) Cross section of a GaAs MESFET; (b) top view, showing drain, gate, and source contacts. devices, and DC biasing considerations. The design of amplifiers and oscillators relies primarily on the terminal characteristics of the transistor, and these can be expressed either in terms of the two-port S parameters of the device, or in terms of the component values of an equivalent circuit. We will use the 5 parameter method tor most of our design work, as this is a procedure that is both accurate and convenient, although it does have the drawback of requiring knowledge of the transistor S parameters (usually through measurement) over the frequency band of interest. This is usually not a serious problem unless a very wide frequency range is being considered, since the S parameters of microwave transistors typically change fairly slowly with frequency. In contrast, the use of a good transistor equivalent circuit model involves only a few circuit parameters which are generally stable over a wide frequency range. An equivalent circuit model can also provide a closer linkage between the operation of the device and its physical parameters, and can be more useful for some design problems. Field Effect Transistors (FETs) Microwave field effect transistors can be used at frequencies well into the millimeter wave range with high gain and low noise figure, making them the device of choice for hybrid and monolithic integrated circuits at frequencies above 5 to 10 GHz [7]-[9], Figure 10.33 shows the construction of a typical GaAs MESFET. The desirable gain and noise features of the GaAs FET are a result of the higher electron mobility of GaAs compared to silicon, and the absence of shot noise. In operation, electrons are drawn from the source to the drain by the positive supply voltage. An input signal voltage on the gate then modulates these majority electron carriers, producing voltage amplification. The maximum frequency of operation is limited by the gate length; presently manufactured FETs have gate lengths on the order of 0.3 to 0.6 fim, with corresponding upper frequency limits of 100 to 50 GHz, A small-signal equivalent circuit for a microwave FET is shown in Figure 10.34, for a common-source configuration. The components and typical values for this circuit model are listed below: R, (series gate resistance) = 10, RdS (drahvto-source resistance) = 400 £2 Cg3 (gate-to-source capacitance) = 0.3 pF Cds (drain-to-source capacitance) = 0.12 pF Cgd (gate-to-drain capacitance) = 0.01 pF %„, (transconductance) = 40 mS This model does not include package parasitics, which typically introduce small series resistances and inductances at the three terminals due to ohmic contacts and bonding leads. The dependent current generator gmVc depends on the voltage across the gate-to-source capacitor C^, leading to a value of |52i | > 1 under normal operating conditions (where 524 Chapter 10: Noise and Active RF Components Source FIGURE 10.34 Small-signal equivalent circuit for a microwave FET in (he common-source configuration, port 1 is at the gate, and port 2 is at the drain). The reverse signal path, given by St2, is due solely to the capacitance Cgd. As can be seen from the above data, this is typically a very small capacitor which can often be ignored in practice. In this case, Sl2 = 0, and the device is said to be unilateral. The equivalent circuit model of Figure 10.34 can be used to determine the upper frequency of operation for the transistor. The short-circuit current gain, Gf\ is defined as the ratio of drain to gate current when the output is short-circuited. For the unilateral case, where Cgd is assumed to be zero, this can be derived as u 8m 1 8m h h (10.77) The upper frequency limit, fj, is the frequency where the short-circuit current gain is unity, thus we have that 8» (10.78) 277" C^j For proper operation, the transistor must be DC biased at an appropriate operating point. This depends on the application (low-noise, high-gain, high-power), the class of the amplifier (class A, class AB, class B), and the type of transistor (FET, HBT, HEMT). Figure 10.35a shows a typical family of DC versus curves for a GaAs FET. For low-noise design, the drain current is generally chosen to be about 15% of Tfa (the saturated drain-to-source current). High-power circuits generally use higher values of drain current. DC bias voltage must be applied to the gate and drain, without disturbing the RF signal paths. This can be 10 ft - OV -IV - V = -2V is -3V 3 J FIGURE 10.35 (a) DC characteristics of a GaAs FET; (b) biasing and decoupling circuit for a GaAs FET. 10.4 RF Transistor Characteristics 525 Emitter t3;?se Emitter Base ;- \ P base t -0.1 urn N collector I -15 lim N+ collector substrate I ~200 flm FIGURE I0.36 (a) Cross section of a microwave silicon bipolar transistor; (b) top view, showing base and emitter contacts. done as shown in Figure 10.35b, which shows the biasing and decoupling circuitry for a dual polarity supply. The RF chokes provide a very low DC resistance for biasing, and a very high impedance at RF frequencies to prevent the microwave signal from being shorted by the bias supply. Similarly, the input and output decoupling capacitors block DC from the input and output lines, while allowing passage of microwave signals. There are many other types of bias circuits that provide compensation for temperature and device variations, and that can work with single-polarity power supplies. Bipolar Junction Transistors (BJTs) Microwave bipolar transistors are usually of the npn type, and are often preferred over GaAs FETs at frequencies below 2 to 4 GHz because of higher gain and lower cost. Bipolar transistors are subject to shot noise as well as thermal noise effects, so their noise figure is not as good as that of FETs, Figure 10.36 shows the construction of a typical silicon bipolar transistor. In contrast to the FET, the bipolar transistor is current driven, with the base current modulating the collector current. The upper frequency limit of the bipolar transistor is controlled primarily by the base length, which is on the order of 0.1 ^m. A small-signal equivalent circuit model for a microwave bipolar transistor is shown in Figure 10.37, for a common emitter configuration. The components of this circuit, along with typical values, are listed below: Ri, (base resistance) — 7 Q Rx (equivalent n resistance) = 110 H Cn (equivalent n capacitance) = 18 pF Cc (collector capacitance) = 18 pF gm (transconductance) ss 900 mS Observe that the transconductance is much higher than that of the GaAs FET, leading to higher power gain at lower frequencies. The larger capacitances in the bipolar transistor Base Collector Emitter FIGURE 10.37 Simplified hybrid-:? equivalent circuit for a microwave bipolar transistor in the common emitter configuration. 526 IB-],0 mA IB = 0.15 mA /s = 0.50 mA ;fl= 0.25 mA (b) FIGURE 10.38 (a) DC characteristics of a silicon bipolar transistor; (b) biasing and decoupling circuit for a bipolar transistor. 10.5 model serve to reduce the gain at higher frequencies. The model in Figure 10.37 is popular because of its similarity to the FET equivalent circuit, but more sophisticated equivalent circuits may be advantageous for use over wide frequency ranges [7]—[9]. In addition, this model does not include parasitic resistances and inductances due to the base and emitter leads. The equivalent circuit of Figure 10.37 can be used to estimate the upper frequency limit, fj, where the short-circuit current gain is unity. The result is similar to that found above for the FET: fT = (10.79) 2nLn Figure 10.38a shows typical DC operating characteristics for a bipolar transistor. As with the FET, the biasing point for a bipolar transistor depends on the application and type of transistor, with low collector currents generally giving the best noise figure, and higher collector currents giving the best power gain. Figure 10.38b shows a typical bias and decoupling circuit for a bipolar transistor that requires only a single polarity supply. MICROWAVE INTEGRATED CIRCUITS The trend of any maturing electrical technology is toward smaller size, lighter weight, lower cost, and increased complexity. Microwave technology has been moving in this direction for the last 10-20 years, with the development of microwave integrated circuits [2]. This 10,5 Microwave Integrated Circuits 527 technology serves to replace bulky and expensive waveguide and coaxial components with small and inexpensive planar components, and is analogous to the digital integrated circuitry that has led to the rapid increase in sophistication of computer systems. Microwave integrated circuitry (MIC) can incorporate transmission lines, discrete resistors, capacitors, and inductors, and active devices such as diodes and transistors. MIC technology has advanced to the point where complete microwave subsystems, such as receiver front ends and radar transmit/receive modules, can be integrated on a chip that is only a few square millimeters in size. There are two distinct types of microwave integrated circuits. Hybrid MICs have one layer of metallization for conductors and transmission lines, with discrete components (resistors, capacitors, transistors, diodes, etc.) bonded to the substrate. In a thin-film hybrid MIC, some of the simpler components are deposited on the substrate. Hybrid MICs were first developed in the 1960s, and still provide a very flexible and cost-effective means for circuit implementation. Monolithic microwave integrated circuits (MMICs) are a more recent development, where the active and passive circuit elements are grown on the substrate. The substrate is a semiconductor material, and several layers of metal, dielectric, and resistive films are used. Below we will briefly describe these two types of MICs, in terms of the materials and fabrication processes that are required and the relative merits of each type of circuitry. Hybrid Microwave Integrated Circuits Material selection is an important consideration for any type of MIC; characteristics such as electrical conductivity, dielectric constant, loss tangent, thermal transfer, mechanical strength, and manufacturing compatability must be evaluated. Generally the substrate material is of primary importance. For hybrid MICs, alumina, quartz, and Teflon fiber are commonly used for substrates. Alumina is a rigid ceramic-Iike material with a dielectric constant of about 9-10. A high dielectric constant is often desirable for lower frequency circuits because it results in a smaller circuit size. At higher frequencies, however, the substrate thickness must be decreased to prevent radiation loss and other spurious effects; then the transmission lines (typically microstrip, slotline, or coplanar waveguide) can become too narrow to be practical. Quartz has a lower dielectric constant ("--4) which, with its rigidity, makes it useful for higher frequency (>20 GHz) circuits. Teflon and similar types of soft plastic substrates have dielectric constants ranging from 2 to 10, and can provide a large substrate area at a low cost, as long as rigidity and good thermal transfer are not required. Transmission line conductors for hybrid MICs are typically copper or gold. Computer-aided design (CAD) tools are used extensively for microwave integraTed" circuit design, optimization, layout, and mask generation. Commonly-used software packages include CADENCE (Cadence Design Systems, Inc.), ADS (Agilent Technologies, Inc.), Microwave Office (Applied Wave Research, Inc.), and SERENADE or DESIGNER (Ansoft, Inc.). The mask itself may be made on mbylith (a soft mylar film), usual)^\iata magnified scale (2x, 5x, 10x, etc.) for a high accuracy. Then an actual-size mask is- n^jjde*j on a thin sheet of glass or quartz. The metalized substrate is coated wim photoresist, coveted1-' with the mask, and exposed to a light source. The substrate can then be etched to remove the unwanted areas of metal. Plated-through, or via, holes can be made by evaporating a layer of metal inside a hole tiiat has been drilled in the substrate. Finally, the discrete components are soldered or wire-bonded to the conductors. This is generally the most labor-inlfensive part of hybrid MIC fabrication, and therefore the most expensive part of the process. Then the MIC can be tested. Often provision is made for variations in component values and other circuit tolerances by providing tuning or trim stubs that can be manually trimmed for each circuit. This increases circuit yield, but also increases cost since trirnming 528 Chapter 10: Noise and Active RF Components Wilkinson Chip Hyhiid divider resistor Ground Chip FET Ceramic Choke plane capacitor chip substrate inductor FIGURE 10.39 Layout of a hybrid microwave integrated circuit. involves labor at a highly skilled level. The layout of a typical hybrid MIC circuit is shown in Figure 10.39. A photograph of a hybrid MIC module is shown in Figure 10.40. Monolithic Microwave Integrated Circuits Progress in GaAs material processing and device development since the late 1970s has led to the feasibility of the monolithic microwave integrated circuit, where all passive and FIGURE 10.40 Photograph of one of the 25,344 hybrid integrated TVR modules used in Raytheon's Ground Based Radar system. This X-band module contains phase shifters, amplifiers, switches, couplers, a ferrite circulator, and associated control and bias circuitry. Courtesy of Raytheon Company, Lexington. Mass. 10.5 Microwave Integrated Circuits 529 active components required for a given circuit can be grown or implanted in the substrate. Potentially, the MMIC can be made at low cost because the manual labor required for fab-heating hybrid MICs is eliminated. In addition, a single wafer can contain a large number of circuits, all of which can be processed and fabricated simultaneously. The substrate of an MMIC must be a semiconductor material to accommodate the fabrication of active devices; the type of devices and the frequency range dictate the type of substrate material. Thus, silicon bipolar transistors can be used up to several gigahertz, silicon-on-sapphire (SOS) MESFETs can be used up to several gigahertz, and submicron gate-length GaAs FETs have been used up to 60 GHz. The GaAs FET is a very versatile circuit element, finding applications in low-noise amplifiers, high-gain amplifiers, broadband amplifiers, mixers, oscillators, phase shifters, and switches. Thus, GaAs is probably the most common substrate for MMlCs, but silicon, silicon-on-sapphire, and indium phosphide (InP) are also used. Transmission lines and other conductors are usually made with gold metallization. To improve adhesion of the gold to the substrate, a thin layer of chromium or titanium is generally deposited first. These metals are relatively lossy, so the gold layer must be made at least several skin depths thick to reduce attenuation. Capacitors and overlaying lines require insulating dielectric films, such as SiO, Si02, SJ2N4, and Ta^Oj. These materials have a high dielectric constant and low loss, and are compatible with integrated circuit processing. Resistors require the deposition of lossy films; NiCr, Ta, Ti, and doped GaAs are commonly used. Designing an MMIC requires extensive use of CAD software, for circuit design and optimization as well as mask generation. Careful consideration must be given to the circuit design to allow for component variations and tolerances, and the fact that circuit trimming after fabrication will be difficult, or impossible (and defeats the goal of low-cost production). Thus, effects such as transmission line discontinuities, bias networks, spurious coupling, and package resonances must be taken into account After the circuit design has been finalized, the masks can be generated. One or more masks are generally required for each processing step. Processing begins by forming an active layer in the semiconductor substrate for the necessary active devices; this can be done by ion implantation or by epitaxial techniques. Then active areas are isolated by etching or additional implantation, leaving mesas for the active devices. Next, ohmic contacts are made to the active device areas by alloying a gold or gold/germanium layer onto the substrate. FET gates are then formed with a titanium/platinum/gold compound deposited between the source and drain areas. At this time the active device processing has been essentially completed, and intermediate tests can be made to evaluate the wafer. If it meets specifications, the next step is to deposit the first layer of metallization for contacts, transmission lines, inductors, and other conducting areas. Then resistors are formed by depositing resistive films, and the dielectric films required for capacitors and overlays are deposited. A second layer of metallization completes the formation of capacitors and any remaining interconnections. The final processing steps involve the bottom, or backside, of the substrate. First it is lapped to the required thickness, then via holes are formed by etching and plating, Via holes provide ground connections to the circuitry on the top side of the substrate, and provide a heat dissipation path from the active devices to the ground plane. After the processing has been completed, the individual circuits can be cut from the wafer, and tested. Figure 10.41 shows the structure of a typical MMIC, and Figure 10.42 shows a photograph of an X-band monolithic integrated power amplifier. Monolithic microwave integrated circuits are not without some disadvantages, when compared with hybrid MICs or other type of circuitry. First, MMICs tend to waste large areas of relatively expensive semiconductor substrate for components such as transmission lines and hybrids. Also, the processing steps and required tolerances for an MMIC are very critical, 530 Chapter 10: Noise and Active RF Components Air Microstrip MIM Inductor bridge input line capacitor / plane kok FET resistor subsume metauzation FIGURE 10.41 Layout of a monolithic microwave integrated circuit resulting in low yields. These factors tend to make MMICs expensive, especially when made in small quantities (less than several hundred). MMICs generally require a more thorough design procedure to include effects such as component tolerances and discontinuities, and debugging, tuning, or trimming after fabrication is difficult. Because their small size limits heat dissipation, MMICs cannot be used for circuits requiring more than moderate power FIGURE 10.42 Photograph of a monolithic integrated X-band power amplifier. This circuit uses eight heterojunction bipolar transistors with power dividers/combiners at the input and output to produce 5 watts. Courtesy of M. Adlerslein and R. Wohlert Raytheon Company, Lexington, Mass. 10.5 Microwave Integrated Circuits 531 levels. And high-Q resonators and filters are difficult to implement in MM1C form because of the inherent resistive losses in MMIC materials. Besides the obvious features of small size and weight, MMlCs have some unique advantages over other types of circuits. Since it is very easy to fabricate additional FETs in an MMIC design, circuit flexibility and performance can often be enhanced with little additional cost. Also, monolithic ally integrated devices have much less parasitic reactance than discrete packaged devices, so MMIC circuits can often be made with broader bandwidth than hybrid circuits. And MMICs generally give very reproducible results, especially for circuits from the same wafer. POINT OF INTEREST: RF MEMS Switch Technology An exciting new field is the use of mkromacbining techniques to form suspended or movable structures in a silicon substrate that can be used for microwave resonators, antennas, and switches. A micromachined RF switch, having a mechanically movable contact, is an example of a micro-electro-mechanical system (MEMS), where the unique properties of silicon can be used to construct extremely small devices that employ miniaturized mechanical components such as levers, gears, motors, and actuators. RF MEMS switches are one of the most promising applications of this new technology. A MEMS switch can be made in several different configurations, depending on the signal path (capacitive or direct contact), the actuation mechanism (electrostatic, magnetic, or thermal), the pull-back mechanism (spring or active), and the type of structure (cantilever, bridge, lever arm, or rotary). Oce popular configuration for microwave switches is shown below, where the capacitance of the signal path is switched between a low capacitance state and a high capacitance state by moving a flexible conductive membrane through the application of a DC control voltage. Low-capacitance High-capacitance (open circuit) (closed circuit) MEMS switches have very good loss characteristics, very low power consumption, wide bandwidth, and (unlike diode or transistor switches) exhibit virtually no mtórmodulatíon distortion or other nonlinear effects. The table below compares some of the key parameters of MEMS switches with popular solid-state switch technology over the 10-20 GHz band. Switch Insertion Switching DC Switching technology loss Isolation power voltage speed PIN Diode 0.1-0.8 dB 25-45 dB 1-5 mW 1-10 V 1-5 nS FET 0.5-1,0dB 20-50 dB l-5mW 1-10 V 2-10 nS MEMS 0.1-1.0 dB 25-60 dB I /iW 10-20V >30/tS Probably the most significant drawbacks of RF MEMS switches are the relatively slow switching time and potential lifetime limitations; both of these are a result of the mechanical nature of the device. One of the most important applications foreseen for MEMS switches is to low-cost switched-line-length phase shifters, which are required in large numbers for phased array antennas. 532 Chapter 10: Noise and Active RF Components REFERENCES [1] M. E. Hines, "The Virtues of Nonhnearity—-Detection, Frequency Conversion, Parametric Amplification and Harmonic Generation," IEEE Trans. Microwave Theory and Techniques, vol. MTT-32, pp. 1097-1104, September 1984, [2] D. N. McQuiddy, Jr., J. W. Wassel, J. B. Lagrange, and W, R, Wisseman, "Monolithic Microwave Integrated Circuits: An Historical Perspective," IEEE Trans, Microwave Theory and Techniques, vol. MTT-32, pp. 997-1008, September 1984. [3] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive, Volume I, Microwave Remote Sensing, Fundamentals and Radiornetry. Addison-Wesley, Reading, Mass, 1981. [4] S. A. Maas, Microwave Mixers, Artech House, Dedham, Mass, 1986. [5] S. Y. Yngvesson, Microwave Semiconductor Devices, Kluwer Academic Publishers, 1991, [6] I. Bahl and P. Bhartia, Microwave Solid-State Circuit Design, Wiley Interscience, N.Y., 1988. [7] G. D. Vendelin, A, M. Pavio, and U. L. Rohde, Microwave Circuit Design Using Linear and Nonlinear Techniques, Wiley, N.Y., 1990. [8] G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd edition, Prentice-Hall, NJ., 1997. [9] K. Chang, ed.. Handbook of Microwave and Optical Components, vol. 2, Wiley Interscience, N.Y.. 1990. 10.1 The Y-factor method is used to measure the noise figure of an amplifier, with a noise source having an ENR = 20 dB, and a cold load at a temperature of 77 K. If the v-factor ratio is measured to be 16.285 dB, what is the noise figure of the amplifier? 10.2 Assume that measurement error introduces an uncertainly of AY into the measurement of Y in a F-factor measurement. Derive an expression for the normalized error, ATf/T^, of the equivalent noise temperature in terms of AF/ Y and the temperatures J[, Tj, and Tf. Minimize this result with respect to Tt to obtain an expression for Te in terms of T\ and T2 that will result in minimum error. 10.3 It is necessary to connect an antenna to a low-noise receiver with a transmission line. The frequency is 10 GHz, and the distance is 2 meters. The choices are: copper X-band waveguide, RG-8/U coaxial cable, or copper circular waveguide with an inner diameter of 2.0 cm. Which type of line should be used for the best noise figure? Disregard impedance mismatch. 10.4 A certain transmission fine has a noise figure F = 1 dB at a temperature T$ = 290 K. Calculate and plot the noise figure of this line as its physical temperature ranges from T — 0 K to 1000 K. 10£ An amplifier with a bandwidth of 1 GHz has a gain of 15 dB and a noise temperature of 250 K. If the 1 dB compression point occurs for an input power level of —10 dBm, what is the linear dynamic range of the amplifier? 10.6 An amplifier with a gain of 12 dB, a bandwidth of 150 MHz, and a noise figure of 4.0 dB feeds a receiver with a noise temperature of 900 K. Find the noise figure of the overall system, 10.7 A PCS cellular receiver front-end circuit is shown below. The operating frequency is 1805-1880 MHz, and die physical temperature of the system is 300 K. A noise source with JV; = —95 dBm is applied to the receiver input, (a) What is the equivalent noise temperature of the source over the operating bandwidth? (b) What is the noise figure (in dB) of the amplifier? 180K Problems 533 10.8 Consider the wireless local area network (WLAN) receiver front-end shown below, where the bandwidth of the bandpass filter is 100 MHz centered at 2.4 GHz. If the system is at room temperature, find me noise figure of the overall system. What is the resulting signal-to-noise ratio at the output, if the input signal power level is -90 dBm? Can the components be rearranged to give a better noise figure? IL = 1.5 dB G= 10 dB F= 2 dB G = 20 dB F = 2 dB 10.9 A two-way power divider has one output port terminated in a matched load, as shown below. Find the noise figure of me resulting two-port network if the divider is (a) an equal-split two-way resistive divider, flj) a two-way Wilkinson divider, and (c) a 3 dB quadrature hybrid. Assume the divider in each case is matched, and at room temperature. Power -■— divider —1 AYvV 10.10 Show that, for fixed loss L >■ 1, the equivalent noise temperature of a mismatched lossy hne given in (10.33) is inimmized when \TS \ = 0. 10.11 A lossy line at temperature T feeds an amplifier with noise figure F", as shown below. If an impedance mismatch T is present at die input of the amplifier, find the overall noise figure of the system. 10,12 A balanced amplifier circuit is shown below. The two amplifiers are identical, each with power gain G and noise figure F. The two quadrature hybrids are also identical, with an insertion loss from the input to either output of L > 1 (not including the 3 dB power division factor). Derive an expression for the overall noise figure of the balanced amplifier. What does mis result reduce to when the hybrids are lossless? L G, F L 10.13 A receiver subsystem has a noise figure of 6 dB, a 1 dB compression point of 21 dBm (referenced to output), a gain of 30 dB, and a third-order intercept point of 33 dBm (referenced to output). If the subsystem is fed with a noise source wiui Ns = -105 dBm, and the desired output SNR is 8 dB, find the linear and spurious free dynamic ranges of the subsystem. Assume a system bandwidth of 20 MHz. 10.14 In practice, the third-order intercept point is extrapolated from measured data taken at input power levels well below P3. For the spectrum analyzer display shown on (he next page, where AP is the 534 Chapter 10: Noise and Active RF Components difference in power between PW| and ,_(,„, show that the third-order intercept point is given by Pi = — | AP. Is this referenced at the input or output? A. AP 10.15 A two-tone input with a 6 dB difference in the two signal levels is applied to a nonlinear component. What is the relative power ratio of the resulting two third-order intermodulation products Itoy — a>i and 2i — a>\, if i are close together? 10.16 Find the third-order intercept point for the problem of Example 10.5 when the positions of the amplifier and mixer are reversed. 10.17 A diode has the following parameters: Cj = 0.1 pF, Rs = 15 £2, I? = 0.1 fiA, and Lp = Cp = 0. Compute the open-circuit voltage sensitivity at 10 GHz for /q = 0, 20, and 50 ft A. Assume a = 1/(25 inV), and neglect the effect of bias current on the junction capacitance. 10.18 A single-pole, single-throw switch uses a PIN diode in a shunt configuration. The frequency is 4 GHz, Z<> = 50 fl, and the diode parameters are C} = 0.5 pF, Rr = 0.5 U, Rf = 03ri, Li = 0.3 nH. Find the electrical length of an open-circuited shunt stub, placed across the diode, to minimize the insertion loss for the ON state of rhe switch. Calculate the resulting insertion losses for the ON and OFF states. 10.19 A single-pole, single-throw switch is constructed using two identical PIN diodes in the arrangement shown below. In the ON state, the series diode is forward biased and the shunt diode is reversed biased; and vice versa for the OFF state. If / = 6 GHz, Z0 = 50 n, Cj = 0.1 pF, Rr = 0.5 S2, Rj =03 Q, and Li = 0.4 nH, determine the insertion losses for the ON and OFF states. +r- 2c 10.20 Consider the loaded-line phase shifter shown below. If Z» = 50 Si, find the necessary stub lengths for a differential phase shift of 45'\ and calculate the resulting insertion loss for both states of the phase shifter. Assume all lines are lossless, and that the diodes can be approximated as ideal shorts or opens. A/4- Problems 535 10.21 Use the equivalent circuit of Figure 10.37 to derive the expression for the short-circuit current gain of a bipolar transistor. Assume a unilateral device, where C = 0, 10.22 Derive expressions for the y parameters of an FET using the unilateral equivalent circuit model. Evaluate these parameters at 5 GHz for the following FET characteristics: R = 7 Q, = 400 Q, CSI = 0.3 pF, Qs = 0,12 pFt Csd = 0, gm = 30 mS. Convert the y parameters to S parameters for a 50 Q system impedance, and find the unilateral transducer gain assuming conjugately matched source and load impedances. Microwave Amplifier Design Amplification is one of the most basic and prevalent microwave circuit functions in modern RF and microwave systems. Early microwave amplifiers relied on tubes, such as klystrons and traveling-wave lubes, or solid-state reflection amplifiers based on the negative resistance characteristics of tunnel or varactor diodes. But due to the dramatic improvements and innovations in solid-state technology that have occurred since the 1970s, most RF and microwave amplifiers today use transistor devices such as Si or SiGe BJTs, GaAs HBTs, GaAs or InP FETs, or GaAs HEMTs [l]-[4]. Microwave transistor amplifiers are rugged, low-cost, reliable, and can be easily integrated in both hybrid and monolithic integrated circuitry. As discussed in more detail in Chapter 10, transistor amplifiers can be used at frequencies in excess of 100 GHz in a wide range of applications requiring small size, low-noise figure, broad bandwidth, and low to medium power capacity. Although microwave tubes are still required for very high power and/or very high frequency applications, continuing improvement in the performance of microwave transistors is steadily reducing the need for microwave tubes. Our discussion of transistor amplifier design will rely on the termina] characteristics of transistors, as represented by either S parameters or one of the equivalent circuit models introduced in the previous chapter. We will begin with some general definitions of two-port power gains that are useful for amplifier design, and then discuss the subject of stability. These results will then be applied to single-stage transistor amplifiers, including designs for maximum gain, specified gain, and low noise figure. Broadband balanced and distributed amplifiers are discussed in Section 11.4. We conclude with a brief treatment of transistor power amplifiers. 11.1 TWO-PORT POWER GAINS In this section we develop several expressions for the gain and stability of a general two-port amplifier circuit in terms of the S parameters of the transistor. These results will be used in the following sections for amplifier and oscillator design. 536 11.1 Two-Port Power Gains 537 + V, r ■uul FIGURE 11.1 A rwo-port network with general source and load impedances. Definitions ol Two-Port Power Gains Consider an arbitrary two-port network [S] connected to source and load impedances Zs and Zi, respectively, as shown in Figure 11.1, We will derive expressions for three types of power gain in terms of the S parameters of the two-port network and the reflection coefficients, Vs and rt, of the source and load. • Power Gain = G = PL/P\n is the ratio of power dissipated in the load ZL to the power delivered to the input of the two-port network. This gain is independent of Z$, although some active circuits are strongly dependent on Zs- • Available Gain = GA — iVn/Pavs is the ratio of the power available from the two-port network to the power available from the source. This assumes conjugate matching of both the source and the load, and depends on Z$ but not Zi. • Transducer Power Gain = GT = Pl/P^ is the ratio of the power delivered to the load to the power available from the source. This depends on both Zs and Zi. These definitions differ primarily in the way the source and load are matched to the two-port device; if the input and output are both conjugately matched to the two-port, then the gain is maximized and G — GA = G>. With reference to Figure 11.1, the reflection coefficient seen looking toward the load is Zi — Zq rt"Z, + Z0' while the reflection coefficient seen looking toward the source is Zs - Z0 Z5 + Z0' (11.1a) (11.1b) where Zq is the characteristic impedance reference for the $ parameters of the two-port network. In general, the input impedance of the terminated two-port network will be mismatched with a reflection coefficient given by Tin, which can be determined using a signal flow graph (see Example 4.7), or by the following analysis. From the definition of the S parameters that Vz+ = TL V{, we have v- = su + suv? = $n v,+ + snrL v-, Eliminating V{ from (11.2a) and solving for V~/ Vt+ gives „ V[ c £12% Tl _ Z-m - Zq (11.2a) (11.2b) (11.3a) where Zm is the impedance seen looking into port 1 of the terminated network, Siinilarly, 538 Chapter 11: Microwave Amplifier Design the reflection coefficient seen looking into port 2 of the network when port 1 is terminated by Z$ is p v2~ Si2s2i rs t oat — TTP — ^22 + *2 (11.3b) By voltage division, Vi = Vs Zs + Z-u = v1+ + v,- = v+d + n«). Using Z-a — Zq i + r„ í-n, from (11.3a) and solving for V,+ in terms of Vs gives v+ =vs (i - r5) 1 2(1-^^)' (11.4) If peak values are assumed for all voltages, the average power delivered to the network is P. -_Lj^|2(i-ir. |2)-lMJi^IiL in ~ 2Z01 v* IP 11 sz0 11 -rsrm? where (11.4) was used. The power delivered to the load is (i-irin|2) (11.5) r-\2 Pi. = 2Z0 (11.6) Solving for V2~ from (11.2b), substituting into (11.6), and using (11.4) gives |v+|2 |53li2 (1 - \rL\2) 1 vs\2 \s2l\2 (1 - ird2) 11 - rs\2 Pl = 2z0 n-S22rj: 8Z0 11 -s22rL\2\i-rsru (11.7) The power gain can then be expressed as \s2i\2 (1-irj2) (i - ir^i2) 11 - 522ri.i 2 ■ (11.8) The power available from the source, i*avs, is the maximum power that can be delivered to the network. This occurs when the input impedance of the terminated network is conju-gately matched to the source impedance, as discussed in Section 2.6. Thus, from (11.5), Pavs — Piti \vs\2 u-rs[2 szo (i-irs|2)' (11.9) Similarly, the power available from the network, Pavat is the maximum power that can be delivered to the load. Thus, from (1 i.7), P*vn — Pi _ \Vs\2\S2i\2{i-\r™\2)\i-rs\: r,=r^ 8Z0 |i-522r*ut|2|i-r5rin|2 (11.10) 11.1 Two-Port Power Gains 539 In (I MO), must be evaluated for Ti = r*ut. From (11.3a), it can be shown that li-r^j2' |i~SiiTM2(i-|r°u.l2)2 which reduces (11.10) to p Ws\2 iWu-r5|2 szo n-5nr5|2(i-|rout|2)- U1'U) Observe that Pavs and Pavn have been expressed in terms of the source voltage, Vs, which is independent of the input or load impedances. There would be confusion if these quantities were expressed in terms of V,+, since V* is different for each of the calculations of P^, Pivst and Pgvn. Using (11.11) and (11.9), the available power gain is then q ,_Pavn_ iWyl-IPsI2) A Pavs u-Snr^Hi-ir^i2)' From (11.7) and (11.9), the transducer power gain is Gr = ^ = l^l2(i-|r,P)(i-|r,P) p^ H-r5r]nj2|i-^rt|2 A special case of the transducer power gain occurs when both the input and output are matched for zero reflection (in contrast to conjugate matching). Then Vi = Ts = 0, and (11,13)reduces to Gr = ]S2J|2. (11.14) Another special case is the unilateral transducer power gain, Gtu, where Sn = 0 (or is negligibly small). This nonreciprocal characteristic is common to many practical amplifier circuits. From (11.3a), t$n ss Sn when = 0, so (11.13) gives the unilateral transducer gain as |52lP(l-|rp)(l_|rLp} |l-5nrs|2!l- 522r£|2 EXAMPLE 11.1 COMPARISON OF POWER GAIN DEFINITIONS A microwave transistor has the following S parameters at 10 GHz, with a 50 ii reference impedance: Sn =0.45Zi50* Si 2 =0,01^10° Ski = 2.05/10° S22 = 0-40/-150° The source impedance is 2S — 20 A and the load impedance is Zl = 30 £2. Compute the power gain, the available gain, and the transducer power gain. S40 Chapter 11: Microwave Amplifier Design Solution From 0 l Ta,b) the reflection coefficients at the source and load are Zs - Zn 20 - 50 ZL + Z0 30 + 50 From (11.3a,b) the reflection coefficients seen looking at the input and output of the terminated network are r - SnSu rL nA.AW> , {0.0l^l0g)(2.05ZlQ°X-0.250) r'" = S" + = mSM + 1 - (0.40tl50^)(-0.250) = 0.455 d5Qa, r , Sl2S2lrs _ TW (0.01^IQg)(2.05ZlO°)(-0.429) r-'-^ + T3^r7-0-40^ + 1-<0.45Z1500)(-0.429) = 0.408 t!5T\ Then from (11.8) the power gain is Iftil'Q -|rj2)__(2.05)2 [1 - (O.250)2]_ C ■ (1 - |rin|2) |1 - S22TL\2 J1 - (Q.40/-1500) (-0.250)1' [1 - (0.455)2! = 5.94. From (11.12) the available power gain is |52il: (1 - ir5|2) (2.05)2 [1 - (0.429)2] G A = |1 - 5„rs|2 (1 - ir^l2) J i _ (0.45^=150°) (-0.429)|2 [1 - (0.408)2] = 5.85. From (11.13) the transducer power gain is |52,l2(i-irs|2)(i-|rj2) Gj = |i -r^rinpn -s22rL\2 _(2.05)2 [1 - (0.429)2][1 - (0.250)2]_ 11 - (0.40/-150°) (-0.250)|2 11 - (-0.429) (0 4SV-150") | = 5.49. Further Discussion of Two-Port Power Gains A single-stage microwave transistor amplifier can be modeled by the circuit of Figure 11.2, where a matching network is used on both sides of the transistor to transform the input and 2u ^r^VVVv— Input matching circuit Transistor [S] Output matching circuit GL r 1 ii m FIGURE 11.2 The general transistor amplifier circuit. 11.1 Two-Port Power Gains 541 output impedance Z0 to the source and load impedances Z$ and ZL. The most useful gain definition for amplifier design is the transducer power gain of (11.13), which accounts for both source and load mismatch. Thus, from (11.13), we can define separate effective gain factors for the input (source) matching network, the transistor itself, and the output (load) matching network as follows: M - 1 ml S\ G0 = |S21|2, (11.16b) Gl = --TT-zr-ä- (11.16c) Then the overall transducer gain is Gj = G$GqGl - The effective gains from G$ and Gl are due to the impedance matching of the transistor to the impedance Zr> If the transistor is unilateral, so that Si2 = 0 or is small enough to be ignored, then (11.3) reduces to Tj„ = S\ \. rou, = S22, and the unilateral transducer gain reduces to Gju — GsGqGl, where Gs = --1 (11.17a) 1 - \rs\2 \i-snrs[ G0 = |521|2. (11.17b) GL =-L^L^. (11.17c) 11 - s22rL\2 The above results have been derived using the S parameters of the transistor, but it is possible to obtain alternative expressions for gain in terms of the equivalent circuit parameters of the transistor. As an example, consider the evaluation of the unilateral transducer gain for a conjugately matched GaAs FET using the equivalent circuit of Figure 10.34 (with Cget = 0). To conjugately match the transistor we choose source and load impedances as shown in Figure 11.3. Setting the series source inductive reactance X = l/coCgs will make Z\„ = Zj, and setting the shunt load inductive susceptance B = -wQ, will make Zqm = £j£; this effectively eliminates the reactive elements from the FET equivalent circuit. Then by voltage division Vc = Vs/2jcoRjCgS, and the gain can be easily evaluated as \8»Vc\2 g2mRäs WM2 (Um iWm Wftsa *SW"J' 1 ' } Gju = — - *avs where the last step has been written in terms of the cutoff frequency, fti from (10.78). This FIGURE 11.3 Unilateral FET equivalent circuit and source and load terminations for the calculation of unilateral transducer power gain. Chapter 11: Microwave Amplifier Design FIGURE 11,4 Photograph of a low noise MMIC amplifier using three HEMTs with coplanar waveguide circuitry. The amplifier has a gain of 20 dB from 20 to 24 GHz. The contact pads on the left and right of the chip are for RF input and output, with DC bias connections at the top. Chip dimensions are 1.1 x 2-0 mm. Courtesy of E, W. Jackson and B. Hou of the University of Massachusetts-Amherst, and J. Wendler of M/A-COM, Lowell, Mass. shows the interesting result that the gain of a conjugately matched FET amplifier drops off as l//2, or 6 dB per octave. A photograph of a MMIC low-noise amplifier is shown in Figure 11.4. STABILITY We now discuss the necessary conditions for a transistor amplifier to be stable. In the circuit of Figure 11.2, oscillation is possible if either the input or output port impedance has a negative real part; this would then imply that \ria\ > 1 or ir^J > 1. Because Tin and Tout depend on the source and load matching networks, the stability of the amplifier depends on and Vi as presented by the matching networks. Thus, we define two types of stability: • Unconditional stability. The network is unconditionally stable if |Tic | < 1 and |r0uil < 1 for all passive source and load impedances (i.e., \ rs\ < 1 and [TlI < 1). • Conditional stability: The network is conditionally stable if | r;n| < 1 and | Fom I < 1 only for a certain range of passive source and load impedances. This case is also referred to as potentially unstable. Note that the stability condition of an amplifier circuit is usually frequency dependent, since the input and output matching networks generally depend on frequency, Thus it is possible for an amplifier to be stable at its design frequency, but unstable at other frequencies. Careful amplifier design should consider this possibility. We must also point out that the following discussion of stability is limited to two-port amplifier circuits of the type 11.2 Stability 543 shown in Figure 11.2, and where the 5 parameters of the active device can be measured without oscillations over the frequency band of interest. The rigorous general treatment of stability requires that the network S parameters (or other network parameters) have no poles in the right-half complex frequency plane, in addition to the conditions that |Fj„| < 1 and I Tout | < 116]. This can be a difficult assessment in practice, but for the special case considered here, where the S parameters are known to be pole-free (as confirmed by measurability), the following stability conditions are adequate. Stability Circles Applying the above requirements for unconditional stability to (11.3) gives the following conditions that must be satisfied by Fs and fx if the amplifier is to be unconditionally stable: 1^1 = S22 + i - s22rL i-surs < 1. (11.19a) (11.19b) If the device is unilateral (S12 = 0), these conditions reduce to the simple results that I Si 11 < 1 and IS221 < 1 are sufficient for unconditional stability. Otherwise, the inequalities of (11.19) define a range of values for Tjy and FL where the amplifier will be stable. Finding this range for Ts and rL can be facilitated by using the Smith chart, and plotting the input and output stability circles. The stability circles are defined as the loci in the p£ (or F$) plane for which |r\n| = 1 (or |rgut| = 1). The stability circles then define the boundaries between stable and potentially unstable regions of rs and rL. Ts and rL must lie on the Smith chart (\Ts\ < 1, jT^I < 1 for passive matching networks). We can derive the equation for the output stability circle as follows. First use (11.19a) to express the condition that |Fjn| = 1 as 512^21 Ti. = 1, or 1 - &nrL \Su(i-s2ZrL) + slisllrL\ = \i (11,20) Now define A as the determinant of the scattering matrix: A = S11S22 — S]2$2|. Then we can write the above result as (11.21) (11.22) is,i-ATii ot tt-s12rL\. Now square both sides and simplify to obtain is„i2 + |A|2irL|2 - (ArLsr, + A*r£sn) = 1 + |S22i2irL|2 - (^r£ + %Jvj (is22|2- |A|3)rtr*-(S22 - AStvr, - (s*22 - A*sn)rL = |Sui2 - l (S22 - AS^Wi + (S|2 - A*Sjj)r* |5]112 - 1 IS22P-IAP |s22l2-|Ap (11.23) S44 Chapter 11: Microwave Amplifier Design Next, complete the square by adding IS22 - AS*, |2/(|S22l2 - | A|2)2 to both sides: (3a- AS?,)* |522|^-|Ap 1. Similarly, we will have I Tin I < 1 on one side of the output stability circle, and |J\j, | > 1 on the other side. So we now need to determine which areas on the Smith chart represent the stable region, for which |rin| < 1 and |r0ut| < 1. Consider the output stability circles plotted in the rL plane for |5\, | < 1 and \Su\ > 1, as shown in Figure 11.5, If we set ZL = Z0, then Tl — 0 and (11.19a) shows that |rto| \Su I- Now if I Si 11 < 1, then ir^l < 1, so TL = 0 must be in a stable region, This means that the center of the Smith chart (Tz, = 0) is in the stable region, so all of the Smith chart (ITL J < 1) that is exterior to the stability circle defines the stable range for Ti. This region is IW-1 \S2i-AS*u\2 |S22P-|A|2 + (jS22p-|Af2)3, CS22- A5n)* |522l2-|Ap $12$21 \s22\2~\A\2 Cs Rs \Su\2-\Af Si2S2i |S,i12HA|2 FIGURE 11.5 Output stability circles for a conditionally stable device, (a) | S] 11 < 1. (b) | S\ 11 > 1. 11.2 Stability 545 shaded in Figure 11.5a. Alternatively, if we set Zi = Z0 but have \Su \ > 1, then |riQ| > 1 for rL — 0 and the center of the Smith chart must be in an unstable region. In this case the stable region is the inside region of the stability circle that intersects the Smith chart, as illustrated in Figure 11.5b. Similar results apply to the input stability circle. If the device is unconditionally stable, the stability circles must be completely outside (or totally enclose) the Smith chart. We can state this result mathematically as \\CL\-RL\> 1, for|5,,| 1, for|522| 1 or 11 > 1, the amplifier cannot be unconditionally stable because we can always have a source or load impedance of Zo leading to rs = 0 or rL = 0, thus causing I Tin I > 1 or I rout I > 1. If the device is only conditionally stable, operating points for Ts and Ti must be chosen in stable regions, and it is good practice to check the stability at several frequencies near the design frequency. If it is possible to accept a design with less than maximum gain, a transistor can usually be made to be unconditionally stable by using resistive loading. Tests for Unconditional Stability The stability circles discussed above can be used to determine regions for r$ and F^ where the amplifier circuit will be conditionally stable, but simpler tests can be used to determine unconditional stability. One of these is the K — A test, where it can be shown that a device will be unconditionally stable if Roiiet's condition, defined as along with the auxiliary condition that [Af:=t%%-%%| < l, (H.29) are simultaneously satisfied. These two conditions are necessary and sufficient for unconditional stability, and are easily evaluated. If the device S parameters do not satisfy the K — A test, the device is not unconditionally stable, and stability circles must be used to determine if there are values of Fj and Ti for which the device will be conditionally stable. Also, recall that we must have jSu | < 1 and IS221 < 1 if the device is to be unconditionally stable. While the K - A test of (11.2SH11.29) is a mathematically rigorous condition for unconditional stability, it cannot be used to compare the relative stability of two or more devices since it involves constraints on two separate parameters. Recently, however, a new criterion has been proposed [71 that combines the S parameters in a test involving only a single parameter, fi, defined as =-LrM—> i. (ii.3o) Thus, if a, > 1, the device is unconditionally stable. In addition, it can be said that larger values of li imply greater stability. 546 Chapter 11: Microwave Amplifier Design We can derive the jx-test of (1 J.30) by starting with the expression from (11.3b) for om- r _ S- ~ FfMI A — 5nroilt Taking the magnitude of both sides gives S22 — fct,. = L A — 5) [rout Squaring both sides and expanding gives \rMX\2(i - \sn\2) + r011t(A*s,i - s*22) + - s22) = ia\2 - |522|: Next, divide by 1 - |S,i|2 to obtain (A*S„ - S*21)rw + (ASt1 - S22W* \A\2 - \S22\2 l-|5,il2 Now complete the square by adding --—-tUv- to both sides: (1-15,112)2 TcrtJI + A5U - 523 1-|5mI2 2 \A\2-\S22\2 + \A*Sn-S*n\2 15,25,, \2 1-|W {l-\Su\2T o-i5upr (11.32) This equation is of the form |rout — C\ = /?, which represents a circle with center C and radius R in the P^, plane. Thus the center and radius of the mapped \TS\ = 1 circle are given by S22 - as;} l-|5uP |5p„S3I| R - (11.33a) (11.33b) l-|Snl2' If points within this circular region are to satisfy |rout| < 1, then we must have that |CT + *<1. (11.34) Substituting (11.33) into (11.34) gives |522 -ASri| + |Sl252l| 1. 11.2 Stability 547 The K — A test of (11.28M11.29) can be derived from a similar starting point, or more simply from the fi-test of (11.30). Rearranging (11.30) and squaring gives m - A^l2 < (1 - ISiil2 - l%Í2il)2. (11.35) It can be verified by direct expansion that \S12 - ASM|2 = \SÍ2S2Ú2 + (1 - l5,,|2)(iS22|2 - |A|2), so (11.35) expands to |Si2S2i|2+(l " |Snl2)(|522l3 - |A|2) <(1 - |S,il3){l - \St,\2 - 2|S,2S21i) + |Sl252J |2. Simplifying gives |S22|:-|A|2 ' In addition to (11.28), the K - A test also requires an auxiliary condition to guarantee unconditional stability. Although we derived Rollet's condition from the necessary and sufficient result of the /i.-test, the squaring step used in (11.35) introduces an ambiguity in the sign of the right-hand side, thus requiring an additional condition. This can be derived by requiring that the right-hand side of (11.35) be positive before squaring. Thus, \Sl2S2i\ < 1- I Si 112. Because similar conditions can be derived for the input side of the circuit, we can interchange Sn and S22 to obtain the analogous condition that |S1252l| 1 - IS22I2- Adding these two inequalities gives 2|5i252il <2 - |Sn|2 - |i?22l2- From the triangle inequality we know that |A| = \S\]S22 - SX2S2\ I < \SUS21\ + so we have that |A| < \SU\\S22\ + 1 - ^|5n|2 - -\Šié < 1 - i(|5„|2 - |S22|2) < 1, which is identical to (11.29). EXAMPLE 11.2 TRANSISTOR STABILITY The 5 parameters for the HP HFET-102 GaAs FET at 2 GHz with a bias voltage Vgj. = 0 are given as follows (Zo = 50 £2): Sn = n RQ4/-60.Ó0 521 =3.122/1.23.6°, S,2 = 0.020^624°, 522 = 0.781 Z=2L6°. 548 Chapter 11: Microwave Amplifier Design 11.3 Determine the stability of this transistor by using the k — a test and the ^t-test, and plot the stability circles on a Smith chart. Solution From (11.28) and (11.29) we compute k and |AJ as IAI = \SnS22 - Sl2S2i\ = \0.696£z$l°\ =0.696, f=l-|W-|^ + |AP=0 607 2IS12S21I Thus we have | A| = 0.696 < 1, but K < 1, so the unconditional stability criteria of (11.28M11-29) is not satisfied, and the device is potentially unstable. The stability of this device could also be evaluated using the ^-test, for which (11.30) gives ll = 0.86, again indicating potential instability. The centers and radii of the stability circles are given by (11.25) and (11.26): (s22 - aswut .t RL= l^'l =0.50, \$2l\2 ~ |A|2 This data can be used to plot the input and output stability circles, as shown in Figure 11.6. Since |5i 11 < 1 and IS22I < 1» the central part of the Smith chart represents the stable operating region for Ts and VL. The unstable regions are darkened. ■ SINGLE-STAGE TRANSISTOR AMPLIFIER DESIGN Design for Maximum Gain (Conjugate Matching) After the stability of the transistor has been determined, and the stable regions for r$ and Tl have been located on the Smith chart, the input and output matching sections can be designed. Since Go of (J1.16b) is fixed for a given transistor, the overall gain of the amplifier will be controlled by the gains, G$ and G^, of the matching sections. Maximum gain will be realized when these sections provide a conjugate match between the amplifier source or load impedance and the transistor. Because most transistors appear as a significant impedance mismatch (large \S\\ \ and IS22I), the resulting frequency response will be narrowband. In the next section we will discuss how to design for less than maximum gain, with a corresponding improvement in bandwidth. Broadband amplifier design will be discussed in Section 11.4. With reference to Figure 11.2 and our discussion in Section 2.6 on conjugate impedance matching, we know that maximum power transfer from the input matching network to the transistor will occur when n. - r* 1 in — 1 S' (11.36a) 11,3 Síngle-Stage Transistor Amplifier Design 549 FIGURE 11.6 Stability circles for Example 11.2. and the maximum power transfer from the transistor to the output matching network will occur when rout = r*. ni 36b) Then, assuming lossless matching sections, these conditions will maximize the overall transducer gain. From (11.13), this maximum gain will be given by %■ = iHr^'^i-sJ^- (1L37) In the general case with a bilateral transistor, Tin is affected by rom, and vice versa, so that the input and output sections must be matched simultaneously. Using (11.36) in (11.3) gives the necessary equations: T's = S„ + p^, (11.38a) i—put $ 550 Chapter 11: Microwave Amplifier Design We can solve for F$ by first rewriting these equations as follows: r — c* _i_ *r&***t 1 í - j>n + r? = where A = S\] $22 — S|2S2i- Substituting this expression for T£ into the expression for and expanding gives TsQ - |S22|2) + T2(A5|2 - Su) = r5(A^5^ - IS,,!2 - ASf2S2+t) + 5ri(l-|522|2) + ^252*1^2- Using the result that A(SU522 — S[2£2i) = |A|2 allows this to be rewritten as a quadratic equation for rs: - AS^r2 + (|Aj2 - \SU I2 + IS22I2 - l)r5 + (S*n - A*S22) = 0. (1139) The solution is rs =-S—--- (11.40a) Similarly, the solution for can be written as bz± Jb} -4\C2\2 TL =--- (11.40b) The variables B\, C\, B2, C2 are defined as B, = 1 + |SH|2-|522|2-|A|2, (11.41a) b2 = l + \S22\2-\Sli\2-\&\2, (11.41b) Ci = Sn-AS^, (11.41c) C2 = S22-A5U. (11.41d) Solutions to (11.40) are only possible if the quantity within the square root is positive, and it can be shown that this is equivalent to requiring K > 1. Thus unconditionally stable devices can always be conjugately matched for maximum gain, and potentially unstable devices can be conjugately matched if K > 1 and |A| < 1. The results are much simpler for the unilateral case. When S(2 = 0, (11.38) shows that !> = and = S^, and then maximum transducer gain of (11.37) reduces to ^-vš&^p&sř (11-42> The maximum transducer power gain given by (11.37) occurs when the source and load are conjugately matched to the transistor, as given by the conditions of (11.36). If the transistor 11.3 Single-Stage Transistor Amplifier Design 551 is unconditionally stable, so that K > 1, the maximum transducer power gain of (11.37) can be simply rewritten as follows: This result can be obtained by substituting (11.40) and (11.41) for Ts and TL into (11.37) and simplifying. The maximum transducer power gain is also sometimes referred to as the matched gain. The maximum gain does not provide a meaningful result if the device is only conditionally stable, since simultaneous conjugate matching of the source and load is not possible if K < 1 (see Problem 11.7). In this case a useful figure of merit is the maximum stable gain, defined as the maximum transducer power gain of (11.43) with K = 1. Thus, G^ = }^. (li-44) \Sl2\ The maximum stable gain is easy to compute, and offers a convenient way to compare the gain of various devices under stable operating conditions. EXAMPLE 11J CONJUGATELY MATCHED AMPLIFIER DESIGN Design an amplifier for maximum gain at 4.0 GHz using single-stub matching sections. Calculate and plot the input return loss and the gain from 3 to 5 GHz. The GaAs FET has the following S parameters (Z0 = 50 £2): /(GHz) Su S2} Si2 S22 3,0 n.«n/-sy 2.$6ᥠ0.03 iM Q7fi/-4F 4.0 n 77./-116° 2.60/76* 0.03/57° 0.73Z=54° 5.0 f)fifi/-142a 2.39/54* 0.03/62" 0.72^=68° Solution We first check the stability of the transistor by calculating A and K at 4.0 GHz: A = SnS22 - Si2S2i = 0.48S7-162°, K =-21^1-=L195- Since [A] < 1 and K > 1, the transistor is unconditionally stable at 4.0 GHz. There is no need to plot the stability circles. For maximum gain, we should design the ínatcMng sections for a conjugate match to tlie transistor. Thus, = rfn and p£ = r*^, and Fs, rL can be determined from (11.40): B] ± J Br - 4|Ci |2 b2±Jb2-4\c2\* rL =-y—-= 0.876/61°. 552 Chapter 11: Microwave Amplifier Design Then the effective gain factors of (1 IT6) can be calculated as ^ = TTTr7r4'17 = 6-20dB- G0 = |S2, |2 = 6.76 = 8.30 dB, Ct = rr^r^ = L67 = 2-22dB' 11 - m™bf So the overall transducer gain will be GiW, = 6-20 + 8-30 + 2-22 = 167 dB The matching networks can easily be determined using the Smith chart. For the input matching section, we first plot Ts, as shown in Figure 11,7a. The impedance, Z5, represented by this reflection coefficient is the impedance seen looking into the matching section toward the source impedance, Zo, Thus, the matching section must transform Z0 to the impedance Zs. There are several ways of doing this, but FIGURE 11.7 Circuit design and frequency response for the transistor amplifier of Example 11.3. (a) Smith chart for the design of the input matching network. 11.3 Single-Stage Transistor Amplifier Design 553 we will use an open-circuited shunt stub followed by a length of line. Thus we convert to the nonnalized admittance ySi and work backward (toward the load on the Smith chart) to find that a line of length 0.120k will bring us to the I -j- jb circle. Then we see that the required stub admittance is + j 3.5, for an open-circuited stub length of 0.206X. A similar procedure gives a line length of Q.2Q6X and a stub length of 0.206A for the output matching circuit. The final amplifier circuit is shown in Figure 11.7b, This circuit only shows the RF components; the amplifier will also require some bias circuitry. The return loss and gain were calculated using a CAD package, interpolating the necessary S parameters from the table on page 551. The results are plotted in Figure 11.7c, and show the expected gain of 16.7 dB at 4.0 GHz, with a very good return loss. The bandwidth where the gain drops by 1 dB is about 2.5%. ■ Constant Gain Circles and Design for Specified Gain In many cases it is preferable to design for less than the maximum obtainable gain, to improve bandwidth or to obtain a specific value of amplifier gain. This can be done by designing the input and output matching sections to have less than maximum gains; in other words, mismatches are purposely introduced to reduce the overall gain. The design procedure is facilitated by plotting constant gain circles on the Smith chart, to represent loci of Fj and ft that give fixed values of gain (Gs and Gl )- To simplify our discussion, we will 554 Chapter 11: Microwave Amplifier Design only ireat the case of a unilateral device; the more general case of a bilateral device must sometimes be considered in practice and is discussed in detail in references [1], [2], and [3]. In many practical cases |S]2| is small enough to be ignored, and the device can then be assumed to be unilateral. This gready simplifies the design procedure. The error in the transducer gain caused by approximating \S\2\ as zero is given by the ratio Gj/Gju- It can be shown that this ratio is bounded by < < t.-rrrT' (11.45) (1 + t/)2 Gjy (1-C/)2' where U is defined as the unilateral figure of merit, Usually an error of a few tenths of a dB or less justifies the unilateral assumption. The expression for G $ and G i for the unilateral case are given by (11.17a) and (11.17c); r i - ir*!2 Gs = Gi = i-inj2 u-522rL|2' These gains are maximized when Y$ — S*, and Tl = S*z, resulting in the maximum values given by Gs =---s, (11.47a) GL =--—r. (11.47b) Now define normalized gain factors gs and gi as Then we have that 0 < gs < 1, and 0 < gt < 1. For fixed values of gs and gL, (11.48) represents circles in the Fj or V& plane. To show this, consider (11.48a), which can be expanded to give ^|i-snr5|2 = (i-|rs|2)(i-|5n|2), (gs\Su\2 +1 - !5i!i2)|r5|2 - gs(Surs + surs) = 1 - |5,,i2 - 8s, r gs(snrs + s*ur*s) _ i-\sup-gs 5 5 l-(l-^)|5u|2 l-(l-^)|5n|2' Now add (g||Sn!2)/[l - (1 - gs)\Sn |2]2 to both sides to complete the square: [1-(1-^)|5„|2]2 1 -d -£s)|Siil2 11.3 Single-Stage Transistor Amplifier Design 555 Simplifying gives r5- l-(l-^)|Snl2 -gs)\Su\2 ' (11.50) which is the equation of a circle with its center and radius given by c'-i-(.-^.p- wm ynrH0-|5„P) 5, l-0-te)|Snl* The results for the constant gain circles of the output section can be shown to be, -£^- (1152a) l-d-^)|S22|2 v-slv-1-221/ (H.52b) I — (1 - gL)\Sll\ The centers of each family of circles He along straight lines given by the angle of Sn or •$22- Note that when gs (or gz.) = 1 (maximum gain), the radius R$ (or RL) — 0, and the center reduces to Sf, (or S£>), as expected. Also, it can be shown that the 0 dB gain circles (Gs = 1 or Gi = 1) will always pass through the center of the Smith chart. These results can be used to plot a family of circles of constant gain for the input and output sections. Then F$ and T^ can be chosen along these circles to provide the desired gains. The choices for rs and rL are not unique, but it makes sense to choose points close to the center of the Smith chart to minimize the mismatch and thus maximize the bandwidth. Alternatively, as we will see in the next section, the input network mismatch can be chosen to provide a low-noise design. EXAMPLE 11.4 AMPLIFIER DESIGN FOR SPECIFIED GAIN Design an amplifier to have a gain of 11 dB at 4.0 GHz. Plot constant gain circles for Gs = 2 dB and 3 dB, and GL = 0 dB and 1 dB. Calculate and plot the input return loss and overall amplifier gain from 3 to 5 GHz. The FET has the following 5 parameters (Z0 = 50 Q): f (GHz) £„ Sn S,2 Sn 3 0.80^90* 2.8il00° 0 0.66^50" 4 n 7s/-120" 2.5z800 0 0,60/=2T 5 0 71/-140° 2-3^60" 0 os»/-85° Solution Since Sn = 0 and \Sw\ < 1 and \S22\ < 1, die transistor is unilateral and unconditionally stable. From (11.47) we calculate the maximum matching section gains as G.II»1 = r-^ = 2.29 = 3.6 dB, G^ = j~f=-ij = 1.56 = 1.9 dB. 1 ~ \S22r 556 Chapter 11: Microwave Amplifier Design The gain of the mismatched transistor is Gc = \S2]\2 = 6.25 = 8.0 dB, so the maximum unilateral transducer gain is Gtu^ = 3.6 + 1.9 + 8.0 = 13.5 dB. Thus we have 2.5 dB more available gain than is required by the specifications. We use (11.48), (11.51), and (11.52) to calculate the following data for the constant gain circles: Gs = 3dB gs = 0.875 Cs = 0.706/120» = 0.166 Gs = 2dB gs = 0.691 Cs = o.627 fjffl = 0,294 GL - 1 dB gL = 0.806 cL = 0.520Z7_0G = 0.303 GL = 0dB gL = 0.640 Ct = 0.44O/W Rl = 0,440 The constant gain circles are shown in Figure 11.8a. We choose Gs = 2 dB and GL = I dB, for an overall amplifier gain of II dB. Then we select Ts and Ft («) FIGURE 11.8 Circuit design and frequency response for the transistor amplifier of Example 11.4. (a) Constant gain circles. 11.3 Single-Stage Transistor Amplifier Design 557 0.045 A 50 il -10 4.0 Frequency (GHz) (c) FIGURE U.8 Continued (b) RF circuit. (c) Transducer gain and return loss. 5.0 along these circles as shown, to minimize the distance from the center of the chart (this places r$ and fL along the radial lines at 120° and 70°, respectively). Thus, rs = 0.33zJ20° and VL = Q,22lTLT, and the matching networks can be designed using shunt stubs as in Example 11.3. The final amplifier circuit is shown in Figure 11.8b. The response was calculated using CAD software, with interpolation of the given 5 parameter data. The results are shown in Figure 11.8c, where it is seen the desired gain of 11 dB is achieved at 40 GHz. The bandwidth over which the gain varies by i 1 dB or less is about 25%, which is considerably belter than the bandwidth of the maximum gain design in Example 11,3. The return loss, however, is not very good, being only about 5 dB at the design frequency. This is due to the deliberate mismatch introduced into the matching sections to achieve the specified gain. ■ Low-Noise Amplifier Design Besides stability and gain, another important design consideration for a microwave amplifier is its noise figure. In receiver applications especially, it is often required to have a preamplifier with as low a noise figure as possible since, as we saw in Section 10.1, the first stage of a receiver front end has the dominant effect on the noise performance of the overall system. Generally it is not possible to obtain both minimum noise figure and maximum gain for an 556 Chapter 11: Microwave Amplifier Design amplifier, so some sort of compromise must be made. This can be done by using constant gain circles and circles of constant noise figure to select a usable trade-off between noise figure and gain. Here we will derive the equations for constant noise figure circles, and show how they are used in transistor amplifier design. As derived in references [4] and [5], the noise figure of a two-port amplifier can be expressed as g$ where the following definitions apply: Ys = G$ + JBS = source admittance presented to transistor. Kopt = optimum source admittance that results in minimum noise figure. Fmin = minimum noise figure of transistor, attained when Ys = Kopt. r?,v = equivalent noise resistance of transistor, Gs = real part of source admittance. Instead of the admittance Ys and Kqp(, we can use the reflection coefficients r$ and T^, where r«* = TTTTE- (ll-54b) ^0 1 + 1 opt Vs is the source reflection coefficient defined in Figure 11.1. The quantities F^, ropt, and Rjv are characteristics of the particular transistor being used, and are called the noise parameters of the device; they may be given by the manufacturer, or measured. Using (11.54), the quantity | Ys - Y^\2 can be expressed in terms of rs and ropt: Also, GS = Re = A ip& + pm = (11.56) 2Z0Vi+r5 i+r*/1 z0|i + rs|2 ' Using these results in (11.53) gives the noise figure as F-Fmirt + ^-(i-irsP)|i-rrQptp- (1L57) For a fixed noise figure, F, we can show that this result defines a circle in the r$ plane. First define the noise figure parameter, N, as N = \rS - ropt|2 F-Fmin 2 11.3 Single-Stage Transistor Amplifier Design 559 which is a constant, for a given noise figure and set of noise parameters. Then rewrite (11.58) as (rs-ropt)(r^-r:pt) = ^(i-|r5|2), rsr; - (rsr* + r^ropt) + r^r' = n - n\rs\2, r r+ (rsr^ + r^,) N-\rapt\ i si s--—-= —1- .' -' JV + 1 N + l Now add {V^/iN + l)2 to both sides to complete the square to obtain I s — N + l m +1) (11,59) This result defines circles of constant noise figure with centers at C> = N + ť (11.60a) and radii of Rf = ' N + 1 (11.60b) EXAMPLE 11 .5 LOW-NOISE AMPLIFIER DESIGN A GaAs FET is biased for minimum noise figure, and has the following S parameters and noise parameters at 4 GHz (Z0 — 50£2): Su = 0.6/-60°, £2I = l.9M°, Sl2 = 0.05z26°f S22 = 0.5^60"; *U - 1.6 dB, ropt = 0.62Zl00°, Rjv = 20 Q. For design purposes, assume the device is unilateral, and calculate the maximum error in Or resulting from this assumption. Then design an amplifier having a 2.0 dB noise figure with the maximum gain that is compatible with this noise figure. Solution We first compute the unilateral figure of merit from (11.46): V = \S\2S2\S\1S22\ (i-|S„i2)(i-iW) = 0.059. Then from (11.45) the ratio GjjGjjj is bounded as 1 GT 1 Gf < - < or (i + i/)2 Gm a-m1' 0.891 < Ä < 1-130. G tu 560 Chapter 11: Microwave Amplifier Design In dB, this is -0.50 < GT - GTu < 0-53 dB, where GT and Gjy are now in dB. Thus, we should expect less than about ±0.5 dB error in gain. Next, we use (11.58) and (11.60) to compute the center and radius of the 2 dB noise figure circle: u - F ~ Fmin 11 j. p i2 - s 0.0986, 0 = -^£L =0.56il0Q^ Af + 1 vMtf + i-ir^p) «F ----- N + l This noise figure circle is plotted in Figure 11.9a. Minimum noise figure (F^ — 1.6 dB) occurs for rs = = 0.62^100°. Next we calculate data for several input section constant gain circles. From (11-51), GS(W) Ss 1.0 0.805 0.300 1.5 0.904 0.56z60° 0.205 1.7 0.946 0.58^60° 0.150 These circles are also plotted in Figure 11.9a. We see that the Gs = 1.7 dB gain circle just intersects the F = 2 dB noise figure circle, and that any higher gain will result in a worse noise figure. From the Smith chart the optimum solution is then rs = 0.53/25*, yielding Gs = 1.7 dB and F = 2.0 dB. For the output section we choose Fj, = = 0.5/60° for a maximum GLof GL a -—\—- = 1.33 = 1.25 dB, The transistor gain is G0 = \S2i\2 =3.6.1 =5.58 dB, so the overall transducer gain will he Gjd = Gs + G0-\-GL = 8.53 dB. 4(20/50) 1 = 0.24. A complete AC circuit for the amplifier, using open-circuited shunt stubs in the matching sections, is shown in Figure 11.9b. A computer analysis of the circuit gave a gain of 8.36 dB. ■ 11.4 Broadband Transistor Amplifier Design 561 50 n i—AA/W—o— 50 a o ' 11 ' 0.25 A I 1 ~ 0.I3ÖA 50ft (bl FIGURE 11.9 Circuit design for the transistor amplifier of Example 11.5. (a) Constant gain and noise figure circles. (b> RF circuit. BROADBAND TRANSISTOR AMPLIFIER DESIGN The ideal microwave amplifier would have constant gain and good input matching over the desired frequency bandwidth. As the examples of the last section have shown, conjugate matching will give maximum gain only over a relatively narrow bandwidth, while designing for less than maximum gain will improve the gain bandwidth, but the input and output ports of the amplifier will be poorly matched. These problems are primarily a result of the fact 562 Chapter 11: Microwave Amplifier Design that microwave transistors typically are not well matched to 50 ÍÍ, and large impedance mismatches are governed by the Bode-Fano gain-bandwidth criterion discussed in Section 5.9. Another factor is that IS21I decreases with frequency at the rate of 6 dB/octave. For these reasons, special consideration must be given to the problem of designing broadband amplifiers. Some of the common approaches to this problem are listed below; note in each case that an improvement in bandwidth is achieved only at the expense of gain, complexity, or similar factors. • Compensated matching networks: Input and output matching sections can be designed to compensate for the gain rolloff in (S^il, but generally at the expense of the input and output match. • Resistive matching networks: Good input and output matching can be obtained by using resistive matching networks, with a corresponding loss in gain and increase in noise figure. • Negative feedback: Negative feedback can be used to flatten the gain response of the transistor, improve the input and output match, and improve the stability of the device. Amplifier bandwidths in excess of a decade are possible with this method, at the expense of gain and noise figure. • Balanced amplifiers: Two amplifiers having 90° couplers at their input and output can provide good matching over an octave bandwidth, or more. The gain is equal to that of a single amplifier, however, and the design requires two transistors and twice the DC power. • Distributed amplifiers: Several transistors are cascaded together along a transmission line, giving good gain, matching, and noise figure over a wide bandwidth. The circuit is large, and does not give as much gain as a cascade amplifier with the same number of stages. Below we discuss in detail the operation of balanced and distributed amplifiers. Balanced Amplifiers As we saw in Example 11.4, a fairly flat gain response can be obtained if the amplifier is designed for less than maximum gain, but the input and output matching will be poor. The balanced amplifier circuit solves this problem by using two 90* couplers to cancel input and output reflections from two identical amplifiers. The basic circuit of a balanced amplifier is shown in Figure 11.10. The first 90° hybrid coupler divides the input signal into two equal-amplitude components with a 90° phase difference, which drive the two amplifiers. The second coupler recombines the amplifier outputs. Because of the phasing properties of the hybrid coupler, reflections from the amplifier inputs cancel at the input to the hybrid, FIGURE 11.10 A balanced amplifier using 90" hybrid couplers. 11.4 Broadband Transistor Amplifier Design 563 resulting in an improved impedance match; a similar effect occurs at the output of the balanced amplifier. The gain bandwidth is not improved over that of the single amplifier sections. This type of circuit is more complex than a single-stage amplifier since it requires two hybrid couplers and two separate amplifier sections, but it has a number of interesting advantages; • The individual amplifier stages can be optimized for gain flatness or noise figure, without concern for input and output matching. • Reflections are absorbed in the coupler terminations, improving input/output matching as well as the stability of the individual amplifiers, ■ The circuit provides a graceful degradation of a —6 dB loss in gain if a single amplifier section fails. • Bandwidth can be an octave or more, primarily limited by the bandwidth of the coupler. In practice, balanced MMIC amplifiers often use Lange couplers, which are broadband and very compact, but quadrature hybrids and Wilkinson power dividers (with an extra 90° line on one arm) can also be used. If we assume ideal hybrid couplers, then with reference to Figure 11.10 the voltages incident at the amplifiers can be written as Kri = ^i+< tll61a) V*l = ^\ (11.61b) where V,+ is the incident input voltage. Then the output voltage can be found as (11.62) where (11.61) was used. Then we can write S21 as S2i = ^ = ^-iGA + GB), (11.63) which shows that the overall gain of the balanced amplifier is the average of the individual amplifier voltage gains. The total reflected voltage at the input can be written as *T = g=$fi + ^Vbx = gf-M + ^r*v*+i * !*?0r< - no. (n.64) Then we can write Su as Sn = ^T = ^(rA-rB). (11.65) If the amplifiers are identical, then Ga — Gb and TA = r$, and (11.65) shows that S\ 1 =0, and (11.63) shows that the gain of the balanced amplifier will be the same as the gain of an individual amplifier. If one amplifier fails, the overall gain will drop by 6 dB, with the remaining power lost in the coupler terminations. It can also be shown that the noise figure of the balanced amplifier is F = (FA + F#)/2, where FA and FB are the noise figures of the individual amplifiers. 534 Chapter 11: Microwave Amplifier Design EXAMPLE 11.6 PERFORMANCE AND OPTIMIZATION OF A BALANCED AMPLIFIER Use the amplifier of Example 11.4 in a balanced amplifier configuration operating from 3 to 5 GHz. Use quadrature hybrids, and plot the gain and return loss over this frequency range. Using microwave CAD software, optimize the amplifier matching networks to give 10 dB gain over this band. Solution The amplifier of Example 11.4 was designed for a gain of 11 dB at 4 GHz. As seen from Figure 1 l.Sc, the gain varies by a few dB from 3 to 5 GHz, and the return loss is no better than 5 dB. We can design a quadrature hybrid, according to the discussion in Section 7.5, to have a center frequency of 4 GHz. Then the balanced amplifier configuration of Figure 11.10 can be modeled using a microwave CAD package, with the results shown in Figure 11.11. Note the dramatic improvement in return loss over the band, as compared with the result for the original amplifier in Figure 11.8c. The input matching is best at 4 GHz since this was the design frequency of the coupler; a coupler with better bandwidth will give improved results at the band edges. Also observe that the gain at 4 GHz is still 11 dB, and that it drops by a few dB at the band edges. Most modem microwave CAD software packages have an optimization feature with which a small set of design variables can be adjusted to optimize a particular performance variable. In the present example, we will reduce the gain specification to 10 dB, and allow the CAD software to adjust the four transmission line stub and line lengths in the amplifier circuit of Figure 11.8b to give the best fit to this gain over the frequency range 3 to 5 GHz. Both amplifiers in the balanced circuit remain identical, so we should still see the improved input matching. The results of this optimization are shown in Figure 11.11, where it can be seen that the gain response is much flatter over the operating band. The input match is still very good in the vicinity of the center frequency, with a slightly worse result 3.0 3,S 4,0 4.5 5,0 Frequency (GHz) FIGURE 11.11 Gain and return loss, before and after optimization, for the balanced amplifier of Example 11.6. 11.4 Broadband Transistor Amplifier Design 565 ai the low-frequency end. The optimized stub and line lengths for the amplifier matching networks are listed below; Matching Network Before After Parameter Optimization Optimization Input section stub length 0.100A 0.109A Input section line length 0V179A 0.I13A, Output section line length 0.045k 0,134A. Output section stub length 0.432X 0.46U These represent fairly small deviations from the lengths in the original matching networks, ■ Distributed Amplifiers The concept of the distributed amplifier dates back to the 1940s. when it was used in the design of broadband vacuum tube amplifiers. With recent advances in microwave integrated circuit and device processing technology, the distributed amplifier has found new applications in broadband microwave amplifiers. Bandwidths in excess of a decade are possible, with good input and output matching. Distributed amplifiers are not capable of very high gains or very low noise figure, however, and generally are larger in size than an amplifier having comparable gain over a narrower bandwidth. The basic configuration of a microwave distributed amplifier is shown in Figure 11,12. A cascade of N identical FETs have their gates connected to a transmission line having a characteristic impedance Zs, with a spacing of while the drains are connected to a transmission line of characteristic impedance Zrf, with a spacing %, The operation of the distributed amplifier is very similar to that of the muhihole waveguide coupler discussed in Section 7.4. The input signal propagates down the gate fine, with each FET tapping off some of the input power. The amplified output signals from the FETs form a traveling wave on the drain line. The propagation constants and lengths of the gate and drain lines are chosen for constructive phasing of the output signals, and the termination impedances on the lines serve to absorb waves traveling in the reverse directions. The gate and drain capacitances of the FET effectively become part of the gate and drain transmission lines, while the gate and drain resistances introduce loss on these lines. This type of circuit is also known as a traveling wave amplifier. Here we will analyze the distributed amplifier in terms of the loaded gate and drain transmission lines [8], although it is also possible to apply the concept of image parameters [9], or to simply use CAD software. An analytical treatment has the advantage of illustrating FIGURE 11.12 Configuration of an A1-stage distributed amplifier. 566 Chapter 11: Microwave Amplifier Design m FIGURE 11.13 (a) Transmission line circuit for the gate line of the distributed amplifier; (b) equivalent circuit of a single unit cell of the gate line. the underlying principles of operation of the amplifier, while the numerical CAD approach is recommended for better accuracy and optimization capabilities. The first step in the analysis of the distributed amplifier is to employ the unilateral (Cgd = 0) version of the FET equivalent circuit to decompose the circuit of Figure 11.12 into separate loaded transmission lines for the gate and drain terminals. These are shown in Figures 11.13 and 11.14. The gate and drain transmission lines are typically microstrip; the ground conductors are not shown in Figure 11.12, but they are in Figures 11,13 and 11.14. The gate and drain lines are isolated except for the coupling through the dependent current sources, where ldn = gmVcn, and are matched at both ends. Figures 11.13b and 11.14b show the equivalent circuits for a single unit cell from the gate and drain lines, respectively. FIGURE 11.14 (a) Transmission line circuit for the drain line of the distributed amplifier; (b) equivalent circuit of a single unit cell of die drain line. 11.4 Broadband Transistor Amplifier Design 567 Lg and Cg are the inductance and capacitance per unit length of the gate transmission line, while R;£s and Cgf/tg represent the equivalent per unit length loading due to the FET input resistance Rt and gate-to-source capacitance Cgi. Similar definitions apply to the quantities Ld, Cj, Rdfid, and Cjsftd for the drain line. Thus we have taken the lumped loading of each FET and distributed its circuit parameters over the transmission lines of each unit cell. This approximation is generally valid when the electrical lengths of the unit cells are small. We can now use basic transmission line theory to find the effective characteristic impedance and propagation constants of the gate and drain lines. For the gate line, the series impedance and shunt admittance per unit length can be written as Z = jt»Lg, (11.66a) Y = jo>Cg+ .J**"'*' ■ (H.66b) 1 + jcoRjCgs If we assume that loss can be neglected for the calculation of characteristic impedance, as discussed r jection 2.7, then we have Z , = ./?= / L& (11.67) Vy \jcs + cgs/ts For the calculation of the propagation constant we retain the resistive term, since this will lead to attenuation: Yg = + Jßg - *JŽT = / jíúLg \+jo>RicJ If we assume small loss, such that y/Ld(Cd + Cdl/t4). (H-71) 2Rds tj 566 Chapter 11: Microwave Amplifier Design For an incident input voltage, V;, the voltage on the gate-to-source capacitance of the nth FET can be written as tea = far*4***, far \r ) 01-72) for a phase reference at the first FET, The factor in parentheses in (11.72) accounts for voltage division between Rt and Cg$; for typical FET parameters o>fi/Cg} gtg = 0.184 and a^d = 0.083. 18 16 14 12 m to c O 6 i JV=3 JV=4 \ 0 2 4 6 8 10 12 14 16 16 Frequency (GHz) FIGURE 11.IS Gain versus frequency for the distributed amplifier of Example 11.7. 570 Chapter 11: Microwave Amplifier Design The optimum size is then _ lnCofrVa^) _ In (0.184/0.083) = * ("'79) where G is the power gain of the amplifier. Silicon transistor amplifiers in the cellular telephone band of 800-900 MHz band have power added efficiencies on the order of 80%, but efficiency drops quickly with increasing frequency. Power amphfiers are often designed to provide the best efficiency, even if this means that the resulting gain is less than the maximum possible. Another useful parameter for power amplifiers is the compressed gain, G\, defined as the gain of the amplifier at the ldB compression point. Thus, if Go is the small-signal 11.5 Power Amplifiers 571 (linear) power gain, we have Gi(dB) = (?o(dB)-l. (11.80) As we have seen in Chapter 10, nonlinearities can lead to the generation of spurious frequencies and intermodulation distortion. This can be a serious issue in wireless transmitters, especially in a multicarrier system, where spurious signals may appear in adjacent channels. Linearity is also critical for nonconstant envelope modulations, such as amplitude shift keying and higher quadrature amplitude modulation methods. Class A amplifiers are inherently linear circuits, where the transistor is biased to conduct over the entire range of the input signal cycle, Because of this, class A amplifiers have a theoretical maximum efficiency of 50%. Most small-signal and low-noise amplifiers operate as class A circuits. In contrast, the transistor in a class B amplifier is biased to conduct only during one-half of the input signal cycle. Usually two complementary transistors are operated in a class B push-pull amplifier to provide amplification over the entire cycle. The theoretical efficiency of a class B amplifier is 78%. Class C amplifiers are operated with the transistor near cutoff for more than half of the input signal cycle, and generally use a resonant circuit in the output stage to recover the fundamental. Class C amplifiers can achieve efficiencies near 100%, but can only be used with constant envelope modulations. Higher classes, such as class D, E, F, and S, use the transistor as a switch to pump a highly resonant tank circuit, and may achieve very high efficiencies. The majority of communications transmitters operating at UHF frequencies or above rely on class A, AB, or B power amplifiers because of the need for low distortion products. Large-Signal Characterization of Transistors A transistor behaves linearly for signal powers well below the ldB compression point (Pi), and so the small-signal 5-parameters should not depend on either the input power level or the output termination impedance. But for power levels comparable to or greater than Pi, where the nonlinearity of the transistor becomes apparent, the measured S parameters will depend on input power level and the output termination impedance (as well as frequency, bias conditions, and temperature). Thus large-signal S parameters are not uniquely defined and do not satisfy linearity, and cannot be used in place of small-signal parameters. (For device stability calculations, however, small-signal S parameters can generally be used with good results.) A more useful way to characterize transistors under large-signal operating conditions is to measure the gain and output power as a function of source and load impedances. One way of doing this is to determine the large-signal source and load reflection coefficients, rSP and r^p, that maximize power gain for a particular output power (often chosen as P\), and versus frequency. Table ll.l shows typical large-signal source and load reflection coefficients for a typical NPN silicon bipolar power transistor, along with the small-signal S parameters. TABLE 11.1 Small-Signal S Parameters and Large-Signal Reflection Coefficients (Silicon Bipolar Power Transistor) /(MHz) Sj, 5, i L S22 Gp(dB) 800 0.76476* 4.10; 0.065, 0.35 /-163* 0.856/-167" 0.455 13.5 900 0.76ZÍ7T 3.42; IT 0.073, 0.35 f-lW 0.747 tzXLT 0.478 ZÍ6T 12.0 1000 0.76d6T 3.08; l69' 0.079, 0 797/-187° 0.49 i dM' 10.0 Another way of characterizing the large-signal behavior of a transistor is to plot contours of constant power output on a Smith chart as a function of the load reflection coefficient, Vu>, with the transistor conjugately matched at its input. These are called load-pull contours, and can be obtained using an automated measurement set-up with computer-controlled electromechanical stub tuners. A typical set of load-pull contours is shown in Figure 11.16. Load-pull contours are similar in function to the constant gain contours of Section 11.3, but are not perfect circles due to the nonlinearities of the device. Nonlinear equivalent circuit models can also be developed and used to predict the large-signal performance of FETs and BJTs [10]. The dominant nonlinear parameters for a microwave FET are Cgs, gm, Cgd, and R&. An important consideration in modeling large-signal transistors is the fact that most parameters are dependent on temperature, which of course increases with output power. Equivalent circuit models can be very useful when combined with computer-aided design software. Design of Class A Power Amplifiers In this section we will discuss the use of large-signal parameters for the design of class A amplifiers. Since class A amplifiers are ideally linear, it is sometimes possible to use small-signal 5 parameters for design, but better results are usually obtained if large-signal parameters are available. As with small-signal amplifier design, the first step is to check the stability 11.6 Power Amplifiers S73 of the device. Since instabilities begin at low signal levels, small-signal S parameters can be used for this purpose. Stability is especially important for power amplifiers, as high-power oscillations can easily damage active devices and related circuitry. The transistor should be chosen on the basis of frequency range and power output, ideally with about 20% more power capacity than is required by the design. Silicon bipolar transistors have higher power outputs than GaAs FETs at frequencies up to a few GHz, and are generally cheaper. Good thermal contact of the transistor package to a heat sink is essential for any amplifier with more than a few tenths of a watt power output. Input matching networks are generally designed for maximum power transfer (conjugate matching), while output matching networks are designed for maximum output power (as derived from T^). The optimum values of source and load reflection coefficients are different from those obtained from small-signal $ parameters via (11.40). Low-loss matching elements are important for good efficiency, particularly in the output stage, where currents are highest. Internally matched chip transistors are sometimes available, and have the advantage of reducing the effect of parasitic package reactances, thus improving efficiency and bandwidth. EXAMPLE 11.8 DESIGN OF A CLASS A POWER AMPLIFIER Design a power amplifier at 900 MHz using a Motorola MRF858S NPN silicon bipolar transistor with an output power of 3 W Design input and output impedance matching sections for the amplifier, find the required input power, and compute the power added efficiency. Use the given S parameters to compute the source and load reflection coefficients for conjugate matching, and compare to the actual large-signal values for and Tsp. The small-signal S parameters of the MRF858S transistor at 900 MHz are: Sn = 0.940 Zl64°, Si2 = 0.031 S2\ = 1.222Z43% S22 = 0.570 /-165° For an emitter-collector voltage Vce = 24 V and a collector current of Ic — 0.5 A, the output power at the 1 dB compression point is 3.6 W, and the power gain is 12 dB, The source and load impedances are Zia = 1.2 + ;3.5 £2, and Zout = 9.0 + j 14.5 Q. Solution We begin by establishing the stability of the device. Using the small-signal S parameters in (11,28) and (11.29) gives |A| = \S[[S22 — SuS2] | = 1(0.940 Zl64a)(0.570 /-16S°) - (0.031 M°)(1222 z43°)| ~ 0.546 K l-lSiil2-|aal2 + |A|2 = I — (0.940)2 - (0.570)2 + (0.546)2 = 2|5,2S2il 2(0.031X1.222) showing that the device is unconditionally stable. Converting the large-signal input and output impedances to reflection coefficients gives 574 Chapter 11: Microwave Amplifier Design 0.110 ). FIGURE 11.17 RF circuit for (he amplifier of Example 11.8. Using the small-signal S parameters in (11.40) to find the source and load reflection coefficients for conjugate matching gives fi]±v/B2-4|Clp rs = -^—----= 0.963 /-166 2Ci rL--^-J--= 0.712 Zl34c. 2C2 Note that these values approximately satisfy the relationships of (11.36), that rs = r£ and VL — r*M, but not exactly, due to the fact that the 5 parameters used to calculate r$ and p£ do not apply for large power levels. Thus we should use the given large-signal reflection coefficients, and let r5 = r* = 0.953 t=m\ ri = 1^ = 0.716 441°. Then the input and output matchuig networks can be designed as usual. The complete AC amplifier circuit is shown in Figure 11.17. For an output power of 3 W, the required input drive power is Pi„(dBm) = /V(dBm) - G/dB) = 10 log(3000) - 12 = 22.8 dBm = 189 mW. Then the power added efficiency of the amplifier can be found from (11.79) to be Pom-Pin 3.0-0.189 ^ rip ac =----—-= 23.4%. ,PAE Poc (24)(0.5) ■ REFERENCES f 1] G. D. Vendelín, A.M. Pavio, and U. L. Rohde, Microwave Circuit Design Using Linear and Nonlinear Techniques, Wiley, N.Y., 1990. [2] G, Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd edition, Prentice Hall, N.J., 1997. [3] I. Bahl and P Bhartia, Microwave Solid-State Circuit Design, Wiley Interscience, N.Y., 1988. [4] K. Chang, ed., Handbook of Microwave and Optical Components, vol. 2, Wiley Interscience, N.Y, 1990. [5] C, Gentile, Microwave Amplifiers and Oscillators, McGraw-Hill, NY., 1987. [6] M. Ohtomo, "Proviso on the Unconditional Stability Criteria for Linear Iwoports," IEEE Trans. Microwave Theory and Techniques, vol. MTT-43, pp. 1197-1200. May 1995. Problems 575 [7] M. L. Edwards and J. H. Sinksy, "A New Criteria for Linear 2-Port Stability Using a Single Geometrically Derived Parameter," IEEE Trans. Microwave Theory and Techniques, vol, MTT-40, pp. 2803-2811, December 1992. [81 Y. Ayasli, R. L. Mozzi, J. L. Vorhous, L. D. Reynolds, and R. A. Pucel, "A Monolithic GaAs 1-13 GHz Traveling-Wave Amplifier," IEEE Trans. Microwave Theory and Techniques, vol. MTT-30, pp. 976-981, July 1982. [9] J. B. Beyer, S. N. Prasad, R. C. Becker, J. E. Nordman, and G. K. Hohenwarter, "MESFET Distributed Amplifier Design Guidelines," IEEE Trans. Microwave Theory and Techniques, vol. MTT-32, pp. 268-275, March 1984. [10] W. R. Curtice and M. EUenberg. "A Nonlinear GaAs FET Model for Use in the Design of Output Circuits for Power Amplifiers," IEEE Trans. Microwave Theory and Techniques, vol. MTT-33, pp. 1383-1394. December 1985. PROBLEMS 11.1 Consider the microwave network shown below, consisting of a 50 £2 source, a 50 £2, 3 dB matched attenuator, and a 50 £2 load, (a) Compute the available power gain, the transducer power gain, and the actual power gain, (b) How do these gains change if the load is changed to 25 £2? (c) How do these gains change if the source impedance is changed to 25 £27 ZL = 50 ii, 25 ÍÍ 11.2 An amplifier uses a transistor having the following S parameters (Z<> = 50 £2): 5M = 0.61 /-170", 511 = 0.06 lM', S2I=23 lW, S22 = 0.72 /-2S°. The input of the transistor is connected to a source with Vj = 2 V (peak) and Z$ = 25 £2, and the output of the transistor is connected to a load of ZL = 100 £2. (a) What is the power gain, the available gain, the transducer power gain, and the unilateral transducer power gain? (b) Compute the available power from the source, and the power delivered to the load. 11.3 A microwave transistor has the following S parameters: S\i = 0.34/-170^ $2X — 4,3^80°, S\2 = 0.06/70° t and $12 = 0A5L=2T. Determine the stability, and plot the stability circles if the device is potentially unstable. 11.4 Repeat Problem 11.3 for the following transistor S parameters: Su — O.S^JJT, s2l = 5.1ZSQ', 512 - 0,3^20°, and S22 = dip./-40° 11J Use the /i-parameter test to determine which of (he following devices are unconditionally stable, and of those, which has the greatest stability: Device SM S[2 £3) S22 A n.M/-l7W 0.06/70° 4.3 0.45^25* B 0.75^60= 0.2/70° 5.0/90° 0.51/60° C rnw/-140° 0.04/60" 2.4/50" 0.70/^65° 11.6 Show that for a unilateral device, where Sl2 = 0, the /t-parameter test of {11.30) implies that | .Si j I -c 1 and ISVjI < 1 for unconditional stability. 11.7 Prove that the condition for a positive discriminant in (11.40a), that is, B2 > 4|C| \2, is equivalent to the condition that K2 > 1. 576 Chapter 11: Microwave Amplifier Design 11.8 Design an amplifier for maximum gain at 5.0 GHz with a GaAs FET thai has the following 5 parameters (Z0 = 50 £2): 5M = n*S/-l40J ^ = 2.4/50°, jfc = 0.04/60°, S2i = 0.70/-65°. Design matching sections using open-circuited shunt stubs. 11.9 Design an amplifier with maximum GTu using a transistor with the following 5 parameters (Zo = 50ft) at 6.0 GHz: Su =0.61 /-HO", = 2.24/32°, Sl2 = 0, S22 = 0.72z=S33, Design L-section matching sections using lumped elements. 11.10 Design an amplifier to have a gain of 10 dB at 6.0 GHz, using a transistor with the following S para meters (Z0 = 50 ft): Sn = 0.61/^170", S2\ = 2.24/32°, $ia = o, S22 = 0,72/^83°. Plot (and use) constant gain circles for Gs = 1 dB and Gr, = 2 dB. Use matching sections with open-circuited shunt stubs. 11.11 Compute the unilateral figure of merit for the transistor of Problem 11.3. What is the maximum error in the transducer gain if an amplifier is designed assuming the device is unilateral? 11.12 Show that the 0 dB gain circle for Gs (Gs = 1), defined by (11.51), will pass through the center of the Smith chart. 11.13 A GaAs FET has the following scattering and noise parameters at 8 GHz (Z0 — 50 ft): Sn = 0.7/-H0", 5,2 = 0.02/60°, = l,sm\ S21 = 0.&^W, Fmin - 2.5 dB, T^, = 0.70zi2Q\ Rh = 15 ft. Design an amplifier with minimum noise figure, and maximum possible gain. Use open-circuited shunt stubs in the matching sections, 11.14 A GaAs FET has the following scattering and noise parameters at 6 GHz (Zo = 50 ft): Su = ft.fi/-6r Si, = 2.0/M°, S,2 = 0, ^ = n7/-60g = 2.0dBT r0pl = 0.62dOQ°, R„ = 20 ft. Design an amplifier to have a gain of 6 dB, and the rninimum noise figure possible with this gain. Use open-circuited shum stubs in the matching sections. 11.15 Repeat Problem 11.14, but design the amplifier for a noise figure of 2.5 dB, and the maximum possible gain that can be achieved with this noise figure. 11.16 Repeat the analysis of the balanced amplifier of Example 11.6 using a 3 dB coupled line hybrid coupler. Use CAD software to optimize the input and output matching networks of the amplifiers to obtain a flat 10 dB gain response from 3 to 5 GHz, and compare the results with those obtained using the quadrature hybrid. 11.17 If the individual amplifier stages in a balanced amplifier have mismatches of and Ts at their output ports, show that (he output mismatch of the balanced amplifier is S22 = (Va — Fb)/2. 11.18 Derive the result for the optimum size of a distributed amplifier given in (11.77). 11.19 Consider a distributed amplifier using FETs having the following parameters: R( =5 ft, Rdi = 200 ft, Cgs = 0.35 pF, and g„ = 40 mS. Calculate and plot the gain from 2 to 20 GHz, for N = 4 and N = 16 sections, Find the optimum value of N that will give maximum gain at 18 GHz. 11.20 Use the transistor data given in Table 11.1 to design a power amplifier at 1 GHz with a power output of 1 W. Design the input and output matching circuits using the given large-signal reflection coefficients. Compute the required input power level. Chapter Twelve Oscillators and Mixers RF and microwave oscillators are universally found in all modern radar and wireless communications systems to provide signal sources for frequency conversion and carrier generation, A solid-state oscillator uses an active nonlinear device, such as a diode or transistor, in conjunction with a passive circuit to convert DC to a sinusoidal steady-state RF signal, Basic transistor oscillator circuits can generally be used at low frequencies, often with crystal resonators to provide improved frequency stability and low noise performance. At higher frequencies, diodes or transistors biased to a negative resistance operating point can be used with cavity, transmission line, or dielectric resonators to produce fundamental frequency oscillations up to 100 GHz. Alternatively, frequency multipliers can be used to produce power at millimeter wave frequencies. Because of the requirement of a nonlinear active device, the rigorous analysis and design of oscillator circuits is very difficult, and usually carried out today with sophisticated CAD tools. In this chapter we begin with art overview of low-frequency transistor oscillator circuits, including the well-known Hartley and Colpitts configurations, as well as crystal controlled oscillators. Next we consider oscillators for use at microwave frequencies, which differ from their lower frequency counterparts primarily due to different transistor characteristics and the ability to make practical use of negative resistance devices and high-Q microwave resonators. We also discuss the important topic of oscillator phase noise. Finally, an introduction to frequency multiplication techniques is given. A related topic is that of frequency conversion, or mixing, so we also discuss in this chapter the fundamental operations of frequency up-conversion and down-conversion. Detectors and single-ended mixers using both diodes andFETs are discussed, along with some specialized mixer circuits. Important considerations for oscillators used in RF and microwave systems include the following: • furring range (specified in MHz/V for voltage tuned oscillators) • frequency stability (specified in PPM/°C) • AM and FM noise (specified in dBc/Hz below carrier, offset from carrier) • harmonics (specified in dBc below carrier) 577 578 Chapter 12i Oscillators and Mixers Typical frequency stability requirements can range from 2 PPM/°C to 0.5 PPMTC, while phase noise requirements may range from —80 dBc/Hz to —110 dBc/Hz at a 10 kHz offset from the carrier. 12,1 RF OSCILLATORS In the most general sense, an oscillator is a nonlinear circuit that converts DC power to an AC waveform. Most RF oscillators provide sinusoidal outputs, which minimizes undesired harmonics and noise sidebands. The basic conceptual operation of a sinusoidal oscillator can be described with the linear feedback circuit shown in Figure 12.1. An amplifier with voltage gain A has an output voltage V0. This voltage passes through a feedback network with a frequency dependent transfer function H(to), and is added to the input V; of the circuit. Thus the output voltage can be expressed as VM) = AVi(), (12.1) which can be solved to yield the output voltage in terms of the input voltage as 1 - AH (to) If the denominator of (12.2) becomes zero at a particular frequency, it is possible to achieve a non-zero output voltage for a zero input voltage, thus forming an oscillator. This is known as the Nyquisi criterion, or the Barkhausen criterion. In contrast to the design of an amplifier, where we design to achieve maximum stability, oscillator design depends on an unstable circuit. The oscillator circuit of Figure 12.1 is useful conceptually, but provides little helpful information for the design of practical transistor oscillators. Thus we consider next a general analysis of transistor oscillator circuits. General Analysis There are a large number of possible RF oscillator circuits using bipolar or field-effect transistors in either common emitter/source, base/gate, or collector/drain configurations. Various types of feedback networks lead to the well-known Hartley, Colpitis, Clapp, and Pierce oscillator circuits [l]-[3]. All of these variations can be represented by the general oscillator circuit shown in Figure 12.2, Hi*) FIGURE 12.1 Block diagram of a sinusoidal oscillator using an amplifier with a frequency-dependent feedback path. 12.1 RF Oscillators 579 feedback network V BJTorFET Collector/ drain v. Emitter/ source FIGURE 122 General circuit for a transistor oscillator. The transistor may be either a bipolar junction transistor or a field effect transistor. This circuit can be used for common emitter/source, base/gate, or collector/drain configurations by grounding either V2l V|, or V4, respectively. Feedback is provided by connecting node Vít to V4. The equivalent circuit on the right-hand side of Figure 12.2 is used to model either a bipolar or a field-effect transistor. As discussed in Chapter 10, we have assumed here a unilateral transistor, which is usually a good approximation in practice. We can simplify the analysis by assuming real input and output admittances of the transistor, defined as gt and G0, respectively, with a transistor transconductance gm. The feedback network on the left side of the circuit is formed from three admittances in a bridged-T configuration. These components are usually reactive elements (capacitors or inductors) in order to provide a frequency selective transfer function with high q, A common emitter/source configuration can be obtained by setting V2 = 0, while common base/gate or common collector/drain configurations can be modeled by setting either V\ — 0 or V4 = 0, respectively. As shown, the circuit of Figure 12.2 does not include a feedback path—this can be achieved by connecting node V3 to node V4. Writing Kirchhoff's equation for the four voltage nodes of the circuit of Figure 12.2 gives the following matrix equation: " {yx + y3 + g<) -(>,+£,-) —SŠ 0 rVi" -(Yi+g,+gm) (Ti + y2 + gi + gl> + gm) -Y2 -g0 v2 -fa -Yi (y2 + y3) 0 0 g0_ V4 = 0 (12.3) Recall from circuit analysis that if the ftfl node of the circuit is grounded, so that % = 0, the matrix of (12,3) will be modified by elirmnating the ith row and column, reducing the order of the matrix by one. Additionally, if two nodes are connected together, the matrix is modified by adding the corresponding rows and columns. Oscillators Using a Common Emitter BJT As a specific example, consider an oscillator using a bipolar junction transistor in a common emitter configuration. In this case we have % = 0, with feedback provided from the collector, so that Vj = K». In addition, the output admittance of the transistor is negligible, so we set g0 = 0. These conditions serve to reduce the matrix of (12.3) to the following: TKj + Y3 + Qt) ■IP]-* (12.4) where V = V3 = V4. 580 Chapter 12: Oscillators and Mixers If the circuit is to operate as an oscillator, then (12.4) must be satisfied for nonzero values of V\ and V, so the determinant of the matrix must be zero. If the feedback network consists only of lossless capacitors and inductors, then Yy, Y2i and Y$ must be imaginary, so we let Yj = jB^ Y2 = jB2, and Yj = jBz. Also, recall that the transconductance gm and transistor input conductance G, are real. Then the determinant of (12.4) simplifies to G, + + B3) -JBs gm - jB3 j(B2 + By) = 0 (12.5) Separately equating the real and imaginary parts of the determinant to zero gives two equations: — + — + — =0, (12.6a) B\ B2 B3 i + (,+l)i=ft (l2'6b> If we convert susceptances to reactances, and let X} = \/By, X2 = l/B:, and X3 = l/B3. then (12.6a) can be written as X1+X2 + X3=0. (12,7a) Using (12.6a) to eliminate £3 from (12.6b) reduces that equation to the following; ti = ^-X2. (12.7b) G, Since gm and G,- are positive, (12.7b) implies that X\ and X2 have the same sign, and therefore are either both capacitors, or both inductors. Equation (12.7a) then shows that X$ must be opposite in sign from Xi and X2, and therefore the opposite type of component. This conclusion leads to two of the most commonly used oscillator circuits. If X[ and X2 are capacitors and X$ is an inductor, we have a Colpitts oscillator. Let Xt = — 1/tooCy, X2 = -l/tuoCo, and X3 = coaLi. Then (12.7a) becomes — + ^)+wol3 = 0, coo \Ci CzJ which can be solved for the frequency of oscillation, too, as Using these same substitutions in (12.7b) gives a necessary condition for oscillation of the Colpitts circuit as t? = f£. (12.9) Cj G, The resulting common-emitter Colpitts oscillator circuit is shown in Figure 12.3a. Alternatively, if we choose X j and X2 to be inductors, and X3 to be a capacitor, then we have a Hartley oscillator. Let X] = co^L], X2 = coaL2y and X3 = -1 /0J0C3, Then (12.7a) becomes + Li)--l— =0. 12.1 RF Oscillators 581 VAA/-°vÄ. if c v v v (a) (b) FIGURE 123 Transistor oscillator circuits using a common-emitter BJT. (a) Colpitis oscillator. (b) Hartley oscillator. which can be solved for 0*0 to give 1 <0:: = Cid) +L2) (12.10) These same substitutions used in (12.7b) gives a necessary condition for oscillation of the Hartley circuit as The resulting common-emitter Hartley oscillator circuit is shown in Figure 12.3b. (12.11) Oscillators Using a Common Gate FET Next consider an oscillator using an FET in a common gate configuration. In this case Vi = 0, and again V3 — V4 provides the feedback path. For an FET the input admittance can be neglected, so we set G, = 0. Then the matrix of (123) reduces to UYl + Y2 + gln + G„) -(y2 + G,) irvvi where V = V3 = V4. Again we assume the feedback network is composed of lossless reactive elements, so that Y], Y2, and Sg can be replaced with their susceptances. Setting the determinant of (12.12) to zero then gives (ft™ + G„) + j(Bi + B2) -G0- jB2 ~(Gc + gm) -A G° + f&l + BŮ = 0. (12.13) Equating the real and imaginary parts to zero gives two equations: (12.14a) (12.14b) 582 Chapter 12: Oscillators and Mixers As before, let X\ i X2, and X3 be the reciprocals of the corresponding susceptances. Then (12.14a) can be rewritten as Xl+X2 + X3^0. (12.15a) Using (12.14a) to eliminate B$ from (12.14b) reduces that equation to ^ = f^. (12.15b) X] G0 Since gm and G0 are positive, (12.15b) shows that X\ and X2 must have the same sign, while (12,15a) indicates that X3 must have the opposite sign. If X\ and X2 are chosen to be negative, then these elements will be capacitive and X3 will be inductive. This corresponds to a Colpitis oscillator, Since (12,15a) is identical to (12,7a), its solution gives the result for the resonant frequency for the common gate Colpitis oscillator as -mm which is identical to the result obtained in (12.8) for the common emitter Colpitts oscillator. This is because the resonant frequency is determined by the feedback network, which is identical in both cases. The further condition for oscillation given by (12.15b) reduces to % = Jr.. (12.17) C2 Go If we choose X\ and X2 to be positive (inductive), then X3 will be capacitive, and we have a Hartley oscillator. The resonant frequency of the common gate Hartley oscillator is given by ^■{mh^f (,2-18) which is identical to the result of (12.10) for the common emitter Hartley oscillator. Equation (12.15b)reduces to — = (12.19) L] Go The circuits for common gate Colpitts and Hartley oscillators are similar to the circuits shown in Figure 12.3, if the BJT is replaced with an FET device. Practical Considerations It must be emphasized that the above analysis is based on very idealized assumptions, and in practice successful oscillator design requires attention to factors such as the reactances associated with the input and output transistor ports, the variation of transistor properties with temperature, transistor bias and decoupling circuitry, and the effect of inductor losses. For these purposes computer-aided design software can be very helpful [3]. The above analysis can be extended to account for more realistic feedback network inductors having series resistance, which invariably occurs in practice. For example, consider the case of a common emitter BJT Colpitts oscillator, with the impedance of the inductor given by Z3 = 1/ y3 = R + jatLs. Substituting into (12.4) and setting the real and 12,1 RF Oscillators 583 imaginary parts of the determinant to zero gives the following result for resonant frequency: OJ0 This equation is similar to the result of (12.8) for the lossless inductor, except tliat CJ is defined as d C =---. (12.21) The corresponding condition for oscillation is This result sets the maximum value of the series resistance /?; the left side of (12.22) should generally be chosen to be less than the right hand side to ensure oscillation. EXAMPLE 12.1 COLPITTS OSCILLATOR DESIGN Design a 50 MHz Colpitts oscillator using a transistor in a common emitter configuration with B = gm/Gi = 30, and a transistor input resistance of Rj — 1/G* = 1200 £i. Use an inductor with L3 = 0.10 /iH, with a Q of 100. What is the minimum Q of the inductor for which oscillation will be sustained? Solution From (12.20) the series combination of Cj and C2 is found to be CIC2 1 1 Cj +C2 u>\U (2^)2(50 x HWO.l x 10"6) = 100 pF. This value can be obtained in several ways, but here we will choose C\ — C2 = 200 pF From Chapter 6 we know that the Q of an inductor is related to its series resistance by Q — ioL/R, so the series resistance of the 0.1 [tH inductor is _ (27t)(50 x 106)(0.1 x IP"6) R = ~Q~ =-100 ~ 031 R* Then (12.21) gives C, as C, = c;) = J?in(/, jto) + jXin(7, jco). The device is temiinated with a passive load impedance, ZL = RL + jXi. Applying Kirchhoff's voltage law gives (Zt + Zk)/ = 0. (12.24) If oscillation is occurring, such that the RF current / is nonzero, then the following conditions must be satisfied; «1+^= 0, (12.25a) XL + XtD = 0. (12.25b) Since the load is passive, Ri > 0 and (12.25a) indicates that Ria < 0. Thus, while a positive resistance implies energy dissipation, a negative resistance implies an energy source. The condition of (12.25b) controls the frequency of oscillation. The condition in (12.24), that ^ Negative i resistance I device (ZD (Z\B> FIGURE 12.6 Circuit for a one-port negative-resistance oscillator. 586 Chapter 12: Oscillators and Mixers ZL — — Zjj, for steady-state oscillation, implies that the reflection coefficients Ti and T^ are related as r, = Zl — Zp _ — Zin — Zp _ Zjg + Zp _ 1 Zl + Zo — Zjn + Zo Zin — Zo rin (12.26) The process of oscillation depends on the nonlinear behavior of Zin, as follows. Initially, it is necessary for the overall circuit to be unstable at a certain frequency, that is, R\„ (/, jto) + Rl < 0. Then any transient excitation or noise will cause an oscillation to build up at the frequency, a>. As / increases, R^il, Joj) must become less negative until the current /<> is reached such that Ria(In> fton) + Rl = 0, and Xm(Ia, jcoo) + Xiijm) = 0. Then the oscillator is running in a stable state. The final frequency, a*), generally differs from the startup frequency because Xin is current dependent, so that Xm(I, ja>) ^ Xjn(/o, j. If we let Zr(/, s) = Zin(I, s) + Zi(s), then we can write a Taylor series for Zj(l, s) about the operating point la, at® as ZT(I, s) = Zj-(/o,50) + dZr dZr so, fa di 51 = 0, (12.27) .50./0 since ZT(l; s) must still equal zero if oscillation is occurring. In (12.27), so — j: 5s — Sa + jSco = ~dZTßI dZT/ds SI = -j(dZrßl){dZ*Tjdw)sj *,./<> \dZT/dto\2 (12.28) Now if the transient caused by SI and So) is to decay, we must have Sa < 0 when SI > 0. Equation (12.28) then implies that dzT dz H > 0. (12.29) For a passive load, 3RL/dI = dXjJdl = dRL/du> = 0, so (12.29) reduces to ÖRm Ö dXia 3Rm -(XL + Xia)--— —— > U. 31 Stü dl da) (12.30) As discussed above, we usually have that 3RinJ3I > 0 [4]. So (12.30) can be satisfied if d(Xi 4- Xin)/da) >- 0, which implies that a high- Q circuit will result in maximum oscillator stability. Cavity and dielectric resonators are often used for this purpose. Effective oscillator design requires the consideration of several other issues, such as the selection of an operating point for stable operation and maximum power output, frequency-pulling, large-signal effects, and noise characteristics. But we must leave these topics to more advanced texts [5]. 12.2 Microwave Oscillators 587 0.254 A 50 -Q 50 ÍÍ 0.308 A Diotte rf rin = 1.25/40° m lžu FIGURE 12.7 Load matching circuit for the one-pott oscillator of Example 12.2. EXAMPLE 1A2 NEGATIVE-RESISTANCE OSCILLATOR DESIGN A one-port oscillator uses a negative-resistance diode having Tin = 1.25^0° (Z0 = 50 ň) at its desired operating point, for / = 6 GHz. Design a load matching network for a 50 Q load impedance. Solution From either the Smith chart (see Problem 12.6), or by direct calculation, we find • the input impedance as ZLn = -44 + jl23C2. Then, by (12,25), the load impedance must be ZL = 44- jl23fl. A shunt stub and series section of line can be used to convert 50 ÍÍ to Z/,, as shown in the circuit of Figure 12.7. ■ Transistor Oscillators In a transistor oscillator, a negative-resistance one-port network is effectively created by terminating a potentially unstable transistor with an impedance designed to drive the device in an unstable region. The circuit model is shown in Figure 128; the actual power output port can be on either side of the transistor. In the case of an amplifier, we preferred a device with a high degree of stability—ideally, an unconditionally stable device. For an oscillator, we require a device with a high degree of instability. Typically, common source or common gate FET configurations are used (common emitter or common base for bipolar devices), Negative I resistance J Load network (tuning) ! Transistor IS] Terminating network Tl Tin *"o«t FIGURE 12.8 Circuit for a two-port transistor oscillator. 588 Chapter 12: Oscillators and Mixers often with positive feedback to enhance the instability of the device. After the transistor configuration is selected, the output stability circle can be drawn in the Tr plane, and rT selected to produce a large value of negative resistance at the input to the transistor. Then the load impedance Zr, can be chosen to match Zin. Because such a design uses the small-signal 5 parameters, and because Ria will become less negative as the oscillator power builds up, it is necessary to choose Ri so that Ri + Rm < 0. Otherwise, oscillation will cease when the increasing power increases /fjn to the point where Rl + R™ > 0. In practice, a value of Rl^^ (12.31a) is typically used. The reactive part of Zl is chosen to resonate the circuit, XL = -Xia. (12.31b) When oscillation occurs between the load network and the transistor, oscillation will simultaneously occur at the output port, which we can show as follows. For steady-state oscillation at the input port, we must have r^r^ = I, as derived in (12.26). Then from (11.3a) (after replacing with Tr), we have -1 =rin = Su + Si2S2,Tt = 5"-AI> (i2.32) Ti 1 — $22^1 1 — Sil^t where A = Si]S22 — S]2S2i > Solving for FT gives rr= 1~5"r\ (11.33) Then from (11.3b) (after replacing r$ with F^), we have that r„ = & + -5!^ = ^l£, (12.34) 1 — int l J — ^11* l which shows that rvrom = 1, and hence Zj = — Z^. Thus, the condition for oscillation of the terminating network is satisfied. Note that the appropriate 5 parameters to use in the above development are generally the large-signal parameters of the transistor. EXAMPLE 12.3 TRANSISTOR OSCILLATOR DESIGN Design a transistor oscillator at 4 GHz using a GaAs FET in a common gate configuration, with a 5 nH inductor in series with the gate to increase the instability. Choose a terminating network to match to a 50 Ci load, and an appropriate tuning network. The S parameters of the transistor in a common source configuration are (Zo — 50 Q): 5,| = 0 79/-116° S2l = 2.60Z26°, Sl2 = 0.03/52°, S22 = 0.73^54°. Solution The first step is to convert the common source S parameters to the S parameters that apply to the transistor in a common gate configuration with a series inductor, (See Figure 12.9a.) This is most easily done using a microwave CAD package. The new S parameters are S'n =2,18Zz35*, ^ = 2.75/96°, S']2= 1.26Z1T, 522 = 0.52Z!55o. 12.2 Microwave Oscitlators 589 FIGURE 12.9 Circuit design for the transistor oscillator of Example 12J. (a) Oscillator circuit, (b) Smith chart for determining rT. Note that |5J,| is significantly greater than |$n |, which suggests that the configuration of Figure 12.9a is more unstable than the common source configuration. Calculating the output stability circle {Vj plane) parameters from (11.25) gives CT (s22 - A'S'^y 27 '2 _ •A \S22\2-\&'\2 = 1.08Z330, = 0.665. 590 Chapter 12: Oscillators and Mixers Since \S\ 11 = 2.18 > 1, the stable region is inside this circle, as shown in the Smith chart in Figure 12.9b. There is a great amount of freedom in our choice for Fr, but one objective is to make |FiD| large. Thus we try several values of FT located on the opposite side of the chart from the stability circle, and select Vt = 0 SQ/—104°. Then we can design a single-stub matching network to convert a 50 £2 load to Zj = 20 — _/35 £2, as shown in Figure 12.9a. For the given value of Fr, we calculate as or Zin = -84 - j 1.9 Q. Then, from (11.86), we find t% as ZL = ^-jXm=2S + jh9U. Using RID/3 should ensure enough instability for the startup of oscillation. The easiest way to implement the impedance ZL is to use a 90 Q load with a short length of line, as shown in die figure. It is likely that the steady-state oscillation frequency will differ from 4 GHz because of the nonlinearity of the transistor parameters. ■ Dielectric Resonator Oscillators As we saw from the result of (12.30), oscillator stability is enhanced with the use of a high-g tuning network. The Q of a resonant network using lumped elements or microstrip lines and stubs is typically limited to a few hundred (see Examples 6.1 and 6.2), and while waveguide cavity resonators can have (?s of I04 or more, they are not well-suited for integration in miniature microwave integrated circuitry. Another disadvantage of metal cavities is the significant frequency drift caused by dimensional expansion due to a variation in temperature. The dielectric cavity resonator discussed in Section 6.5 overcomes most of these disadvantages, as it can have an unloaded Q as high as several thousand, is compact and easily integrated with planar circuitry, and can be made from ceramic materials that have excellent temperature stability. For these reasons, transistor dielectric resonator oscillators (DROs) are becoming increasingly common over the entire microwave and millimeter wave frequency range. A dielectric resonator is usually coupled to an oscillator circuit by positioning it in close proximity to a microstrip line, as shown in Figure 12.10a. The resonator operates FIGURE 12.10 (a) Geometry of a dielectric resonator coupled to a microstripline; (b) equivalent circuit. 12.2 Microwave Oscillators 591 in the TEq\s mode, and couples to the fringing magnetic field of the microstrip line. The strength of coupling is determined by the spacing, d, between the resonator and microstrip line. Because coupling is via the magnetic field, the resonator appears as a series load on the microstrip line, as shown in the equivalent circuit of Figure 12.10b. The resonator is modeled as a parallel RLC circuit, and the coupling to the feedline is modeled by the turns ratio, N, of the transformer. Using the result of (619) for the impedance of a parallel RLC resonator, we can express the equivalent series impedance, Z, seen by the microstrip line as Z= g*_, (12.35) where Q = R/coqL is the unloaded resonator Q,coq — 1/-/LC is the resonant frequency, and Au> = 01-100. The coupling factor, defined in (6.76), between the resonator and the feedline is the ratio of the unloaded to external Q, and can be found as Q = R/ojqL = N^R 8 Qe Rl/N2ojoL 2Z0 ' K ' } where Rl = 2Zo is the load resistance for a feedline with source and termination resistances Zo In some cases the feedline is terminated with an open-circuit A./4 from the resonator to maximize the magnetic field at that point; in this case RL = Zo and the coupling factor is twice the value given in (12.36). The reflection coefficient seen on the terminated microstrip fine looking toward the resonator can be written as (ZP + JVaJt)-Z0= N2R = g {Z0 + N2R) + Zq 2Zq + N2R \+g- This allows the coupling coefficient to be found from g = T/(l - T) after the simple procedure of measuring Y at resonance; the resonant frequency and Q can also be found by measurement. Alternatively, these quantities can be calculated using approximate analytical solutions [6]. Note that this procedure leaves a degree of freedom between N and R, since only the product N2R is uniquely determined. There are many oscillator configurations using common source (emitter), common gate (base), or common drain (collector) connections of either FET or bipolar transistors, in addition to the optional use of series or shunt elements to increase the instability of the device [4]-[5]. A dielectric resonator can be incorporated into the circuit to provide frequency stability using either the parallel feedback arrangement of Figure 12.11a, or the series feedback technique shown in Figure 12.1 lb. The parallel configuration uses a FIGURE 12.11 (a) Dielectric resonator oscillator using parallel feedback; (b) dielectric resonator oscillator using series feedback. 592 Chapter 12: Oscillators and Mixers resonator coupled to two microstrip lines, functioning as a high-Q bandpass filter that couples a portion of the transistor output back to its input. The amount of coupling is controlled by the spacing between the resonator and the lines, and the phase is controlled by the length of the lines. The series feedback configuration is simpler, using only a single microstrip feedline, but typically does not have a tuning range as wide as that obtained with parallel feedback. Design of an oscillator using parallel feedback is most conveniently done using a microwave CAD package, but a dielectric resonator oscillator using series feedback can be designed using the same procedure that was discussed in the previous section on two-port oscillators. EXAMPLE 12.4 DIELECTRIC RESONATOR OSCILLATOR DESIGN A wireless local area network application requires a local oscillator operating at 2.4 GHz. Design a dielectric resonator oscillator using the series feedback circuit of Figure 12.1 lb with a bipolar transistor having the following S parameters (Z0 = 50a): Sn = 1.8430°, Sl2 = 0.4^5°, % = 3.8^6°, S22 = 0.7t«o. Determine the required coupling coefficient for the dielectric resonator, and a microstrip matching network for the termination network. The termination network should include the output load impedance. Plot the magnitude of rw versus A///o, for small variations in frequency RLCabout the design value, assuming an unloaded resonator Q of 1000. Solution The DRO circuit is shown in Figure 12.12a. The dielectric resonator is placed A./4 from the open end of the microstrip line; the line length lr can be adjusted to match the phase of the required value of YL. In contrast to the oscillator of the previous example, the output load impedance for this circuit is part of the terminating network. The stability circles for the load and termination sides of the transistor can be plotted if desired, but are not necessary to the design, since we will begin by choosing Yi to provide a large value of | roul |. From (12.34) we have p „ S\iSi\YL 1 out = >>22 + i-snrt' which indicates that we can maximize Youi by making 1 — Sn Yi close to zero. Thus we choose Yl = Q.fi/-13Q°. which gives rout = 10.7/132ť. This corresponds to an impedance „ „I + T^ enl +10.7/132° FIGURE 12.12 (a) Circuit for the dielectric resonator of Example 12.4, 12.2 Microwave Oscillators 593 12.0 -0.05 -0.04 -0.03 -0.02 -0.03 0,00 0.01 0.02 0.03 0.04 0.05 4fV/o (percent) FIGURE 12.12 Continued, (b) ir^J vs. frequency in Example 12.4. Applying the analogous startup condition of (12,31) for the termination side gives the required termination impedance as -R jXaat = 5.5-j6A £2. The termination matching network can now be designed using a Smith chart. The shortest transmission line length for matching Zj to the load impedance Z0 is it = 0.48Ik, and the required open-circuit stub length is ts = 0.307A. Next we match TL to the resonator network. From (12.35) we know that the equivalent impedance of the resonator seen by the microstrip line is real at the resonant frequency, so the phase angle of the reflection coefficient at this point, V'L, must be either zero or 180°. For an undercoupled parallel RLC resonator, R < Zo, so the proper phase will be 180°, which can be achieved by transformation through the line length lr. The magnitude of the reflection coefficient is unchanged, so we have the relation = rLe2^' = (0.6^13QV^' = 0.6^80°, which gives tr = 0.431A. The equivalent impedance of the resonator at resonance is then Z'L = Zo i-n = 12.5 £1. The coupling coefficient can be found using (12.36), with a factor of two to account for the A/4 stub termination, as 8 = N2R 12.5 Zo 50 = 0.25. The variation of |rout I with frequency will give an indication of the frequency stability of the oscillator. We can calculate rout from (12.34), after first using (12.35) to compute Z'L, T'L, and then transforming down the line of length tr to obtain Yi. 594 Chapter 12: Oscillators and Mixers The electrical line length can be approximated as constant for the small changes in frequency associated with this calculation. A short computer program or a microwave CAD package can be used to generate data for -0.01 < A///0 < 0.01, which is shown in the graph of Figure 12.12b. Observe that |F0Ut | decreases rapidly with a change in frequency as small as a few hundredths of a percent, demonstrating the sharp selectivity that can be obtained with a dielectric resonator. ■ The noise produced by an oscillator or other signal source is important in practice because it may severely degrade the performance of a radar or communication receiver system. Besides adding to the noise level of the receiver, a noisy local oscillator will lead to down-conversion of undesired nearby signals, thus limiting the selectivity of the receiver and how closely adjacent channels may be spaced. Phase noise refers to the short-term random fluctuation in the frequency (or phase) of an oscillator signal. Phase noise also introduces uncertainty during the detection of digitally modulated signals. An ideal oscillator would have a frequency spectrum consisting of a single delta function at its operating frequency, but a realistic oscillator will have a spectrum more like that shown in Figure 12.13. Spurious signals due to oscillator harmonics or intermodulation products appear as discrete spikes in the spectrum. Phase noise, due to random fluctuations caused by thermal and other noise sources, appears as a broad continuous distribution localized about the output signal. Phase noise is defined as the ratio of power in one phase modulation sideband to the total signal power per unit bandwidth (one Hertz) at a particular offset, fm, from the signal frequency, and is denoted as C{fm). It is usually expressed in decibels relative to the carrier power per Hertz of bandwidth (dBc/Hz). A typical oscillator phase noise specification for an FM cellular radio, for example, may be -110 dBc/Hz at 25 kHz from the carrier. In the following sections we show how phase noise may be represented, and present a widely used model for characterizing the phase noise of an oscillator. Representation of Phase Noise In general, the output voltage of an oscillator or synthesizer can be written as where A(t) represents the amplitude fluctuations of the output, and B(t) represents the phase variation of the output waveform, Of these, amplitude variations can usually be well-controlled, and generally have less impact on system performance. Phase variations may be discrete (due to spurious mixer products or harmonics), or random in nature (due to thermal or other random noise sources). Note from (12.38) that an instantaneous phase variation is indistinguishable from a variation in frequency. 12.3 OSCILLATOR PHASE NOISE vo(f) ■= W + A(0]oos[oW + (?(/)], (12.3$) Random phase ■* variation Discrete spurious jS>sr signal FIGURE 12.13 Outpu« spectrum of a typical RF oscillator. 12.3 Oscillator Phase Noise 596 Small changes in the oscillator frequency can be represented as a frequency modulation of the carrier by letting B(t) = -/-sinter = Bp sin wmt, (12.39) where fm = com/2Tc is the modulating frequency. The peak phase deviation is 6P — A f/fm (also called the modulation index). Substituting (12.39) into (12.38) and expanding gives v0(t) = V0[cos m0t co&(0p sin 0t - y [cos(tt)0 + (dm )t - cos(a>0 - o)m)t] J (12.41) This expression shows that small phase or frequency deviations in the output of an oscillator result in modulation sidebands at co0 ± located on either side of the carrier signal at 0. When these deviations are due to random changes in temperature or device noise, the output spectrum of the oscillator will take the form shown in Figure 12.13. According to the definition of phase noise as the ratio of noise power in a single sideband to the carrier power, the waveform of (12.41) has a corresponding phase noise of 1 (Y^pY p % \ ? ) o1 a2 at) = — = \ 7 = = (12.42) where Brms = 9pj-f2 is the mis value of the phase deviation. The two-sided power spectral density associated with phase noise includes power in both sidebands: w«) = 2C(fm) = j = 0;,„,- 02.43) While noise generated by passive or active devices can be interpreted in terms of phase noise by using the same definition. From Chapter 10 we know that the noise power at the output of a noisy two-port network is kT^BFC, where To = 290 K, B is the measurement bandwidth, F is the noise figure of the network, and G is the gain of the network. For a 1 Hertz bandwidth, the ratio of output noise power density to output signal power gives the power spectral density as kT F S${fm) = ^-. (12.44) where Pe is the input signal (carrier) power. Note that the gain of the network cancels in this expression. Leeson's Model for Oscillator Phase Noise In this section we present Leeson's model for characterizing the power spectral density of oscillator phase noise [21, [7J. As in Section 12.1, we will model the oscillator as an amplifier with a feedback path, as shown in Figure 12.14, If the voltage gain of the amplifier is included 596 Chapter 12: Oscillators and Mixers Noise-free amplifier MM FIGURE 12.14 Feedback amplifier model for characterizing oscillator phase noise. in the feedback transfer function H(oj), then the voltage transfer function for the oscillator circuit is 1 - H{oi) If we consider oscillators that use a high-g resonant circuit in the feedback loop (e.g., Colpitis, Hartley, Clapp, and similar oscillators), then H(a>) can be represented as the voltage transfer function of a parallel RLC resonator: H(co):--J-- - 1 , (12.46) 1 + / to wq\ 1 + 2jQAco/coc (— - — \ \coq co} where coq is the resonant frequency of the oscillator, and Aw = o> — coq is the frequency offset relative to the resonant frequency. Since die input and output power spectral densities are related by the square of the magnitude of the voltage transfer function [S], we can use (I2.45)-(12.46) to write 1 - H(a>) 1+42^ = (1 + 4e^)^) = (1 + ä)5ö(ö>) (l2'47) where S(,(io) is the input power spectral density, and S^ico) is the output power spectral density. In (12.47) we have also defined toh = too/2Q as the half-power (3 dB) bandwidth of the resonator. The noise spectrum of a typical transistor amplifier with an applied sinusoidal signal at /o is shown in Figure 12.15. Besides kTB thermal noise, transistors generate additional noise mat varies as 1/f at frequencies below the frequency /„. This 1//, or flicker, noise is likely caused by random fluctuations of the carrier density in the active device. Due to the nonlinearity of the transistor, the 1// noise will modulate the applied signal at /0, and I q fa /o FIGURE 12,15 Noise power versus frequency for an amplifier with an applied input signal. 12.3 Oscillator Phase Noise 597 FIGURE 12.16 Idealized power spectra] density of amplifier noise, including 1// and thermal components. appear as l/f noise sidebands around fa. Since the 1// noise component dominates the phase noise power at frequencies close to the carrier, it is important to include it in our model. Thus we consider an input power spectral density as shown in Figure 12.16, where K/Af represents the 1// noise component around the carrier, and kToF/Po represents thermal noise. Thus the power spectral density applied to the input of the oscillator can be written as (12-48) Po V Aw/ where K is a constant accounting for the strength of the 1 // noise, and l uo\ Kco^ / Ko>acolh j4_ + Ka>a A \ Ato* Ato2 Aco J (12.49) This result is sketched in Figure 12.17. There are two cases, depending on which of the middle two terms of (12.49) is more significant. In either case, for frequencies close to the carrier at /o, the noise power decreases as l//3, or —18 dB/octave. If the resonator has a relatively low Q, so that its 3 dB bandwidth fh > fa, then for frequencies between fa and FIGURE 12.17 Power spectral density of phase noise at the output of an oscillator, (a) Response for fh > fa (low Q). (b) Response for fh > fa (high Q). 596 Chapter 12: Oscillators and Mixers fh the noise power drops as I //2, or —12 dB/octave, If the resonator has a relatively high Q, so that fh < fai then for frequencies between ft, and /„ the noise power drops as 1//, or —6 dB/octave. At higher frequencies the noise is predominantly thermal, constant with frequency, and proportional to the noise figure of the amplifier. A noiseless amplifier with F = 1 (0 dB) would produce a minimum noise floor of kTo = -174 dBm/Hz. In accordance with Figure 12.13, the noise power is greatest at frequencies closest to the carrier frequency, but (12.49) shows that the l//3 component is proportional to \/Q2, so that better phase noise characteristics close to the carrier are achieved with a high-O; resonator. Finally, recall from (12.43) that the single-sideband phase noise will be one-half of the power spectral density of (12.49). These results give a reasonably good model for oscillator phase noise, and quantitatively explain the roll-off of noise power with frequency offset from the carrier. The effect of phase noise in a receiver is to degrade both the signal-to-noise ratio (orbit error rate) and the selectivity [9]. Of these, the impact on selectivity is usually the most severe. Phase noise degrades receiver selectivity by causing down conversion of signals located nearby the desired signal frequency. The process is shown in Figure 12. IS. A local oscillator at frequency /o is used to down convert a desired signal to an IF frequency. Due to phase noise, however, an adjacent undesired signal can be down converted to the same IF frequency due to the phase noise spectrum of the local oscillator. The phase noise that leads to this conversion is located at an offset from the carrier equal to the IF frequency from the undesired signal. This process is called reciprocal mixing. From this diagram, it is easy to see that the maximum allowable phase noise is order to achieve an adjacent channel rejection (or selectivity) of S dB (S > 0) is given by £(fm) = C (dBm) - S (dB) - / (dBm) - 10 log(fi), (dBc/Hz), (12.50) where C is the desired signal level (in dBm), / is the undesired (interference) signal level (in dBm), and B is the bandwidth of the IF filter (in Hz). EXAMPLE 12.5 GSM RECEIVER PHASE NOISE REQUIREMENTS The GSM cellular standard requires a minimum of 9 dB rejection of interfering signal levels of -23 dBm at 3 MHz from the carrier, -33 dBm at 1.6 MHz from the carrier, and —43 dBm at 0.6 MHz from the carrier, for a carrier level of —99 dBm. Determine the required local oscillator phase noise at these carrier frequency offsets. The channel bandwidth is 200 kHz. I J Unwanted signal V ' F V D s t e.sire .s?n a' / p Desired LO \ Phase noise Noisy LO H-fF-H 0 H-/F-H fa FIGURE 12,18 Illustrating how local oscillator phase noise can lead to the reception of undesired signals adjacent to the desired signal. 12.4 Frequency Multipliers 599 Solution From (12.50) we have C(fm) = C (dBm) - S (dB) - / (dBm) - 101og(B) = -99 dBm - 9 dB - / (dBm) - I01og(2 x 10s). The table below lists the required LO phase noise as computed from the above expression: frequency offset interfering signal £>(fm) fm (MHz) level (dBm) dBc/Hz 3.0 -23 -138 1.6 -33 -128 0.6 -43 -118 This level of phase noise requires a phase-locked synthesizer. Bit errors in GSM systems are usually dominated by the reciprocal mixing effect, while errors due to thermal antenna and receiver noise are generally negligible. ■ FREQUENCY MULTIPLIERS As frequency increases into the millimeter wave range it becomes increasingly difficult to build fundamental frequency oscillators with good power, stability, and noise characteristics. An alternative approach is to generate a harmonic of a lower frequency oscillator through the use of a frequency multiplier. As we have seen in Section 10.2, a nonlinear element may generate many harmonics of an input sinusoidal signal, so frequency multiplication is a natural occurrence in circuits containing diodes and transistors. Designing a good quality frequency multiplier, however, is a difficult task that generally requires nonlinear analysis, matching at multiple frequencies, stability analysis, and thermal considerations. We will discuss some of the general operational principles and properties of diode and transistor frequency multipliers, and refer the reader to the literature for more practical details [10]. Frequency multiplier circuits can be categorized as reactive diode multipliers, resistive diode multipliers, or transistor multipliers. A reactive diode multiplier uses either a varactor or a step-recovery diode biased to present a nonlinear junction capacitance. Since losses in such diodes are small, conversion efficiencies (the fraction of RF input power that is converted to the desired harmonic) can be relatively high. In fact, as we will show, ideal (lossless) reactive multipliers can achieve a theoretical conversion efficiency of 100%. Varactor multipliers are most useful for low harmonic conversion (multiplier factors of 2 to 4), while step-recovery diodes are able to generate more power at higher harmonics. Resistive multipliers exploit the nonlinear I-V characteristic of a forward biased Schottky barrier detector diode. We will show that resistive multipliers have conversion efficiencies that decrease as the square of the harmonic number, and so these multipliers are only useful for low multiplication factors. Transistor multipliers can use both bipolar and FET devices, and can provide conversion gains. Transistor multipliers are limited by their cutoff frequency, however, and therefore are generally not useful at very high frequencies. A disadvantage of frequency multipliers is that noise levels are increased by the multiplication factor. This is because frequency multiplication is effectively a phase multiplication process as well, so phase noise variations get multiplied in the same way that frequency is multiplied. The increase in noise level is given by 20 log n, where n is the multiplication 600 Chapter 12: Oscillators and Mixers FIGURE 12.19 Conceptual circuit for the derivation of the Manky-Rowe relations. factor. Thus a frequency doubler will increase die fundamental oscillator noise level by at least 6 dB, while a frequency tripler will lead to an increase of at least 9.5 dB. Reactive diode multipliers typically add little additional noise of their own, since varactors and step-recovery diodes have very low series resistances, but resistive diode multipliers can generate significant additional noise power. Reactive Diode Multipliers (Manley-Rowe Relations) We begin our discussion with the Manley-Rowe relations, which result from a very general analysis of power conservation associated with frequency conversion in a nonlinear reactive element [11]. Consider the circuit of Figure 12.19, where two sources at frequencies co\ and drive a nonlinear capacitor, C. The circuit also shows ideal bandpass filters to conceptually isolate powers in all harmonics of the form no>\ + mo>i- Since the capacitor is nonlinear, its charge Q can be expressed as a power series in terms of the capacitor voltage, v: Q = aa + a\v +a2v2 +a^v3 H---- As in Section 10.2, this nonlinear relationship implies the generation of all frequency products of the form nan +ma>i. Thus we can write the capacitor voltage as a Fourier series of the form v(t)= £ V„meiin^+mi^', (12.51) fi=-0Om=- Similarly, the capacitor charge and current can be written as co 00 Q(0= £ £ (12.52) i=—\ and are not multiples of each other, there is no average power due to interacting harmonics. Then the average 12.4 Frequency Multipliers 601 power (ignoring a factor of 4) at frequency ±\nco] + mcoi\ is given as Pmn — 2Re( Vnmlnm) = VnmIRm -f \'nmInm = Vnmlf}m + V-n.-m1 ~n.-m ~ P~n,-m- (12.54) Conservation of power can then be expressed as 00 oq Now multiply (12,55) by---to obtain mf ■ £ ffia +0), 3T E =o. (12.56) Using (12.54) and the fact that Inm = /t«a*i + mci)2)Ö„ffl gives H—-oc- m=-oc OS 00 +Q* E £ m(-yvfl«e;n,-jv_Jf,_1,e:J,i_m) = o (12.57) The double summation factors in (12.57) do not depend on o>\ or o>2, since we can always adjust the external circuitry so that all Vnm remain constant, and the Q„m will remain constant as well since the capacitor charge depends directly on the voltage. Thus each summation in (12.56) must be identically zero: 00 00 y y _n£™--^ H2,58ä) oo 00 y y mPnm = Q Some simplification can be carried out by eliminating the negative indices of one summation by using the fact that P-„-m = Pnm- For example, from (12.58a), oo co p 00 00 p 00 co _„p 'lrnm \ ' V""r™ V 1 \ ' fir-n,—m ET nrnm _ y-i U r„m y, yy _■ 00 OO n This results in the usual form for the Manley-Rowe relations: 00 00 p y y nem = ^ y p mP„m = o m The Manley-Rowe relations express power conservation for any lossless nonlinear reactance, and can be useful for harmonic generation, parametric amplifiers, and frequency converters at RF, microwave, and optical frequencies to predict the maximum possible power gain and conversion efficiency. 602 Chapter 12: Oscillators and Mixers rAAAV- input fa JL Low-pass filler Diode Bandpass filter output FIGURE 12.20 Block diagram of a diode frequency multiplier. Reactive frequency multipliers involve a special case of the Manley-Rowe relations, since only a single source is used. If we assume a source at frequency co\, then setdng m = 0 in (12.59a) gives rt=l or äs ^P«o=-/>io, 02.60) where P„o represents the power associated with the nth harmonic (the DC term for n = 0 is zero). In practice, Pw > 0 because this represents power delivered by the source, while the summation in (12.60) represents the total power contained in all the harmonics of the input signal, as generated by the nonlinear capacitor. If all harmonics but the nth are terminated with lossless reactive loads, the power balance of (12.60) reduces to, 10 = 1, (12.61) indicating that it is theoretically possible to achieve 100% conversion efficiency for any harmonic. Of course, in practice, losses in the diode and matching circuitry serve to reduce the achievable efficiency substantially. A block diagram of a diode frequency multiplier is shown in Figure 12,20, An input signal of frequency fo is applied to the diode, which is terminated with reactive loads at all frequencies except n fG, the desired harmonic. If the diode junction capacitance has a square-law I-V characteristic, it is often necessary to terminate unwanted harmonics with short circuits if harmonics higher than the second are to be generated. This is because voltages at higher harmonics may not be generated unless lower harmonic currents are allowed to flow. These currents are commonly referred to as idler currents. For example, a varactor tripler will generally require terminations to allow idler currents at 2/q. Typical conversion efficiencies for varactor multipliers range from 50-80% for doublers and triplers at 50 GHz. The upper frequency limit is controlled mainly by fc, the cutoff frequency of the diode, which depends on the series resistance and dynamic junction capacitance. Typical varactor cutoff frequencies can exceed 1000 GHz. but efficient frequency multiplication requires that nfc, « fc. Resistive Diode Multipliers Resistive multipliers generally use forward-biased Schottky-barrier detector diodes to provide a nonlinear 1-V characteristic. Resistive multipliers are less popular than reactive multipliers because their efficiencies are lower, especially for higher harmonic numbers. But resistive multipliers offer better band widths, and more stable operation, than reactive 12.4 Frequency Multipliers 603 0 HO DC 2w iL 3tt» FIGURE 12.21 Conceptual circuit for the derivation of power relations in a resisUve frequency multiplier. multipliers. In addition, at high millimeter wave frequencies even the best varactor diodes begin to exhibit resistive properties. Since a resistive frequency multiplier is not lossless, the Manley-Rowe relations do not apply. But we can derive a similar set of relations for a nonlinear resistor, and demonstrate an important result for frequency conversion using nonlinear resistors. Consider the resistive multiplier circuit shown in Figure 12,21. We have simplified the analysis by specializing to the frequency multiplier case by considering only a single source frequency—the more general case of two frequency sources is treated in [12]. For a source frequency a>, the nonlinear resistor generates harmonics of the form nco, so the resistor voltage and current can be written as a Fourier series: v{t) = £ Vme m=—cc 00 K0= £ A, lit) tme m=—oa The Fourier coefficients are deterrnined as T jmisit (12.62a) (12.62b) = — f V T J (t)e-immdt, 1=0 T L = | j i(ty-^!dt. 1=0 (12.63a) (12.63b) Since v(t) and i(0 are real functions, we must have Vm — V*m and Im — I*m. The power associated with the mth harmonic is (ignoring a factor of 4) Pm=2Rz{Vml*m) = Vmrm + V*Im-Now multiply Vm of (12.63a) by —m2!^ and sum: (12.64) - E ^vmi: = — / m £ m2>. *e~jmwtdt. i=0 m=-00 Next, use the result that d2i(t) dt2 00 00 m=—00 m=—o3 (12.65) 604 Chapter 12: Oscillators and Mixers to write (12.65) as m=-oo J d2i(t) , dt2 t=0 Into mm at f=0 dv(t)di(t) ~dt dt~ dt. (12.66) Since v(t) and i(t) are periodic functions (period F), we have i/(0) = u(T) and i(0) = i(T). Derivatives of /(/) have the same periodicity, so the second to last term in (12.66) vanishes. In addition, we can write 3v(j)di(i) _ dv(t) dt dt ~ dt t Equation (12.66) then reduces to di dv(t) _ di_ (dv(t)\2 dv dt ~ dv \ dt ) = —oc ' 111=0 m=Q r=0 r or ^ m 2ttw J dv\ dt / (12.67) For positive nonlinear resistors (defined as having an /-V curve whose slope is always positive), the integrand of (12.67) will always be positive. Thus (12.67) can be reduced to (12.68) m=0 If all harmonics are terminated in reactive loads except for to (the fundamental) and mto (the desired harmonic), (12.68) reduces to P +m2Pm > 0. The power P{ > 0 is delivered by the source, while Pm < 0 represents harmonic power supplied by the device. The maximum theoretical conversion efficiency is then given as 1 < — - m2 (12.69) This result indicates that the efficiency of a resistive frequency multiplier drops as the square of the multiplication factor. The performance of diode frequency multipliers can often be improved by using two diodes in a balanced configuration. This can lead to increased output power, improved input impedance characteristics, and the rejection of certain (all even or all odd) harmonics. Two diodes can be fed using a quadrature hybrid, or two diodes can be configured in an antipar-allet arrangement (back-to43ack with reversed polarities). The antiparallel configuration will reject all even harmonics of the input frequency. Transistor Multipliers Compared to diode frequency multipliers, transistor multipliers offer better bandwidth and the possibility of conversion efficiencies greater than 100% (conversion gain). FET multipliers also require less input and DC power than diode multipliers. In the past, before 12.4 Frequency Multiptiers 605 solid-state amplifiers were available at millimeter wave frequencies, high power diode multipliers were one of the few ways of generating millimeter wave power. Today, however, it is possible to generate the required frequency at low power, then amplify that signal to the desired power level using transistor amplifiers. This approach results in better efficiency, lower DC power requirements, and allows the separate optimization of signal generation and amplification functions. Transistor multipliers are well-suited for this application. There are several nonlinearities that exist in a FET device that can be used for harmonic generation: the transconductance near pinch-off, the output conductance near pinch-off, the rectifying properties of the Schottky gate, and the varactor-like capacitances at the gate and drain. For frequency doubler operation, the most useful of these is the rectification property, where the FET is biased to conduct only during the positive half of the input signal waveform. This results in operation similar to a class B amplifier, and provides a multiplier circuit that is useful for low-power output (typically less than 10 dBm) at frequencies up to 60 to 100 GHz. Bipolar transistors can also be used for frequency multiplication, with the capacitance of the collector-base junction providing the necessary nonlinearity. The basic circuit of a class B FET frequency multiplier is shown in Figure 12.22. A unilateral device is assumed here to simplify the analysis. The source is a generator of frequency 6Dq, with period T = Itt/íůo, and matched to the FET with the source impedance Rs + jXs. The drain of the FET is terminated with a load impedance Rl + jXi> which is chosen to form a parallel RLC resonator with at the desired harmonic frequency, híoq. The gate is biased at a DC voltage of Vgs < 0, while the drain is biased at Vdd > 0. The operation of the FET multiplier can be understood with the help of the waveforms shown in Figure 12.23. As seen in Figure I2 23ar the FET is biased below the turn-on voltage, V[, so the transistor does not conduct until the gate voltage exceeds V,. The resulting drain current is shown in Figure 12.23b, and is seen to be similar in form to a half-wave rectified version of the gate voltage. This waveform is rich in harmonics, so the drain resonator can be designed to present a short circuit at the fundamental and all undesired harmonics, and an open circuit at the desired harmonic frequency. The resulting drain voltage for n = 2 is shown in Figure 12.23c. We can make an approximate analysis of the FET multiplier by representing the drain current in terms of a Fourier series. If we assume that the drain current waveform is a half-cosine function of the form id(t) = JTt max COS- for |f | •< t/2 t (12.70) for r/2 < |(| < T/2, where r is the duration of the drain current pulse, the Fourier series can be found as 2nnt 1„ cos K=0 ij(t) = Y,In cos (12.71) FIGURE 12.22 Circuit diagram of an FET frequency multiplier. The transistor is modeled using a unilateral equivalent circuit. 606 Chapter 12: Oscillators and Mixers V5S m FIGURE 12.23 Voltage and currents in the FET multiplier (doubler) circuit of Figure 12.22. (a) Gate voltage when the transistor is biased just below pinch-off. (b) Drain current, which conducts when the gate voltage is above the threshold voltage, (c) Drain voltage when the load resonator is tuned to the second harmonic. with the Fourier coefficients given by fo^i*m% (12.72a) «W> kmxJ*£*gSJ*L fo,„>0. (12.72b) jzT 1 — (2«r/ry The coefficient /„ represents the drain current of harmonic frequency hojq, so maximizing multiplier efficiency involves maximizing ln. Since (12.72b) clearly shows that the maximum value of In decreases with n, circuits of this type are generally limited to frequency doublers or triplers. For a given value of n, the maximum value of /,,/Anax depends on the ratio t/T: for n — 2 the optimum occurs at i/T = 0.35, while for n = 3 the optimum occurs at t/T = 0.22. Because of device and biasing constraints, however, the designer usually has very little control of the pulse width t, and practical values of t/T are usually greater than optimum. Examination of Figure .12.23a shows that the normalized pulse 12.4 Frequency Multipliers 607 duration is related to the gate voltages Vlt Vga^R, and Vgmia as cos — = 2Vf~^max~Vrgmk. (12.73) The gate bias voltage satisfies the relation that Vw = (Vgmax- Vgmin)/2, (12.74) and the peak value of the AC component of the gate voltage (frequency (uq) is given by V, = Vsnuw - V„. (12.75) Then the input power delivered to the FET can be expressed as 2%J ' W-j/tooCtA2' Pi, = -\h\2Rt = „,„ ' ' „. (12.76) If the source is conjugately matched to the transistor, the input power will be equal to the available power, Pavan, On the load side, the peak value of the AC component of the drain voltage (frequency ncwo) is given by Vl = h h = t$m - iwft (12-77> assuming resonance of and Qs. This gives the optimal load resistance as D Yd max — K/fnin ,nlm /if, = -—-, (IZ./Q) 4* ft Then the output power at the harmonic nu)q is 1 2 Finally, the conversion gain is given as Pavail Gr = —!-. (12.80) EXAMPLE 12,6 FET FREQUENCY DOUBLER DESIGN A 12-24 GHz frequency doubler is designed using an FET with the following parameters: V, = -2.0 V, Rj = 10 Si, Cgs - 0.20 pF, Cds =0.15 pF, and Rds = 40 Q. Assume the operating point of the transistor is chosen so that Vg ^ = 0.2 V, Vsmn = -6.0 V, =5.0 V, Vtm = 1.0 V, and /max = 80 mA. Find the conversion gain of the multiplier. Solution We first use (12.74) and (12.75) to find the peak value of the AC input voltage, The gate bias voltage is Vgs =*i%m - Vsmjn)/2 = (0.2 - 6.0)/2 = -2.9 V, and the peak AC input voltage is V„ = W- Vss = 0.2 + 2.9 = 3.1V, 608 Chapter 12; Oscillators and Mixers Then the input power is given by (12-76): |V,I2*,- (3.1)2(10) P,n^ 12.5 21 ft- - jfaxiCss\2 2[(10)3 + (1/2tt(12 x 10*)(0.2 x 10"12))2] = 10,7 mW. The pulse width is found from (12.73) as at 2V, - Vgmsx - Vgmj0 2(-2.0) - 0.2 + 6.0 ^ cos -— =----— =--—-= 0.29, T Vgm-Vgito 0.2 + 6.0 for I = 0.406. T Then the load current for the second harmonic is given by (12.72b): h = ^£**2m = 0.262/„x = 2, .0 m. 7iT 1 — (4r/T)- The load resistance required to match the transistor is found from (12.78): * 2/2 2(0.021) The output power at 24 GHz is given by (12.79): h - ]:\h\2RL = ^(0.021)2(95.2) = 21.0mW, Finally, the conversion gain is, assuming the input is conjugately matched, ^2 21.0 . _ G, = -JL. = —- = 2.9 dB. Pavait 10.7 The load reactance required to resonate the second harmonic is Xt = 1 /2waC(iS = 44.2 £2, which corresponds to an inductance of 0.293 nH. ■ OVERVIEW OF MICROWAVE SOURCES A source of microwave power is essential for any microwave system. Communication and radar systems generally use a relatively high-power source for the transmitter, and one or more low-power sources for local oscillator and down conversion functions in the receiver. Radar transmitters are often operated in a pulsed mode, and peak powers that are much greater than the continuous power rating of the source can then be attained. Electronic warfare systems use sources in much the same way as a radar system, with the additional requirement for tunability over a wide bandwidth. Radiometer and radio astronomy receiver systems require low-power sources for local oscillators (although it can be argued that the primary source of microwave power for such systems is the radiation emitted from the hot body under observation). Test and measurement systems usually require a low-power microwave source, often tunable over a wide bandwidth. And the microwave oven, that most common of all microwave systems, requires a single-frequency high-power source, At present, these requirements are met with a variety of solid-state and microwave tube sources. Generally the division is between solid-state sources for low power and low frequencies, and tubes for high power and/or high frequencies. Figure 12.24 illustrates the power versus frequency performance for these two types of sources. Solid-state sources have the advantages of small size, ruggedness, low cost, and compatability with microwave 12.5 Overview of Microwave Sources 609 integrated circuits, and so are usually preferred whenever they can meet the necessary power and frequency requirements. But very high power applications are dominated by microwave tubes, and even though the power and frequency performance of solid-state sources is steadily improving, it appears that the need for microwave tubes will not be eliminated any time soon. Here we will briefly describe and summarize the performance of several of the most common types of solid-state and microwave tube sources. Solid-State Sources Solid-state microwave sources can be categorized as two-terminal devices (diodes), or three-terminal devices (transistor oscillators). The most common diode sources are the Gunn diode and the IMPATT diode, both of which directly convert a DC bias to RF power in the frequency range of about 2 to 100 GHz. The Gunn diode is a transferred-electron device mat uses a bulk semiconductor (usually GaAs or InP), as opposed to a pn junction [13]. This effect leads to a negative-resistance characteristic that can be employed with an external resonator to produce a stable oscillator. EXT to RF efficiencies are generally less than 10%. Figure 12.25 shows the power (continuous and pulsed) versus frequency performance for a variety of commercially available Gunn sources. Gunn diodes can also be used as negative-resistance reflection-type amplifiers. Figure 12.26 shows a photograph of two commercially available Gunn diode sources. The IMPATT diode uses a reverse-biased pn junction to generate microwave power [13]. The materia] is usually silicon or gallium arsenide, and the diode is operated with a relatively high voltage (70-100 V) to achieve a reverse-biased avalanche breakdown current. When coupled with a high-Q resonator and biased at an appropriate operating point, a negative-resistance effect can be achieved at the RF operating frequency, and oscillation will occur. IMPATT sources are generally more noisy than sources using Gunn diodes, but are capable of higher powers and higher DC to RF conversion efficiencies. IMPATTs also have better temperature stability than Gunn diodes. Figure 12.27 shows the power versus frequency performance for typical commercial IMPATT sources. IMPATT diodes can also be used as negative-resistance amplifiers. Transistor oscillators generally have lower frequency and power capabilities compared to Gunn or IMPATT sources, but offer several advantages over diodes. First, oscillators using GaAs FETs are readily compatible with MIC or MMIC circuitry, allowing easy integration 610 Chapter 12: Oscillators and Mixers I0O00 0 1 10gij ino • 29% RCA • 32% RCA I.q 0.01 I5%CA RCA 5% CA* \ 10%CA' \ Pdsed \ Pf2= 105WGHz2 \ \ \ • \ 6%CA \ \ \ \ o Hitachi o 5% 6% Phillips o MA 11% BTLo 12.5% Varian o Lo \ 3% Pies 6.7% Van an 3% Plessey 4.3% Variati j Varian InP * , cw, \ Pf5 = 4.6xlO-WGH2J .\ Varian ^ \ 4.7% Varian InP .äB. ~ o \ Hughes 2,8% __\ i \_\_ 10 100 Frequency (GHz) _pf__ 9.5 X 10-3.5 x 10* 1.1 x 104 1000 FIGURE 12.27 Power versus frequency performance of IMPATT diodes. 612 Chapter 12: Oscillators and Mixers a magnetically-biased YIG sphere. Thus, a voltage-controlled oscillator (VCO) can be made by using a reverse-biased varactor diode in the tank circuit of a transistor oscillator, In a YIG-tuned oscillator (YTO), a single-crystal YIG sphere is used to control the inductance of a coil in the tank circuit of the oscillator. Since YIG is a ferrimagnetic material, its effective permeability can be controlled with an external DC magnetic bias field, thus controlling the oscillator frequency. YIG oscillators can be made to tune over a decade or more of bandwidth, while varactor-tuned oscillators are limited to a tuning range of about an octave. YIG-tuned oscillators, however, cannot be tuned as fast as varactor oscillators. In many applications the RF power requirement exceeds the power capacity of a single solid-state source, But because of the many advantages offered by solid-state sources, substantial effort has been directed toward increasing output power through the use of various power combining techniques. Thus, the outputs of two or more sources are combined in phase, effectively multiplying the output power of a single source by the number of individual sources being used. In principle, an unlimited amount of RF power can be generated in this manner; in practice, however, factors such as high-order modes and combiner losses limit the multiplication factor to about 10-20 dB. Power combining can be done by combining powers at the device level or at the circuit level. In some applications, power can be combined spatially by using an array of antennas, where each element is fed with a separate source. At the device level, several diode or transistor junctions are essentially connected in parallel over an electrically small region, and used as a single device. This technique is thus limited to a relatively few device junctions. At the circuit level, the power output from N devices can be combined with an iV-way combiner. The combining circuit may be an AT-way Wilkinson-type network, or a similar type of planar combining network. Resonant cavities can also be used for this purpose. These various techniques all have their own advantages and disadvantages in terms of efficiency, bandwidth, isolation between sources, and circuit complexity. Microwave Tubes The first truly practical microwave source was the magnetron tube, developed in England in the 1930s, and later providing the impetus for the development of microwave radar during World War II. Since then, a large variety of microwave tubes have been designed for the generation and amplification of microwave power. In recent years, solid-state devices have been progressively filling the roles that were once reserved for microwave tubes, generally with a multitude of advantages. But tubes are still essential for the generation of very high powers (10 kW to 10 MW), and for the higher millimeter wave frequencies (100 GHz and higher). Here we will provide a brief overview of some of the most common microwave tubes, and their basic characteristics. Several of these tubes are not actually sources by themselves, but are high-power amplifiers. Such tubes are used in conjunction with lower power sources (often solid-state sources) in transmitter systems. There is a wide variety of tube geometries, as well as a wide variety of principles on which tube operation is based, but all tubes have several common features. First, all tubes involve the interaction of an electron beam with an electromagnetic field, inside a glass or metal vacuum envelope. Thus, a way must be provided for RF energy to be coupled outside the envelope; this is usually accomplished with transparent windows or coaxial coupling probes or loops. Next, a hot cathode is used to generate a stream of electrons by thermionic emission. Cathodes are usually fabricated from a barium oxide-coated metal surface, or an impregnated tungsten surface. The electron stream is then focused into a narrow beam by a focusing anode with a high voltage bias. Alternatively, a solenoidal electromagnet can be used to focus the electron beam. For pulsed operation, a beam modulating electrode is 12.5 Overview of Microwave Sources 613 used between the cathode and anode. A positive bias voltage will attract electrons from the cathode, and rum the beam on, while a negative bias will turn the beam off. After the electron beam leaves the region of the tube where the desired interaction with the RF field takes place, a collector element is used to provide a complete current path back to the cathode power supply. The assembly of the cathode, focusing anode, and modulating electrode is called the electron gun. Because of the requirement for a high vacuum, and the need to dissipate large amounts of heat, microwave tubes are generally very large and bulky. In addition, tubes often require large, heavy biasing magnets, and high voltage power supplies. Factors to consider when choosing a particular type of tube include power output, frequency, bandwidth, tuning range, and noise. Microwave tubes can be grouped into two categories, depending on the type of electron beam-field interaction. In linear-beam, or "O," type tubes the electron beam traverses the length of the tube, and is parallel to the electric field. In the crossed-field, or "m," type tube the focusing field is perpendicular to the accelerating electric field. Microwave tubes can also be classified as either oscillators or amplifiers. The klystron is a linear-beam tube that is widely used as both an amplifier and an oscillator. In a klystron amplifier, the electron beam passes through two or more resonant cavities. The first cavity accepts an RF input and modulates the electron beam by bunching it into high-density and low-density regions. The bunched beam then travels to the next cavity, which accentuates the bunching effect. At the final cavity the RF power is extracted, at a highly amplified level. Two cavities can produce up to about 20 dB of gain, while using four cavities (about the practical limit) can give 80-90 dB gain. Klystrons are capable of peak powers in the megawatt range, with RF output/DC input power conversion efficiencies of 30-50%. The reflex klystron is a single-cavity klystron tube which operates as an oscillator by using a reflector electrode after the cavity to provide positive feedback via the electron beam. It can be tuned by mechanically adjusting the cavity size. The major disadvantage of klystrons is their narrow bandwidth, which is a result of the high- Q cavities required for electron bunching. Klystrons have very low AM and FM noise levels. The narrow bandwidth of the klystron amplifier is overcome in the traveling wave tube (TWT). The TWT is a linear-beam amplifier dial uses an electron gun and a focusing magnet to accelerate a beam of electrons through an interaction region. Usually the interaction region consists of a slow-wave helix structure, with an RF input at the electron gun end, and an RF output at the collector end. The helical structure slows down the propagating RF wave so that it travels at the same velocity as the wave and beam travel along the interaction region, and amplification is effected. Then the amplified signal is coupled from the end of the helix. The TWT has the highest bandwidth of any amplifier tube, ranging from 30 to 120%; this makes it very useful for electronic warfare systems, which require high power over broad bandwidths. It has a power rating of several hundred watts (typically), but this can be increased to several kilowatts by using an interaction region consisting of a set of coupled cavities; the bandwidth will be reduced, however. The efficiency of the TWT is relatively small, typically ranging from 20 to 40%. A variation of the TWT is the backward wave oscillator (BWO). The difference between a TWT and the BWO is that in the BWO, the RF wave travels along the helix from the collector toward the electron gun. Thus the signal for amplification is provided by the bunched electron beam itself, and oscillation occurs. A very useful feature of the BWO is that its output frequency can be tuned by varying the DC voltage between the cathode and the helix; tuning ranges of an octave or more can be achieved. The power output of the BWO, however, is relatively low (typically less than 1 W), so these tubes are generally being replaced with solid-state sources, 614 Chapter 12: Oscillators and Mixers Another type of linear-beam oscillator tube is the extended interaction oscillator (EIO)-The EIO is very similar to a klystron, and uses an interaction region consisting of several cavides coupled together, with positive feedback to support oscillation. It has a narrow tuning bandwidth, and a moderate efficiency, but it can supply high powers at frequencies up to several hundred GHz. Only the gyratron can deliver more power. Crossed-field tubes include the magnetron, the crossed-field amplifier, and ihs, gyratron. As previously mentioned, the magnetron was the first high-power microwave source. It consists of a cylindrical cathode surrounded by a cylindrical anode with several cavity resonators along the inside of its periphery. A magnetic bias field is applied parallel to the cathode-anode axis. In operation, a cloud of electrons is formed which rotates around the cathode in the interaction region. As with linear-beam devices, electron bunching occurs, and energy is transferred from the electron beam to the RF wave. RF power can be coupled out of the tube with a probe, loop, or aperture window. Magnetrons are capable of very high power outputs—on the order of several kilowatts. And the magnetron has an efficiency of 80% or more. A significant disadvantage, however, is that they are very noisy, and cannot maintain frequency or phase coherency when operated in a pulsed mode. These factors are important for high-performance pulsed radars, where processing techniques operate on a sequence of returned pulses. (Modern radars of this type today generally use a stable low-noise solid-state source, followed by a TWT for power amplification.) The application of magnetrons is now primarily for microwave cooking. The crossed-field amplifier (CFA) has a geometry similar to a TWT, but employs a crossed-field interaction that is similar to that of Uie magnetron. The RF input is applied to a slow-wave structure in the interaction region of the CFA, but the electron beam is deflected by a negatively biased electrode to force the beam perpendicular to the slow-wave structure. In addition, a magnetic bias field is applied perpendicular to this electric field, and perpendicular to the electron beam direction. The magnetic field exerts a force on the electron beam that counteracts the field from the sole. In die absence of an RF input, the electric and magnetic fields are adjusted so that their effects on the electron 1 mWi-J-1-L-1-1-1 0.3 1 3 10 30 100 300 Frequency (GHz) FIGURE 12.28 Power versus frequency performance of microwave oscillator tubes. 12.6 Mixers 615 beam cancel, leaving the beam to travel parallel to the slow-wave structure. Applying an RF field causes velocity modulation of the beam, and bunching occurs. The beam is also periodically deflected toward the slow-wave circuit, producing an amplified signal. Crossed-field amplifiers have very good efficiencies—up to 80%, but the gain is limited to 10-15 dB. Also, the CEA has a noisier output than either a klystron amplifier or TWT. Its bandwidth can be up to 40%, Another crossed-field tube is the gyratron, which can be used as an amplifier or an oscillator. This tube consists of an electron gun with input and output cavities along the axis of the electron beam, similar to a klystron amplifier. But the gyratron also has a solenoidal bias magnet that provides an axial magnetic field. This field forces the electrons to travel in tight spirals down the length of the tube. The electron velocity is high enough so that relativistic effects are important. Bunching occurs, and energy from the transverse component of the electron velocity is coupled to the RF field. A significant feature of the gyratron is that the frequency of operation is determined by the bias field strength and the electron velocity, as opposed to the dimensions of the tube itself. This makes the gyratron especially useful for millimeter wave frequencies; it offers the highest output power (10-100 kW) of any tube in this frequency range. It also has a high efficiency for tubes in the millimeter wave range. The gyratron is a relatively new type of tube, but it is rapidly replacing tubes such as reflex klystrons and EIOs as sources of millimeter wave power. Figures 12,28 and 12.29 summarize the power versus frequency performance of microwave tube oscillators and amplifiers. MIXERS A mixer is a three-port device that uses a nonlinear or time-varying element to achieve frequency conversion. As introduced in Section 10.3, an ideal mixer produces an output consisting of the sum and difference frequencies of its two input signals. Operation of 616 Chapter 12: Oscillators and Mixers practical RF and microwave mixers is usually based on the nonlinearity provided by either a diode or a transistor. As we have seen, a nonlinear component can generate a wide variety of harmonics and other products of input frequencies, so filtering must be used to select the desired frequency components. Modern microwave systems typically use several mixers and filters to perform the functions of frequency up-conversion and down-conversion between baseband signal frequencies and RF carrier frequencies. We begin by discussing some of the important characteristics of mixers, such as image frequency, conversion loss, noise effects, and intermodulation distortion. Next we discuss the operation of single-ended mixers, using either a single diode or a FET as the nonlinear element. The balanced diode mixer circuit is then described, followed by a brief description of more specialized mixer circuits. Mixer Characteristics The symbol and functional diagram for a mixer are shown in Figure 1230. The mixer symbol is intended to imply that the output is proportional to the product of the two input signals. We will see that this is an idealized view of mixer operation, which in actuality produces a large variety of harmonics and other undesired products of the input signals. Figure 12.30a illustrates the operation ^.frequency up-conversion, as occurs in a transmitter. A local oscillator (LO) signal at the relatively high frequency fio is connected to one of the input ports of the mixer. The LO signal can be represented as VLoU) = cos2jzfLOi. (12.81) A lower frequency baseband or intermediate frequency (IF) signal is applied to the other mixer input. This signal typically contains the information or data to be transmitted, and can be expressed for our purposes as v!F(t) = co$2xfau (12.82) The output of the idealized mixer is given by the product of the LO and IF signals: vrfU) = KvL0(t)v(F(t) = KcQs2irfLOtcos2nfai = —[co$2n(fu> - fa) t + co$2x(fLO + fa)t]f (12.83) where K is a constant accounting for the voltage conversion loss of the mixer. The RF output is seen to consist of the sum and differences of the input signal frequencies: fsF = fw ± fa. (12.84) The spectra of the input and output signals are shown in Figure 12.30a, where we see that the mixer has the effect of modulating die LO signal with the IF signal. The sum and difference frequencies at /to ± fa are called the sidebands of the carrier frequency /to, with fix) 4- fa being the upper sideband (USB), and fyy — fa being the lower sideband (LSB). A double-sideband (DSB) signal contains both upper and lower sidebands, as in (12.83), while a single-sideband (SSB) signal can be produced by filtering or by using a single-sideband mixer. Conversely, Figure 12.30b shows the process of frequency down-conversion, as used in a receiver In this case an RF input signal of the form VRF (12.87) gives the RF input frequency that will be down-converted to the IF frequency as fitF — flX) + flF> (12.88a) since the insertion of (12.88a) into (12.87) yields fIF (after low-pass filtering). Now consider the RF input frequency given by flM = fa — ftF- (12.88b) Insertion of (12.88b) into (12.87) yields - fjF (after low-pass filtering). Mathematically, this frequency is identical to ffF because the Fourier spectrum of any real signal is symmetric 618 Chapter 12: Oscillators and Mixers about zero frequency, and thus contains negative frequencies as well as positive. The RF frequency defined in (12.88b) is called the image response. The image response is important in receiver design because a received RF signal at the image frequency of (12.88b) is indistinguishable at the IF stage from the desired RF signal of frequency (12.88a), unless steps arc taken in the RF stages of the receiver to preselect signals only within the desired RF frequency band. The choice of which RF frequency in (12.88) is the desired and which is the image response is arbitrary, depending on whether the LO frequency is above or below the desired RF frequency. Another way of viewing this difference is to note that Jif in (12,88) may be negative. Observe that the desired and image frequencies of (12.88a) and (12.88b) are separated by 2 Another implication of (12.87) and the fact that f[F may be negative is that there are two local oscillator frequencies that can be used for a given RF and IF frequency: fLO = fRF±flFt (12.89) since taking the difference frequency of far with these two LO frequencies gives ± ftp-These two frequencies correspond to the upper and lower sidebands when a mixer is operated as an upconverter. In practice, most receivers use a local oscillator set at the upper sideband, fuo = /rf + fiF, because this requires a smaller LO tuning ratio when the receiver must select RF signals over a given band. Conversion loss. Mixer design requires impedance matching at three ports, complicated by the fact that several frequencies and their harmonics are involved. Ideally, each mixer port would be matched at its particular frequency (RF, LO, or IF), and undesired frequency products would be absorbed with resistive loads, or blocked with reactive terminations. Resistive loads increase mixer losses, however, and reactive loads can be very frequency sensitive. In addition, there are inherent losses in the frequency conversion process because of the generation of undesired harmonics and other frequency products. An important figure of merit for a mixer is therefore the conversion loss, which is defined as the ratio of available RF input power to the available IF output power, expressed in dB: available RF input power „m Lc = 10log-— - > OdB. (12.90) available IF output power Conversion loss accounts for resistive losses in a mixer as well as loss in the frequency conversion process from RF to IF ports. Conversion loss applies to both up-conversion and down-conversion, even though the context of the above definition is for the latter case. Since the RF stages of receivers operate at much lower power levels than do transmitters, minimum conversion loss is more critical for receivers because of the importance of minimizing losses in the RF stages to maximize receiver noise figure. Practical diode mixers typically have conversion losses between 4 and 7 dB in the 1-10 GHz range, Transistor mixers have lower conversion loss, and may even have conversion gain of a few dB. One factor that strongly affects conversion loss is the local oscillator power level; minimum conversion loss often occurs for LO powers between 0 and 10 dBm. This power level is large enough that the accurate characterization of mixer performance often requires nonlinear analysis. Noise figure. Noise is generated in mixers by the diode or transistor elements, and by thermal sources due to resistive losses. Noise figures of practical mixers range from 1-5 dB, with diode mixers generally achieving lower noise figures than transistor mixers. The noise figure of a mixer depends on whether its input is a single sideband signal or a double sideband signal. This is because the mixer will down-convert noise at both sideband frequencies (since 12,6 Mixers 619 these have the same IF), but the power of a SSB signal is one-half that of a DSB signal (for the same amplitude). To derive the relation between the noise figure for these two cases, first consider a DSB input signal of the form *>DSB(t) = A[cOS((Olo - CQiF) t + COS(£Witf + 0)IF) /]. (12.91) Upon mixing with an LO signal cosw^/ and low-pass filtering, the down-converted IF signal will be AK A K vifU) = -j- cosicojf t) + cos(—oi}Fi) = A K cos a>tf t, (12.92) where if is a constant accounting for the conversion loss for each sideband. The power of the DSB input signal of (12.91) is , A2 A2 , and the power of the output IF signal is A2K2 2 For noise figure, the input noise power is defined as Nj =kToB, where Ta = 290 K and B is the IF bandwidth, The output noise power is equal to die input noise plus Abided - the noise power added by the mixer, divided by the conversion loss (assuming a reference at the mixer input): -%-■ Then using the definition of noise figure gives the DSB noise figure of the mixer as The corresponding analysis for the SSB case begins with a SSB input signal of the form ussb(' ) = A cos(a>LO -ioIF)t. (12.94) Upon mixing with the LO signal cos (oyyt and low-pass filtering, the down-converted IF signal will be AK vlF(t) = — co$(toSFt). (12.95) The power of the SSB input signal of (12.94) is 5< = t> and the power of the output IF signal is _ A2K2 So~ 3 * The input and output noise powers are the same as for the DSB case, so the noise figure for an SSB input signal is FSSB = = — l\ + %A . (12.96) tsSB K2LC V + kT»B J K } 620 Chapter 12: Oscillators and Mixers Comparison with (12.93) shows that the noise figure of the SSB case is twice that of the DSB case; Fssb = 2Fi>sB' (12.97) Other mixer characteristics. Since mixers involve nonlinearity, they will produce inter-modulation products. Typical values of P$ for mixers range from 15-30 dBm. Another important characteristic of a mixer is the isolation between the RF and LO ports. Ideally, the LO and RF ports would be decoupled, but internal impedance mismatches and limitations of coupler performance often result in some LO power being coupled out of the RF port. This is a potential problem for receivers that drive the RF port directly from the antenna, because LO power coupled through the mixer to the RF port will be radiated by the antenna. Because such signals will likely interfere with other services or users, the FCC sets stringent limits on the power radiated by receivers. This problem can be largely alleviated by using a bandpass fitter between the antenna and mixer, or by using an RF amplifier ahead of the mixer. Isolation between the LO and RF ports is highly dependent on the type of coupler used for diplexing these two inputs, but typical values range from 20-40 dB. EXAMPLE 12.7 IMAGE FREQUENCY The IS-54 digital cellular telephone system uses a receive frequency band of 869-894 MHz, with a first rF frequency of 87 MHz, and a channel bandwidth of 30 kHz. What are the two possible ranges for the LO frequency? If the upper LO frequency range is used, determine the image frequency range. Does the image frequency fall within the receive passband? Solution By (12.89), the two possible LO frequency ranges are f 956 to 981 MHz fuo = Irf ± hp = (869 to 894) ± 87 = jlu jrf jsr v ; } 782 to 807 MHz. Using the 956-981 MHz LO, (12.87) gives the IF frequency as ftp = Irf- ftx> = (869 to 894) - (956 to 981) - -87 MHz, so from (12.88b) the RF image frequency range is fm = ho - U = (956 to 981) + 87 = 1043 to 1068 MHz, which is well outside the receive passband. ■ The above treatment of rnixers is idealized because of the assumption that the output was proportional to the product of the input signals, thus producing only sum and difference frequencies (for sinusoidal inputs). We now discuss more realistic mixers, and show that the output does indeed contain a term proportional to the product of the inputs, but many higher order products as well. Single-Ended Diode Mixer A basic diode mixer circuit is shown in Figure 12.3 la. This type of mixer is called a single-ended mixer because it uses a single diode element. The RF and LO inputs are combined 12.6 Mixers 621 Diplexing RF coupler + DC DC block RF choke1 lo input Diode DC block + m Low-pass filter WW -—>~ IF output FIGURE 12.31 (a) Circuit for a single-ended diode mixer, (b) Idealized equivalent circuit. in a diplexer, which superimposes the two input voltages to drive the diode. The diplexing function can be implemented using a directional coupler or hybrid junction to provide combining as well as isolation between the two inputs. The diode may be biased with a DC bias voltage, which must be decoupled from the RF signal paths. This is done by using DC blocking capacitors on either side of the diode, and an RF choke between the diode and the bias voltage source. The AC output of the diode is passed through a low-pass filler to provide the desired IF output voltage. This description is for application as a down-converter, but the same mixer can be used for up-conversion since each port may be used interchangeably as an input or output port. The AC equivalent circuit of the mixer is shown in Figure 12.31b, where the RF and LO input voltages are represented as two series-connected voltage sources. Let the RF input voltage be a cosine wave of frequency a)^: VrfO) = VRF cos &RFt i (12.98) and let the LO input voltage be a cosine wave of frequency (l 4- cos2aW) + 2V^/rVwcos{wftF - tOLo)i + 2V/urVu) cos(rf, T/aio, and corf + colo terms will be blocked by the low-pass filter. This leaves the IF output current as iiF(0 = ^v&mmmm* (12.100 where coif = % — o>w is the IF frequency. The spectrum of the down-converting single-ended mixer is thus identical to that of the idealized mixer shown in Figure 12.30b. Single-Ended FET Mixer There are several FET parameters that offer nonlinearities that can be used for mixing, but the strongest is the transconductance, gmi when the FET is operated in a common source configuration with a negative gate bias. Figure 12.32 shows the variation of transconductance with gate bias for a typical FET When used as an amplifier, the gate bias voltage is chosen near zero, or slightly positive, so the transconductance is near its maximum value, and the transistor operates as a linear device. When the gate bias is near the pinch-off region, where the transconductance approaches zero, a small positive variation of gate voltage can cause a large change in transconductance, leading to a nonlinear response. Thus the LO voltage can be applied to the gate of the FET to pump the transconductance to switch the FET between high and low transconductance states, thus providing the desired mixing function. The circuit for a single-ended FET mixer is shown in Figure 12.33. A diplexing coupler is again used to combine the RF and LO signals at the gate of the FET. An impedance matching network is also usually required between the inputs and the FET, which typically presents a very low input impedance. RF chokes are used to bias the gate at a negative voltage near pinch-off, and to provide a positive bias for the drain of the FET. A bypass capacitor at the drain provides a return path for the LO signal, and a low-pass filter provides the final IF output signal. 12.6 Mixers 623 RF input RF/LO FET r dip lexer and LO input matching Í RF C choke J. -DC bias FIGURE 1233 Circuit for a single-ended FET mixer. Our analysis of the mixer of Figure 12.33 follows the original work described in reference [14], The simplified equivalent circuit is shown in Figure 12.34, and is based on the unilateral equivalent circuit of a FET introduced in Section 10.4. The RF and LO input voltages are given in (12.98) and (12.99). Let Zg = Rg + jKs be the Thevenin source impedance for the RF input port, and Z£ = RL + jXL be the Thevenin source impedance at the IF output port. These impedances are complex to allow a conjugate match at the input and output ports for maximum power transfer. The LO port has a real generator impedance of Zo, since we are not concerned with maximum power transfer for the LO signal. Since the FET transconductance is driven by the LO signal, its time variation can be expressed as a Fourier series in terms of harmonics of the LO: (12.102) «=i Because we do not have an explicit formula for the transconductance, we cannot calculate directly the Fourier coefficients of (12.102), but must rely on measurements for these values. As we will see, the desired down-conversion result is due solely to the n = 1 term of the Fourier series, so we only need the gl coefficient. Measurements typically give a value in the range of 10 mS for g\. The conversion gain of the FET mixer can be found as GL = avail Wrf\2 ylF 'RF (12.103) where Vjf is the IF drain voltage, and the impedances Zs and ZL are chosen for maximum power transfer at the RF and IF ports. The RF frequency component of the phasor voltage across the gate-to-source capacitance is given in terms of the voltage divider between Zg, Rir ^ z v; =ti a v FIGURE 12,34 Equivalent circuit for the FET mixer of Figure 12.33. 624 Chapter 12: Oscillators and Mixers and Css: P*.-=—%--^-. (12.1(M, Multiplying the transconductance of (12.102) by v^F(t) — V*F cosa>/{Ft gives terms of the form gmV)vf(t) = gQ cos coRFt + 2g i V** cos (ORft cos + ■ ■.. (12.105) The down-converted IF frequency component can be extracted from the second term of (12.105) using the usual trigonometric identity: gmiOv™(0|w = gi V?Fcos V Rd + ZL } 1 + }(oRFCgARi + ZR) \Rd + ZLJ where (12.104) has been used. Using this result in (12.103) gives the conversion gain (before conjugate matching) as not matched _ / 28lRj V \OJRFCgsJ R? RL [(Rd + Rü2 + XU We now conjugately match the RF and IF ports to maximize the conversion gain. Thus we let Rg = Rj, Xg = ~l/a>RFCg3, Ri = Rj, and Xc — 0, which reduces the above result to Gc = A n . (12.108) The quantities Rd> Rt> and CSii are all parameters of the FET. Practical mixer circuits generally use matching circuits to transform the FET impedance to 50 ST2 for the RF, LO, and IF ports, EXAMPLE 12.8 MIXER CONVERSION GAIN A single-ended FET mixer is to be designed for a wireless local area network receiver operating at 2.4 GHz. The parameters of the FET are: Rd = 300 &. Rj = 10 , Cgs = 0,3 pF, and g\ = 10 mS. Calculate the maximum possible conversion gain. Solution This is a straightforward application of the formula for conversion gain given in (12.108): 0c = -4^__, <"» » 10"3>;(3°°> . 36.6 = 15.6dB. ' *4Cl,«, 4(2^(2.4 x 10»)»(10) Note that this value does not include losses due to the necessary impedance matching networks. ■ 12.6 Mixers 62S KF input LO inpui 3 1& 'i '2 -90° 3 dB 90° hybrid (a) 3 dB 180* hybrid (b) Low-pass filler IF output Low-pass filter FIGURE 12.35 Balanced mixer circuits, (a) Using a 90" hybrid, (b) Using a 180" hybrid. Balanced Mixer RF input matching and RF-LO isolation can be improved through the use of a balanced mixer, which consists of two single-ended mixers combined with a hybrid junction. Figure 12.35 shows the basic configuration, with either a 90° hybrid (Figure 12.35a), or a 1 SO* hybrid (Figure 12.35b). As we will see, a balanced mixer using a 90c hybrid junction will ideally lead to a perfect input match at the RF port over a wide frequency range, while the use of a 180° hybrid will ideally lead to perfect RF-LO isolation over a wide frequency range. In addition, both mixers will reject all even order intermodulation products. Figure 12.36 shows a photograph of a microstrip circuit mat contains several balanced mixers. We can analyze the performance of a balanced mixer using the small-signal approach that was used for the single-ended diode mixer. Here we will concentrate on the balanced mixer with a 90° hybrid, shown in Figure 12.35a, and leave the 180" hybrid case as a problem. As usual, let the RF and LO voltages be defined as and VRA*) = Vrf COS CDfurt. f£.o(0Vtť>C0Sťt)^oř. (12.109) (12.110) From Section 7.5, the scattering matrix for the 90° hybrid junction is "0 j 1 0' j 0 0 1 10 0; 0 1 j 0 (12.111) where the ports are numbered as shown in Figure 12.35a. The total RF and LO voltages applied to the two diodes can then be written as u,(f) = it% cm((oRFt -90°) + VuDCOs&iot - 180°)] V2 = A=[Vrfsina)RFt - VlocosaiLOtJ. V2 (12.112a) 626 Chapter 12: Oscillators and Mixers FIGURE 12.36 Photograph of a 35 GHz microstrip monopulse radar receiver circuit. Three balanced mixers using ring hybrids are shown, along with three stepped-impedance low-pass filters, and six quadrature hybrids. Eight feedlines are aperture coupled to microstrip antennas on the reverse side, The circuit also contains a Gunn diode source for the local oscillator. Courtesy of Millitecji Corporation, Northampton, Mass. v2(t) = —[VRfCOsdoRFt - 180°) + VLoCQsiowi - 90°)] 72 1 = ~-[-VRF cos tosFt + VLO sin oiiotl (12,112b) Using only the quadratic term from the small-signal diode approximation of (10.60) gives the diode approximation of (10.60) gives the diode currents as t'i(r) = Kv\ = —[vjfr sin2 toRFi - 2VrfVlo sin corf cos coLOt + v^cos2 6>u)t], (12.113a) i2(/) = -Kv] = — [Yjfecos2 toRpi -IVrfVlocoso>RFswo>LOt + V[0 sin2towt], (12.113b) 12.6 Mixers 627 where the negative sign on i2 accounts for the reversed diode polarity, and K is a constant for the quadratic term of the diode response. Adding these two currents at the input to the low-pass filter gives —K i"i(0 + hit) = — [VlpcosIojrf14 2V>f Vio sin o)tr-t - v£, cos2ojlo t], where the usual trigonometric identities have been used, and o>if — u>rf — wyo is the IF frequency, Note that the DC components of the diode currents cancel upon combining. After low-pass filtering, the IF output is itF(t) = -KVRF Vio sin co,rt, (12.114) as desired. We can also calculate the input match at the RF port, and the coupling between the RF and LO ports. If we assume the diodes are matched, and that each exhibits a voltage reflection coefficient T at the RF frequency, then the phasor expression for the reflected RF voltages at the diodes will be -jTVrf V2 V'r, = TV, = Jjr", (12.115a) Vr, = T V2 = VV*F. (12.115b) a/2 These reflected voltages appear at ports 2 and 3 of the hybrid, respectively, and combine to form the following outputs at the RF and LO ports: vsf^^±_Y^=ArVRF+l-rVRF = 0, (12.116a) •J2 V2 2 2 V^ = ~jt ~J7f = l2JrVRF + = JTV,iF' (12"116b) Thus we see that the phase characteristics of the 90° hybrid lead to perfect cancellation of reflections at the RF port. The isolation between the RF and LO ports, however, is dependent on the matching of the diodes, which may be difficult to maintain over a reasonable frequency range. Image Reject Mixer We have already discussed the fact that two distinct RF input signals at frequencies ojrf ss? (x>iq ±ojif will down-convert to the same IF frequency when mixed with (£>w- These two frequencies are the upper and lower sidebands of a double-sideband signal. The desired response can be arbitrarily selected as either the LSB (tow — io 4- wifX assuming a positive IF frequency. The image reject mixer, shown in Figure 12.37, can be used to isolate these two responses into separate output signals. The same circuit can also be used for up-conversion, in which case it is usually called a single-sideband modulator. In this case, the IF input signal is delivered to either the LSB or the USB port of the IF hybrid, and the associated single sideband signal is produced at the RF port of the mixer. We can analyze the image reject mixer using the small-signal approximation. Let the RF input signal be expressed as VRF(t) = Vu CQ$(t$LO + Ct)jF) t+VL COS(/f) r, (12.1 17) 628 Chapter 12: Oscillators and Mixers RF input ©_-.0 Z05 90° RF hybrid Low-pass filter Low-pass filter FIGURE 12,37 Circuit for an image reject mixer. ® g) >l LSB ®_^ USB 90° IF „B hybrid where Vv and Vj, represent the amplitudes of the upper and lower sidebands, respectively. Using the 5-matrix given in (12.111) for the 90° hybrid gives the RF voltages at the diodes as vA{t) =5 —-{% COSiiOLOt + {Ft - 180Q>] V2 — [Vy cos(ft>lo + <*>tp) t + VL co$(cow - (0!f) t]. s/2 (12.118b) After mixing with the LO signal of (12.110) and low-pass filtering, the IF inputs to the IF hybrid are 4(0 = ^|(VV - Vijsinw/fi, (Vy + V^)cosw/f t, (12.119a) (12.119b) where K is the mixer constant for the squared term of the diode response. The phasor representation of the IF signals of (12.119) is Combining these voltages in the IF hybrid gives the following outputs: (12.120a) (12.120b) (12.121a) (12.121b) 12.6 Mixers 629 which we see are the separate sidebands of the down-converted input signal of (12.117). These outputs can be expressed in time-domain form as vm=KVwVL (12.122a) v2(t)= smatpt, (12.122b) which clearly shows the presence of a 90° phase shift between the two sidebands. Also note that the image rejection mixer does not incur any additional losses beyond the usual conversion losses of the single rejection rnixer. A practical difficulty with image rejection mixers is in fabricating a good hybrid at the relatively low IF frequency. Losses, and hence noise figure, are also usually greater than for a simpler mixer. Other Mixers There are a number of other mixer circuits that provide various advantages in terms of bandwidth, harmonic generation, and intermodulation products. The double-balanced mixer of Figure 12.38 uses two hybrid junctions or transformers, and provides good isolation between all three ports, as well as rejection of all even harmonics of the RF and LO signals. This leads to very good conversion loss, but less than ideal input matching at me RF port. The double-balanced mixer also provides a higher third-order intercept point than either a single-ended mixer or a balanced mixer. The mixer shown in Figure 12.39 uses two FETs in a differential amplifier configuration. The balun (balanced-to-unbalanced) networks on the LO and IF ports provide a transition between a two-wire line that is balanced with respect to ground and a single line that is unbalanced relative to ground. Baluns may be implemented with center-tapped transformers, or with 180c hybrid junctions. The differential mixer operates as an alternating switch, with the LO turning the top two FETs on and off with alternate half-cycles of the LO. These FETs are biased slightly above pinch-off, so each FET will be conducting for slightly more than half of each LO cycle. Thus, one of the upper FETs is always conducting, and the lower FET will remain in saturation. The RF and LO ports require impedance matching, and the IF output circuit must provide a return path to ground for the LO signal. Figure 12.40 shows the circuit for an antiparallel diode mixer, which is often used for subharraonically pumped millimeter wave frequency conversion. The back-to-back diodes function as a frequency doubler, thus requiring an LO frequency of one-half the usual value. FIGURE 12,38 Double balanced mixer circuit. 630 Chapter 12: Oscillators and Mixers FIGURE 1239 A differential FET mixer. Lowpass filter for IF RF input iL "3 ťor -L Bandpass filter for RF output Lowpass filter forLO and 11 ■ FIGURE 12.40 Subharmonically pumped mixer using an antrparallel diode pair. The diode nonlinearity operates as a resistive frequency multiplier to generate the second harmonic of the LO to mix with the RF input to produce the desired output frequency. The antiparallel diode pair has a symmetric I-V characteristic that suppresses the fundamental mixing product of die RF and LO input signals, leading to better conversion loss. Table 12.1 summarizes the characteristics of several of the mixers that we have discussed. TABLE 12.1 Mixer Characteristics Mixer type Number of diodes RF input match RF-LO isolation Conversion loss Third-order intercept Single-ended 1 Poor Fair Good Fair Balanced (90*) 2 Good Poor Good Fair Balanced (180*) 2 Fair Excellent Good Fair Double balanced 4 Poor Excellent Excellent Excellent Image reject 2 or 4 Good Good Good Good REFERENCES [ 1 ] L. E, Larson, RF and Microwave Circuit Design for Wireless Communications, Artech House, 1996. [2] J. R. Smith, Modern Communication Circuits, 2nd edition, McGraw-Hill, N.Y., 1998. [31 U, L, Rohde, Microwave and Wireless Synthesizers: Theory and Design. Wiley Interscience, N.Y., 1997. [4] I. Bahl and P. Bhartia, Microwave Solid-State Circuit Design, Wiley Interscience, N.Y., 1988. Problems 631 [5] G. D. Vendelin, A. M. Pavio, and U. L. Rohde, Microwave Circuit Design Using Linear and Nonlinear Techniques, Wiley, N.Y., 1990. [6] Y. Komatsu and Y. Murakami, "Coupling Coefficient Between Microstrip Line and Dielectric Resonator," IEEE Trans. Microwave Theory and Techniques, vol, MTT-31, pp. 34—40, January 1983. [7] D. B, Leeson, "A Simple Model of Feedback Oscillator Noise Spectrum,''' Proc. IEEE, vol. 54, pp. 329-330, 1966. [81 A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 2nd edition, Addison- Wesley, Reading, MA, 1994, [9] M. K. Nezami, "Evaluate the Impact of Phase Noise on Receiver Performance," Microwaves & RF Magazine,^. 1-11, June 1998. [10] S. A. Maas, Nonlinear Microwave Circuits, IEEE Press, N.Y, 1997. [11] R. E. Collin, Foundations for Microwave Engineering, 2nd edition, McGraw-Hill, N.Y., 1992. [12] R. H, Pantell, "General Power Relationships for Positive and Negative Resistive Elements," Proc. [RE, pp. 1910-1913, December 1958. [ 13] S. Y. Yngvesson, Microwave Semiconductor Devices, Khrwer Academic Publishers, 1991 -{14] R. A. Pucel, D. Masse, and R. Bera, 'Terformance of GaAs MESFET Mixers at X Band." IEEE Trans. Microwave Theory and Techniques, vol. MTT-24, pp. 351-360, June 1976. PROBLEMS 12.1 Derive the admittance matrix representation of the transistor oscillator circuit given in (12.3). 12.2 Derive the results in (12.20)-(12,22) for a Colpitts oscillator using a common emitter transistor with an inductor having a series resistance R. 12.3 Design a common emitter Colpitts oscillator operating at 30 MHz, using a transistor with 8 =40 and Rj = 800 Q. Select reasonable values for the inductor and the two capacitors. Determine the minimum value of the inductor Q in order to sustain oscillations. 124 A particular quartz crystal operating at 10 MHz has equivalent circuit parameters of R = 30 ň, C = 21 fF, and Co = 5.5 pF (1 rF =s 10"15 F). What is the value of the inductance in the equivalent circuit? What is the Q of this crystal? What is the percentage difference between the series and parallel resonant frequencies? 12J> For either the one-port negative resistance oscillator of Figure 12.6, or the two-port transistor oscillator of Figure 12.8, show that rLrm = 1 for steady-state oscillation. 12.6 Prove that the standard Smith chart can be used for negative resistances by plotting 1/ V* (instead of T). Then the resistance circle values are read as negative, while the reactance circles are unchanged. 12.7 Design a transistor oscillator at 6 GHz using an FET in a common source configuration driving a 50 Í2 load on the drain side. The S parameters are (Z0 = 50 Sí): Sn = 0.9/-.15Q', $u = 2.6z50°, Sn = 0.2/~15°, S22 = ft 105° Calculate and plot the output stability circle, and choose TT for |Tjn| » 1- Design load and terminating networks, 12.8 Repeat the oscillator design of Example 12.4 by replacing the dielectric resonator and microstrip feedline with a single-stub tuner to match ?L to a 50 Í2 load. Find the Q of the tuner and 50 ň load, then compute and plot |r0ll[ | versus A///0. Compare with die result in Figure 12.12b for the dielectric resonator case. 12.9 Repeat the dielectric oscillator design of Example 12.4 using a GaAs FET having the following 5 parameters: SM = 1.2/150*, Sí2 - 0.2ÚM, % = 3.7^72°, % = 1,3 12.10 An oscillator uses an amplifier with a noise figure of 6 dB and a resonator having a Q of 500, and produces a 100 MHz output at a power level of 10 dBm. If the measured fa is 50 kHz, plot the spectra] density of the output noise power, and determine the phase noise (in dBc/Hz) at the following frequencies: (a) at 1 MHz from the carrier; (b) at 10 kHz from the carrier (assume K = I). 12.11 Repeat Problem 12.10 for fB = 200 kHz. 12.12 Derive Equation (12.50) giving the required phase noise for a specified receiver selectivity. 12.13 Find the necessary local oscillator phase noise specification if an 860 MHz cellular receiver with a 30 kHz channel spacing is required to have an adjacent channel rejection of 80 dB, assuming the interfering channel is at the same level as the desired channel. The final IF voice bandwidth is 12 kHz, 632 Chapter 12i Oscillators and Mixers 12.14 Apply the Manley-Rowe relations to an up-converting mixer. Assume a nonlinear reactance is excited at frequencies f{ (RF) and f2 (LO), and terminated with open circuits at all other frequencies except f2 = f \ + f2. Show that the maximum possible conversion gain is given by — P] ]/P]q = 1 + ti>2liF)t + cos(«)££> -i- ojjf)!] is applied to a mixer witb an LO voltage given by (12.99). Derive the output of the mixer after low-pass filtering. 12.17 A diode has an t-Vcharacteristic given by i(t) = l$S™& — 1). Letf(r) = O.lcos&^r +0.] cos^r, and expand i{i) in a power series in t>, retaining only the v, v2, and terms. For /f = 1 A, find the magnitudes of the current at each frequency. 12.18 An RF input signal at 900 MHz is down-converted in a mixer to an IF frequency of SO MHz. What are the two possible LO frequencies, and the corresponding image frequencies? 12.19 Consider a diode mixer with a conversion loss of 5 dB and a noise figure of 4 dB, and a FET mixer with conversion gain of 3 dB and a noise figure of 8 dB. If each of these mixers is followed by an LF amplifier having a gain of 30 dB and a noise figure FA, as shown below, calculate and plot the overall noise figure for both amplifier-mixer configurations for FA = 0 to 10 dB. Diode mixer Lc = 5 dB FJř-4dB Amp G = 30 dB ^ = 0—10 dB Gc = 3 dB Fy = 8 dB G = 30 dB 10 dB FA = 0 12.20 Let Tssb be the equivalent noise temperature of a mixer receiving a SSB signal, and T^se be the temperature when it receives a DSB signal. Compute the output noise powers in each case, and show that Tsss — 27"«®, and that therefore F^b — IFosg. Assume that the conversion gains for the signal and its image are identical. 12.21 If the noise power Nj = kTB is applied at the RF input port of a mixer having noise figure F (DSB) and conversion loss Lc, what is the available output noise power at the IF port? Assume the mixer is at a physical temperature 7Jj. 12.22 A phase detector produces an output signal proportional to the phase difference between two RF input signals. Let these input signals be expressed as V\ — i*o cos tot. v2 = v(, cos((Uf + $). If these two signals are applied to a single-balanced mixer using a 90" hybrid, show that the IF output signal, after low-pass filtering, is given by i = kv^ sin#, where k is a constant. If the miser uses a 180^ hybrid, show that the corresponding output signal is given by I = kv2. cos 9. 12.23 Analyze a balanced mixer using a 180° hybrid junction. Find the output IF current, and the input reflections at the RF and LO ports. Show that this mixer suppresses even harmonics of the LO. Assume that the RF signal is applied to the sum port of the hybrid, and that the LO signal is applied to the difference pott. 12.24 For an image rejection mixer, let the RF hybrid have a dissipative insertion loss of Lg, and the IF hybrid have a dissipative insertion loss of Ls. If the component single-ended mixers each have a conversion loss Lr and noise figure F, derive expressions for the overall conversion loss and noise figure of the image rejection mixer. Chapter Thirteen Introduction to Microwave Systems A microwave system consists of passive and active microwave components arranged to perform a useful function. Probably the two most important examples are microwave radar systems and microwave communication systems, but there are many others. In this chapter we will describe the basic operation of several types of microwave systems to give a general overview of the application of microwave technology, and to show how many of the subjects of earlier chapters fit into the overall scheme of complete microwave systems. An important component in any radar or communication system is the antenna, so we will discuss some of the basic properties of antennas in Section 13,1. Then communication, radar, and radiometry systems are treated as important applications of microwave technology. Propagation effects, biological effects, and other miscellaneous applications are also briefly discussed, All of the above topics are of sufficient depth that many books have been written for each. Thus our purpose here is not to give a complete and thorough treatment of these subjects, but instead to introduce these topics as a way of placing the other material in this book in the larger systems context. The interested reader is referred to the references at the end of the chapter for more complete treatments. SYSTEM ASPECTS OF ANTENNAS In this section we describe some of the basic characteristics of antennas that will be needed for our study of microwave radar, communications, and remote sensing systems. We are interested here not in the detailed electromagnetic theory of antenna operation, but rather the systems aspect of the operation of an antenna in terms of its radiation patterns, directivity, gain, efficiency, and noise characteristics. References [11—[21 can be reviewed for a more in-depth treatment of the fascinating subject of antenna theory and design. Figure 13,1 shows some of the different types of antennas that have been developed for commercial wireless systems. 633 634 Chapter 13: Introduction to Microwave Systems FIGURE 13.1 Photograph of various millimeter wave antennas. Clockwise from top: a high-gain 38 GHz reflector antenna with radome, a prime-focus parabolic antenna, a corrugated conical horn antenna, a 38 GHz planar microstrip array, a pyramidal horn antenna with a Gunn diode module, and a multibeam reflector antenna. Courtesy of H. Syrigos. Alpha Industries, Inc., Wobum, Mass. An antenna can be viewed as a device that converts a guided electromagnetic wave on a transmission line to a plane wave propagating in free space. Thus, one side of an antenna appears as an electrical circuit element, while the other side provides an interface with a propagating plane wave. Antennas are inherendy bi-directional, in that they can be used for both transmit and receive functions. Figure 13.2 illustrates the basic operation of transmit and receive antennas. The transmitter can be modeled as a Thevenin source consisting of a voltage generator and series impedance, delivering a transmit power Pt to the transmit antenna. The transmit antenna radiates a spherical wave which, at large distances, approximates a plane wave, at least over a localized area. The receive antenna intercepts a portion of the propagating wave, and delivers a receive power Pr to the receiver load impedance. A wide variety of antennas have been developed for different applications, as summarized in the following categories: • Wire antennas include dipoles, monopoles, loops, sleeve dipoles, Yagi-Uda arrays, and related structures. Wire antennas generally have low gains, and are most often 13.1 System Aspects of Antennas 635 Spherical Plane wave we :} Transmit antenna Receive antenna FIGURE 13 J Basic operation of transmit and receive antennas. used at lower frequencies (HF to UHF). They have the advantages of light weight, low cost, and simple design. • Aperture antennas include open-ended waveguides, rectangular or circular homs, reflectors, and lenses. Aperture antennas are most commonly used at microwave and millimeter wave frequencies, and have moderate to high gains. • Printed antennas include printed slots, printed dipoles, and micros trip patch antennas. These antennas can be made with photolithographic methods, with both radiating elements and associated feed circuitry fabricated on dielectric substrates. Printed antennas are most often used at microwave and millimeter wave frequencies, and can be easily arrayed for high gain. • Array antennas consist of a regular arrangement of antenna elements with a feed network. Pattern characteristics such as beam pointing angle and sidelobe levels can be controlled by adjusting the amplitude and phase distribution of the array elements. An important type of array antenna is the phased array, where variable phase shifters are used to electronically scan the main beam of the antenna. Fields and Power Radiated by an Antenna While we do not require detailed solutions to Maxwell's equations for our purposes, we do need to be familiar with the far-zone electromagnetic fields radiated by an antenna. We consider an antenna located at the origin of a spherical coordinate system. At large distances, where the localized near-zone fields are negligible, the radiated electric field of an arbitrary antenna can be expressed as E (r, 0, ) = \dFe{0,4>) + 0/>(£>, #1.—— V/m, (13.1) where E is the electric field vector, § and 4> are unit vectors in the spherical coordinate system, r is the radial distance from the origin, and k$ = Itz/X is the free-space propagation constant, with wavelength k = c/f. Also defined in (13.1) are the pattern functions, F$(&,). The interpretation of (13.1) is that this electric field propagates in the radial direction, with a phase variation of e~^r and an amplitude variation of \ jr. The electric field may be polarized in either the § or ^ directions, but not in the radial direction, since this is a TEM wave. The magnetic fields associated with the electric field of (13.1) can be found from (1.76) as (13.2a) (13.2b) 636 Chapter 13: Introduction to Microwave Systems where 170 = 377 ft, the wave impedance of free-space. Note that the magnetic field vector is also polarized only in the transverse directions. The Poynting vector for this wave is given by (1.90) as: S = E x H* W/m2, (13.3) and the time-average Poynting vector is S^g = ±Re{S} = *Re{£ x /?♦} W/m2. (13.4) We mentioned earlier that at large distances the near fields of an antenna are negligible, and the radiated electric field can be written as in (13.1). We can give a more precise meaning to this concept by defining the far-field distance as the distance where the spherical wave front radiated by an antenna becomes a close approximation to the ideal planar phase front of a plane wave. This approximation applies over the aperture area of the antenna, and so depends on the maximum dimension of the antenna. If we call this dimension D, then the far-field distance is defined as 2D2 Rir = —— m. (13.5) This result is derived from die condition that the actual spherical wave front radiated by the antenna departs less than jt/8 = 22.5° from a true plane wave front over the maximum extent of the antenna. For electrically small antennas, such as short dipoles and small loops, this result may give a far-field distance that is too small; in this case, a minimum value of Rjf=:2\ should be used. EXAMPLE 13.1 FAR-FIELD DISTANCE OF AN ANTENNA A parabolic reflector antenna used for reception with the Direct Broadcast System (DBS) is 18" in diameter, and operates at 12.4 GHz, Find the operating wavelength, and the far-field distance for this antenna. Solution The operating wavelength at 12.4 GHz is , c 3 x 10s k=f = l2A^W = 2A2cm- The far-field distance is found from (13.5), after converting 18" to 0.457 m: 2D2 2(0.457)? R,t = --= „ - = 17.3 m. * k 0.0242 The actual distance from a DBS satellite to earth is about 36,000 km, so it is safe to say that the receive antenna is in the far-field of the transmitter. ■ Next we define the radiation intensity of the radiated electromagnetic field as, 2 U($, |2] W, 2no 2*io where (13.1), (13.2), and (13.4) were used. The units of the radiation intensity are watts, or watts per unit solid angle, since the radial dependence has been removed. The radiation 13.1 System Aspects of Antennas 637 intensity gives the variation in radiated power versus position around the antenna. We can find the total power radiated by the antenna by integrating the Poynting vector over the surface of a sphere of radius r that encloses the antenna. This is equivalent to integrating the radiation intensity over a unit sphere: 2jt it 2x \d= j j Stiv3 ■ rr2sm0ded(f> = J J U(8,=0$=0 ^=0 0=0 Antenna Pattern Characteristics The radiation pattern of an antenna is a plot of the magnitude of the far-zone field strength versus position around the antenna, at a fixed distance from the antenna. Thus the radiation pattern can be plotted from the pattern functions F#(8, )uwoded 0=0 0=0 where (13,7) has been used for the radiated power. Directivity is a dimensionless ratio of power, and is usually expressed in dB as £>(dB) = 10 log (D). An antenna that radiates equally in all directions is called an isotropic antenna. Applying the integral identity that In sin 0d$d4> = 4jt 7! in to the denominator of (13.8) for U(9,$)= 1 shows that the directivity of an isotropic element is D = 1, or 0 dB. Since the minimum directivity of any antenna is unity, directivity is sometimes stated as relative to the directivity of an isotropic radiator, and written as dBi. Directivities of some typical antennas are 2.2 dB for a wire dipole, 7.0 dB for a microstrip patch antenna, 23 dB for a waveguide horn antenna, and 35 dB for a parabolic reflector antenna. BeamwidUi and directivity are both measures of the focusing ability of an antenna: an antenna pattern with a narrow main beam will have a high directivity, while a pattern with a wide beam will have a lower directivity. We might therefore expect a direct relation between beamwidth and directivity, but in fact there is not an exact relationship between these two quantities, This is because beamwidth is only dependent on the size and shape of the main beam, whereas directivity involves integration of the entire radiation pattern. Thus it is possible for many different antenna patterns to have the same beamwidth, but quite different directivities due to differences in sidelobes or the presence of more than one main beam. With this qualification in mind, however, it is possible to develop approximate relations between beamwidth and directivity that apply with reasonable accuracy to a large number of practical antennas. One such approximation that works well for antennas with pencil beam patterns is the following: 32,400 D=—-. (13.9) where 0\ and $i are the beamwidths in two orthogonal planes of the main beam, in degrees. This approximation does not work well for omnidirectional patterns because there is a well-defined main beam in only one plane for such patterns. 13.1 System Aspects of Antennas 639 EXAMPLE 13.2 PATTERN CHARACTERISTICS OF A DIPOLE ANTENNA The far-zone electric field radiated by an electrically small wire dipole antenna on the z-axis is given by Ee(r, $, )smedQd

=ú 0=0 where the required integral identity is listed in Appendix D. Since = C, the directivity reduces to 3 D = - = 1.76dB. ■ 2 Antenna Gain and Efficiency Resistive losses, due to non-perfect metals and dielectric materials, exist in all antennas. Such losses result in a difference between the power delivered to the input of an antenna and the power radiated by that antenna. As with other electrical components, we can define the radiation efficiency of an antenna as the ratio of the desired output power to the supplied input power: flw fld floss , floss >,~ . A, = — =-p-" = 1 - ~* (13.10) where Pr(ui is the power radiated by the antenna, fln is the power supplied to the input of the antenna, and P[oss is the power lost in the antenna. Note that there are other factors that can contribute to the effective loss of transmit power, such as impedance mismatch at the input to the antenna, or polarization mismatch with the receive antenna. But these losses are external to the antenna, and could be eliminated by the proper use of matching networks, or the proper choice and positioning of the receive antenna. Therefore losses of 640 Chapter 13: Introduction to Microwave Systems this type should not be attributed to the antenna itself, as are dissipative losses due to metal conductivity or dielectric loss within the antenna-Recall that antenna directivity is a function only of the shape of the radiation partem (the radiated fields) of an antenna, and is not affected by losses in the antenna. In order to account for the fact that an antenna with radiation efficiency less than unity will not radiate all of its input power, we define antenna gain as the product of directivity and efficiency: G-n^D. (13.11) Thus, gain is always less than or equal to directivity. Gain can also be computed directly, by replacing Prad in the denominator of (13.8) with Fj„, since by the definition of radiation efficiency in (13,10) we have Pm)s'm0ded(f> =0 9=0 where TBiO, )&m0d8(t4> f sm0d0 + / 0.01 sin _ -lOcM0|Jf - 0.1 cos0|ff - cosflgff _ 0.00152 + 0,0134 + 0.866 _ -cos 0fj*-0.01 cos0|*? ~ 0.0102 ~~ In this example most of the noise power is collected dirough the sidelobe region of the antenna. ■ 13.1 System Aspects of Antennas 645 Background temperature Te<6,4>) Antenna Lossy line Tp,T Receiver FIGURE 13,8 A receiving antenna connected to a receiver through a lossy transmission line. An impedance mismatch exists between the antenna and the line. The more general problem of a receiver with a lossy transmission line and an antenna viewing a background noise temperature distribution Tjs can be represented by the system shown in Figure 13.8. The antenna is assumed to have a radiation efficiency nrad, and the line a power loss factor of L > 1, with both at physical temperature Tp, We also include the effect of an impedance mismatch between the antenna and the transmission line, represented by the reflection coefficient T, The equivalent noise temperature seen at the output terminals of the transmission line consists of three contributions: noise power from the antenna due to internal noise and the background brightness temperature, noise power generated from the lossy line in the forward direction, and noise power generated by the lossy line in the backward direction and reflected from the antenna mismatch toward the receiver. The noise due to the antenna is given by (13.18), but reduced by the loss factor of the line, 1/L, and the reflection mismatch factor, (1 — |r|2). The forward noise power from the lossy line is given by (10,15), after reduction by the loss factor, 1/L. The contribution from the lossy line reflected from die mismatched antenna is given by (10.15), after reduction by the power reflection coefficient |T j2 and the loss factor, 1/L2 (since the reference point for the back-directed noise power from the lossy line given by (10.15) is at the output terminals of the line). Thus the overall system noise temperature seen at the input to the receiver is given by Ts = -fo - \r\2) + (L - rrS + (L - i)^|r|2 = ° ~p\«tTb + (1 - w)Tp] + [l + Tp. (13.19) Observe that for a lossless line (L = 1) the effect of an antenna mismatch is to reduce the system noise temperature by the factor (1 - |T|2). Of course, the received signal power will be reduced by the same amount. Also note that for the case of a matched antenna (T = 0), (13,19)reduces to ^=^[^ + (1-^] + ^^, (13.20) as expected for a cascade of two noisy components. Finally, it is important to realize the difference between radiation efficiency and aperture efficiency, and their effects on antenna noise temperature. While radiation efficiency accounts for resistive losses, and thus involves the generation of thermal noise, aperture efficiency does not. Aperture efficiency applies to the loss of directivity in aperture antennas, such as reflectors, lenses, or horns, due to feed spillover or suboptimum aperture excitation, 646 Chapter 13: Introduction to Microwave Systems and by itself does not lead to any additional effect on noise temperature that would not be included through the pattern of the antenna. The antenna noise temperature defined above is a useful figure of merit for a receive antenna because it characterizes the total noise power delivered by the antenna to the input of a receiver. Another useful figure of merit for receive antennas is the G/T ratio, defined as G/T(dB) = iQIog -=r dB/K, (13.21) where G is the gain of the antenna, and TA is the antenna noise temperature. This quantity is important because, as we will see in Section 13.2, the signal-to-noise ratio at the input to a receiver is proportional to GiTA. GIT can often be maximized by increasing the gain of the antenna, since this increases the numerator and usually minimises reception of noise from hot sources at low elevation angles. Of course, higher gain requires a larger and more expensive antenna, and high gain may not be desirable for applications requiring omnidirectional coverage (e.g., cellular telephones or mobile data networks), so often a compromise must be made. Finally, note that the dimensions given in (13.21) for 10]og(G/T) are not actually decibels per degree Kelvin, but this is the nomenclature that is commonly used for this quantity. WIRELESS COMMUNICATION SYSTEMS Wireless communications involves the transfer of information between two points without direct connection. While this may be accomplished using sound, infrared, optical, or radio frequency energy, most modern wireless systems rely on RF or microwave signals, usually in the UHF to millimeter wave frequency range. Because of spectrum crowding, and the need for higher data rates, the trend is to higher frequencies, so that the majority of wireless systems today operate at frequencies ranging from about 800 MHz to a few gigahertz. RF and microwave signals offer wide band widths, and have the added advantage of being able to penetrate fog, dust, foliage, and even buildings and vehicles to some extent. Historically, wireless communication using RF energy began with the theoretical work of Maxwell, followed by the experimental verification by Hertz of electromagnetic wave propagation, and the commercial development of practical radio systems by Marconi in the early part of the 20th century. Today, wireless systems include broadcast radio and television, cellular telephone systems, Direct Broadcast Satellite (DBS) television service, Wireless Local Area Networks (WLANs), paging systems, Global Positioning Satellite (GPS) service, and Radio Frequency Identification (RF1D) systems. These systems promise to provide, for the first time in history, worldwide connectivity for voice, video, and data communications. One way to categorize wireless systems is according to the nature and placement of the users. In a point-to-point radio system a single transmitter communicates with a single receiver. Such systems generally use high-gain antennas in fixed positions to maximize received power and minimize interference with other radios that may be operating nearby in the same frequency range. Point-to-point radios are generally used for dedicated data communications by utility companies and for connection of cellular phone sites to a central switching office. Point-to-multipoint systems connect a central station to a large number of possible receivers. The most common examples are commercial AM and FM radio and broadcast television, where a central transmitter uses an antenna with a broad beam to reach many listeners and viewers. MultipoinMo-multipoint systems allow simultaneous communication between individual users (who may not be in fixed locations). Such systems generally do not connect two users directly, but instead rely on a grid of base stations to provide the desired interconnections between users. Cellular telephone systems and some types of wireless local area networks (WLANs) are examples of this type of application. 13.2 Wireless Communication Systems 647 Another way to characterize wireless systems is in terms of the directionality of communication. In a simplex system, communication occurs only in one direction, from the transmitter to the receiver. Examples of simplex systems include broadcast radio, television, and paging systems. In a half-duplex system, communication may occur in two directions, but not simultaneously. Early mobile radios and citizens band radio are examples of duplex systems, and generally rely on a "push-to-talk" function so that a single channel can be used for both transmitting and receiving at different intervals. Full-duplex systems allow simultaneous two-way transmission and reception. Examples include cellular telephone and point-to-point radio systems. Full-duplex transmission clearly requires a duplexing technique to avoid interference between transmitted and received signals. This can be done by using separate frequency bands for transmit and receive {frequency division duplexing), or by allowing users to transmit and receive only in certain predefined time intervals {time division duplexing). While most wireless systems are ground based, there is also interest in the use of satellite systems for voice, video, and data communications. Satellite systems offer the possibility of communication with a large number of users over wide areas, perhaps including the entire planet. Satellites in a geosynchronous earth orbit (GEO) are positioned approximately 36,000 km above the Earth, and remain in a fixed position relative to the surface. Such satellites are useful for point-to-point radio links between widely separated stations, and are commonly used for television and data communications throughout the world At one time transcontinental telephone service relied on such satellites, but undersea fiber optics cables have largely replaced satellites for transoceanic connections as being more economical, and avoiding the annoying delay caused by the very long round trip path between the satellite and the Earth. Another drawback of GEO satellites is that their high altitude greatly reduces the received signal strength, making it impractical for two-way communication with very small transceivers. Low earth orbit (LEO) satellites orbit much closer to the Earth, typically in the range of 500 to 2000 km. The shorter path length allows communication between LEO satellites and handheld radios, but satellites in LEO orbits are visible from a given point on the ground for only a short time, typically between a few minutes to about 20 minutes. Effective coverage therefore requires a large number of satellites in different orbital planes. The Flips Formula A general radio system link is shown in Figure 13.9, where the transmit power is pt, the transmit antenna gain is gt, the receive antenna gain is Gr, and the received power (deli vered to a matched load) is pr. The transmit and receive antennas are separated by the distance r, From (13.6)-( 13,7), the power density radiated by an isotropic antenna (d = 1 = 0 dB) at a distance r is given by = \TrJ Wml (13'22) This result reflects the fact that we must be able to recover all of the radiated power by integrating over a sphere of radius r surrounding the antenna; since the power is distributed 648 Chapter 13: Introduction to Microwave Systems isotropically, and the area of a sphere is 4xR2, (13.22) follows. If the transmit antenna has a directivity greater than 0 dB, we can find the radiated power density by multiplying by the directivity, since directivity is defined as the ratio of the actual radiation intensity to the equivalent isotropic radiation intensity. Also, if the transmit antenna has losses, we can include the radiation efficiency factor, which has the effect of converting directivity to gain. Thus, the general expression for the power density radiated by an arbitrary transmit antenna is G,P, AnR2 S^g = W/m2. (13.23) If this power density is incident on the receive antenna, we can use the concept of effective aperture area, as defined in (13.14), to find the received power: Next, (13.15) can be used to relate the effective area to the directivity of the receive antenna. Again, the possibility of losses in the receive antenna can be accounted for by using the gain (rather than the directivity) of the receive antenna. So the final result for the received power is (4tt R)2 This result is known as the Friis radio link formula, and it addreses the fundamental question of how much power is received by a radio antenna. In practice, the value given by (13.24) should be interpreted as the maximum possible received power, as there are a number of factors that can serve to reduce the received power in an actual radio system. These include impedance mismatch at either antenna, polarization mismatch between the antennas, propagation effects leading to attenuation or depolarization, and multlpath effects that may cause partial cancellation of the received field. Observe in (13.24) that the received power decreases as 1 /R2 as the separation between transmitter and receiver increases. This dependence is a result of conservation of energy, While it may seem to be prohibitively large for large distances, in fact the space decay of \/R2 is much belter than the exponential decrease in power due to losses in a wired communications link. This is because the attenuation of power on a transmission line varies as e_laz (where ot is the attenuation constant of the line), and at large distances the exponential function decreases faster than an algebraic dependence like i/R2. Thus for long distance communications, radio links will perform better than wired links. This conclusion applies to any type of transmission line, including coaxial lines, waveguides, and even fiber optic lines. (It may not apply, however, if the communications link is land or sea-based, so that repeaters can be inserted along the link to recover lost signal power.) As can be seen from the Friis formula, received power is proportional to the product P,GS. These two factors, the transmit power and transmit antenna gain, characterize the transmitter, and in the main beam of the antenna the product PtGt can be interpreted equivalently as the power radiated by an isotropic antenna with input power PtGt. Thus, this product is defined as the effective Isotropic radiated power (EIRP): EIRP= PtG, W. (13.25) For a given frequency, range, and receiver antenna gain, the received power is proportional to the EIRP of the transmitter, and can only be increased by increasing the EIRP. This can be done by increasing the transmit power, or the transmit antenna gain, or both. The derivation of the Friis formula given above assumed that the transmit and receive antennas were impedence matched to the transmitter and receiver, respectively. As with 13.2 Wireless Communication Systems 649 any RF or microwave system, impedance mismatch will reduce the power delivered from a source to a load by the factor (1 — |T|2), where r is the reflection coefficient between the source and the load. In a radio link there is the possibility of an impedance mismatch between the transmitter and the transmit antenna, as well as between the receive antenna and the receiver. Thus the Friis formula of (13.24) can be multiplied by the impedance mismatch factor, nimp, defined as to account for the reduction in received power due to impedance mismatch effects at the transmitter and receiver. In (13.26) T, is the reflection coefficient at the transmitter, and F% is the reflection coefficient at the receiver. Note that impedance mismatch is not included in me definition of antenna gain. This is because mismatch is dependent on the external source or load impedances to which the antenna is connected, and thus is not a property of the antenna itself. It is always possible to match an antenna to a given source or load by using an appropriate external tuning network. A final consideration to note in connection with the Friis link formula is that maximum transmission between transmitter and receiver requires that both antennas be polarized in the same direction. If a transmit antenna is vertically polarized, for example, maximum power will be delivered to a vertically polarized receiving antenna, while zero power would be delivered to a horizontally polarized receive antenna. Polarization matching of antennas is therefore critical for optimum communications system performance. The Direct Broadcast System (DBS) operates at 12.2-12.7 GHz, with a transmit earner power of 120 W, a transmit antenna gain of 34 dB, an IF bandwidth of 20 MHz, and a worst-case slant angle (30°) distance from the geosynchronous satellite to earth of 39,000 km. The 18'' receiving dish antenna has a gain of 33.5 dB and sees an average background brightness temperature of T& = 50 K., with a receiver low-noise block (LNB) having a noise figure of I. I dB. The overall system is shown in Figure 13.10. Find (a) the EIRP of the transmitter, (b) GIT for the receive antenna and LNB system, (c) the received carrier power at the receive antenna terminals, and (d) the carrier-to-noise ratio (CNR) at the output of the LNB. Solution First we convert quantities in dB to numerical values; ^ = (iHr,l2)0-|r,|2), (13.26) EXAMPLE 13.4 ANALYSIS OF DBS SYSTEM 34 dB = 2512 LI dB = 1.29 33.5 dB = 2239 DBS receiver FIGURE 13.10 Diagram of the DBS system for Example 13,4, 650 Chapter 13: Introduction to Microwave Systems We will take the operating frequency to be 12,45 GHz, so the wavelength is 0.0241 m. (a) The EIRP of the transmitter is found from (13.25): EIRP = P,G, = (120)(2512) - 3.01 x 105 W = 54.8 dBm. (b) To find GIT we first find the noise temperature of the antenna and LNB cascade, referenced at the input of the LNB: Te = TA + Tm = Tb + (F- l)7b = 50+ (1.29 - 1)(290) = 134 K. Then GIT for the antenna and LNB is 2239 G/T(dB) = 10log = 12.2 dB/K. (c) The received carrier power is found from the Friis formula of (13.24); PtGtGrK2 _ (3.01 x 105)(2239)(0.0241)2 ' ~ (4wR)2 ~ (4jr)3(3.9 x 107)2 = 1.63 x 10"12 W = -117.9 dBW. (d) Then the CNR at the output of the LNB is PrGrMB L63 x 10"J2 CNR —_ —_—_= 44 i — 16 4 dB kTgBGuiB (1-38 x 10"23)(134)(20 x 106) Note that G^b, the gain of the LNB module, cancels in the ratio for the output CNR. A CNR of 16 dB is adequate for good video quality with the error-corrected digital modulation used in the DBS system. ■ Radio Receiver Architectures The receiver is usually the most critical component of a wireless system, having the overall purpose of reliably recovering the desired signal from a wide spectrum of txansmitung sources, interference, and noise. In this section we will describe some of the critical requirements for radio receiver design, and summarize some of the most common types of receiver architectures. The well-designed radio receiver must provide several different functions: • high gain (—100 dB) to restore the low power of the received signal to a level near its original baseband value. • selectivity, in order to receive the desired signal while rejecting adjacent channels, image frequencies, and interference. • down-conversion from the received RF frequency to an IF frequency for processing. • detection of the received analog or digital information. • isolation from the transmitter to avoid saturation of the receiver. Because the typical signal power level from the receive antenna may be as low as — 100 to — 120 dBm, the receiver may be required to provide gain as high as 100 to 120 dB. This much gain should be spread over the RF, IF, and baseband stages to avoid instabilities and possible oscillation; it is generally good practice to avoid more than about 50-60 dB of gain at any one frequency band. The fact that amplifier cost generally increases with frequency is a further reason to spread gain over different frequency stages. In principle, selectivity can be obtained by using a narrow bandpass filter at the RF stage of the receiver, but the bandwidth and cutoff requirements for such a filter are usually 13.2 Wireless Communication Systems 6S1 Filler tuning FIGURE 13,11 Block diagram of a tuned radio frequency receiver. impractical to realize at RF frequencies. It is more effective to achieve selectivity by down-converting a relatively wide RF bandwidth around the desired signal, and using a sharp-cutoff bandpass filter at the IF stage to select only the desired frequency band. In addition, many wireless systems use a number of narrow but closely spaced channels which must be selected using a tuned local oscillator, while the IF passband is fixed. The alternative of using an extremely narrow band electronically tunable RF filter is not practical. Tuned radio frequency receiver. One of the earliest types of receiving circuits to be developed was the tuned radio frequency (TRF) receiver. As shown in Figure 13.11, a TRF receiver employs several stages of RF amplification along with tunable bandpass filters to provide high gain and selectivity. Alternatively, filtering and amplification may be combined by using amplifiers with a tunable bandpass response. At relatively low broadcast radio frequencies, such filters and amplifiers have historically been tuned using mechanically variable capacitors or inductors. But such tuning is very difficult because of the need to tune several stages in parallel, and selectivity is poor because the passband of such filters is fairly broad. In addition, all the gain of the TRF receiver is achieved at the RF frequency, limiting the amount of gain that can be obtained before oscillation occurs, and increasing the cost and complexity of the receiver. Because of these drawbacks TRF receivers are seldom used today, and are an especially bad choice for higher RF or microwave frequencies. Direct conversion receiver. The direct conversion receiver, shown in Figure 13.12, uses a mixer and local oscillator to perform frequency down-conversion with a zero IF frequency. The local oscillator is set to the same frequency as the desired RF signal, which is then converted directly to baseband. For this reason, the direct conversion receiver is sometimes called a homodyne receiver. For AM reception the received baseband signal would not require any further detection. The direct conversion receiver offers several advantages over the TRF receiver, as selectivity can be controlled with a simple low-pass baseband filter, and gain may be spread through the RF and baseband stages (although it is difficult to obtain stable high gain at very low frequencies). Direct conversion receivers are simpler and less costly than superheterodyne receivers, since there is no IF amplifier, IF bandpass filter, or IF local oscillator required for final down conversion. Another important advantage of direct conversion is that there is no image frequency, since the mixer difference frequency Demod FIGURE 13.12 Block diagram of a direct-conversion receiver. 652 Chapter 13; Introduction to Microwave Systems V RF amP Miser BpF IF apip fa FIGURE 13.13 Block diagram of a single-conversion superheterodyne receiver. is effectively zero, arid the sum frequency is twice the LO and easily filtered. But a serious disadvantage is that the LO must have a very high degree of precision and stability, especially for high RF frequencies, to avoid drift of the received signal frequency This type of receiver is often used with Doppler radars, where the exact LO can be obtained from the transmitter, but a number of newer wireless systems are being designed with direct conversion receivers. Superheterodyne receiver. By far the most popular type of receiver used today is the superheterodyne circuit, shown in Figure 13.13. The block diagram is similar to the direct conversion receiver, but the IF frequency is now nonzero, and generally selected to be between the RF frequency and baseband. A midrange IF allows the use of sharper cutoff filters for improved selectivity, and higher IF gain through the use of an IF amplifier. Tuning is conveniently accomplished by varying the frequency of the local oscillator so that the IF frequency remains constant. The superheterodyne receiver represents the culmination of over 50 years of receiver development, and is used in the majority of broadcast radios and televisions, radar systems, cellular telephone systems, and data communications systems, At microwave and millimeter wave frequencies it is often necessary to use two stages of down conversion to avoid problems due to LO stability. Such a dual-conversion superheterodyne receiver employs two local oscillators, two mixers, and two IF frequencies to achieve down-conversion to baseband. Noise Characterization of a Microwave Receiver Let us now analyze the noise characteristics of a complete antenna-transmission line-receiver front end, as shown in Figure 13,14. In this system the total noise power at the output of the receiver, N0, will be due to contributions from the antenna pattern, the loss in the antenna, the loss in the transmission line, and from the receiver components. This noise power will determine the minimum detectable signal level for the receiver and, for a given transmitter power, the maximum range of the communication link. Background Antenna Receiver FIGURE 13.14 Noise analysis of a microwave receiver front end, including antenna and transmission line contributions. 13.2 Wireless Communication Systems 653 The receiver components in Figure 13.14 consist of an RF amplifier with gain Grf and noise temperature Trf, a mixer with an RF-to-D? conversion loss factor Lm and noise temperature 7V, and an IF amplifier with gain Gjp and noise temperature Tn?. The noise effects of later stages can be ignored, since the overall noise figure is dominated by the characteristics of the first few stages. The component noise temperatures can be related to noise figures as T = (F — l)7b. From (10.22) the equivalent noise temperature of the receiver can be found as tw-.f^^ + SskL. 03.27) The transmission line connecting the antenna to the receiver has a loss Lj, and is at a physical temperature Tp. So from (10.15) its equivalent noise temperature is Ttl = (Lr — \)TP. (13.28) Again using (10.22), the noise temperature of the transmission line (TL) and receiver cascade is Ttl+rec — Ttl + ^tTrec = (Lr - l)Tp + LtTkec. (13-29) This noise temperature is defined at the antenna terminals (the input to the transmission line). As discussed in Section 13.1, the entire antenna pattern can collect noise power. If the antenna has a reasonably high gain with relatively low sidelobes, we can assume that all noise power comes via the main beam, so that the noise temperature of the antenna is given by (13,18): TA = rjr^n + (1 - W)7>, (13.30) where nm(i is the efficiency of the antenna, Tp is its physical temperature, and 7^ is the equivalent brightness temperature of the background seen by the main beam. (One must be careful with this approximation, as it is quite possible for the noise power collected by the sidelobes to exceed the noise power collected by the main beam, if the sidelobes are aimed at a hot background.) The noise power at the antenna terminals, which is also the noise power delivered to the transmission line, is = kBTA = kBiij^n + (1 - w)7>l, (13.31) where B is the system bandwidth. If 5, is the received power at the antenna terminals, then the input signal-to-noise ratio at the antenna terminals is S,/Af,. The output signal power is ^ = ^^=SiGs¥S, (13.32) where Gsys has been defined as a system power gain. The output noise power is N0 = [Ni + kBTTL+wsc]G&vs = kB(TA + TYl^recK'sys = kBln^Tt + (1 - j]rad)Tp + (LT- 1)7> + LtTrecIGsys = AtfTsYsGsYs, (13.33) 654 Chapter 13: Introduction to Microwave Systems where Tsys has been defined as the overall system noise temperature. The output signal-to-noise ratio is Sa Si Si A'o kBTsYS kB[rjmdTb + (1 - n^)TP + (LT - l)Tp + Lt-Trbc] ' (13.34) It may be possible to improve this signal-to-noise ratio by various signal processing techniques. Note that it may appear to be convenient to use an overall system noise figure to calculate the degradation in signal-to-noise ratio from input to output for the above system, but one must be very careful with such an approach because noise figure is defined for N; = kToB7 which is not the case here. It is often less confusing to work directly with noise temperatures and powers, as we did above. EXAMPLE 13.5 SIGNAL-TO-NOISE RATIO OF A MICROWAVE RECEIVER A microwave receiver like that of Figure 13.14 has the following parameters: / = 4.0 GHz, Grf = 20 dB, B = 1 MHz, Frf = 3.0 dB, GA = 26 dB, = 6.0 dB, Vrad = 0.90, FM = 7.0 dB, TP = 300 K, = 30 dB, n = 200K, = 1.1 dB. = 1.5 dB, If the received power at the antenna terminals is 5, = -80 dBm, calculate the input and output signal-to-noise ratios. Solution We first convert the above dB quantities to numerical values, and noise figures to noise temperatures: Grf = 102R'10 = 100. = 1030'10 = 1000. LT = 101J/|0 = 1.41, Em = 106/10 = 4.0, Tm = {FM - 1)T0 = (l07/m - l)(290) = 1163 K, ?rf = (FRP-1)71}=(103/10- 1)(290) = 289 K, TW = l)7b = (I0u'lfl- -l)(290)= = 84K. Then from (13.27), (13.28), and (13.30) the noise temperatures of the receiver, transmission line, and antenna are T - T ± & * TxfLm - 289+ 1163 + SSS - 304 K Ttl = (Lt - = (1.41 - 1)300 = 123 K, TA = thM + (1 - VrvMp = 0.9(200) + (1 - 0.9)(300) = 210 K. Then the input noise power, from (13.31), is Ni = kBTA - 1.38 x 10^23(106)(210) = 2.9 x 10"15 W = -115 dBm. 13.2 Wireless Communication Systems 655 So the input signal-to-noise ratio is, & = -80+ 115 = 35 dB. From (13.33) the total system noise temperature is Tsvs = TA + 7Yl + Lt-Trec = 210+123 + (1.41)(304) = 762 K. This result clearly shows the noise contributions of the various components. The output signal-to-noise ratio is found from (13.34) as S0 Si A/?7sys = 1.38 x 10_23(106)(762) = 1.05 x 10_U W = -HOdBm. so — = -80+ 110 = 30 dB. ■ Wireless Systems Finally, we conclude this section with short descriptions of some of the wireless systems in current use. Table 13.1 lists some of the commonly used frequency bands for wireless systems. Cellular telephone systems. Cellular telephone systems were proposed in the 1970s in response to the problem of providing mobile radio service to a large number of users in urban areas. Early mobile radio systems could handle only a very limited number of users due to inefficient use of the radio spectrum and interference between users. The cellular TABLE 13.1 Wireless System Frequencies (T/R — mobile unit transmit/ receive frequency) Wireless System Operating Frequency Advanced Mobile Phone Service {US AMPS) T: 869-894 MHz R: 824-849 MHz Global System Mobile (European GSM) T: 880-915 MHz R: 925-960 MHz Personal Communications Services (PCS) T: 1710-1785 MHz R: 1805-1880 MHz US Paying 931-932 MHz Global Positioning Satellite (GPS) LI: 1575.42 MHz L2: 1227.60 MHz Direct Broadcast Satellite (DBS) 11.7-12.5 GHz Wireless Local Area Networks (WLANs) 902-928 MHz 2.400-2.484 GHz 5.725-5.850 GHz Locai Multipoint Distribution Service (LMDS) 28 GHz US Industrial, Medical, and Scientific bands (ISM) 902-928 MHz 2,400-2.484 GHz 5.725-5.850 GHz 656 Chapter 13: Introduction to Microwave Systems radio concept soives this problem by dividing a geographical area into non-overlapping cells, where each cell has its own transmitter and receiver (base station) to communicate with the mobile users operating in that cell. Each cell site may allow as many as several hundred users to simultaneously communicate with other mobile users, or through the land-based telephone system. Frequency bands assigned to a particular cell can be reused in other non-adjacent cells, The first cellular telephone systems were built in Japan and Europe in 1979 and 1981, and in the US (the Advanced Mobile Phone System (AMPS)) in 1983, These systems used analog FM modulation and divided their allocated frequency bands into several hundred channels, each of which could support an individual telephone conversation. These early services grew slowly at first, because of the initial costs of developing an infrastructure of base stations and the initial expense of handsets, but by the 1990s growth became phenomenal. Because of the rapidly growing consumer demand for wireless telephone service, as well as advances in wireless technology, several second generation standards have been implemented in the US, Europe, and Asia. These standards all employ digital modulation methods and provide better quality service and more efficient use of the radio spectrum than analog systems. Systems in the US use either the IS-136 time division multiple access (TDMA) standard, the 1S-95 code division multiple access (CDMA) standard, or the European Global System Mobile (GSM) system. Many of the new personal communications systems (PCS) in the US have been deployed using the same frequency bands as the AMPS system to take advantage of existing infrastructure. Additional spectrum has also been allocated by the Federal Communications Commission (FCC) around 1.8 GHz, and some of the newer PCS systems use this frequency band. Outside the US, the Global System Mobile (GSM) TDMA system is the most widespread, being used in over 100 countries. The uniformity of a single wireless telephone standard throughout Europe and much of Asia allows travelers to use a single handset throughout these regions. Satellite systems for wireless voice and data. The key advantage of satellite systems is that a relatively small number of satellites can provide coverage to users at any location, including the oceans, deserts, and mountains—areas for which it would otherwise be difficult to provide service. In principle, as few as three geosynchronous satellites can provide complete global coverage, but the very high altitude of the geosynchronous orbit makes it difficult to communicate with hand-held terminals because of very low signal strength. Satellites in lower orbits can provide usable levels of signal power, but many more satellites are then needed to provide global coverage. There are a large number of commercial satellite systems either currently in use, or in the development stage, for wireless communications. These systems generally operate at frequencies above 1 GHz because of available spectrum, the possibility of high data rates, and the fact that such frequencies easily pass through the atmosphere and ionosphere, GEO satellite systems, such as INMARSAT and MS AT, provide voice and low-data rate communications to users with 12" to 18" antennas. These systems are often referred to as very small aperture terminals (VSATs). Other satellite systems operate in medium- or low-earth orbits, to provide mobile telephone and data service to users on a worldwide basis. The Iridium project, financed by a consortium of companies headed by Motorola, was the first commercial satellite system to offer handheld wireless telephone service. It consisted of 66 LEO satellites in near-polar orbits, and connected mobile phone and paging subscribers to the public telephone system through a series of inter-satellite relay links and land-based gateway terminals. Figure 13.15 shows a photo of one of the Iridium phased array antennas. The Iridium system cost was approximately $3.4B, and it began service in 1998. One drawback of using satellites for telephone service is that weak signal levels require a line-of-sight path from the mobile user to the satellite, meaning mat satellite 13.2 Wireless Communication Systems 667 FIGURE 13.15 Photograph of one of the three L-band antenna arrays for a Motorola IRIDIUM communications satellite. The IRIDIUM system consists of 66 satellites in low earth orbit to provide global personal satellite TDMA communications services, including voice, fax, and paging. Courtesy of Raytheon Company. telephones generally cannot be used in buildings, automobiles, or even in many wooded or urban areas. This places satellite phone service at a definite performance disadvantage relative to land-based cellular and PCS wireless phone service. But an even greater problem with satellite phone service is the expense of deploying and maintaining a large fleet of LEO satellites, making it very difficult to compete economically with land-based cellular or PCS service. For these reasons, it appears that the idea of satellite telephone service has been a cosdy mistake. In August 1999 the Iridium LLC company declared bankruptcy, and similar fates have fallen upon other satellite-based phone services, Global positioning satellite system. The Global Positioning Satellite system (GPS) uses 24 satellites in medium earth orbits to provide accurate position information (latitude, longitude, and elevation) to users on land, air, or sea. Originally developed as the NAVSTAR system by the US military, at a cost of about $12B, GPS has quickly become one of the most pervasive applications of wireless technology for consumers and businesses throughout the world. Today, GPS receivers can be found on commercial and private airplanes, boats and ships, and ground vehicles. Advances in technology had led to substantial reductions in size and cost, so that small hand-held GPS receivers can he used by hikers and sportsmen. With differentia] GPS, accuracies on the order of 1 cm can be achieved, a capability that has revolutionized the surveying industry. The GPS positioning system operates by using triangulation with a minimum of four satellites. GPS satellites are in orbits 20,200 km above the Earth, with orbital periods of 12 hours. Distances from the user's receiver to these satellites are found by timing the propagation delay between the satellites and the receiver. The positions of the satellites 658 Chapter 13: Introduction to Microwave Systems (ephemeris) are known to very high accuracy^ and each satellite contains an extremely accurate clock to provide a unique set of timing pulses. A GPS receiver decodes this timing information, and performs the necessary calculations to find die position and velocity of the receiver. The GPS receiver must have a line-of-sight view to at least four satellites in the GPS constellation, although three satellites are adequate if altitude position is known (as in the case of ships at sea). Because of the low gain antennas required for operation, the received signal level from a GPS satellite is very low—typically on the order of — 130 dBm (for a receiver antenna gain of 0 dB). This signal level is usually below the noise power at the receiver, but spread spectrum techniques are used to improve the received signal to noise ratio. GPS operates at two frequency bands: LI, at 1575.42 MHz, and L2, at 1227.60 MHz, transmitting spread spectrum signals with binary phase shift keying modulation. The LI frequency is used to transmit ephemeris data for each satellite, as well as timing codes, which are available to any commercial or public user. This mode of operation is referred to as the Course/Acquisition (C/A) code. In contrast, the L2 frequency is reserved for military use, and uses an encrypted timing code referred to as the Protected (P) code (there is also a P code signal transmitted at the LI frequency). The P code offers much higher accuracy than the C/A code. The typical accuracy that can be achieved with an L! G1JS receiver is about 100 feet. Accuracy is limited by timing errors in the clocks on the satellites and the receiver, as well as error in the assumed position of the GPS satellites. The most significant error is generally caused by atmospheric and ionospheric effects, which introduce small but variable delays in signal propagation from the satellite to the receiver. Wireless local area networks. Wireless local area networks (WLANs) provide connections between computers over short distances. Typical indoor applications may be in hospitals, office buildings, and factories, where coverage distances are usually less than a few hundred feet. Outdoors, in the absence of obstructions and with the use of high gain antennas, ranges up to a few miles can be obtained. Wireless networks are especially useful when it is impossible or prohibitively expensive to place wiring in or between buildings, or when only temporary access is needed between computers. Mobile computer users, of course, can only be connected to a computer network by a wireless link. Currently most commercial WLAN products in the US operate in the Industrial, Scientific, and Medical (ISM) frequency bands, and use either frequency-hopping or direct-sequence spread spectrum techniques in accordance with IEEE Standards 802.11 a, 802.1 lb, 802.1 lg, or the Bluetooth standard. Maximum bit rates range from 1-11 Mbps. Direct broadcast satellite. The US Direct Broadcast Satellite (DBS) system provides television service from geosynchronous satellites directly to home users with a relatively small 18" diameter antenna. Previous to this development satellite TV service required an unsightly dish antenna as large as 6 feet in diameter. This advancement was made possible through the use of digital modulation techniques, which reduce the necessary received signal levels as compared to previous systems which used analog modulation. The DBS system uses quadrature phase shift keying (QPSK) with digital multiplexing and error correction to deliver digital data at a rate of 40 Mbps. Two satellites, DBS-1 and DBS-2, located at 101^2° and 100.8" longitude, each provide 16 channels with 120 W of radiated power per channel. These satellites use opposite circular polarizations to minimize loss due to precipitation, and to avoid interference with each other (polarization duplexing). Point-to-point radio systems. Point-to-point radios are used by businesses to provide dedicated data connections between two fixed points. Electric utility companies use point-to-point radios for transmission of telemetry information for die generation, transmission, and distribution of electric power between power stations and substations. Point-to-point radios 13.3 Radar Systems 659 are also used to connect cellular base stations to the public switched telephone network, and are generally much cheaper than running high-bandwidth coaxial or fiber-optic lines below ground. Such radios usually operate in the 18, 24, or 38 GHz bands, and use a variety of digital modulation methods to provide data rates in excess of 10 Mbps, High gain antennas are typically used io minimize power requirements and avoid interference with other users. 13.3 RADAR SYSTEMS Radar, or Radio Detection And Ranging, is one of the most prevalent applications of microwave technology. In its basic operation, a transmitter sends out a signal which is partly reflected by a distant target, and then detected by a sensitive receiver. If a narrow beam antenna is used, the target's direction can be accurately given by the position of the antenna. The distance to the target is determined by the time required for the signal to travel to the target and back, and the radial velocity of the target is related to the Doppler shift of the return signal. Below are listed some of the typical applications of radar systems. Civilian Applications • Airport surveillance • Marine navigation • Weather radar • Altimetry • Aircraft landing • Burglar alarms • Speed measurement (police radar) • Mapping Military Applications • Air and marine navigation • Detection and tracking of aircraft, missiles, spacecraft • Missile guidance • Fire control for missiles and artillery • Weapon fuses • Reconnaissance Scientific Applications • Astronomy • Mapping and imaging • Precision distance measurement • Remote sensing of natural resources Early radar work in the United States and Britain began in the 1930s using very high frequency (VHF) sources. A major breakthrough occurred in the early 1940s with the British invention of the magnetron as a reliable source of high-power microwaves. Higher frequencies allowed the use of reasonably sized antennas with high gain, allowing mechanical tracking of targets with good angular resolution. Radar was quickly developed in Great Britain and the United States, and played an important role in World Warn. Figure 13.16 shows a photograph of the phased array radar for the PATRIOT missile system. We will now derive the radar equation, which governs the basic operation of most radars, and then describe some of the more common types of radar systems. 660 Chapter 13: Introduction to Microwave Systems FIGURE 13.16 Photograph of the PATRIOT phased array radar. This is a C-band multifunction radar that provides lacdcal air defense, including target search aod tracking, and missile fire control. The phased array antenna uses 5000 ferine phase shifters to electronically scan die antenna beam. Photo provided by Raytheon Company. The Radar Equation Two basic radar systems are illustrated in Figure 13.17; in the monostatic radar the same antenna is used for both transmit and receive, while the bistatic radar uses two separate antennas for these functions. Most radars are of the monostatic type, but in some applications (such as missile fire control) the target is illuminated by a separate transmit antenna. Separate antennas are also sometimes used to achieve the necessary isolation between transmitter and receiver. Here we will consider the monostatic case, but the bistatic case is very similar. If the transmitter radiates a power Pt through an antenna of gain G, the power density incident on the target is, from 13.23, 5, = 4ttR2' (13.35) where R is the distance to the target. It is assumed that the target is in the main beam direction of the antenna. The target will scatter the incident power in various directions; the ratio of the scattered power in a given direction to the incident power density is defined as the radar cross section, o, of the target. Mathematically, er — (13.36) where Ps is the total power scattered by the target. The radar cross section thus has the dimensions of area, and is a property of the target itself. It depends on. the incident and reflection angles, as well as the polarization of the incident wave. ~Z_ "Z_ Receivers processor Receiver/ J processor 0>) FIGURE 13.17 Basic monostatic and bistatic radar systems, (a) Monostatic radar system, (b) Bistatic radar system. Since the target acts as a finite-sized source, the power density of the reradiated field must decay as 1 /4jt R2 away from the target. Thus the power density of the scattered field back at the receive antenna must be P,G2o- (13.37) (4xrR4 ' (13.38) This is the radar equation. Note that the received power varies as 1 /R4, which implies that a high-power transmitter and a sensitive low-noise receiver are needed to detect targets at long ranges. Because of noise received by the antenna and generated in the receiver, there will be some minimum detectable power that can be discriminated by the receiver. If this power is /mim then (13.38) can be rewritten to give the maximum range as Rmta — P,G2ak2 y a (13.39) Signal processing can effectively reduce the minimum detectable signal, and so increase the usable range. One very common processing technique used with pulse radars is pulse integration, where a sequence of N received pulses are integrated over time. The effect is to reduce the noise level, which has a zero mean, relative to the returned pulse level, resulting in an improvement factor of approximately N [5]. Of course, the above results seldom describe the performance of an actual radar system. Factors such as propagation effects, the statistical nature of the detection process, and external interference often serve to reduce the usable range of a radar system. 662 Chapter 13: Introduction to Microwave Systems EXAMPLE 13.6 APPLICATION OF THE RADAR RANGE EQUATION A pulse radar operating at 10 GHz has an antenna with a gain of 28 dB, and a transmitter power of 2 kW (pulse power). If it is desired to detect a target with a cross section of 12 m2, and the minimum detectable signal is /min = -90 dBm, what is the maximum range of the radar? Solution The required numerical values are G = \02^l0 = 63l, Pmin = 10-%/,° mW = 10-12 W, X = 0.03 m. Then the radar range equation of (13.39) gives the maximum range as '(2 x 103)(631)2(12)(.03)211/4 (4jt)3(10-12) = 8114 m. Pulse Radar A pulse radar determines target range by measuring the round-trip time of a pulsed microwave signal. Figure 13.18 shows a typical pulse radar system block diagram. The transmitter portion consists of a single-sideband mixer used to frequency offset a microwave oscillator Transmit/ Power USB Antenna switch an*™»" , . . nu«r Pulse generator Pulse generator Transmit signal Low-noise Mixer IF amplifier amplifier Transmit mode I / Receive mode ■- De lector Vif*? Display amplifier Detected signal Transmitter Clutter Target leakage and return noise FIGURE 13.18 A pulse radar system and timing diagram. 15 13,3 Radar Systems 663 of frequency /0 by an amount equal to the £P frequency. After power amplification, pulses of this signal are transmitted by the antenna. The transmit/receive switch is controlled by the pulse generator to give a transmit pulse width r, with a pulse repetition frequency (PRF) of fr — 1/X- The transmit pulse thus consists of a short burst of a microwave signal at the frequency /0 + fw- Typical pulse durations range from 100 ms to 50 ns; shorter pulses give better range resolution, but longer pulses result in a better signal-to-noise ratio after receiver processing. Typical pulse repetition frequencies range from 100 Hz to 100 kHz; higher PRFs give more returned pulses per unit time, which improves performance, but lower PRFs avoid range ambiguities that can occur when R > cTT(2. In the receive mode, the returned signal is amplified and mixed with the local oscillator of frequency /<, to produce the desired IF signal. The local oscillator is used for both up-conversion in the transmitter as well as down-conversion in the receiver; this simplifies the system and avoids the problem of frequency drift, which would be a consideration if separate oscillators were used. The IF signal is amplified, detected, and fed to a video amplifier/display. Search radars often use a continuously rotating antenna for 360" azimuthal coverage; in this case the display shows a polar plot of target range versus angle. Many modern radars use a computer for the processing of the detected signal and display of target information. The transmit/receive (T/R) switch in the pulse radar actually performs two functions: fotming the transmit pulse train, and switching the antenna between the transmitter and receiver. This latter function is also known as duplexing. In principle, the duplexing function could be achieved with a circulator, but an important requirement is that a high degree of isolation (about 80-100 dB) be provided between the transmitter and receiver, to avoid transmitter leakage into the receiver which would drown the target return (or possibly damage the receiver). As circulators typically achieve only 20-30 dB of isolation, some type of switch, with high isolation, is required. If necessary, further isolation can be obtained by using additional switches along the path of the transmitter circuit. Doppler Radar If the target has a velocity component along the line-of-sight of the radar, the returned signal will be shifted in frequency relative to the transmitted frequency, due to the doppler effect. If the transmitted frequency is f0, and the radial target velocity is v, then the shift in frequency, or the doppler frequency, will be where c is the velocity of light. The received frequency is then f„ ± fd, where the plus sign corresponds to an approaching target and the minus sign corresponds to a receding target. Figure 13.19 shows a basic doppler radar system. Observe that it is much simpler than a pulse radar, since a continuous wave signal is used, and the transmit oscillator can also be used as a local oscillator for the receive mixer, because the received signal is frequency offset by the doppler frequency. The filter following the mixer should have a passband corresponding to the expected minimum and maximum target velocities. It is important that the filter have high attenuation at zero frequency, to eliminate the effect of clutter return and transmitter leakage at the frequency fa, as these signals would down-convert to zero frequency. Then a high degree of isolation is not necessary between transmitter and receiver, and a circulator can be used. This type of filter response also helps to reduce the effect of 1 // noise. The above radar cannot distinguish between approaching and receding targets, as the sign of fj is lost in the detection process. Such information can be recovered, however, by using a mixer that produces separately the upper and lower sideband products. 664 Chapter 13: Introduction to Microwave Systems f a Amentia Circulator Filter Amplifier FIGURE 13.19 Doppler radar system. Since the return of a pulse radar from a moving target will contain a doppler shift, it is possible to determine both the range and velocity (and position, if a narrow beam antenna is used) of a target with a single radar. Such a radar is known as a pulse-doppier radar, and offers several advantages over pulse or doppler radars. One problem with a pulse radar is that it is impossible to distinguish between a true target and clutter returns from the ground, trees, buildings, etc. Such clutter returns may be picked up from the antenna sidelobes. But if the target is moving (e,g., as in an airport surveillance radar application), the doppler shift can be used io separate its return from clutter, which is stationary. Radar Cross Section A radar target is characterized by its radar cross section, as defined in (13.36), which gives the ratio of scattered power to incident power density. The cross section of a target depends on the frequency and polarization of the incident wave, and on the incident and reflected angles relative to the target. Thus we can define a monostatic cross section (incident and reflected angles identical), and a bistatic cross section (incident and reflected angles different). For simple shapes the radar cross section can be calculated as an electromagnetic boundary value problem; more complex targets require numerical techniques, or measurement to find the cross section. The radar cross section of a conducting sphere can be calculated exactly; the monostatic result is shown in Figure 13.20, normalized to na1, the physical FIGURE 13.20 Monostatic radar cross section of a conducting sphere. 13.4 Radiometer Systems 665 TABLE 13.2 Typical Radar Cross Sections Target Missile Person Small plane Bicyle Small boat Fighter plane Bomber Large airliner Truck Bird 0.01 D,5 1. 1-2 2 2 3-8 30-40 100 200 cross-sectional area of the sphere. Note that the cross section increases very quickly with size for electrically small spheres (a A,). This region is called the Rayleigh region, and it can be shown that a varies as (a/kf in this region. (This strong dependence on frequency explains why the sky is blue, as the blue component of sunlight scatters more strongly from atmospheric particles than do the lower frequency red components.) For electrically large spheres, where a ^ a, the radar cross section of the sphere is equal to its physical cross section, no1. This is the optical region, where geometrical optics are valid. Many other shapes, such as flat plates at normal incidence, also have cross sections that approach the physical area for electrically large sizes. Between the Rayleigh region and the optical region is the resonance region, where the electrical size of the sphere is on the order of a wavelength. Here the cross section is oscillating with frequency, due to phase addition and cancellation of various scattered field components. Of particular note is the fact that the cross section may reach quite high values in this region. Complex targets such as aircraft or ships generally have cross sections that vary rapidly with frequency and aspect angle. In military applications it is often desirable to minimize the radar cross section of vehicles, to reduce detectability. This can be accomplished by using radar absorbing materials (lossy dielectrics) in the construction of the vehicle. Table 13,2 lists the approximate radar cross sections of a variety of different targets. A radar system obtains information about a target by transmitting a signal and receiving the echo from the target, and thus can be described as an active remote sensing system. Radiometry, however, is a passive technique which develops information about a target solely from the microwave portion of the blackbody radiation (noise) that it either emits directly or reflects from surrounding bodies. A radiometer is a sensitive receiver specially designed to measure this noise power. Theory and Applications of Radiometry As discussed in Section 10.1. a body in thermodynamic equilibrium at a temperature T radiates energy according to Planck's radiation law. In the microwave region this result reduces to P = kTB, where k is Boltzmann's constant, B is the system bandwidth, and P is the radiated power. This result strictly applies only to a blackbody, which is defined as an 13-4 RADIOMETER SYSTEMS 666 Chapter 13: Introduction to Microwave Systems idealized material which absorbs all incident energy, and reflects none; a blackbody also radiates energy at the same rate as it absorbs energy, thus maintaining thermal equilibrium. A nonideal body will partially reflect incident energy, and so does not radiate as much power as would a blackbody at the same temperature. A measure of the power radiated by a body relative to that radiated by an ideal blackbody at the same temperature is the emissivity, e, defined as P kTB' e — (13.41) where P is the power radiated by the nonideal body, and kTB is the power that would be emitted by a perfect blackbody. Thus, 0 < e < 1, and e = 1 for a perfect blackbody. As we saw in Section 10.1, noise power can also be quantified in terms of equivalent temperature. Thus for radiometric purposes we can define a brightness temperature, TB , as TB = e T, (13.42) where T is the physical temperature of the body. This shows that, radiometrically, a body never looks hotter than its actual temperature, since 0 < e < 1. Now consider Figure 13.21, which shows the antenna of a radiometer receiving noise powers from various sources. The antenna is pointed at a region of the earth which has an apparent brightness temperature Tg. The atmosphere emits radiation in all directions; the component radiated directly toward the antenna is T\r>, while the power reflected from the earth to the antenna is TAfi- There may also be noise powers that enter the sidelobes of the antennas, from the sun or other sources. Thus, we can see that the total brightness temperature seen by the radiometer is a function of the scene under observation, as well as the observation angle, frequency, polarization, attenuation of the atmosphere, and the antenna pattern. The objective of radiometry is to infer information about the scene from the measured brightness temperature, and an analysis of the radiometric mechanisms that relate brightness temperature to physical conditions of the scene. For example, the power reflected from a uniform layer of snow over soil can be treated as plane wave reflection from a multilayer dielectric region, leading to the development of an algorithm that gives the thickness of the snow in terms of measured brightness temperature at various frequencies. Microwave radiometry is a relatively new area of technology, and one which is strongly interdisciplinary, drawing on results from fields such as electrical engineering, oceanography, geophysics, and atmospheric and space sciences, to name a few. On the following page, some of the more typical applications of microwave radiometry are listed. Radiometer antenna Earth FIGURE 13.21 Noise power sources in a typical radiometer application. 13.4 Radiometer Systems 667 Environmental Applications • Measurement of soil moisture * Flood mapping • Snow cover/Ice cover mapping * Ocean surface windspeed • Atmospheric temperature profile * Atmospheric humidity profile Military Applications • Target detection ♦ Target recognition * Surveillance * Mapping Astronomy Applications • Planetary mapping • Solar emission mapping • Mapping of galactic objects * Measurement of cosmological background radiation Figure 13.22 shows a photograph of a radiometer used to measure the water vapor profile of the atmosphere. Total Power Radiometer The aspect of radiometry that is of most interest to the microwave engineer is the design of the radiometer itself. The basic problem is to build a receiver that can distinguish between the desired radiometric noise and the inherent noise of the receiver, even though the radiometric power is usually less than the receiver noise power. Although it is not a very practical instrument, we will first consider the total power radiometer, because it represents a simple and direct approach to the problem and serves to illustrate the difficulties involved in radiometer design. The block diagram of a typical total power radiometer is shown in Figure 13.23. The front end of the receiver is a standard superheterodyne circuit consisting of an RF amplifier, a mixer/local oscillator, and an IF stage. The IF filter determines the system bandwidth, B. The detector is generally a square-law device, so that its output voltage is proportional to the input power. The integrator is essentially a low-pass filter with a cutoff frequency of 1/r, and serves to smooth out short-term variations in the noise power. For simplicity, we assume that the antenna is lossless, although in practice antenna loss will affect the apparent temperature of the antenna, as given in (13.18). If the antenna is pointed at a background scene with a brightness temperature Tg, the antenna power will be PA = kTBB; this is the desired signal. The receiver contributes noise which can be characterized as a power Pr — kT^B at the receiver input, where TR is the overall noise temperature of the receiver. Thus the output voltage of the radiometer is V0 = G{Ta + Ts)kB, (13.43) where G is the overall gain constant of the radiometer. Conceptually, the system is calibrated by replacing the antenna input with two calibrated noise sources, from which the system constants GkB and GTRkB can be determined. (This is similar to the K-factor method for measuring noise temperature.) Then the desired brightness temperature, TB, can be measured with the system. 668 Chapter 13: Introduction to Microwave Systems FIGURE 13.22 Photograph of a multichannel microwave radiometer used to measure the water vapor profile of die atmosphere. This system has one receiver that operates at 36.5 GHz to sense liquid water in the atmosphere, and a second group of receivers operating from 16 to 28 GHz to sample the 22 GHz water vapor resonance. Courtesy of the Microwave Remote Sensing Laboratory, University of Massachusetts at Amherst. Observed scene FIGURE 13.23 Total power radiometer block diagram. 13.4 Radiometer Systems 669 Two types of errors occur with this radiometer. First is an error, A TV in the measured brightness temperature due to noise fluctuations. Since noise is a random process, the measured noise power may vary from one integration period to the next. The integrator (or low-pass filter) acts to smooth out ripples in V0 with frequency components greater than 1 / r. It can be shown that the remaining error is [4] MN = ^^-. (13.44) This result shows that if a longer measurement time, t, can be tolerated, the error due to noise fluctuation can be reduced to a negligible value. A more serious error is due to random variations in the system gain, G. Such variations generally occur in the RF amplifier, mixer, or IF amplifier, over a period of one second or longer. So if the system is calibrated with a certain value of G, which changes by the time a measurement is made, an error will occur, as given in reference [41 as ATG = {TS+TR)^, (13.45) where AG is the rms change in the system gain, G. It will be useful to consider some typical numbers at this time. For example, a 10 GHz total power radiometer may have a bandwidth of 100 MHz, a receiver temperature of 7* = 500 K, an integrator tíme constant of t = 0.01 s, and a system gain variation AG / G = 0,01. If the antenna temperature is TB = 300 K, (13.44) gives the error due to noise fluctuations as, ATjv = 0.8 K, while (13.45) gives the error due to gain variations as A Tq = 8 K. These results, which are based on reasonably realistic data, show that gain variation is the most detrimental factor affecting the accuracy of the total power radiometer. The Dlcke Radiometer We have seen that the dominant factor affecting the accuracy of the total power radiometer is the variation of gain of the overall system. Since such gain variations have a relatively long time constant (> I second), it is conceptually possible to eliminate this error by repeatedly calibrating the radiometer at rapid rate. This is the principle behind the operation of the Dicke null-balancing radiometer. A system diagram is shown in Figure 13,24. The superheterodyne receiver is identical to the total power radiometer, but the input is periodically switched between the antenna and a variable power noise source; this switch is called the Dicke switch. The output of the square-law detector drives a synchronous demodulator, which consists of a switch and a difference circuit. The demodulator switch operates in synchronism with the Dicke switch, so that the output of the subtracter is proportional to the difference between the noise powers from the antenna, Tb, and the reference noise source, Tref- The output of the subtracter is then used as an error signal to a feedback control circuit, which controls the power level of the reference noise source so that V„ approaches zero. In this balanced state, Tb = Trjjf, and Tff can be detenrtiiied from the control voltage, Vc. The square-wave sampling frequency, is chosen to be much faster than the drift time of the system gain, so that this effect is virtually eliminated. Typical sampling frequencies range from 10 to 1000 Hz. A typical radiometer would measure the brightness temperature Tb over a range of about 50-300 K; this then implies that the reference noise source would have to cover this same range, which is difficult to do in practice. Thus, there are several variations on the above design, differing essentially in the way that the reference noise power is controlled or added to the system. One possible method is to use a constant 7^ which is somewhat hotter 670 Chapter 13: Introduction to Microwave Systems Variable power noise source Control voltage RF noise Antenna Feedback control circuit Output voltage DiCkC RF JF amplifier Mjxef IF niter amplifier Detector >-®*i:UW -M- \__J OSC1 Local Uator Synchronous demodulator Square wave i LrL / generator FIGURE 13,24 Balanced Dicke radiometer block diagram. than the maximum TB to be measured. The amount of reference noise power delivered to the system is then controlled by varying the pulse width of the sampling waveform. Another approach is to use a constant reference noise power, and vary the gain of the IF stage during the reference sample time to achieve a null output. Other possibilities, including alternatives to the Dicke radiometer, are discussed in the literature [4]. MICROWAVE PROPAGATION In free-space electromagnetic waves propagate in straight lines without attenuation or other adverse effects. Free-space, however, is an idealization that is only approximated when microwave energy propagates through the atmosphere or in the presence of the earth. In practice the performance of a communication, radar, or radiometry system may be seriously affected by propagation effects such as reflection, refraction, attenuation, or diffraction. Below we discuss some specific propagation phenomenon that can influence the operation of microwave systems, It is important to realize that propagation effects generally cannot be quantified in any exact or rigorous sense, but can only be described in terms of their statistics. Atmospheric Effects The relative permittivity of the atmosphere is close to unity, but is actually a function of air pressure, temperature, and humidity. An empirical result which is useful at microwave frequencies is given by [5] HV 3.8x105V T2 ) (13.46) where P is the barometric pressure in millibars, T is the temperature in kelvin, and V is the water vapor pressure in millibars. This result shows that permittivity generally decreases 13.5 Microwave Propagation 671 FIGURE 13.25 Refraction of radio waves by the atmosphere. (approaches unity) as altitude increases, since pressure and humidity decrease with height faster than does temperature. This change in permittivity with altitude causes radio waves to bend toward the earth, as depicted in Figure 13.25. Such refraction of radio waves can sometimes be useful, since it may extend the range of radar and communication systems beyond the limit imposed by the presence of the earth's horizon. If an antenna is at a height, h, above the earth, simple geometry gives the line-of-sight distance to the horizon as where R is the radius of the earth. From Figure 13.25 we see that the effect of refraction on range can be accounted for by using an effective earth radius kR, where k>l.A value commonly used [5] is k = 4/3, but this is only an average value which changes with weather conditions. In a radar system, refraction effects can lead to errors when determining the elevation of a target close to the horizon. Weather conditions can sometimes produce a temperature inversion, where the temperature increases with altitude. Equation (13.46) then shows that the atmospheric permittivity will decrease much faster than normal, with increasing altitude. This condition can sometimes lead to ducting (also called trapping, or anomalous propagation), where a radio wave can propagate long distances parallel to the earth's surface, via the duct created by the layer of air along the temperature inversion. The situation is very similar to propagation in a dielectric waveguide. Such ducts can range in height from 50-500 feet, and may be near the earth's surface, or higher in altitude. Another atmospheric effect is attenuation, caused primarily by the absorption of microwave energy by water vapor and molecular oxygen. Maximum absorption occurs when the frequency coincides with one of the molecular resonances of water or oxygen, thus atmospheric attenuation has distinct peaks at these frequencies. Figure 13.26 shows the atmospheric attenuation vs. frequency. At frequencies below 10 GHz the atmosphere has very little effect on the strength of a signal. At 22.2 and 183.3 GHz, resonance peaks occur due to water vapor resonances, while resonances of molecular oxygen cause peaks at 60 and 120 GHz. Thus there are "windows" in the millimeter wave band near 35,94, and 135 GHz where radar and communication systems can operate with minimum loss. Precipitation such as rain, snow, or fog will increase the attenuation, especially at higher frequencies. The effect of atmospheric attenuation can be included in system design when using the Friis transmission equation or the radar equation. In some instances the system frequency may be chosen at a point of maximum atmospheric attenuation. Remote sensing of the atmosphere (temperature, water vapor, rain rate) (13.47) 672 Chapter 13: Introduction to Microwave Systems 10 15 20 25 30 40 50 60 70 80 90100 150 200 250 300 400 Frequency (GHz) FIGURE 13.26 Average atmospheric attenuation versus frequency (horizontal polarization). is often done with radiometers operating near 20 or 55 GHz, to maximize the sensing of atmospheric conditions (see Figure 13.22). Another interesting example is spacecraft-to-spacecraft communication at 60 GHz. This millimeter wave frequency has the advantages of a large bandwidth and small antennas with high gains and, since the atmosphere is very lossy at this frequency, the possibilities of interference, jamming, and eavesdropping from earth are greatly reduced. Ground Effects The most obvious effect of the presence of the ground on microwave propagation is reflection from the earth's surface (land or sea). As shown in Figure 13.27, a radar target (or receiver antenna) may be illuminated by both a direct wave from the transmitter and a wave reflected from the ground. The reflected wave is generally smaller in amplitude than the direct wave, because of the larger distance it travels, the fact that it usually radiates from the sidelobe region of the transmit antenna, and because the ground is not a perfect reflector. Nevertheless, the received signal at the target or receiver will be the vector sum of the two wave components and, depending on the relative phases of the two waves, may be greater or less than the direct wave alone. Because the distances involved are usually very large in terms of the electrical wavelength, even a small variation in the permittivity of the atmosphere FIGURE 13.27 Direct and reflected waves over the earth's surface, 13.5 Microwave Propagation 673 can cause fading (long term fluctuations) or scintillation (short term fluctuations) in the signal strength. These effects can also be caused by reflections from inhomogeneities in the atmosphere. In communication systems such fading can sometimes be reduced by making use of the fact that the fading of two communication channels having different frequencies, polarizations, or physical locations is essentially independent. Thus a communication link can reduce fading by combining the outputs of two (or more) such channels; this is called a diversity system. Another ground effect is diffraction, whereby a radio wave scatters energy in the vicinity of the line-of-sight boundary at the horizon, thus giving a range slightly beyond the horizon. This effect is usually very small at microwave frequencies. Of course, when obstacles such as hills, mountains, or buildings are in the path of propagation, diffraction effects can be stronger. In a radar system, unwanted reflections often occur from terrain, vegetation, trees, buildings, and the surface of the sea. Such clutter echoes generally degrade or mask the return of a true target, or show up as a false target, in the context of a surveillance or tracking radar. In mapping or remote sensing applications such clutter returns may actually constitute the desired signal. Plasma Effects A plasma is a gas consisting of ionized particles. The ionosphere consists of spherical layers of atmosphere with particles which have been ionized by solar radiation, and thus forms a plasma region. A very dense plasma is formed on a spacecraft as it reenters the atmosphere, due to the high temperatures produced by friction. Plasmas are also produced by lightning, meteor showers, and nuclear explosions. A plasma is characterized by the number of ions per unit volume; depending on this density and the frequency, a wave might be reflected, absorbed, or transmitted by the plasma medium. An effective permittivity can be defined for a uniform plasma region as ff = fjl--? , (13,48) where «, = /—^ (13.49) is the plasma frequency. In (13.49), q is the charge of the electron, m is the mass of the electron, and N is the number of ionized particles per unit volume. By studying the solution of Maxwell's equations for plane wave propagation in such a medium, it can be shown that wave propagation through a plasma is only possible for oj > wp. Lower frequency waves will be totally reflected. If a magnetic field is present, the plasma becomes anisotropic, and the analysis is more complicated. The earth's magnetic field may be strong enough to produce such an anisotropy in some cases. The ionosphere consists of several different layers with varying ion densities; in order of increasing ion density, these layers are referred to as D, E, F], and F2. The characteristics of these layers depends on seasonal weather and solar cycles, but the average plasma frequency is about 8 MHz. Thus, signals at frequencies less than 8 MHz (e.g., short-wave radio) can reflect off the ionosphere to travel distances well beyond the horizon. Higher frequency signals, however, will pass through the ionosphere. A similar effect occurs with a spacecraft entering the atmosphere. The high velocity of the spacecraft causes a very dense plasma to form around the vehicle. The electron density 674 Chapter 13: Introduction to Microwave Systems is high enough so that, from (13.49), the plasma frequency is very high, thus inhibiting communication with the spacecraft until its velocity has decreased. Besides this blackout effect, me plasma layer may also cause a large impedance mismatch between the antenna and Its feed line. OTHER APPLICATIONS AND TOPICS Microwave Heating To the average consumer, the term "microwave" connotes a microwave oven, which is used in many households for heating food; industrial and medical applications also exist for microwave heating. As shown in Figure 13.28, a microwave oven is a relatively simple system consisting of a high-power source, a waveguide feed, and the oven cavity. The source is generally a magnetron tube operating at 2.45 GHz, although 915 MHz is sometimes used when greater penetration is desired. Power output is usually between 500 and 1500 W. The oven cavity has metallic walls, and is electrically large. To reduce die effect of uneven heating caused by standing waves in the oven, a "mode stirrer," which is just a metallic fan blade, is used to perturb the field distribution inside the oven. The food is also rotated with a motorized platter. In a conventional oven a gas or charcoal fire, or an electric heating element, generates heat outside of the material to be heated. The outside of the material is heated by convection, and the inside of the material by conduction. In microwave heating, by contrast, the inside of the material is heated first. The process through which this occurs primarily involves the conduction losses in materials with large loss tangents [6], [7]. An interesting fact is that the loss tangents of many foods decrease with increasing temperature, so that microwave heating is to some extent self-regulating. The result is that microwave cooking generally gives faster and more uniform heating of food, as compared with conventional cooking. The efficiency of a microwave oven, when defined as the ratio of power converted to heal (in the food) to the power supplied to the oven, is generally less than 50%; this is usually greater than the cooking efficiency of a conventional oven, however. The most critical issue in the design of a microwave oven is safety. Since a very high power source is used, leakage levels must be very small to avoid exposing the user to harmful radiation. Thus the magnetron, feed waveguide, and oven cavity must all be carefully shielded. The door of die oven requires particular attention; besides close mechanical tolerances, the joint around the door usually employs RF absorbing material and a a/4 choke flange to reduce leakage to an acceptable level. "Mode stirrer" Waveguide l_Z Ii Oven c a vily Magnetron Fond Rotating plate Power supply ľlGljRE 13.28 A microwave tivcn. 71 13,6 Other Applications and Topics 675 Power Transfer Electrical power transmission lines are a very efficient and convenient way to transfer energy from one point to another, as they have relatively low loss and initial costs, and can be easily routed, There are applications, however, where it is inconvenient or impossible to use such power lines. In such cases it is conceivable that electrical power can be transmitted without wires by a well-focused microwave beam [8]. One example is the solar satellite power station, where it has been proposed that electricity be generated in space by a large orbiting array of solar cells, and transmitted to a receiving station on earth by a microwave beam. We would thus be provided with a virtually inexhaustible source of electricity. Placing the solar arrays in space has the advantage of power delivery uninterrupted by darkness, clouds, or precipitation, which are problems encountered with earth-based solar arrays. To be economically competitive with other sources, the solar power satellite station would have to be very large. One proposal involves a solar array about 5 x 10 km in size, feeding a 1 km diameter phased array antenna. The power output on earth would be on the order of 5 GW. Such a project is extremely large in terms of cost and complexity. Also of legitimate concern is the operational safety of such a scheme, both in terms of radiation hazards associated with the system when it is operating as designed, as well as the risks involved with a malfunction of the system. These considerations, as well as the political and philosophical ramifications of such a large, centralized power system, have made the future of the solar power satellite station doubtful. Similar in concept, but on a much smaller scale, is the transmission of electrical power from earth to a vehicle such as a small drone helicopter or airplane. The advantages are that such an aircraft could run indefinitely, and very quietly, at least over a limited area. Battlefield surveillance and weather prediction would be some possible applications. The concept has been demonstrated with several projects involving small pilodess aircraft. A very high power pulsed microwave source and a high-gain antenna can be used to deliver an intense burst of energy to a target, and thus used as a weapon. The pulse may be intense enough to do physical damage to the target, or it may act to overload and destroy sensitive electronic systems. Biological Effects and Safety The proven dangers of exposure to microwave radiation are due to thermal effects. The body absorbs RF and microwave energy and converts it to heat; as in the case of a microwave oven, this heating occurs within the body, and may not be felt at low levels. Such heating is most dangerous in the brain, the eye, the genitals, and the stomach organs. Excessive radiation can lead to cataracts, sterility, or cancer. Thus it is important to determine a safe radiation level standard, so that users of microwave equipment will not be exposed to harmful power levels. The most recent U.S. safety standard for human exposure to electromagnetic fields is given by IEEE Standard C95.1-1991. In the RF-microwave frequency range of 100 MHz to 300 GHz, exposure limits are set on the power density (W/cm2) as a function of frequency, as shown in Figure 13.29. The recommended safe power density limit is as low as 0.2 mW/cm2 at the lower end of this frequency range, because fields penetrate the body more deeply at lower frequencies. At frequencies above 15 GHz the power density limit rises to 10 mW/cm2, since most of the power absorption at such frequencies occurs near the skin surface. By comparison, the sun radiates a power density of about 100 mW/cm2 on a clear day, but the effect of this radiation is much less severe than a corresponding level of lower-frequency microwave radiation because the sun heats the outside of the body, with much of the generated heat reabsorbed by the air, while microwave power heats from inside the body. At frequencies below 100 MHz electric and magnetic fields interact with the body 676 Chapter 13: introduction to Microwave Systems 100. rr a I 10.0 0.1 ........ J_I_i ''in J_.......< .......I O.J 1.0 JO. Frequency (GHz) I'.JO. 1000. FIGURE 13.29 IEEE Standard C95,1 -1991 recommended power density limits for human exposure to RF and microwave electromagnetic fields. differently than higher frequency electromagnetic fields, and so separate limits are given for field components at these lower frequencies. Other countries have different standards for microwave exposure limits, some of which are lower than the U.S. limits. Some of these standards are a function of exposure time, with lower power density limits for prolonged exposure. A separate standard applies to microwave ovens in the United States: law requires that all ovens be tested to ensure that the power level at 5 cm from any point on the oven does not exceed 1 mW/cnr. Most experts feel that the above limits represent safe levels, with a reasonable margin. Some researchers, however, feel that health hazards may occur due to nonthermal effects of long-term exposure to even low levels of microwave radiation, EXAMPLE 13.7 POWER DENSITY IN THE VICINITY OF A MICROWAVE RADIO LINK A 6 GHz common-carrier microwave communications link uses a tower-mounted antenna with a gain of 40 dB, and a transmitter power of 5 W. To evaluate the radiation hazard of this system, calculate the power density at a distance of 20 m from the antenna. Do this for a position in the main beam of the antenna, and for a position in the sidelobe region of the antenna. Assume a worst-case sidelobe level of-lOdB. Solution The numerical gain of the antenna is G = 1040''10 = 104. Then from (13.23), the power density in the main beam of the antenna at a distance of R = 20 m is PiriG 5 x 104 5 = = ——r = 10. W/m2 = 1.0 mW/cnr. 4tt R2 4jt(20)2 Problems 677 The worst-case power density in the sidelobe region would be 10 dB below this, or 0.10mW/cm2. Thus we see that the power density in the main beam at 20 m is below the United States standard. The power density in the sidelobe region is well below this limit. These power densities will diminish rapidly with increasing distance, due to the 1/r" dependence. ■ REFERENCES [1] C. A. Balanis, Antenna Theory; Analysis and Design, 2nd edition, Wiley, N.Y., 1997, [2] W. L. Stutzman and G, A. Thiele, Antenna Theory and Design, 2nd edition, Wiley, N.Y, 1998. [3] L. J, rppolito, R. D. Kaul, and R. G. Wallace, Propagation Effects Handbook for Satellite Systems Design, 3rd edition, NASA Publication 1082(03), June 1983. [4] F, T Ulaby, R. K. Moore, and A. K, Fung, Microwave Remote Sensing: Active and Passive, Volume I, Microwave Remote Sensing, Fundamentals and Radiometry. Addison-Wesley, Reading, Mass., 1981. [5] M. I. Skolnik, Introduction to Radar Systems, McGraw-Hill, N.Y., 1962, [6] F. E. Gardiol, Introduction to Microwaves, Artech House, Dedham, Mass., 1984. [7] E. C. Okress, Microwave Power Engineering, Academic Press, N.Y., 1968. [8] W. C. Brown, "The History of Power Transmission by Radio Waves," IEEE Trans. Microwave Theory and Techniques, vol, MTT-32, pp. 1230-1242, September 1984. PROBLEMS 13.1 An antenna has a radiation pattern function given by F${9,4>) = A sine? sin0. Find the main beam position, the 3 dB beam width, and the directivity (in dB) for this antenna. 13.2 A monopole antenna on a large ground plane has a far-field pattern function given by Fff($,4>) = A sin and a noise temperature TA, derive an expression for the maximum allowable amplifier noise figure, F, assuming an amplifier gain G, and a connecting line loss, L, Evaluate this expression for C/N = 32 dB-Hz, GA = 5 dB, TA = 300 K, G = 10 dB, and L = 25 dB. 1-*D> G,F i 13.12 A key premise in many science fiction stories i$ the idea that radio and TV signals from Earth can travel through space and be received by listeners in another star system. Show that this is a fallacy by calculating the maximum distance from Earth where a signal could be received with a signal-to-noise ratio of 0 dB. Specifically, assume TV channel 4, broadcasting at 67 MHz, with a 4 MHz bandwidth, a transmitter power of 1000 W, transmit and receive antenna gains of 4 dB, a cosmic background noise temperature of 4 K, and a perfectly noiseless receiver. How much would this distance decrease if an SNR of 30 dB is required at the receiver? (30 dB is a typical value for good reception of an analog video signal.) Relate these distances to the nearest planet in our solar system. 13.13 Derive the radar equation for the bistatic case, where the transmit and receive antennas have gains of G, and Gr, and are at distances R, and Rr from the target, respectively. 13.14 A pulse radar has a pulse repetition frequency ff = \/Tr, Determine the maximum unambiguous range of the radar. (Range ambiguity occurs when the round-trip time of a return pulse is greater than the pulse repetition time, so it becomes unclear as to whether a given return pulse belongs to the last transmitted pulse, or some earlier transmitted pulse.) 13.15 A doppler radar operating at 12 GHz is intended to detect target velocities ranging from 1 m/sec to 20 m/sec. What is the required passband of the doppler filter? 13.16 A pulse radar operates at 2 GHz and has a per-pufse power of 1 kW. If it is to be used to detect a target with a = 20 m2 at a range of 10 km, what should be the minimum isolation between the transmitter and receiver, so that the leakage signal from the transmitter is at least 1 OdB below the received signal? Assume an antenna gain of 30 dB. 13.17 An antenna, having a gain G, is shorted at its terminals. What is the minimum monostatie radar cross section in the direction of the main beam? 13.18 The atmosphere does not have a definite thickness, since it gradually thins with altitude, with a consequent decrease in attenuation. But if we use a simplified "orange peel" mode), and assume that the atmosphere can be approximated by a uniform layer of fixed thickness, we can estimate the background noise temperature seen through the atmosphere. Thus, let the thickness of the atmosphere be 4000 m and find the maximum distance i to the edge of the atmosphere along the horizon, as Lossy C/N line Receiver Problems 679 shown in the figure below (the radius of the earth is 6400 km), Now assume an average atmospheric attenuation of 0.005 dB/km, with a background noise temperature beyond the atmosphere of 4 K, and find the noise temperature seen on earth by treating the cascade of the background noise with the attenuation of the atmosphere. Do this for an ideal antenna pointing toward the zenith, and toward the horizon. 13.19 A 28 GHz radio link uses a tower-mounted reflector antenna with a gain of 32 dB, and a transmitter power of 5 W. (a) Find the minimum distance within the main beam of the antenna for which the U.S-recommended safe power density limit of 10 mW/cm2 is not exceeded, (b) How does this distance change for a position within the sidelobe region of the antenna, if we assume a worst-case sidelobe level of 10 dB below the main beam)? (c) Are these distances in the far-field region of the antenna? (Assume a circular reflector, with an aperture efficiency of 60%.) 13.20 On a clear day, with the sun direcUy overhead, the received power density from sunlight is about 1300 W/m2. If we make the simplifying assumption that this power is transmitted via a single-frequency plane wave, find the resulting amplitude of the incident electric and magnetic fields. Appendices Appendix A: Prefixes Appendix B: Vector Analysis Appendix C: Bessel Functions Appendix D: Other Mathematical Results Appendix E: Physical Constants Appendix F: Conductivities for Some Materials Appendix G: Dielectric Constants and Loss Tangents for Some Materials Appendix H: Properties of Some Microwave Ferrite Materials Appendix I: Standard Rectangular Waveguide Data Appendix J: Standard Coaxial Cable Data 680 Appendix B Vector Analysis APPENDIX A pnEF|XES Multiplying Factor Prefix Symbol 10!2 tern T 10y giga G 10* mega M 103 w> k 102 hecto h 10' deka da io-' deci d io-2 cenii c IO"3 miili m IO"6 micro IO"9 nano n io-13 pico P IQ"15 femto f APPENDIX B VECT0R ANALYSIS Coordinate Transformations Rectangular to cylindrical: X y z p cos 4> siný 0 $ -sm0 cos 0 i 0 0 1 Rectangular to spherical: 9 3 ř sin 6 cos 4> SÍll Ě? SÍT10 cos cos 0 COS d} cos 8 sm

3

»V J p dp \p dp ) p2 d A dš (divergence theorem) (B.15) v s y (V x A) > dš = j> A ■ dl (Stokes* theorem) (B.16) s c APPENDIX C BESSEL FUNCTIONS Bessel functions are solutions to the differential equation, where k2 is real and n is an integer. The two independent solutions to this equation are called ordinary Bessel functions of the first and second kind, written as Jn{kp) and Y„(kp), and so the general solution to (C.l) is f{p) = AJn(kp)+BYa{kp) (C.2) where A and B are arbitrary constants to be determined from boundary conditions. These functions can be written in series form as = - (k + ln-) J„(x)--> -■- - --> -——— x(l + Í + j + --- + -+l + ^ + --- + —J— J (C.4) \ 2 3 m 2 n+mj 684 Appendices FIGURE C.1 Bessel functions of the first and second kind. where y = 0.5772... is Euler's constant, and x = kp. Note that Y„ becomes infinite at x = 0, due to the In term. From these series expressions, small argument formulas can be obtained as Ycj(x) ~ - In* (C.6) Y»(x) - ^-(n - 1)! i n > 0 (C.7) Large argument formulas can be derived as ^W-^-cos (*—--) (C.8) Figure C. 1 shows graphs of a few of the lowest order Bessel functions of each type. Recurrence formulas relate Bessel functions of different orders; JWffit) = ~Zn(x) - Zn^(x) (CIO) x Z;0r)= ^Z^jO + Z^OO (Cll) Z'„{x) = -Zn(x)-Zn+i(x) x 1 Appendix C Bassel Functions 685 (C.12) Z'n{x)=-[Zn^(x)-Zn+x{x)] (C.13) where Z„ = J„ or Yn. The following integral relations involving Bessel functions are useful: ZUkx) (C.14) Jo (kx)Z„(tx)x dx = k2-e2 2 [kZn(tx)Zn+l(kx) - iZ„(kx)Z„+l(lx)] (C.15) 2 (C.16) where J„(pnm) = 0, and J'„(p'nm) = 0. The zeros of J„(x) and are on the following two pages. Zeros of Bessel Functions of First Kind: Mx) = 0 for 0 < x < 12 n 1 2 3 4 0 2.4048 5.5200 8.6537 11.7951 1 3.8317 7.0155 10.1743 2 5.1356 8.4172 11.6198 3 6.3801 9.7610 4 7.5883 11.0647 5 8.7714 6 9.9361 7 11.0863 Extrema of Besse! Functions of First Kind: dj„(x)/dx = 0 for 0 < x < 12 n 1 2 3 4 0 3.8317 7.0156 10.1735 13.3237 i 1.8412 5.3314 8.5363 11.7060 2 3.0542 6.7061 9.9695 3 4.2012 8.0152 11.3459 4 5.3175 9.2824 5 6.4156 10.5199 6 7.5013 11.7349 7 8.5778 8 9.6474 9 10.7114 10 11.7709 686 Appendices APPENDIX D OTHER MATHEMATICAL RESULTS Useful Integrals Jq f Ja cos —* f Jo Taylor Series MIX -dx - a Jo rtJTx l-ax =/ a Jo nnx i-dx = 0 a sin3 0d6 4 ~ 3 . 2 nnx a sin -dx = —, a 2 tnjrx nnx , sin-sin ——dx = 0, a a forn>l (D.l) for m £ n (D.2) (D.3) (DA) 1 j;=jfo f(x) = f(x0) + (x - XQ)^f dx X2 x^ ^ = i+x+- + - + i+x + x2 + x* + (* - *o)2 d2f 2! dx2 .i=.ijj 1 - A vt+i-i + 5-t+-. for |*| < 1 for |*| < 1 forx > 0 APPENDIX E sinx=x~ + + x2 xA cos* = l-- + - + PHYSICAL CONSTANTS • Permittivity of free-space = €o — 8.854 x 10~12 F/m • Permeability of free-space = mo = 4jt x 10~7 H/m • Impedance of free-space = t/g = 376.7 Q. • Velocity of light in free-space = c = 2.998 x 10s m/s • Charge of electron - q = 1.602 x 10"19 C • Mass of electron = m = 9.107 x 10~M kg • Boltzmann's constant = k = 1.380 x 10~23 J^K • Planck's constant = h — 6.626 x 10~34 J-sec • Gyromagnetic ratio = y - 1.759 x 10" C/Kg (forg = 2) (D.5) /L = 8.0 dB, delay = 60°, (e) T = 0.19/90* 4.17 IL = 6.7 dB, delay = 105° 4.19 Zn = Zn = 2.24 + y'52,2 Q, Zl2 = Z2i = j'44.8 Q 4.24 V-', = l/-90° 4.31 A = 0.082 cm 5.1 {A)b = 0.107,* = 1.78 or b = -0.747, x « -1.78 5,3 = 0.2276X, £ = 0.3776A. or = 0.4059A, £ = 0.1224A. 5.6 = 0.2917^, i = 0.364k or d = 0.4583X, £ = 0.136k 5.9 f, = 0.0B6A, £2 = 0.198X or t{ = 0.315k, t2 = 0.375A 690 Answers to Selected Problems 5.14 error - 4% 5.17 Z, = 1.1067Z0, = 1.355420 5.22 Z] = 1.095Z0, Z2 = 1.363Zu 525 RL < 6.4 dB 6.1 fo » 355.9 MHz, ß = 17.9, = 12.4 6.5 0 = 138 6.9 /ioi = 4.802 GHz, Öioi = 7,251 6.14 a = 2.107 cm, d = 2.479 cm, Q = 1,692 6.18 /b = 7.11 GHz 6.21 (c) /o = 93.8 GHz, ße = 92,500 7.2 RL ts 26 dB, C = 20 dB, D = 6 dB, / = 26 dB 7.7 change = 1.2 dB 7.12 s = 5.28 mm, r0 = 3.77 mm 7.18 s — 0.24 mm, w = 2.1 mm 7.21 s. - 115 mm. w = 1.92 mm. /' = 6.32 nun 7.32 V~ = V7 = V7 = 0, VV" = V5" = -j0.707 8.1 V(z = k/4) = 21.8 V 8.7 Ä = 2.66, C = 0.685, L 8.8 /V = 5 8.9 L, = L5 = 2.28 nH, C2 8.11 attenuation = 11 dB 8.17 ßlx = fil5 = 293\ßt2 8.19 attenuation = 30 dB 8.23 TV = 3 = 1.822 - C4 = 4.18 pF, L3 = 1.75 nH = ßU = 29,4% filz = 43.7° 9.1 (b) M = 0.849/io, * = -0.54O//0 9,4 /, = 3.64 GHz 9.6 L = 1.403 cm 9.8 229. Oe < H0 < 950. Oe 9.12 (a) Ho = 2204 Oe, (b) H0 = 2857 Oe 9.15 A$ = 180° 9.17 L = 44.5 cm 9.18 I =9.2 cm 10.1 /•' = 5.0 dB 10.5 DRt = 75 dB 10.6 FW =4.3 dB 10.9 (a) F = 6 dB, (b) F = 1.76 dB, (c) F = 3 dB 10,13 DRt = 867 dB, DRf = 60,5dB 10.15 ratio = 6 dB 10.16 P3 = 20,SdBm 10.19 ONl /Z, = 0.044 dB, OFF: IL = 18.6 dB 11.1 (b) G/i = 0.5, Gr = 0.444, G = 0.457 11.3 CL = 2.56Z28*, RL = 1,37, K = 1.35 11.11 -0.47 dB < GT - Grv < 0.5 dB 11.19 Nm = 6.33 12.3 0^ = 3.8 12.4 Z, = 9.4 iiiH. Q = 20,000, 0.25% 12.10 (a) C = -181 dBc/Hz, (h) £ = -153 dBc/Hz Index A ABCD parameters, 183-186 table for basic circuits, 185 table for conversions, 187 Admittance inverter, 411-412 Admittance matrix, 170-174 table for conversions, 187 AM modulation, 512 Ampere's law, 7 Amplifier design, 542-574 balanced, 562-565 distributed, 565-570 low-noise, 557-561 maximum gain, 548-553 maximum stable gain, 551 power. 570-574 specified gain, 553-557 stability, 542-548 Anisotropic media, 10-11 Antenna aperture efficiency, 640 directivity, 638 effective aperture area, 640-641 gain, 639-640 G/T.646 noise temperature. 643-644 pattern, 637-638 radiation efficiency, 639 types, 634-635 Aperture coupling, 209-215, 296-298 Aperture efficiency, 640 Attenuation atmospheric, 671-672 transmission line, 79-86 Attenuation constant for circular waveguide, 121, 122-124 coaxial line, 81 dielectric loss, 97-98 microstrip line, 145-146 parallel plate waveguide, 103, 104-105 plane wave in lossy dielectric, 16-19 rectangular waveguide. 111, 115 stripline, 139-140 Attenuator, 175-176 Available power gain, 537-539 B Background noise temperature, 642 Balanced amplifiers, 562-565 Bandpass fillers coupled line, 416-426 coupled resonator, 427-437 lumped element, 401—405 Bandstop filters coupled resonator, 427^431 lumped element, 401-404 BARITT diode, 521-522 Bessel functions, 683-685 zeroes of, 119, 122,685 Bethe hole coupler, 324-327 Binomial coefficients, 247 Binomial filter response, 390, 394-396 Binomial matching transformer, 246-250 Biological effects, 675-677 Bipolar transistors, 522, 525-526 Black body, 665-666 Bloch impedance, 374 Bode-Fano criterion, 261-263 Boltzmann's constant, 489 Boundary conditions, 11-14 Brewster angle, 36 Brightness temperature, 642-643 693 694 Index C Cavity resonators cylindrical cavity, 282-287 dielectric resonator, 287-291 rectangular cavity, 278-282 Cellular telephone systems, 655-656 Characteristic impedance, 51-52 coaxial line, 57 microslrip line, 145 parallel plate line, 100 stripline, 139 Chebyshev filter response, 390-391, 394-396 matching transformers, 250-255 polynomials, 251-252 Chip capacitor, resistor, 227-228 Choke bias, 515,524-525 flange, 117 Circular cavity (see Cavity resonators) Circular polarization, 23-24, 447-449 Circular waveguide, 117-126 attenuation, 121,122-124 cutoff frequency, 120,122-124 propagation constant, 120, 122, 124 table for, 124 Circulator ferrite junction, 478—482 general properties, 310,476-477 Coaxial connectors, 130 Coaxial line attenuation constant, 81, 83-84 characteristic impedance, 57 data for standard lines, 689 distributed line parameters, 54-55 field analysis, 55-57, 126-129 higher-order modes, 127-129 power capacity, 156 Composite filters, 386-389 Compression point, 488, 502 Computer aided design (CAD), 197 Conductivity, 10 table for metals, 687 Conductor loss, 25-27 Conjugate matching, 78-79,548-553 Connectors, coaxial, 130 Constant gain circles, 553-557 Constant-k filters, 380-382, 387 Constant noise figure circles, 557-561 Conversion loss, mixer, 618 Coplanar waveguide, 155-156 Coupled lines, 337-341 characteristic impedance, 338-341 couplers, 341-349 filters, 416-426 Couplers (see Directional couplers) Coupling aperture, 209-215 coefficient, 292 critical, 291-292 resonator, 291-298 Cross guide coupler, 361-362 Current displacement, 7 electric, magnetic, 6, 8-9 Cutoff frequency circular waveguide, 120,122-124 parallel plate waveguide, 101,104 rectangular waveguide, 108,112 Cutoff wavelength, 101, 105, 113,124 D DC block, 515.524-525 Decibel notation, 63-64 Demagnetization factor, 451-453 Detector, 509-513 sensitivity, 512 Dicke radiometer, 669-670 Dielectric constant, table, 687 Dielectric loaded waveguide, 115-116, 150 Dielectric loss, 25-26 Dielectric loss tangent, table, 687 Dielectric resonator oscillators, 590-594 Dielectric resonators, 287-291 Dielectric strength for air, 156-157 Dielectric waveguide, 155 Diode BARTTT, 521-522 detectors, 509-513 Gunn, 521.609-611 IMPATT, 521, 609-611 I-V curve, 510-511 mixer, 620-622 multipliers, 600-604 PIN, 514-515 Schottky, 509-511 Varactor, 520-521 Directional couplers, 311-314 Bethe hole, 324-327 coupled line, 341-349 Lange, 349-352 Moreno cross guide, 361-362 multihole waveguide, 327-332 quadrature, 333-336 Riblet short slot, 362 ring hybrid, 352-357 Schwinger reversed phase, 362 tapered line, 357-360 Directivity antenna, 638 coupler, 313-315 Discontinuities, 197-204 microstrip, 199, 203-204 waveguide, 198 Dispersion, 81,151 Distortionless line, 81-82 Double sideband modulation, 627 Dynamic range, 487, 500-501, 505-507 E Effective aperture area, 640-641 Effective isotropic radiated power (EIRP), 648 Effective permittivity, microstrip, 144 Efficiency aperture, 640 power added, 570 radiation, 639 Electric energy, 24 Electric field, 6 Electric flux density, 6 Electric polarizability, 211 Electric potential, 94-95 Electric susceptibility, 10 Electric wall, 14 Electromagnetic spectrum, 2 Elliptic filter, 391 Emissivity, 666 Energy, electric, magnetic, 24 Energy transmission, 675 E-plane T-junction, 315 Equal ripple filter response, 390-391, 394-396 Equivalent voltages and currents, 162-166 Even-odd mode characteristic impedance, 338-34t Exponential tapered line, 257-258 Extraordinary wave, 459-460 F Fabry-Perot resonator, 306 Far field, 636 Faraday rotation, 455-457 Faraday's law, 7 Ferrite devices circulators, 478-482 gyrator, 475-476 isolators, 465-471 loaded waveguide, 460-465 phase shifters, 471-475 Ferrites, 441 loss in, 449-451 permeability tensor for, 446-447 plane wave propagation in, 454-460 table of properties, 688 Index 696 Field effect transistors, 522-525 Filters bandpass, 401-405, 420-437 bandstop, 401-404, 427-431 composite, 386-389 constant-k, 380-382, 387 coupled line, 416-426 elliptic, 391 high pass, 387,400 high-Z, low-Z, 412-416 implementation, 405-412 linear phase, 391, 396-398 low pass, 380-389,400-401 m-derived, 383-386, 387 scaling, 398-401 transformations, 401-405 Flanges, waveguide, 116-117 Flow graph, 189-192 Frequency bands, 2,655 Frequency multipliers, 599-608 Friis power transmission formula, 647-648 G Gain (also see Power gain) amplifier, 540-542 antenna, 639-640 compression. 501-502 two-port power, 537-542 Global Positioning System (GPS)» 657-658 Group delay, 391 Group velocity, 151-154 for periodic structures, 376 for waveguide. 153-154 G/T.646 Gunn diode, 521,609-611 Gyrator, 475-476 Gyromagnetic ratio, 442 Gyrotropic medium (see Ferrites) H Helmholtz equations, 14-15 Hertz, H.,4 High pass filters constant-k, 382, 387 m-derived, 387 transformation to, 400 High-Z, low-Z filters, 412-416 History, of microwave engineering, 3-5 H-pIane T-junction, 315 Hybrid junctions coupled line, 341-349 quadrature. 333-336 ring (rat-race), 352-357 696 Index Hybrid junctions (Continued) scattering matrix, 333 tapered coupled line, 357-360 waveguide magic-T, 361 I Image frequency, 617-618 Image impedance, 378-380 Image parameters, filter design using, 378-389 Image theory, 42-44 IMPATT diode, 521,609-611 Impedance characteristic, 51-52 concept of, 166-169 image, 378-380 intrinsic, 16 wave, 16,17,95,96 Impedance inverter, 411-412 Impedance matching, 221-223 flode-Fano criterion, 261-263 double stub, 235-240 L-section, 223-227 multisection transformer, 245-255 quarter wave transformer, 73-76, 240-243 single stub, 228-235 tapered line, 255-261 Impedance matrix, 170-174 table for conversions, J 87 Impedance transformers (see Impedance matching) Incremental inductance rule, Wheeler, 84-86 Insertion loss, 63 Insertion loss method for filter design, 389-398 Intermodulation distortion, 502-505 Inverters, admittance, impedance, 411—412 Iris, waveguide, 198 Isolators field displacement, 469-471 resonance, 465-469 J Junction circulator, 478-482 K Kilters equation, 453 Klopfenstein tapered line, 258-261 Klystron, 613-614 Kuroda identities, 406-411 L Lange coupler, 349—352 Line parameters (per unit length), 52-54 Line width, gyromagnetic resonance, 450 Linear dynamic range, 505 Linear phase filter, 391, 396-398 Linearly polarized plane waves, 15-21 Load pull contours, 572 Loaded Q, 271 Loaded waveguide dielectric loading, 115-116, 150 ferrite loading, 460-465 Loss (see also Attenuation constant) conductor, 25-27 dielectric, 25-26 ferrite, 449-451 insertion, 63 return, 59 Loss tangent, 10 table, 687 Lossy transmission lines, 79-86 Low pass filters constant-k, 380-382, 387 high-Z, low-Z, 412-416 m-derived, 383-386, 387 prototype, 400-401 L-section matching, 223-227 M Magic-T, 361 Magnetic energy, 24 Magnetic field, 6 Magnetic flux density, 6 Magnetic polarizabiliry, 211 Magnetic susceptibility, 11 Magnetic wall, 14 Manley-Rowe relations, 600-602 Matched line, 58 Matching (see Impedance matching) Material constants table of conductivities, 687 table of dielectric constants and loss tangents, 687 table of ferrite properties, 688 Maximally flat filter response, 390, 394-396 Maximum power capacity, 156-157 Maximum stable gain, 551 Maxwell, J., 4-5 Maxwell's equations, 5-6, 8 m-derived filters, 383-386, 387 MEMs, 531 Microstrip, 143-149 approximate analysis, 146-148 attenuation, 145-146 characteristic impedance, 145 coupled, 339-340 effective permittivity, 144 propagation constant, 144 Microstrip discontinuities, 199, 203-204 Microwave heating, 674—675 Index 697 Microwave integrated circuits (MIC), 526-531 hybrid, 527-528 monolithic (MMIC), 528-531 Microwave oven, 674-675 Microwave sources, 608-615 Gunn diode, 609-611 IMPATT diode, 609-611 oscillators, 578-594 tubes, 612-615 Microwave tubes, 612-615 backward wave oscillator, 613 crossed-field amplifier, 614-615 extended interaction oscillator, 614 gyratron, 615 klystron, 613-614 magnetron, 612, 614 traveling wave tube, 613 Mixers, 510,615-630 antiparallel diode, 629-630 balanced, 625-627 conversion loss, 618 diode, 620-622 double balanced, 629 FET, 622-624 image rejection, 627-629 image response, 617-618 Modal analysis, 197-203 Modes cavity modes, 278-279, 282-284 circular waveguide, 118-125 parallel plate waveguide, 99-106 rectangular waveguide, 106-116 Modulation, 512-513 Multiple reflections, on quarter wave transformer, 75-76 Multipliers {see Frequency multipliers) N Negative resistance oscillators, 585-587 Neper, 63-64 Network analyzer, 182-183 Noise, 487-489 figure, 493-500 phase, 594-599 sources, 487-489,491 temperature, 489-493 Noise figure, 493-500 circles, 557-561 of cascade, 495-496 of lossy line, 494, 498-499 of mixer, 618-620 of passive network, 497-498 of transistor amplifier, 557-561 O Ohm's law for fields, 10 Open circuit stub, impedance, 61-62 Oscillators crystal, 584-585 dielectric resonator, 590-594 negative resistance, 585-587 transistor, 578-584, 587-590 P Parallel plate waveguide, 98-106 attenuation, 103, 104-105 characteristic impedance (TEM), 100 table for, 105 Passive intermodulation (PIM), 509 Periodic structures analysis, 372-375 k-/?diagram, 375-378 phase and group velocities, 376 Permanent magnets, 453-454 Permeability, 6, 11 tensor, for ferrite, 446-447 Permittivity, 6, 11 of atmosphere, 670 Perturbation theory for attenuation, 83-84 cavity resonance, 298-303 ferrite loaded waveguide, 463-464 Phase constant {see Propagation constant) Phase matching, 36 Phase noise, 594-599 Phase shifters Faraday rotation, 474^475 loaded line, 518-519 reflection, 519-520 Keggia-Spencer, 475 remanent (latching), 471-474 switched line, 517-518 Phase velocity plane wave, 15 transmission line, 52 waveguide, 100, 101, 105, 109,128,139 Phasor notation, 7-8 Physical constants, table, 686 PIN diodes, 514—515 phase shifters, 517-520 switches, 515-517 Plane waves, 15-24 in conducting media, 18-19 in ferrites, 454-460 in lossless dielectric, 15-16 reflection, 27-40 Plasma, 673-674 Polarizability, 211 Polarization, wave, 23 698 Index Power, 24-27 Power added efficiency (PAE), 570 Power amplifiers, 570-574 Power capacity of transmission line, 156-157 Power divider (see also Directional coupler) resistive, 317-318 T-junction, 315-317 Wilkinson, 318-324 Power gain, 537-542 Power loss, 25-27, 32-33, 83 Poynting's theorem, 24-25 Poynting vector, 25 Precession, magnetic dipole, 443-446 Probe coupling, 208-209 Propagation atmospheric effects, 670-672 ground effects, 672-673 plasma effects, 673-674 Propagation constant for circular waveguide, 120, 122. 124 coaxial line, 57 microstrip line, 144 parallel plate guide, 99, 100, 104 plane waves in a good conductor, 17, 19 plane waves in lossless dielectric, 15, 19 rectangular waveguide, 108,111,113 stripune, 139 TEM modes, 94 TM or TE modes, 96-97 Q Q. 268,271 for circular cavity, 284-286 for dielectric resonator, 290 for rectangular cavity, 279-281 for RLC circuit, 268, 270-272 for transmission line resonator, 274. 276, 277 Quadrature hybrid, 333-336 Quarter-wave transformers multiple reflection viewpoint, 75-76 multisection, 245-255 single-section, 73-76, 240-243 R Radar cross section, 664-665 Radar systems, 659-664 Radiation condition, 14 efficiency, 639 hazards, 675-677 patterns, 637-638 Radiometer systems, 665-670 Rat-race (ring hybrid), 352-357 Receivers, 650-655 Reciprocal networks, 171-172, 177-179 Reciprocity theorem, 40-41 Rectangular cavity (see Cavity resonators) Rectangular waveguide, 106-117 attenuation, 110-111,112-115 cutoff frequency, 108, 112 group velocity, 153-154 maximum power capacity, 156-157 phase velocity, 109, 113 propagation constant, 108. Ill, 113 table for, 113 table of standard sizes, 688 Rectification, 509-512 Reflection coefficient, 35, 58 Reflectomeier, 363-365 Remanent magnetization, 472 Resonant circuits, 266-272 Return loss, 59 Richard's transformation, 406 Ridge waveguide, 154-155 Root-finding algorithms, 136-137 S Saturation magnetization, 444 Scattering matrix, 174-182 for circulator, 310, 476-477 for directional coupler, 312-314 for gyrator, 475 for quadrature hybrid, 333 for ring hybrid, 352 generalized, 181-182 shift in reference planes, 180-181 table for conversions, 187 Schwinger reversed phase coupler, 362 Separation of variables, 19-20, 106-108, 118-119, 126-127, 128 Short circuit stub impedance, 60-61 Signal flow graphs, 189-192 Single sideband modulation, 627 Skin depth, 18-19 Slot line, 155-156 Slotted line, 69-73 Small reflection theory, 244-246 Smith chart, 64-69 Snell's law, 36 Sources (see Microwave sources) S parameters (see Scattering matrix) Spectrum analyzer, 514 Spurious free dynamic range, 505 Stability amplifier, 542-548 circles, 543-545 Index 699 Standing wave ratio (SWR), 59 Stepped impedance filters, 412-416 Stripline. 137-143 approximate analysis, 140-143 attenuation, 139-140 characteristic impedance, 139 coupled, 339 propagation constant, 139 Surface current, 8-9, 12-13 Surface impedance, 32-34 Surface resistance, 27, 33 Surface waves at dielectric interface, 38-40 of dielectric slab, 131-136 Switches, PIN diode, 515-517 T Tapered coupled line hybrid, 357-360 Tapered transmission lines exponential taper, 257-258 Klopfenstein taper, 258-261 triangular taper, 258-- ..... Telegrapher equations, 50, 56 /■' * . TE, TM modes ; v attenuation due to dielectric loss, 98 ; \* ■ propagation constant, -96-97 .*';/■ wave impedance, 96-971 -" . .* * TEM waves and modes attenuation due to dielectric loss, 98 plane waves, 15-22 propagation constant, 94 transmission lines, 55-57 wave impedance, 16,17, 57, 95 Terminated transmission line, 57-63 input impedance, 60 reflection coefficient, 58 voltage maxima and minima, 59 Third-order intercept, 504-505 T-junction, 315-317 Total reflection, plane wave, 38-40 Transducer power gain, 537-539 Transistor amplifier, 548-574 characteristics, 522-523 mixer, 622-624 models, 523-526 multipliers, 604-608 oscillator, 578-584, 587-590 types, 522-526 Transmission coefficient, 63 Transmission line equations, 50-51 input impedance, 60 junctions, 63 parameters, 52-54 Transmission line resonators, 272-277 Transmission lines coaxial, 54-57,126-129 microstrip, 143-149 parallel plate, 55, 98-106 stripline, 137-143 two-wire, 55 Transverse resonance method, 149-150 Traveling wave amplifier (see Amplifier design) Traveling waves plane waves, 15 on transmission lines, 51 TRL calibration, 193-196 Two-port networks, equivalent circuits, 186-189 Two-port power gains (see Power gain) V Varactor diode, 520-521 Velocity (see Wave velocities) Voltage standing wave ratio (see Standing wave ratio) W Wave equation, 15, 17,19 Wave velocities group, 151-154, 376 phase, 15, 52, 100,101, 105, 109, 376 Waveguide (see Rectangular waveguide; Circular waveguide; Loaded waveguide; Parallel plate waveguide) Waveguide components, 107 directional couplers, 324-332 discontinuities, 198 isolators, 465-471 magic-T, 361 phase shifters, 471 -475 T-junctions, 315 Waveguide excitation by apertures, 209-215 arbitrary sources, 206-209 current sheets, 204-206 TJ Unilateral device, 524 Unilateral figure of merit, 554 Unilateral transducer power gain, - -- 537-539 Unit element, 407 Unit matrix, 177 , Unitary matrix, 178 '■Unloaded Q,27l 700 Index Waveguide flanges, 116-117 Waveguide impedance, 96-97 Wavelength in free-space, 15—16 for waveguide, 101, 105, 109,113, 124 on transmission line, 52 Wheeler incremental inductance rule, 84-86 Wilkinson power divider, 318-324 Wireless systems, 646-647, 655-659 Y YIG-tuned oscillator, 610-612 Y-parameters (see Admittance matrix) Z Z-parameters (see Impedance matrix) 42 Solutions Manual for Microwave Engineering 3/e David Pozar 3/23/04 Solutions Manual for Microwave Engineering Third Edition Contained here are solutions for all of the end-of-chapter problems in the third edition of Microwave Engineering. Some of these problems require the derivation of theoretical results, but many are design oriented. Some of these problems are easy, while others are lengthy and challenging. Many of the matching, coupler, filter, and amplifier design problems ask for a CAD analysis of the final circuit, where it is presumed that the student has access to a microwave CAD software tool, such as Ansoft's Serenade, or similar. There are several such packages that are available for free download on the Internet. The Wiley web site contains Serenade files for the problems and examples amenable to CAD analysis. Working problems is a critical part of the learning process for engineering students, and these problems have been developed to give students practice in applying the basic concepts of microwave engineering, as well as practice in the analysis and design of practical microwave circuits and components. These problems can be used as assigned homework problems, exam problems, or as supplemental problems for students to work out on their own. The present edition features many new and revised problems, but if additional problems are needed, it should be easy for the instructor to derive new problems from those given in the text. Also new to this edition is the inclusion of short answers to many of the problems at the back of the text. The majority of these solutions have been checked with known results, compared with independent solutions by others or, in the case of design problems, verified by computer simulation. Such results usually have a check mark to indicate that they have a high (but not perfect!) likelihood of correctness. Nevertheless, there are undoubtedly some errors that remain, and the author will be grateful if such mistakes are brought to his attention. David Pozar Amherst Chapter i (Vv . 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