pg3085

File Attachment

Jarry_New_Thumbjpg

MICROWAVE AMPLIFIERAND ACTIVE CIRCUITDESIGN USING THE REALFREQUENCY TECHNIQUE

Pierre Jarry and Jacques N Beneat

Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved

Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada

MATLAB and Simulink are registered trademarks of The MathWorks Inc See wwwmathworkscomtrademarks for a list of additionaltrademarks The MathWorks Publisher Logo identifies books that contain MATLABreg content Used with permission TheMathWorks does not warrant the accuracy of the text or exercises in this book or in the software downloadable from httpwwwwileycomWileyCDAWileyTitleproductCd-047064477Xhtml and httpwwwmathworkscom matlabcentralfileexchangeterm=authorid3A80973 The bookrsquos or downloadable softwarersquos use or discussion of MATLABreg software or related products doesnot constitute endorsement or sponsorship by The MathWorks of a particular use of the MATLABreg software or related products

For MATLABreg and Simulinkreg product information or information on other related products please contactThe MathWorks Inc3 Apple Hill DriveNatick MA 01760-2098 USATel 508-647-7000Fax 508-647-7001E-mail infomathworkscomWeb wwwmathworkscom

No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronicmechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108 of the 1976 United StatesCopyright Act without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy feeto the Copyright Clearance Center Inc 222 Rosewood Drive Danvers MA 01923 (978) 750-8400 fax (978) 750-4470 or on the web atwwwcopyrightcom Requests to the Publisher for permission should be addressed to the Permissions Department JohnWiley amp Sons Inc111 River Street Hoboken NJ 07030 (201) 748-6011 fax (201) 748-6008 or online at httpwwwwileycomgopermissions

Limit of LiabilityDisclaimer of Warranty While the publisher and author have used their best efforts in preparing this book they makeno representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim anyimplied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representativesor written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with aprofessional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damagesincluding but not limited to special incidental consequential or other damages

For general information on our other products and services or for technical support please contact our Customer Care Department within theUnited States at (800) 762-2974 outside the United States at (317) 572-3993 or fax (317) 572-4002

Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronicformats For more information about Wiley products visit our web site at wwwwileycom

MATLABreg is a trademark of The MathWorks Inc and is used with permission The MathWorks doesnot warrant the accuracy of the text or exercises in this book This bookrsquos use or discussion ofMATLABreg software or related products does not constitute endorsement or sponsorship by TheMathWorks of a particular pedagogical approach or particular use of the MATLABreg software

Library of Congress Cataloging-in-Publication Data

Names Jarry Pierre 1946ndash author | Beneat Jacques 1964ndashTitle Microwave amplifier and active circuit design using the real frequency technique Pierre Jarry and Jacques N BeneatDescription Hoboken John Wiley amp Sons Inc 2016 | Includes bibliographical references and indexIdentifiers LCCN 2015040505 | ISBN 9781119073208 (hardback)Subjects LCSH Microwave amplifiersndashDesign and construction | Electric filters ActivendashDesign and construction |BISAC TECHNOLOGY amp ENGINEERING Microwaves

Classification LCC TK78712 J37 2016 | DDC 621381325ndashdc23LC record available at httplccnlocgov2015040505

Set in 1012pt Times by SPi Global Pondicherry India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

MICROWAVE AMPLIFIERAND ACTIVE CIRCUITDESIGN USING THE REALFREQUENCY TECHNIQUE

Pierre Jarry and Jacques N Beneat

Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved

Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada

MATLAB and Simulink are registered trademarks of The MathWorks Inc See wwwmathworkscomtrademarks for a list of additionaltrademarks The MathWorks Publisher Logo identifies books that contain MATLABreg content Used with permission TheMathWorks does not warrant the accuracy of the text or exercises in this book or in the software downloadable from httpwwwwileycomWileyCDAWileyTitleproductCd-047064477Xhtml and httpwwwmathworkscom matlabcentralfileexchangeterm=authorid3A80973 The bookrsquos or downloadable softwarersquos use or discussion of MATLABreg software or related products doesnot constitute endorsement or sponsorship by The MathWorks of a particular use of the MATLABreg software or related products

For MATLABreg and Simulinkreg product information or information on other related products please contactThe MathWorks Inc3 Apple Hill DriveNatick MA 01760-2098 USATel 508-647-7000Fax 508-647-7001E-mail infomathworkscomWeb wwwmathworkscom

No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronicmechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108 of the 1976 United StatesCopyright Act without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy feeto the Copyright Clearance Center Inc 222 Rosewood Drive Danvers MA 01923 (978) 750-8400 fax (978) 750-4470 or on the web atwwwcopyrightcom Requests to the Publisher for permission should be addressed to the Permissions Department JohnWiley amp Sons Inc111 River Street Hoboken NJ 07030 (201) 748-6011 fax (201) 748-6008 or online at httpwwwwileycomgopermissions

Limit of LiabilityDisclaimer of Warranty While the publisher and author have used their best efforts in preparing this book they makeno representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim anyimplied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representativesor written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with aprofessional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damagesincluding but not limited to special incidental consequential or other damages

For general information on our other products and services or for technical support please contact our Customer Care Department within theUnited States at (800) 762-2974 outside the United States at (317) 572-3993 or fax (317) 572-4002

Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronicformats For more information about Wiley products visit our web site at wwwwileycom

MATLABreg is a trademark of The MathWorks Inc and is used with permission The MathWorks doesnot warrant the accuracy of the text or exercises in this book This bookrsquos use or discussion ofMATLABreg software or related products does not constitute endorsement or sponsorship by TheMathWorks of a particular pedagogical approach or particular use of the MATLABreg software

Library of Congress Cataloging-in-Publication Data

Names Jarry Pierre 1946ndash author | Beneat Jacques 1964ndashTitle Microwave amplifier and active circuit design using the real frequency technique Pierre Jarry and Jacques N BeneatDescription Hoboken John Wiley amp Sons Inc 2016 | Includes bibliographical references and indexIdentifiers LCCN 2015040505 | ISBN 9781119073208 (hardback)Subjects LCSH Microwave amplifiersndashDesign and construction | Electric filters ActivendashDesign and construction |BISAC TECHNOLOGY amp ENGINEERING Microwaves

Classification LCC TK78712 J37 2016 | DDC 621381325ndashdc23LC record available at httplccnlocgov2015040505

Set in 1012pt Times by SPi Global Pondicherry India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved

Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada

MATLAB and Simulink are registered trademarks of The MathWorks Inc See wwwmathworkscomtrademarks for a list of additionaltrademarks The MathWorks Publisher Logo identifies books that contain MATLABreg content Used with permission TheMathWorks does not warrant the accuracy of the text or exercises in this book or in the software downloadable from httpwwwwileycomWileyCDAWileyTitleproductCd-047064477Xhtml and httpwwwmathworkscom matlabcentralfileexchangeterm=authorid3A80973 The bookrsquos or downloadable softwarersquos use or discussion of MATLABreg software or related products doesnot constitute endorsement or sponsorship by The MathWorks of a particular use of the MATLABreg software or related products

For MATLABreg and Simulinkreg product information or information on other related products please contactThe MathWorks Inc3 Apple Hill DriveNatick MA 01760-2098 USATel 508-647-7000Fax 508-647-7001E-mail infomathworkscomWeb wwwmathworkscom

No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronicmechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108 of the 1976 United StatesCopyright Act without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy feeto the Copyright Clearance Center Inc 222 Rosewood Drive Danvers MA 01923 (978) 750-8400 fax (978) 750-4470 or on the web atwwwcopyrightcom Requests to the Publisher for permission should be addressed to the Permissions Department JohnWiley amp Sons Inc111 River Street Hoboken NJ 07030 (201) 748-6011 fax (201) 748-6008 or online at httpwwwwileycomgopermissions

Limit of LiabilityDisclaimer of Warranty While the publisher and author have used their best efforts in preparing this book they makeno representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim anyimplied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representativesor written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with aprofessional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damagesincluding but not limited to special incidental consequential or other damages

For general information on our other products and services or for technical support please contact our Customer Care Department within theUnited States at (800) 762-2974 outside the United States at (317) 572-3993 or fax (317) 572-4002

Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronicformats For more information about Wiley products visit our web site at wwwwileycom

MATLABreg is a trademark of The MathWorks Inc and is used with permission The MathWorks doesnot warrant the accuracy of the text or exercises in this book This bookrsquos use or discussion ofMATLABreg software or related products does not constitute endorsement or sponsorship by TheMathWorks of a particular pedagogical approach or particular use of the MATLABreg software

Library of Congress Cataloging-in-Publication Data

Names Jarry Pierre 1946ndash author | Beneat Jacques 1964ndashTitle Microwave amplifier and active circuit design using the real frequency technique Pierre Jarry and Jacques N BeneatDescription Hoboken John Wiley amp Sons Inc 2016 | Includes bibliographical references and indexIdentifiers LCCN 2015040505 | ISBN 9781119073208 (hardback)Subjects LCSH Microwave amplifiersndashDesign and construction | Electric filters ActivendashDesign and construction |BISAC TECHNOLOGY amp ENGINEERING Microwaves

Classification LCC TK78712 J37 2016 | DDC 621381325ndashdc23LC record available at httplccnlocgov2015040505

Set in 1012pt Times by SPi Global Pondicherry India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

Contents

Foreword viiPreface ixAcknowledgments xiii

1 Microwave Amplifier Fundamentals 111 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25

188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

2 Introduction to the Real Frequency Technique Multistage LumpedAmplifier Design 3721 Introduction 3722 Multistage Lumped Amplifier Representation 3823 Overview of the RFT 4024 Multistage Transducer Gain 4125 Multistage VSWR 4326 Optimization Process 44

261 Single-Valued Error and Target Functions 44262 LevenbergndashMarquardtndashMore Optimization 46

27 Design Procedures 4828 Four-Stage Amplifier Design Example 4929 Transistor Feedback Block for Broadband Amplifiers 57

291 Resistive Adaptation 57292 Resistive Feedback 57293 Reactive Feedback 57294 Transistor Feedback Block 58

210 Realizations 592101 Three-Stage Hybrid Amplifier 592102 Two-Stage Monolithic Amplifier 622103 Single-Stage GaAs Technology Amplifier 64

References 64

3 Multistage Distributed Amplifier Design 6731 Introduction 6732 Multistage Distributed Amplifier Representation 6833 Multistage Transducer Gain 7034 Multistage VSWR 7135 Multistage Noise Figure 7336 Optimization Process 7437 Transistor Bias Circuit Considerations 7538 Distributed Equalizer Synthesis 78

381 Richardrsquos Theorem 78382 Stub Extraction 80383 Denormalization 82384 UE Impedances Too Low 83385 UE Impedances Too High 85

39 Design Procedures 88310 Simulations and Realizations 92

3101 Three-Stage 2ndash8 GHz Distributed Amplifier 923102 Three-Stage 115ndash15 GHz Distributed Amplifier 943103 Three-Stage 115ndash15 GHz Distributed Amplifier (Noncommensurate) 943104 Three-Stage 5925ndash6425 GHz Hybrid Amplifier 96

References 99

iv Contents

4 Multistage Transimpedance Amplifiers 10141 Introduction 10142 Multistage Transimpedance Amplifier Representation 10243 Extension to Distributed Equalizers 10444 Multistage Transimpedance Gain 10645 Multistage VSWR 10946 Optimization Process 11047 Design Procedures 11148 Noise Model of the Receiver Front End 11449 Two-Stage Transimpedance Amplifier Example 116References 118

5 Multistage Lossy Distributed Amplifiers 12151 Introduction 12152 Lossy Distributed Network 12253 Multistage Lossy Distributed Amplifier Representation 12754 Multistage Transducer Gain 13055 Multistage VSWR 13256 Optimization Process 13357 Synthesis of the Lossy Distributed Network 13558 Design Procedures 14159 Realizations 144

591 Single-Stage Broadband Hybrid Realization 144592 Two-Stage Broadband Hybrid Realization 145

References 149

6 Multistage Power Amplifiers 15161 Introduction 15162 Multistage Power Amplifier Representation 15263 Added Power Optimization 154

631 Requirements for Maximum Added Power 154632 Two-Dimensional Interpolation 156

64 Multistage Transducer Gain 15965 Multistage VSWR 16266 Optimization Process 16367 Design Procedures 16468 Realizations 166

681 Realization of a One-Stage Power Amplifier 166682 Realization of a Three-Stages Power Amplifier 167

69 Linear Power Amplifiers 172691 Theory 172692 Arborescent Structures 175693 Example of an Arborescent Linear Power Amplifier 176

References 179

7 Multistage Active Microwave Filters 18171 Introduction 18172 Multistage Active Filter Representation 182

vContents

73 Multistage Transducer Gain 18474 Multistage VSWR 18675 Multistage Phase and Group Delay 18776 Optimization Process 18877 Synthesis Procedures 18978 Design Procedures 19579 Simulations and Realizations 198

791 Two-Stage Low-Pass Active Filter 198792 Single-Stage Bandpass Active Filter 200793 Single-Stage Bandpass Active Filter MMIC Realization 202

References 206

8 Passive Microwave Equalizers for Radar Receiver Design 20781 Introduction 20782 Equalizer Needs for Radar Application 20883 Passive Equalizer Representation 20984 Optimization Process 21285 Examples of Microwave Equalizers for Radar Receivers 213

851 Sixth-Order Equalizer with No Transmission Zeros 213852 Sixth-Order Equalizer with Two Transmission Zeros 214

References 217

9 Synthesis of Microwave Antennas 21991 Introduction 21992 Antenna Needs 21993 Antenna Equalizer Representation 22194 Optimization Process 22295 Examples of Antenna-Matching Network Designs 223

951 Mid-Band Star Antenna 223952 Broadband Horn Antenna 224

References 227

Appendix A Multistage Transducer Gain 229Appendix B LevenbergndashMarquardtndashMore Optimization Algorithm 239Appendix C Noise Correlation Matrix 245Appendix D Network Synthesis Using the Transfer Matrix 253Index 271

vi Contents

Foreword

It has been a privilege for me to read through the manuscript of the Microwave Amplifier andActive Circuit Design Using the Real Frequency Technique I found that the authors have beenvery thorough in putting together this outstanding book containing a unique blend of theoryand practice I sensed that the authors are very passionate about the design of microwave circuitswhich is also evidenced from their other recent booksThe book provides an extensive use of an impedance matching methodology known as the real

frequency technique (RFT) in numerous applications The topics treated include the RFT itselfdesign of a wide variety of multistage amplifiers active filters passive equalizers for radar pulseshaping and antenna impedance matching applications All topics are self-contained and alsoinclude practical aspects Extensive analysis and optimization methods for these topics are dis-cussed The design techniques are well explained by means of solved examplesThe book is divided into nine chapters covering the basics of amplifiers an overview of RFT

multistage distributed amplifiers the use of RFT to design trans-impedance microwave amplifiersthe optimization of equalizers employing lossy distributed networks the use of RFT to designmultistage power amplifiers the design of multistage active filters the design of equalizers forradar pulse shaping and antenna impedance matching To solve impedance matching-relateddesign problems using RFT from specifications to realization of the end product the book pro-vides a unique integration of analysisoptimization aspects I found that the book is well balancedand treats the material in depth With emphasis on theory design and practical aspects applied tonumerous day-to-day applications the book is suitable for graduate students teachers and designengineersCongratulations Profs Jarry and Beneat on this excellent book that I am confident will be very

well received in the RF and microwave community for many years to come

Inder J BahlRoanoke VA

November 2015

Preface

Microwave and radio-frequency (RF) amplifiers play an important role in communication sys-tems and due to the proliferation of radar satellite and mobile wireless systems there is a needfor design methods that can satisfy the ever-increasing demand for accuracy reliability and fastdevelopment times This book provides an original design technique for a wide variety of multi-stage microwave amplifiers and active filters and passive equalizers for radar pulse shaping andantenna return loss applications This technique is referred to as the real frequency technique(RFT) It has grown out of the authors own research and teaching and as such has a unity of meth-odology and style essential for a smooth readingThe book is intended for researchers and RF and microwave engineers but is also suitable for an

advanced graduate course in the subject area Furthermore it is possible to choose material fromthe book to supplement traditional courses in microwave amplifier designEach chapter provides complete representation and characterization of the multistage or passive

equalizer structure as well as the design methodology We hope that this will provide theresearcher with a set of approaches that heshe could use for current and future microwave amp-lifier designs We also emphasize the practical nature of the subject by summarizing the designsteps and giving numerous examples of amplifier realizations and measured responses so thatRF and microwave engineers can have an appreciation of each amplifier in view of their needsThis approach we believe has produced a coherent practical and real-life treatment of the sub-ject The book is therefore theoretical but also experimental with over 18 microwave amplifierrealizationsThe book is divided into nine chapters In Chapter 1 recalls fundamental equations and defin-

itions useful for understanding the design of the microwave amplifiers presented hereChapter 2 provides a complete description of the RFT as it is first used to design multistage

lumped amplifiers The chapter starts with the multistage amplifier representation made of field-effect transistors (FETs) and equalizers defined using their scattering matrices In this introductorychapter the equalizers are assumed to be realizable as lumped ladder structures with transmissionzeroes at 0 or infinity The equalizers are optimized to improve the overall transducer gain and volt-age standing wave ratio (VSWR) The complete expressions for multistage transducer gain andVSWR are provided and further explained in Appendix A The RFT uses a progressive optimizationof the equalizers leading to a small number of parameters to optimize simultaneously This is a great

advantage of the RFT over typical design techniques that require simultaneous optimization of allunknown parametersThe RFT uses a practical implementation of the LevenbergndashMarquardt optimization method

and is summarized in Chapter 2 and detailed in Appendix B The optimization is performed overhundreds of frequency data points and is numerical in nature so that measured scattering param-eters of the transistors can be used The chapter then presents a complete step-by-step example ofthe design of a four-stage amplifier Intermediary and final results are provided A first extensionof the technique is shown It consists in using a transistor feedback circuit instead of the standalonetransistor to increase the useful bandwidth of the amplifier The chapter concludes with reportingtwo realizations designed using this technique in the range of 0ndash67 GHz and a realizationintended for 45 MHz to 65 GHz operationChapter 3 extends the RFT to the case multistage distributed amplifiers The equalizers are made

of quarter-wave transmission lines A new variable and formalism are introduced that are bettersuited for the distributed medium The modified RFT is able to optimize and synthesize the trans-mission lines for transducer gain VSWR and also for multistage noise figure This chapter showshow the effects of the bias circuit can be combined with the scattering parameters of the transistorto present a more accurate representation of the structure A method is provided to solve the real-ization problem of having too high or too low characteristic impedance requirements The chapterconcludes with realizations of a 115ndash15 GHz a 2ndash8 GHz and a 5925ndash6425 GHz amplifierChapter 4 presents the modifications to the RFT to design trans-impedance microwave ampli-

fiers that are used for example in the case of photodiodes acting as high impedance current sourcesIt must provide a flat gain for a load charge impedance of 50Ω Contrary to the previous cases theRFT performs progressive optimization of the equalizers from load to source (ie right to left)It uses admittance and impedance matrices rather than solely relying on scattering matrix repre-sentations A technique based on peaking inductor is described and it is sued to reduce the noise atthe input of the amplifier critical for the overall noise figure Results for lumped and distributedexamples from 3 to 7 GHz are givenIn Chapter 5 a method is presented for optimizing equalizers made of a lossy distributed net-

work Compared to the previous RFT for broadband multistage amplifiers parallel resistors at thegate and drain of the transistors become part of the RFT optimization process The chapter sets thefoundations for this new medium and topology An image network technique is used for definingthe properties of the equalizers as well as the synthesis technique needed for transmission lineswith lossy junctions The chapter presents two hybrid realizations of broadband lossy distributedamplifiers The first is optimized for gain and VSWR from 01 to 5 GHz and the second from 01 to9 GHzIn Chapter 6 it is shown how the RFT can be used in the case of multistage power amplifiers

It is shown how added power is optimized in addition to gain and VSWR The technique requiresinterpolating large-signal scattering parameters in two dimensions frequency and power Theoptimization is done from load to source as was the case in Chapter 4 The technique is usedin the realization of a single-stage 225 GHz power amplifier and in the case of a three-stage2245 GHz power amplifier The chapter also provides a method and examples for the designof linear multistage power amplifiers using arborescent structuresChapter 7 describes how to use the RFT to design multistage active filters In this chapter the

transducer gain and VSWR are optimized in the passband while finite transmission zeros can beplaced in the stopband to provide the desired selectivity of the filter The technique also optimizesthe group delay in the passband an important feature for radar and digital communications Threeactive filter examples are presented The first is a low-pass case with two transmission zeroes usingtwo transistors and three equalizers operating in the 01ndash5 GHz band The second is a band-pass

x Preface

case with four transmission zeroes using one transistor and two equalizers with a 3ndash7 GHz passband The third is a band-pass case with two transmission zeroes with a 77ndash81 GHz pass bandThis last case was realized in monolithic microwave integrated circuit (MMIC) technologyChapter 8 shows the flexibility of the RFT to solve a variety of microwave circuit design prob-

lems In this chapter the RFT is modified to optimize arbitrary responses using a structure madesolely of passive equalizer In this case a transistor is shown to act as a ldquodummy passive devicerdquowhen providing well-chosen scattering parameter data to represent it Therefore with very littlemodification the optimization capabilities of the traditional RFT can be used for passive struc-tures This chapter shows how the RFT no longer optimizes a constant transducer gain but ratheroptimizes a specific forward transmission curve for radar applications provided as a set of datapoints in a frequency bandTwo examples are provided The first equalizer has an order six and used no transmission zeroes

and the second equalizer has an order six and used two transmission zeroes to improve the opti-mization results in the stopbandFinally Chapter 9 describes a possible method for the synthesis of microwave antennas The

RFT is used to optimize the matching network placed between antenna and receivingtransmittingcircuitry The optimization focuses on improving the return loss in the receiving and transmit-ting bandsThe organization of the book is summarized in Figure 01 After a review chapter with important

amplifier and microwave notations and equations the central element is the RTF The RFT isintroduced in Chapter 2 for the design of multistage lumped amplifiers and is then modified tosolve numerous problems in microwave amplifier design (Chapters 3ndash7) and adapted to solvemore general microwave circuit design problems (Chapters 8 and 9)

Chapter 2

RFT core

lumped multistage

Chapter 9

Antenna design

Chapter 8

Misc equalizers

Chapter 7

Active filters

Chapter 6

Power amplifiers

Chapter 5

Lossy amplifiers

Chapter 4

Trans-impedance

Chapter 3

Distributed

Real frequency technique

Microwave amplifiers

Figure 01 Summary of the organization of the book

xiPreface

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

2 Introduction to the Real Frequency Technique Multistage LumpedAmplifier Design 3721 Introduction 3722 Multistage Lumped Amplifier Representation 3823 Overview of the RFT 4024 Multistage Transducer Gain 4125 Multistage VSWR 4326 Optimization Process 44

261 Single-Valued Error and Target Functions 44262 LevenbergndashMarquardtndashMore Optimization 46

27 Design Procedures 4828 Four-Stage Amplifier Design Example 4929 Transistor Feedback Block for Broadband Amplifiers 57

291 Resistive Adaptation 57292 Resistive Feedback 57293 Reactive Feedback 57294 Transistor Feedback Block 58

210 Realizations 592101 Three-Stage Hybrid Amplifier 592102 Two-Stage Monolithic Amplifier 622103 Single-Stage GaAs Technology Amplifier 64

References 64

3 Multistage Distributed Amplifier Design 6731 Introduction 6732 Multistage Distributed Amplifier Representation 6833 Multistage Transducer Gain 7034 Multistage VSWR 7135 Multistage Noise Figure 7336 Optimization Process 7437 Transistor Bias Circuit Considerations 7538 Distributed Equalizer Synthesis 78

381 Richardrsquos Theorem 78382 Stub Extraction 80383 Denormalization 82384 UE Impedances Too Low 83385 UE Impedances Too High 85

39 Design Procedures 88310 Simulations and Realizations 92

3101 Three-Stage 2ndash8 GHz Distributed Amplifier 923102 Three-Stage 115ndash15 GHz Distributed Amplifier 943103 Three-Stage 115ndash15 GHz Distributed Amplifier (Noncommensurate) 943104 Three-Stage 5925ndash6425 GHz Hybrid Amplifier 96

References 99

iv Contents

4 Multistage Transimpedance Amplifiers 10141 Introduction 10142 Multistage Transimpedance Amplifier Representation 10243 Extension to Distributed Equalizers 10444 Multistage Transimpedance Gain 10645 Multistage VSWR 10946 Optimization Process 11047 Design Procedures 11148 Noise Model of the Receiver Front End 11449 Two-Stage Transimpedance Amplifier Example 116References 118

5 Multistage Lossy Distributed Amplifiers 12151 Introduction 12152 Lossy Distributed Network 12253 Multistage Lossy Distributed Amplifier Representation 12754 Multistage Transducer Gain 13055 Multistage VSWR 13256 Optimization Process 13357 Synthesis of the Lossy Distributed Network 13558 Design Procedures 14159 Realizations 144

591 Single-Stage Broadband Hybrid Realization 144592 Two-Stage Broadband Hybrid Realization 145

References 149

6 Multistage Power Amplifiers 15161 Introduction 15162 Multistage Power Amplifier Representation 15263 Added Power Optimization 154

631 Requirements for Maximum Added Power 154632 Two-Dimensional Interpolation 156

64 Multistage Transducer Gain 15965 Multistage VSWR 16266 Optimization Process 16367 Design Procedures 16468 Realizations 166

681 Realization of a One-Stage Power Amplifier 166682 Realization of a Three-Stages Power Amplifier 167

69 Linear Power Amplifiers 172691 Theory 172692 Arborescent Structures 175693 Example of an Arborescent Linear Power Amplifier 176

References 179

7 Multistage Active Microwave Filters 18171 Introduction 18172 Multistage Active Filter Representation 182

vContents

73 Multistage Transducer Gain 18474 Multistage VSWR 18675 Multistage Phase and Group Delay 18776 Optimization Process 18877 Synthesis Procedures 18978 Design Procedures 19579 Simulations and Realizations 198

791 Two-Stage Low-Pass Active Filter 198792 Single-Stage Bandpass Active Filter 200793 Single-Stage Bandpass Active Filter MMIC Realization 202

References 206

8 Passive Microwave Equalizers for Radar Receiver Design 20781 Introduction 20782 Equalizer Needs for Radar Application 20883 Passive Equalizer Representation 20984 Optimization Process 21285 Examples of Microwave Equalizers for Radar Receivers 213

851 Sixth-Order Equalizer with No Transmission Zeros 213852 Sixth-Order Equalizer with Two Transmission Zeros 214

References 217

9 Synthesis of Microwave Antennas 21991 Introduction 21992 Antenna Needs 21993 Antenna Equalizer Representation 22194 Optimization Process 22295 Examples of Antenna-Matching Network Designs 223

951 Mid-Band Star Antenna 223952 Broadband Horn Antenna 224

References 227

Appendix A Multistage Transducer Gain 229Appendix B LevenbergndashMarquardtndashMore Optimization Algorithm 239Appendix C Noise Correlation Matrix 245Appendix D Network Synthesis Using the Transfer Matrix 253Index 271

vi Contents

Foreword

It has been a privilege for me to read through the manuscript of the Microwave Amplifier andActive Circuit Design Using the Real Frequency Technique I found that the authors have beenvery thorough in putting together this outstanding book containing a unique blend of theoryand practice I sensed that the authors are very passionate about the design of microwave circuitswhich is also evidenced from their other recent booksThe book provides an extensive use of an impedance matching methodology known as the real

frequency technique (RFT) in numerous applications The topics treated include the RFT itselfdesign of a wide variety of multistage amplifiers active filters passive equalizers for radar pulseshaping and antenna impedance matching applications All topics are self-contained and alsoinclude practical aspects Extensive analysis and optimization methods for these topics are dis-cussed The design techniques are well explained by means of solved examplesThe book is divided into nine chapters covering the basics of amplifiers an overview of RFT

multistage distributed amplifiers the use of RFT to design trans-impedance microwave amplifiersthe optimization of equalizers employing lossy distributed networks the use of RFT to designmultistage power amplifiers the design of multistage active filters the design of equalizers forradar pulse shaping and antenna impedance matching To solve impedance matching-relateddesign problems using RFT from specifications to realization of the end product the book pro-vides a unique integration of analysisoptimization aspects I found that the book is well balancedand treats the material in depth With emphasis on theory design and practical aspects applied tonumerous day-to-day applications the book is suitable for graduate students teachers and designengineersCongratulations Profs Jarry and Beneat on this excellent book that I am confident will be very

well received in the RF and microwave community for many years to come

Inder J BahlRoanoke VA

November 2015

Preface

Microwave and radio-frequency (RF) amplifiers play an important role in communication sys-tems and due to the proliferation of radar satellite and mobile wireless systems there is a needfor design methods that can satisfy the ever-increasing demand for accuracy reliability and fastdevelopment times This book provides an original design technique for a wide variety of multi-stage microwave amplifiers and active filters and passive equalizers for radar pulse shaping andantenna return loss applications This technique is referred to as the real frequency technique(RFT) It has grown out of the authors own research and teaching and as such has a unity of meth-odology and style essential for a smooth readingThe book is intended for researchers and RF and microwave engineers but is also suitable for an

advanced graduate course in the subject area Furthermore it is possible to choose material fromthe book to supplement traditional courses in microwave amplifier designEach chapter provides complete representation and characterization of the multistage or passive

equalizer structure as well as the design methodology We hope that this will provide theresearcher with a set of approaches that heshe could use for current and future microwave amp-lifier designs We also emphasize the practical nature of the subject by summarizing the designsteps and giving numerous examples of amplifier realizations and measured responses so thatRF and microwave engineers can have an appreciation of each amplifier in view of their needsThis approach we believe has produced a coherent practical and real-life treatment of the sub-ject The book is therefore theoretical but also experimental with over 18 microwave amplifierrealizationsThe book is divided into nine chapters In Chapter 1 recalls fundamental equations and defin-

itions useful for understanding the design of the microwave amplifiers presented hereChapter 2 provides a complete description of the RFT as it is first used to design multistage

lumped amplifiers The chapter starts with the multistage amplifier representation made of field-effect transistors (FETs) and equalizers defined using their scattering matrices In this introductorychapter the equalizers are assumed to be realizable as lumped ladder structures with transmissionzeroes at 0 or infinity The equalizers are optimized to improve the overall transducer gain and volt-age standing wave ratio (VSWR) The complete expressions for multistage transducer gain andVSWR are provided and further explained in Appendix A The RFT uses a progressive optimizationof the equalizers leading to a small number of parameters to optimize simultaneously This is a great

advantage of the RFT over typical design techniques that require simultaneous optimization of allunknown parametersThe RFT uses a practical implementation of the LevenbergndashMarquardt optimization method

and is summarized in Chapter 2 and detailed in Appendix B The optimization is performed overhundreds of frequency data points and is numerical in nature so that measured scattering param-eters of the transistors can be used The chapter then presents a complete step-by-step example ofthe design of a four-stage amplifier Intermediary and final results are provided A first extensionof the technique is shown It consists in using a transistor feedback circuit instead of the standalonetransistor to increase the useful bandwidth of the amplifier The chapter concludes with reportingtwo realizations designed using this technique in the range of 0ndash67 GHz and a realizationintended for 45 MHz to 65 GHz operationChapter 3 extends the RFT to the case multistage distributed amplifiers The equalizers are made

of quarter-wave transmission lines A new variable and formalism are introduced that are bettersuited for the distributed medium The modified RFT is able to optimize and synthesize the trans-mission lines for transducer gain VSWR and also for multistage noise figure This chapter showshow the effects of the bias circuit can be combined with the scattering parameters of the transistorto present a more accurate representation of the structure A method is provided to solve the real-ization problem of having too high or too low characteristic impedance requirements The chapterconcludes with realizations of a 115ndash15 GHz a 2ndash8 GHz and a 5925ndash6425 GHz amplifierChapter 4 presents the modifications to the RFT to design trans-impedance microwave ampli-

fiers that are used for example in the case of photodiodes acting as high impedance current sourcesIt must provide a flat gain for a load charge impedance of 50Ω Contrary to the previous cases theRFT performs progressive optimization of the equalizers from load to source (ie right to left)It uses admittance and impedance matrices rather than solely relying on scattering matrix repre-sentations A technique based on peaking inductor is described and it is sued to reduce the noise atthe input of the amplifier critical for the overall noise figure Results for lumped and distributedexamples from 3 to 7 GHz are givenIn Chapter 5 a method is presented for optimizing equalizers made of a lossy distributed net-

work Compared to the previous RFT for broadband multistage amplifiers parallel resistors at thegate and drain of the transistors become part of the RFT optimization process The chapter sets thefoundations for this new medium and topology An image network technique is used for definingthe properties of the equalizers as well as the synthesis technique needed for transmission lineswith lossy junctions The chapter presents two hybrid realizations of broadband lossy distributedamplifiers The first is optimized for gain and VSWR from 01 to 5 GHz and the second from 01 to9 GHzIn Chapter 6 it is shown how the RFT can be used in the case of multistage power amplifiers

It is shown how added power is optimized in addition to gain and VSWR The technique requiresinterpolating large-signal scattering parameters in two dimensions frequency and power Theoptimization is done from load to source as was the case in Chapter 4 The technique is usedin the realization of a single-stage 225 GHz power amplifier and in the case of a three-stage2245 GHz power amplifier The chapter also provides a method and examples for the designof linear multistage power amplifiers using arborescent structuresChapter 7 describes how to use the RFT to design multistage active filters In this chapter the

transducer gain and VSWR are optimized in the passband while finite transmission zeros can beplaced in the stopband to provide the desired selectivity of the filter The technique also optimizesthe group delay in the passband an important feature for radar and digital communications Threeactive filter examples are presented The first is a low-pass case with two transmission zeroes usingtwo transistors and three equalizers operating in the 01ndash5 GHz band The second is a band-pass

x Preface

case with four transmission zeroes using one transistor and two equalizers with a 3ndash7 GHz passband The third is a band-pass case with two transmission zeroes with a 77ndash81 GHz pass bandThis last case was realized in monolithic microwave integrated circuit (MMIC) technologyChapter 8 shows the flexibility of the RFT to solve a variety of microwave circuit design prob-

lems In this chapter the RFT is modified to optimize arbitrary responses using a structure madesolely of passive equalizer In this case a transistor is shown to act as a ldquodummy passive devicerdquowhen providing well-chosen scattering parameter data to represent it Therefore with very littlemodification the optimization capabilities of the traditional RFT can be used for passive struc-tures This chapter shows how the RFT no longer optimizes a constant transducer gain but ratheroptimizes a specific forward transmission curve for radar applications provided as a set of datapoints in a frequency bandTwo examples are provided The first equalizer has an order six and used no transmission zeroes

and the second equalizer has an order six and used two transmission zeroes to improve the opti-mization results in the stopbandFinally Chapter 9 describes a possible method for the synthesis of microwave antennas The

RFT is used to optimize the matching network placed between antenna and receivingtransmittingcircuitry The optimization focuses on improving the return loss in the receiving and transmit-ting bandsThe organization of the book is summarized in Figure 01 After a review chapter with important

amplifier and microwave notations and equations the central element is the RTF The RFT isintroduced in Chapter 2 for the design of multistage lumped amplifiers and is then modified tosolve numerous problems in microwave amplifier design (Chapters 3ndash7) and adapted to solvemore general microwave circuit design problems (Chapters 8 and 9)

Chapter 2

RFT core

lumped multistage

Chapter 9

Antenna design

Chapter 8

Misc equalizers

Chapter 7

Active filters

Chapter 6

Power amplifiers

Chapter 5

Lossy amplifiers

Chapter 4

Trans-impedance

Chapter 3

Distributed

Real frequency technique

Microwave amplifiers

Figure 01 Summary of the organization of the book

xiPreface

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

4 Multistage Transimpedance Amplifiers 10141 Introduction 10142 Multistage Transimpedance Amplifier Representation 10243 Extension to Distributed Equalizers 10444 Multistage Transimpedance Gain 10645 Multistage VSWR 10946 Optimization Process 11047 Design Procedures 11148 Noise Model of the Receiver Front End 11449 Two-Stage Transimpedance Amplifier Example 116References 118

5 Multistage Lossy Distributed Amplifiers 12151 Introduction 12152 Lossy Distributed Network 12253 Multistage Lossy Distributed Amplifier Representation 12754 Multistage Transducer Gain 13055 Multistage VSWR 13256 Optimization Process 13357 Synthesis of the Lossy Distributed Network 13558 Design Procedures 14159 Realizations 144

591 Single-Stage Broadband Hybrid Realization 144592 Two-Stage Broadband Hybrid Realization 145

References 149

6 Multistage Power Amplifiers 15161 Introduction 15162 Multistage Power Amplifier Representation 15263 Added Power Optimization 154

631 Requirements for Maximum Added Power 154632 Two-Dimensional Interpolation 156

64 Multistage Transducer Gain 15965 Multistage VSWR 16266 Optimization Process 16367 Design Procedures 16468 Realizations 166

681 Realization of a One-Stage Power Amplifier 166682 Realization of a Three-Stages Power Amplifier 167

69 Linear Power Amplifiers 172691 Theory 172692 Arborescent Structures 175693 Example of an Arborescent Linear Power Amplifier 176

References 179

7 Multistage Active Microwave Filters 18171 Introduction 18172 Multistage Active Filter Representation 182

vContents

73 Multistage Transducer Gain 18474 Multistage VSWR 18675 Multistage Phase and Group Delay 18776 Optimization Process 18877 Synthesis Procedures 18978 Design Procedures 19579 Simulations and Realizations 198

791 Two-Stage Low-Pass Active Filter 198792 Single-Stage Bandpass Active Filter 200793 Single-Stage Bandpass Active Filter MMIC Realization 202

References 206

8 Passive Microwave Equalizers for Radar Receiver Design 20781 Introduction 20782 Equalizer Needs for Radar Application 20883 Passive Equalizer Representation 20984 Optimization Process 21285 Examples of Microwave Equalizers for Radar Receivers 213

851 Sixth-Order Equalizer with No Transmission Zeros 213852 Sixth-Order Equalizer with Two Transmission Zeros 214

References 217

9 Synthesis of Microwave Antennas 21991 Introduction 21992 Antenna Needs 21993 Antenna Equalizer Representation 22194 Optimization Process 22295 Examples of Antenna-Matching Network Designs 223

951 Mid-Band Star Antenna 223952 Broadband Horn Antenna 224

References 227

Appendix A Multistage Transducer Gain 229Appendix B LevenbergndashMarquardtndashMore Optimization Algorithm 239Appendix C Noise Correlation Matrix 245Appendix D Network Synthesis Using the Transfer Matrix 253Index 271

vi Contents

Foreword

It has been a privilege for me to read through the manuscript of the Microwave Amplifier andActive Circuit Design Using the Real Frequency Technique I found that the authors have beenvery thorough in putting together this outstanding book containing a unique blend of theoryand practice I sensed that the authors are very passionate about the design of microwave circuitswhich is also evidenced from their other recent booksThe book provides an extensive use of an impedance matching methodology known as the real

frequency technique (RFT) in numerous applications The topics treated include the RFT itselfdesign of a wide variety of multistage amplifiers active filters passive equalizers for radar pulseshaping and antenna impedance matching applications All topics are self-contained and alsoinclude practical aspects Extensive analysis and optimization methods for these topics are dis-cussed The design techniques are well explained by means of solved examplesThe book is divided into nine chapters covering the basics of amplifiers an overview of RFT

multistage distributed amplifiers the use of RFT to design trans-impedance microwave amplifiersthe optimization of equalizers employing lossy distributed networks the use of RFT to designmultistage power amplifiers the design of multistage active filters the design of equalizers forradar pulse shaping and antenna impedance matching To solve impedance matching-relateddesign problems using RFT from specifications to realization of the end product the book pro-vides a unique integration of analysisoptimization aspects I found that the book is well balancedand treats the material in depth With emphasis on theory design and practical aspects applied tonumerous day-to-day applications the book is suitable for graduate students teachers and designengineersCongratulations Profs Jarry and Beneat on this excellent book that I am confident will be very

well received in the RF and microwave community for many years to come

Inder J BahlRoanoke VA

November 2015

Preface

Microwave and radio-frequency (RF) amplifiers play an important role in communication sys-tems and due to the proliferation of radar satellite and mobile wireless systems there is a needfor design methods that can satisfy the ever-increasing demand for accuracy reliability and fastdevelopment times This book provides an original design technique for a wide variety of multi-stage microwave amplifiers and active filters and passive equalizers for radar pulse shaping andantenna return loss applications This technique is referred to as the real frequency technique(RFT) It has grown out of the authors own research and teaching and as such has a unity of meth-odology and style essential for a smooth readingThe book is intended for researchers and RF and microwave engineers but is also suitable for an

advanced graduate course in the subject area Furthermore it is possible to choose material fromthe book to supplement traditional courses in microwave amplifier designEach chapter provides complete representation and characterization of the multistage or passive

equalizer structure as well as the design methodology We hope that this will provide theresearcher with a set of approaches that heshe could use for current and future microwave amp-lifier designs We also emphasize the practical nature of the subject by summarizing the designsteps and giving numerous examples of amplifier realizations and measured responses so thatRF and microwave engineers can have an appreciation of each amplifier in view of their needsThis approach we believe has produced a coherent practical and real-life treatment of the sub-ject The book is therefore theoretical but also experimental with over 18 microwave amplifierrealizationsThe book is divided into nine chapters In Chapter 1 recalls fundamental equations and defin-

itions useful for understanding the design of the microwave amplifiers presented hereChapter 2 provides a complete description of the RFT as it is first used to design multistage

lumped amplifiers The chapter starts with the multistage amplifier representation made of field-effect transistors (FETs) and equalizers defined using their scattering matrices In this introductorychapter the equalizers are assumed to be realizable as lumped ladder structures with transmissionzeroes at 0 or infinity The equalizers are optimized to improve the overall transducer gain and volt-age standing wave ratio (VSWR) The complete expressions for multistage transducer gain andVSWR are provided and further explained in Appendix A The RFT uses a progressive optimizationof the equalizers leading to a small number of parameters to optimize simultaneously This is a great

advantage of the RFT over typical design techniques that require simultaneous optimization of allunknown parametersThe RFT uses a practical implementation of the LevenbergndashMarquardt optimization method

and is summarized in Chapter 2 and detailed in Appendix B The optimization is performed overhundreds of frequency data points and is numerical in nature so that measured scattering param-eters of the transistors can be used The chapter then presents a complete step-by-step example ofthe design of a four-stage amplifier Intermediary and final results are provided A first extensionof the technique is shown It consists in using a transistor feedback circuit instead of the standalonetransistor to increase the useful bandwidth of the amplifier The chapter concludes with reportingtwo realizations designed using this technique in the range of 0ndash67 GHz and a realizationintended for 45 MHz to 65 GHz operationChapter 3 extends the RFT to the case multistage distributed amplifiers The equalizers are made

of quarter-wave transmission lines A new variable and formalism are introduced that are bettersuited for the distributed medium The modified RFT is able to optimize and synthesize the trans-mission lines for transducer gain VSWR and also for multistage noise figure This chapter showshow the effects of the bias circuit can be combined with the scattering parameters of the transistorto present a more accurate representation of the structure A method is provided to solve the real-ization problem of having too high or too low characteristic impedance requirements The chapterconcludes with realizations of a 115ndash15 GHz a 2ndash8 GHz and a 5925ndash6425 GHz amplifierChapter 4 presents the modifications to the RFT to design trans-impedance microwave ampli-

fiers that are used for example in the case of photodiodes acting as high impedance current sourcesIt must provide a flat gain for a load charge impedance of 50Ω Contrary to the previous cases theRFT performs progressive optimization of the equalizers from load to source (ie right to left)It uses admittance and impedance matrices rather than solely relying on scattering matrix repre-sentations A technique based on peaking inductor is described and it is sued to reduce the noise atthe input of the amplifier critical for the overall noise figure Results for lumped and distributedexamples from 3 to 7 GHz are givenIn Chapter 5 a method is presented for optimizing equalizers made of a lossy distributed net-

work Compared to the previous RFT for broadband multistage amplifiers parallel resistors at thegate and drain of the transistors become part of the RFT optimization process The chapter sets thefoundations for this new medium and topology An image network technique is used for definingthe properties of the equalizers as well as the synthesis technique needed for transmission lineswith lossy junctions The chapter presents two hybrid realizations of broadband lossy distributedamplifiers The first is optimized for gain and VSWR from 01 to 5 GHz and the second from 01 to9 GHzIn Chapter 6 it is shown how the RFT can be used in the case of multistage power amplifiers

It is shown how added power is optimized in addition to gain and VSWR The technique requiresinterpolating large-signal scattering parameters in two dimensions frequency and power Theoptimization is done from load to source as was the case in Chapter 4 The technique is usedin the realization of a single-stage 225 GHz power amplifier and in the case of a three-stage2245 GHz power amplifier The chapter also provides a method and examples for the designof linear multistage power amplifiers using arborescent structuresChapter 7 describes how to use the RFT to design multistage active filters In this chapter the

transducer gain and VSWR are optimized in the passband while finite transmission zeros can beplaced in the stopband to provide the desired selectivity of the filter The technique also optimizesthe group delay in the passband an important feature for radar and digital communications Threeactive filter examples are presented The first is a low-pass case with two transmission zeroes usingtwo transistors and three equalizers operating in the 01ndash5 GHz band The second is a band-pass

x Preface

case with four transmission zeroes using one transistor and two equalizers with a 3ndash7 GHz passband The third is a band-pass case with two transmission zeroes with a 77ndash81 GHz pass bandThis last case was realized in monolithic microwave integrated circuit (MMIC) technologyChapter 8 shows the flexibility of the RFT to solve a variety of microwave circuit design prob-

lems In this chapter the RFT is modified to optimize arbitrary responses using a structure madesolely of passive equalizer In this case a transistor is shown to act as a ldquodummy passive devicerdquowhen providing well-chosen scattering parameter data to represent it Therefore with very littlemodification the optimization capabilities of the traditional RFT can be used for passive struc-tures This chapter shows how the RFT no longer optimizes a constant transducer gain but ratheroptimizes a specific forward transmission curve for radar applications provided as a set of datapoints in a frequency bandTwo examples are provided The first equalizer has an order six and used no transmission zeroes

and the second equalizer has an order six and used two transmission zeroes to improve the opti-mization results in the stopbandFinally Chapter 9 describes a possible method for the synthesis of microwave antennas The

RFT is used to optimize the matching network placed between antenna and receivingtransmittingcircuitry The optimization focuses on improving the return loss in the receiving and transmit-ting bandsThe organization of the book is summarized in Figure 01 After a review chapter with important

amplifier and microwave notations and equations the central element is the RTF The RFT isintroduced in Chapter 2 for the design of multistage lumped amplifiers and is then modified tosolve numerous problems in microwave amplifier design (Chapters 3ndash7) and adapted to solvemore general microwave circuit design problems (Chapters 8 and 9)

Chapter 2

RFT core

lumped multistage

Chapter 9

Antenna design

Chapter 8

Misc equalizers

Chapter 7

Active filters

Chapter 6

Power amplifiers

Chapter 5

Lossy amplifiers

Chapter 4

Trans-impedance

Chapter 3

Distributed

Real frequency technique

Microwave amplifiers

Figure 01 Summary of the organization of the book

xiPreface

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

73 Multistage Transducer Gain 18474 Multistage VSWR 18675 Multistage Phase and Group Delay 18776 Optimization Process 18877 Synthesis Procedures 18978 Design Procedures 19579 Simulations and Realizations 198

791 Two-Stage Low-Pass Active Filter 198792 Single-Stage Bandpass Active Filter 200793 Single-Stage Bandpass Active Filter MMIC Realization 202

References 206

8 Passive Microwave Equalizers for Radar Receiver Design 20781 Introduction 20782 Equalizer Needs for Radar Application 20883 Passive Equalizer Representation 20984 Optimization Process 21285 Examples of Microwave Equalizers for Radar Receivers 213

851 Sixth-Order Equalizer with No Transmission Zeros 213852 Sixth-Order Equalizer with Two Transmission Zeros 214

References 217

9 Synthesis of Microwave Antennas 21991 Introduction 21992 Antenna Needs 21993 Antenna Equalizer Representation 22194 Optimization Process 22295 Examples of Antenna-Matching Network Designs 223

951 Mid-Band Star Antenna 223952 Broadband Horn Antenna 224

References 227

Appendix A Multistage Transducer Gain 229Appendix B LevenbergndashMarquardtndashMore Optimization Algorithm 239Appendix C Noise Correlation Matrix 245Appendix D Network Synthesis Using the Transfer Matrix 253Index 271

vi Contents

Foreword

It has been a privilege for me to read through the manuscript of the Microwave Amplifier andActive Circuit Design Using the Real Frequency Technique I found that the authors have beenvery thorough in putting together this outstanding book containing a unique blend of theoryand practice I sensed that the authors are very passionate about the design of microwave circuitswhich is also evidenced from their other recent booksThe book provides an extensive use of an impedance matching methodology known as the real

frequency technique (RFT) in numerous applications The topics treated include the RFT itselfdesign of a wide variety of multistage amplifiers active filters passive equalizers for radar pulseshaping and antenna impedance matching applications All topics are self-contained and alsoinclude practical aspects Extensive analysis and optimization methods for these topics are dis-cussed The design techniques are well explained by means of solved examplesThe book is divided into nine chapters covering the basics of amplifiers an overview of RFT

multistage distributed amplifiers the use of RFT to design trans-impedance microwave amplifiersthe optimization of equalizers employing lossy distributed networks the use of RFT to designmultistage power amplifiers the design of multistage active filters the design of equalizers forradar pulse shaping and antenna impedance matching To solve impedance matching-relateddesign problems using RFT from specifications to realization of the end product the book pro-vides a unique integration of analysisoptimization aspects I found that the book is well balancedand treats the material in depth With emphasis on theory design and practical aspects applied tonumerous day-to-day applications the book is suitable for graduate students teachers and designengineersCongratulations Profs Jarry and Beneat on this excellent book that I am confident will be very

well received in the RF and microwave community for many years to come

Inder J BahlRoanoke VA

November 2015

Preface

Microwave and radio-frequency (RF) amplifiers play an important role in communication sys-tems and due to the proliferation of radar satellite and mobile wireless systems there is a needfor design methods that can satisfy the ever-increasing demand for accuracy reliability and fastdevelopment times This book provides an original design technique for a wide variety of multi-stage microwave amplifiers and active filters and passive equalizers for radar pulse shaping andantenna return loss applications This technique is referred to as the real frequency technique(RFT) It has grown out of the authors own research and teaching and as such has a unity of meth-odology and style essential for a smooth readingThe book is intended for researchers and RF and microwave engineers but is also suitable for an

advanced graduate course in the subject area Furthermore it is possible to choose material fromthe book to supplement traditional courses in microwave amplifier designEach chapter provides complete representation and characterization of the multistage or passive

equalizer structure as well as the design methodology We hope that this will provide theresearcher with a set of approaches that heshe could use for current and future microwave amp-lifier designs We also emphasize the practical nature of the subject by summarizing the designsteps and giving numerous examples of amplifier realizations and measured responses so thatRF and microwave engineers can have an appreciation of each amplifier in view of their needsThis approach we believe has produced a coherent practical and real-life treatment of the sub-ject The book is therefore theoretical but also experimental with over 18 microwave amplifierrealizationsThe book is divided into nine chapters In Chapter 1 recalls fundamental equations and defin-

itions useful for understanding the design of the microwave amplifiers presented hereChapter 2 provides a complete description of the RFT as it is first used to design multistage

lumped amplifiers The chapter starts with the multistage amplifier representation made of field-effect transistors (FETs) and equalizers defined using their scattering matrices In this introductorychapter the equalizers are assumed to be realizable as lumped ladder structures with transmissionzeroes at 0 or infinity The equalizers are optimized to improve the overall transducer gain and volt-age standing wave ratio (VSWR) The complete expressions for multistage transducer gain andVSWR are provided and further explained in Appendix A The RFT uses a progressive optimizationof the equalizers leading to a small number of parameters to optimize simultaneously This is a great

advantage of the RFT over typical design techniques that require simultaneous optimization of allunknown parametersThe RFT uses a practical implementation of the LevenbergndashMarquardt optimization method

and is summarized in Chapter 2 and detailed in Appendix B The optimization is performed overhundreds of frequency data points and is numerical in nature so that measured scattering param-eters of the transistors can be used The chapter then presents a complete step-by-step example ofthe design of a four-stage amplifier Intermediary and final results are provided A first extensionof the technique is shown It consists in using a transistor feedback circuit instead of the standalonetransistor to increase the useful bandwidth of the amplifier The chapter concludes with reportingtwo realizations designed using this technique in the range of 0ndash67 GHz and a realizationintended for 45 MHz to 65 GHz operationChapter 3 extends the RFT to the case multistage distributed amplifiers The equalizers are made

of quarter-wave transmission lines A new variable and formalism are introduced that are bettersuited for the distributed medium The modified RFT is able to optimize and synthesize the trans-mission lines for transducer gain VSWR and also for multistage noise figure This chapter showshow the effects of the bias circuit can be combined with the scattering parameters of the transistorto present a more accurate representation of the structure A method is provided to solve the real-ization problem of having too high or too low characteristic impedance requirements The chapterconcludes with realizations of a 115ndash15 GHz a 2ndash8 GHz and a 5925ndash6425 GHz amplifierChapter 4 presents the modifications to the RFT to design trans-impedance microwave ampli-

fiers that are used for example in the case of photodiodes acting as high impedance current sourcesIt must provide a flat gain for a load charge impedance of 50Ω Contrary to the previous cases theRFT performs progressive optimization of the equalizers from load to source (ie right to left)It uses admittance and impedance matrices rather than solely relying on scattering matrix repre-sentations A technique based on peaking inductor is described and it is sued to reduce the noise atthe input of the amplifier critical for the overall noise figure Results for lumped and distributedexamples from 3 to 7 GHz are givenIn Chapter 5 a method is presented for optimizing equalizers made of a lossy distributed net-

work Compared to the previous RFT for broadband multistage amplifiers parallel resistors at thegate and drain of the transistors become part of the RFT optimization process The chapter sets thefoundations for this new medium and topology An image network technique is used for definingthe properties of the equalizers as well as the synthesis technique needed for transmission lineswith lossy junctions The chapter presents two hybrid realizations of broadband lossy distributedamplifiers The first is optimized for gain and VSWR from 01 to 5 GHz and the second from 01 to9 GHzIn Chapter 6 it is shown how the RFT can be used in the case of multistage power amplifiers

It is shown how added power is optimized in addition to gain and VSWR The technique requiresinterpolating large-signal scattering parameters in two dimensions frequency and power Theoptimization is done from load to source as was the case in Chapter 4 The technique is usedin the realization of a single-stage 225 GHz power amplifier and in the case of a three-stage2245 GHz power amplifier The chapter also provides a method and examples for the designof linear multistage power amplifiers using arborescent structuresChapter 7 describes how to use the RFT to design multistage active filters In this chapter the

transducer gain and VSWR are optimized in the passband while finite transmission zeros can beplaced in the stopband to provide the desired selectivity of the filter The technique also optimizesthe group delay in the passband an important feature for radar and digital communications Threeactive filter examples are presented The first is a low-pass case with two transmission zeroes usingtwo transistors and three equalizers operating in the 01ndash5 GHz band The second is a band-pass

x Preface

case with four transmission zeroes using one transistor and two equalizers with a 3ndash7 GHz passband The third is a band-pass case with two transmission zeroes with a 77ndash81 GHz pass bandThis last case was realized in monolithic microwave integrated circuit (MMIC) technologyChapter 8 shows the flexibility of the RFT to solve a variety of microwave circuit design prob-

lems In this chapter the RFT is modified to optimize arbitrary responses using a structure madesolely of passive equalizer In this case a transistor is shown to act as a ldquodummy passive devicerdquowhen providing well-chosen scattering parameter data to represent it Therefore with very littlemodification the optimization capabilities of the traditional RFT can be used for passive struc-tures This chapter shows how the RFT no longer optimizes a constant transducer gain but ratheroptimizes a specific forward transmission curve for radar applications provided as a set of datapoints in a frequency bandTwo examples are provided The first equalizer has an order six and used no transmission zeroes

and the second equalizer has an order six and used two transmission zeroes to improve the opti-mization results in the stopbandFinally Chapter 9 describes a possible method for the synthesis of microwave antennas The

RFT is used to optimize the matching network placed between antenna and receivingtransmittingcircuitry The optimization focuses on improving the return loss in the receiving and transmit-ting bandsThe organization of the book is summarized in Figure 01 After a review chapter with important

amplifier and microwave notations and equations the central element is the RTF The RFT isintroduced in Chapter 2 for the design of multistage lumped amplifiers and is then modified tosolve numerous problems in microwave amplifier design (Chapters 3ndash7) and adapted to solvemore general microwave circuit design problems (Chapters 8 and 9)

Chapter 2

RFT core

lumped multistage

Chapter 9

Antenna design

Chapter 8

Misc equalizers

Chapter 7

Active filters

Chapter 6

Power amplifiers

Chapter 5

Lossy amplifiers

Chapter 4

Trans-impedance

Chapter 3

Distributed

Real frequency technique

Microwave amplifiers

Figure 01 Summary of the organization of the book

xiPreface

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

Foreword

It has been a privilege for me to read through the manuscript of the Microwave Amplifier andActive Circuit Design Using the Real Frequency Technique I found that the authors have beenvery thorough in putting together this outstanding book containing a unique blend of theoryand practice I sensed that the authors are very passionate about the design of microwave circuitswhich is also evidenced from their other recent booksThe book provides an extensive use of an impedance matching methodology known as the real

frequency technique (RFT) in numerous applications The topics treated include the RFT itselfdesign of a wide variety of multistage amplifiers active filters passive equalizers for radar pulseshaping and antenna impedance matching applications All topics are self-contained and alsoinclude practical aspects Extensive analysis and optimization methods for these topics are dis-cussed The design techniques are well explained by means of solved examplesThe book is divided into nine chapters covering the basics of amplifiers an overview of RFT

multistage distributed amplifiers the use of RFT to design trans-impedance microwave amplifiersthe optimization of equalizers employing lossy distributed networks the use of RFT to designmultistage power amplifiers the design of multistage active filters the design of equalizers forradar pulse shaping and antenna impedance matching To solve impedance matching-relateddesign problems using RFT from specifications to realization of the end product the book pro-vides a unique integration of analysisoptimization aspects I found that the book is well balancedand treats the material in depth With emphasis on theory design and practical aspects applied tonumerous day-to-day applications the book is suitable for graduate students teachers and designengineersCongratulations Profs Jarry and Beneat on this excellent book that I am confident will be very

well received in the RF and microwave community for many years to come

Inder J BahlRoanoke VA

November 2015

Preface

Microwave and radio-frequency (RF) amplifiers play an important role in communication sys-tems and due to the proliferation of radar satellite and mobile wireless systems there is a needfor design methods that can satisfy the ever-increasing demand for accuracy reliability and fastdevelopment times This book provides an original design technique for a wide variety of multi-stage microwave amplifiers and active filters and passive equalizers for radar pulse shaping andantenna return loss applications This technique is referred to as the real frequency technique(RFT) It has grown out of the authors own research and teaching and as such has a unity of meth-odology and style essential for a smooth readingThe book is intended for researchers and RF and microwave engineers but is also suitable for an

advanced graduate course in the subject area Furthermore it is possible to choose material fromthe book to supplement traditional courses in microwave amplifier designEach chapter provides complete representation and characterization of the multistage or passive

equalizer structure as well as the design methodology We hope that this will provide theresearcher with a set of approaches that heshe could use for current and future microwave amp-lifier designs We also emphasize the practical nature of the subject by summarizing the designsteps and giving numerous examples of amplifier realizations and measured responses so thatRF and microwave engineers can have an appreciation of each amplifier in view of their needsThis approach we believe has produced a coherent practical and real-life treatment of the sub-ject The book is therefore theoretical but also experimental with over 18 microwave amplifierrealizationsThe book is divided into nine chapters In Chapter 1 recalls fundamental equations and defin-

itions useful for understanding the design of the microwave amplifiers presented hereChapter 2 provides a complete description of the RFT as it is first used to design multistage

lumped amplifiers The chapter starts with the multistage amplifier representation made of field-effect transistors (FETs) and equalizers defined using their scattering matrices In this introductorychapter the equalizers are assumed to be realizable as lumped ladder structures with transmissionzeroes at 0 or infinity The equalizers are optimized to improve the overall transducer gain and volt-age standing wave ratio (VSWR) The complete expressions for multistage transducer gain andVSWR are provided and further explained in Appendix A The RFT uses a progressive optimizationof the equalizers leading to a small number of parameters to optimize simultaneously This is a great

advantage of the RFT over typical design techniques that require simultaneous optimization of allunknown parametersThe RFT uses a practical implementation of the LevenbergndashMarquardt optimization method

and is summarized in Chapter 2 and detailed in Appendix B The optimization is performed overhundreds of frequency data points and is numerical in nature so that measured scattering param-eters of the transistors can be used The chapter then presents a complete step-by-step example ofthe design of a four-stage amplifier Intermediary and final results are provided A first extensionof the technique is shown It consists in using a transistor feedback circuit instead of the standalonetransistor to increase the useful bandwidth of the amplifier The chapter concludes with reportingtwo realizations designed using this technique in the range of 0ndash67 GHz and a realizationintended for 45 MHz to 65 GHz operationChapter 3 extends the RFT to the case multistage distributed amplifiers The equalizers are made

of quarter-wave transmission lines A new variable and formalism are introduced that are bettersuited for the distributed medium The modified RFT is able to optimize and synthesize the trans-mission lines for transducer gain VSWR and also for multistage noise figure This chapter showshow the effects of the bias circuit can be combined with the scattering parameters of the transistorto present a more accurate representation of the structure A method is provided to solve the real-ization problem of having too high or too low characteristic impedance requirements The chapterconcludes with realizations of a 115ndash15 GHz a 2ndash8 GHz and a 5925ndash6425 GHz amplifierChapter 4 presents the modifications to the RFT to design trans-impedance microwave ampli-

fiers that are used for example in the case of photodiodes acting as high impedance current sourcesIt must provide a flat gain for a load charge impedance of 50Ω Contrary to the previous cases theRFT performs progressive optimization of the equalizers from load to source (ie right to left)It uses admittance and impedance matrices rather than solely relying on scattering matrix repre-sentations A technique based on peaking inductor is described and it is sued to reduce the noise atthe input of the amplifier critical for the overall noise figure Results for lumped and distributedexamples from 3 to 7 GHz are givenIn Chapter 5 a method is presented for optimizing equalizers made of a lossy distributed net-

work Compared to the previous RFT for broadband multistage amplifiers parallel resistors at thegate and drain of the transistors become part of the RFT optimization process The chapter sets thefoundations for this new medium and topology An image network technique is used for definingthe properties of the equalizers as well as the synthesis technique needed for transmission lineswith lossy junctions The chapter presents two hybrid realizations of broadband lossy distributedamplifiers The first is optimized for gain and VSWR from 01 to 5 GHz and the second from 01 to9 GHzIn Chapter 6 it is shown how the RFT can be used in the case of multistage power amplifiers

It is shown how added power is optimized in addition to gain and VSWR The technique requiresinterpolating large-signal scattering parameters in two dimensions frequency and power Theoptimization is done from load to source as was the case in Chapter 4 The technique is usedin the realization of a single-stage 225 GHz power amplifier and in the case of a three-stage2245 GHz power amplifier The chapter also provides a method and examples for the designof linear multistage power amplifiers using arborescent structuresChapter 7 describes how to use the RFT to design multistage active filters In this chapter the

transducer gain and VSWR are optimized in the passband while finite transmission zeros can beplaced in the stopband to provide the desired selectivity of the filter The technique also optimizesthe group delay in the passband an important feature for radar and digital communications Threeactive filter examples are presented The first is a low-pass case with two transmission zeroes usingtwo transistors and three equalizers operating in the 01ndash5 GHz band The second is a band-pass

x Preface

case with four transmission zeroes using one transistor and two equalizers with a 3ndash7 GHz passband The third is a band-pass case with two transmission zeroes with a 77ndash81 GHz pass bandThis last case was realized in monolithic microwave integrated circuit (MMIC) technologyChapter 8 shows the flexibility of the RFT to solve a variety of microwave circuit design prob-

lems In this chapter the RFT is modified to optimize arbitrary responses using a structure madesolely of passive equalizer In this case a transistor is shown to act as a ldquodummy passive devicerdquowhen providing well-chosen scattering parameter data to represent it Therefore with very littlemodification the optimization capabilities of the traditional RFT can be used for passive struc-tures This chapter shows how the RFT no longer optimizes a constant transducer gain but ratheroptimizes a specific forward transmission curve for radar applications provided as a set of datapoints in a frequency bandTwo examples are provided The first equalizer has an order six and used no transmission zeroes

and the second equalizer has an order six and used two transmission zeroes to improve the opti-mization results in the stopbandFinally Chapter 9 describes a possible method for the synthesis of microwave antennas The

RFT is used to optimize the matching network placed between antenna and receivingtransmittingcircuitry The optimization focuses on improving the return loss in the receiving and transmit-ting bandsThe organization of the book is summarized in Figure 01 After a review chapter with important

amplifier and microwave notations and equations the central element is the RTF The RFT isintroduced in Chapter 2 for the design of multistage lumped amplifiers and is then modified tosolve numerous problems in microwave amplifier design (Chapters 3ndash7) and adapted to solvemore general microwave circuit design problems (Chapters 8 and 9)

Chapter 2

RFT core

lumped multistage

Chapter 9

Antenna design

Chapter 8

Misc equalizers

Chapter 7

Active filters

Chapter 6

Power amplifiers

Chapter 5

Lossy amplifiers

Chapter 4

Trans-impedance

Chapter 3

Distributed

Real frequency technique

Microwave amplifiers

Figure 01 Summary of the organization of the book

xiPreface

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

Preface

Microwave and radio-frequency (RF) amplifiers play an important role in communication sys-tems and due to the proliferation of radar satellite and mobile wireless systems there is a needfor design methods that can satisfy the ever-increasing demand for accuracy reliability and fastdevelopment times This book provides an original design technique for a wide variety of multi-stage microwave amplifiers and active filters and passive equalizers for radar pulse shaping andantenna return loss applications This technique is referred to as the real frequency technique(RFT) It has grown out of the authors own research and teaching and as such has a unity of meth-odology and style essential for a smooth readingThe book is intended for researchers and RF and microwave engineers but is also suitable for an

advanced graduate course in the subject area Furthermore it is possible to choose material fromthe book to supplement traditional courses in microwave amplifier designEach chapter provides complete representation and characterization of the multistage or passive

equalizer structure as well as the design methodology We hope that this will provide theresearcher with a set of approaches that heshe could use for current and future microwave amp-lifier designs We also emphasize the practical nature of the subject by summarizing the designsteps and giving numerous examples of amplifier realizations and measured responses so thatRF and microwave engineers can have an appreciation of each amplifier in view of their needsThis approach we believe has produced a coherent practical and real-life treatment of the sub-ject The book is therefore theoretical but also experimental with over 18 microwave amplifierrealizationsThe book is divided into nine chapters In Chapter 1 recalls fundamental equations and defin-

itions useful for understanding the design of the microwave amplifiers presented hereChapter 2 provides a complete description of the RFT as it is first used to design multistage

lumped amplifiers The chapter starts with the multistage amplifier representation made of field-effect transistors (FETs) and equalizers defined using their scattering matrices In this introductorychapter the equalizers are assumed to be realizable as lumped ladder structures with transmissionzeroes at 0 or infinity The equalizers are optimized to improve the overall transducer gain and volt-age standing wave ratio (VSWR) The complete expressions for multistage transducer gain andVSWR are provided and further explained in Appendix A The RFT uses a progressive optimizationof the equalizers leading to a small number of parameters to optimize simultaneously This is a great

advantage of the RFT over typical design techniques that require simultaneous optimization of allunknown parametersThe RFT uses a practical implementation of the LevenbergndashMarquardt optimization method

and is summarized in Chapter 2 and detailed in Appendix B The optimization is performed overhundreds of frequency data points and is numerical in nature so that measured scattering param-eters of the transistors can be used The chapter then presents a complete step-by-step example ofthe design of a four-stage amplifier Intermediary and final results are provided A first extensionof the technique is shown It consists in using a transistor feedback circuit instead of the standalonetransistor to increase the useful bandwidth of the amplifier The chapter concludes with reportingtwo realizations designed using this technique in the range of 0ndash67 GHz and a realizationintended for 45 MHz to 65 GHz operationChapter 3 extends the RFT to the case multistage distributed amplifiers The equalizers are made

of quarter-wave transmission lines A new variable and formalism are introduced that are bettersuited for the distributed medium The modified RFT is able to optimize and synthesize the trans-mission lines for transducer gain VSWR and also for multistage noise figure This chapter showshow the effects of the bias circuit can be combined with the scattering parameters of the transistorto present a more accurate representation of the structure A method is provided to solve the real-ization problem of having too high or too low characteristic impedance requirements The chapterconcludes with realizations of a 115ndash15 GHz a 2ndash8 GHz and a 5925ndash6425 GHz amplifierChapter 4 presents the modifications to the RFT to design trans-impedance microwave ampli-

fiers that are used for example in the case of photodiodes acting as high impedance current sourcesIt must provide a flat gain for a load charge impedance of 50Ω Contrary to the previous cases theRFT performs progressive optimization of the equalizers from load to source (ie right to left)It uses admittance and impedance matrices rather than solely relying on scattering matrix repre-sentations A technique based on peaking inductor is described and it is sued to reduce the noise atthe input of the amplifier critical for the overall noise figure Results for lumped and distributedexamples from 3 to 7 GHz are givenIn Chapter 5 a method is presented for optimizing equalizers made of a lossy distributed net-

work Compared to the previous RFT for broadband multistage amplifiers parallel resistors at thegate and drain of the transistors become part of the RFT optimization process The chapter sets thefoundations for this new medium and topology An image network technique is used for definingthe properties of the equalizers as well as the synthesis technique needed for transmission lineswith lossy junctions The chapter presents two hybrid realizations of broadband lossy distributedamplifiers The first is optimized for gain and VSWR from 01 to 5 GHz and the second from 01 to9 GHzIn Chapter 6 it is shown how the RFT can be used in the case of multistage power amplifiers

It is shown how added power is optimized in addition to gain and VSWR The technique requiresinterpolating large-signal scattering parameters in two dimensions frequency and power Theoptimization is done from load to source as was the case in Chapter 4 The technique is usedin the realization of a single-stage 225 GHz power amplifier and in the case of a three-stage2245 GHz power amplifier The chapter also provides a method and examples for the designof linear multistage power amplifiers using arborescent structuresChapter 7 describes how to use the RFT to design multistage active filters In this chapter the

transducer gain and VSWR are optimized in the passband while finite transmission zeros can beplaced in the stopband to provide the desired selectivity of the filter The technique also optimizesthe group delay in the passband an important feature for radar and digital communications Threeactive filter examples are presented The first is a low-pass case with two transmission zeroes usingtwo transistors and three equalizers operating in the 01ndash5 GHz band The second is a band-pass

x Preface

case with four transmission zeroes using one transistor and two equalizers with a 3ndash7 GHz passband The third is a band-pass case with two transmission zeroes with a 77ndash81 GHz pass bandThis last case was realized in monolithic microwave integrated circuit (MMIC) technologyChapter 8 shows the flexibility of the RFT to solve a variety of microwave circuit design prob-

lems In this chapter the RFT is modified to optimize arbitrary responses using a structure madesolely of passive equalizer In this case a transistor is shown to act as a ldquodummy passive devicerdquowhen providing well-chosen scattering parameter data to represent it Therefore with very littlemodification the optimization capabilities of the traditional RFT can be used for passive struc-tures This chapter shows how the RFT no longer optimizes a constant transducer gain but ratheroptimizes a specific forward transmission curve for radar applications provided as a set of datapoints in a frequency bandTwo examples are provided The first equalizer has an order six and used no transmission zeroes

and the second equalizer has an order six and used two transmission zeroes to improve the opti-mization results in the stopbandFinally Chapter 9 describes a possible method for the synthesis of microwave antennas The

RFT is used to optimize the matching network placed between antenna and receivingtransmittingcircuitry The optimization focuses on improving the return loss in the receiving and transmit-ting bandsThe organization of the book is summarized in Figure 01 After a review chapter with important

amplifier and microwave notations and equations the central element is the RTF The RFT isintroduced in Chapter 2 for the design of multistage lumped amplifiers and is then modified tosolve numerous problems in microwave amplifier design (Chapters 3ndash7) and adapted to solvemore general microwave circuit design problems (Chapters 8 and 9)

Chapter 2

RFT core

lumped multistage

Chapter 9

Antenna design

Chapter 8

Misc equalizers

Chapter 7

Active filters

Chapter 6

Power amplifiers

Chapter 5

Lossy amplifiers

Chapter 4

Trans-impedance

Chapter 3

Distributed

Real frequency technique

Microwave amplifiers

Figure 01 Summary of the organization of the book

xiPreface

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

advantage of the RFT over typical design techniques that require simultaneous optimization of allunknown parametersThe RFT uses a practical implementation of the LevenbergndashMarquardt optimization method

and is summarized in Chapter 2 and detailed in Appendix B The optimization is performed overhundreds of frequency data points and is numerical in nature so that measured scattering param-eters of the transistors can be used The chapter then presents a complete step-by-step example ofthe design of a four-stage amplifier Intermediary and final results are provided A first extensionof the technique is shown It consists in using a transistor feedback circuit instead of the standalonetransistor to increase the useful bandwidth of the amplifier The chapter concludes with reportingtwo realizations designed using this technique in the range of 0ndash67 GHz and a realizationintended for 45 MHz to 65 GHz operationChapter 3 extends the RFT to the case multistage distributed amplifiers The equalizers are made

of quarter-wave transmission lines A new variable and formalism are introduced that are bettersuited for the distributed medium The modified RFT is able to optimize and synthesize the trans-mission lines for transducer gain VSWR and also for multistage noise figure This chapter showshow the effects of the bias circuit can be combined with the scattering parameters of the transistorto present a more accurate representation of the structure A method is provided to solve the real-ization problem of having too high or too low characteristic impedance requirements The chapterconcludes with realizations of a 115ndash15 GHz a 2ndash8 GHz and a 5925ndash6425 GHz amplifierChapter 4 presents the modifications to the RFT to design trans-impedance microwave ampli-

fiers that are used for example in the case of photodiodes acting as high impedance current sourcesIt must provide a flat gain for a load charge impedance of 50Ω Contrary to the previous cases theRFT performs progressive optimization of the equalizers from load to source (ie right to left)It uses admittance and impedance matrices rather than solely relying on scattering matrix repre-sentations A technique based on peaking inductor is described and it is sued to reduce the noise atthe input of the amplifier critical for the overall noise figure Results for lumped and distributedexamples from 3 to 7 GHz are givenIn Chapter 5 a method is presented for optimizing equalizers made of a lossy distributed net-

work Compared to the previous RFT for broadband multistage amplifiers parallel resistors at thegate and drain of the transistors become part of the RFT optimization process The chapter sets thefoundations for this new medium and topology An image network technique is used for definingthe properties of the equalizers as well as the synthesis technique needed for transmission lineswith lossy junctions The chapter presents two hybrid realizations of broadband lossy distributedamplifiers The first is optimized for gain and VSWR from 01 to 5 GHz and the second from 01 to9 GHzIn Chapter 6 it is shown how the RFT can be used in the case of multistage power amplifiers

It is shown how added power is optimized in addition to gain and VSWR The technique requiresinterpolating large-signal scattering parameters in two dimensions frequency and power Theoptimization is done from load to source as was the case in Chapter 4 The technique is usedin the realization of a single-stage 225 GHz power amplifier and in the case of a three-stage2245 GHz power amplifier The chapter also provides a method and examples for the designof linear multistage power amplifiers using arborescent structuresChapter 7 describes how to use the RFT to design multistage active filters In this chapter the

transducer gain and VSWR are optimized in the passband while finite transmission zeros can beplaced in the stopband to provide the desired selectivity of the filter The technique also optimizesthe group delay in the passband an important feature for radar and digital communications Threeactive filter examples are presented The first is a low-pass case with two transmission zeroes usingtwo transistors and three equalizers operating in the 01ndash5 GHz band The second is a band-pass

x Preface

case with four transmission zeroes using one transistor and two equalizers with a 3ndash7 GHz passband The third is a band-pass case with two transmission zeroes with a 77ndash81 GHz pass bandThis last case was realized in monolithic microwave integrated circuit (MMIC) technologyChapter 8 shows the flexibility of the RFT to solve a variety of microwave circuit design prob-

lems In this chapter the RFT is modified to optimize arbitrary responses using a structure madesolely of passive equalizer In this case a transistor is shown to act as a ldquodummy passive devicerdquowhen providing well-chosen scattering parameter data to represent it Therefore with very littlemodification the optimization capabilities of the traditional RFT can be used for passive struc-tures This chapter shows how the RFT no longer optimizes a constant transducer gain but ratheroptimizes a specific forward transmission curve for radar applications provided as a set of datapoints in a frequency bandTwo examples are provided The first equalizer has an order six and used no transmission zeroes

and the second equalizer has an order six and used two transmission zeroes to improve the opti-mization results in the stopbandFinally Chapter 9 describes a possible method for the synthesis of microwave antennas The

RFT is used to optimize the matching network placed between antenna and receivingtransmittingcircuitry The optimization focuses on improving the return loss in the receiving and transmit-ting bandsThe organization of the book is summarized in Figure 01 After a review chapter with important

amplifier and microwave notations and equations the central element is the RTF The RFT isintroduced in Chapter 2 for the design of multistage lumped amplifiers and is then modified tosolve numerous problems in microwave amplifier design (Chapters 3ndash7) and adapted to solvemore general microwave circuit design problems (Chapters 8 and 9)

Chapter 2

RFT core

lumped multistage

Chapter 9

Antenna design

Chapter 8

Misc equalizers

Chapter 7

Active filters

Chapter 6

Power amplifiers

Chapter 5

Lossy amplifiers

Chapter 4

Trans-impedance

Chapter 3

Distributed

Real frequency technique

Microwave amplifiers

Figure 01 Summary of the organization of the book

xiPreface

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

case with four transmission zeroes using one transistor and two equalizers with a 3ndash7 GHz passband The third is a band-pass case with two transmission zeroes with a 77ndash81 GHz pass bandThis last case was realized in monolithic microwave integrated circuit (MMIC) technologyChapter 8 shows the flexibility of the RFT to solve a variety of microwave circuit design prob-

lems In this chapter the RFT is modified to optimize arbitrary responses using a structure madesolely of passive equalizer In this case a transistor is shown to act as a ldquodummy passive devicerdquowhen providing well-chosen scattering parameter data to represent it Therefore with very littlemodification the optimization capabilities of the traditional RFT can be used for passive struc-tures This chapter shows how the RFT no longer optimizes a constant transducer gain but ratheroptimizes a specific forward transmission curve for radar applications provided as a set of datapoints in a frequency bandTwo examples are provided The first equalizer has an order six and used no transmission zeroes

and the second equalizer has an order six and used two transmission zeroes to improve the opti-mization results in the stopbandFinally Chapter 9 describes a possible method for the synthesis of microwave antennas The

RFT is used to optimize the matching network placed between antenna and receivingtransmittingcircuitry The optimization focuses on improving the return loss in the receiving and transmit-ting bandsThe organization of the book is summarized in Figure 01 After a review chapter with important

amplifier and microwave notations and equations the central element is the RTF The RFT isintroduced in Chapter 2 for the design of multistage lumped amplifiers and is then modified tosolve numerous problems in microwave amplifier design (Chapters 3ndash7) and adapted to solvemore general microwave circuit design problems (Chapters 8 and 9)

Chapter 2

RFT core

lumped multistage

Chapter 9

Antenna design

Chapter 8

Misc equalizers

Chapter 7

Active filters

Chapter 6

Power amplifiers

Chapter 5

Lossy amplifiers

Chapter 4

Trans-impedance

Chapter 3

Distributed

Real frequency technique

Microwave amplifiers

Figure 01 Summary of the organization of the book

xiPreface

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

It is believed that this book will provide the reader with an appreciation of the RFT as a tool thatcan be applied successfully in many disciplinesWe would like to acknowledge the contributions of our past and present research students

whose collaboration has resulted in much of the material in the book In particular we would liketo mention Professor Eric Kerherve Associate Professors EH El Hendaoui P M Martin and APerennec and Engineers L Courcelle M Hazouard M Lecouve and A OlomoThe book is based on the authorsrsquo research under the sponsorship of France Telecommunica-

tions Research amp Development (CNET) National Center of Spatial Studies (CNES) ALCATELSPACE PHILIPS OMMIC Commissariat Energie Atomique (CEA) PLESSEY (GB)The work resulted in approximately nine contracts with different agencies and companies

Pierre JarryJacques N Beneat

xii Preface

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

Acknowledgments

The authors are deeply indebted to Dr Inder J Bahl (USA) editor of the International Journal ofRF and Microwave Computer-Aided Engineering from Wiley This book could not have beenwritten without his help and he is acknowledged with gratitudeTheir sincere appreciation extends to the Publisher of Wiley-Interscience and all the staff at

Wiley involved in this project for their professionalism and outstanding effortsPierre Jarry thanks his colleagues at the University Bordeaux Sciences including Professor Eric

Kerherve and Assistant Professors Nathalie Deltimple and Jean-Marie Pham He thanks as wellProfessor Yves Garault who introduced Microwave and Telecommunications at the University ofLimoges Finally he expresses his deep appreciation to his wife Roselyne and to his son Jean-Pierre for their tolerance and supportJacques Beneat is very grateful to Norwich University a place conducive to trying and succeed-

ing in new endeavors He particularly thanks the Senior Vice President for Academic Affairs DrGuiyou Huang and the Director of the David Crawford School of Engineering Professor SteveFitzhugh for their encouragements in such a difficult enterprise

Pierre JarryJacques N Beneat

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

1Microwave Amplifier Fundamentals

11 Introduction 212 Scattering Parameters and Signal Flow Graphs 213 Reflection Coefficients 514 Gain Expressions 715 Stability 916 Noise 1017 ABCD Matrix 14

171 ABCD Matrix of a Series Impedance 14172 ABCD Matrix of a Parallel Admittance 15173 Input Impedance of Impedance Loaded Two-Port 15174 Input Admittance of Admittance Loaded Two-Port 16175 ABCD Matrix of the Cascade of Two Systems 16176 ABCD Matrix of the Parallel Connection of Two Systems 17177 ABCD Matrix of the Series Connection of Two Systems 17178 ABCD Matrix of Admittance Loaded Two-Port Connected in Parallel 17179 ABCD Matrix of Impedance Loaded Two-Port Connected in Series 191710 Conversion Between Scattering and ABCD Matrices 19

18 Distributed Network Elements 20181 Uniform Transmission Line 20182 Unit Element 21183 Input Impedance and Input Admittance 22184 Short-Circuited Stub Placed in Series 23185 Short-Circuited Stub Placed in Parallel 24186 Open-Circuited Stub Placed in Series 24187 Open-Circuited Stub Placed in Parallel 25188 Richardrsquos Transformation 25189 Kuroda Identities 28

References 35

Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique First EditionPierre Jarry and Jacques N Beneatcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

11 Introduction

In many high-speed applications there is a need for microwave amplifier circuits For examplesatellite communications can be used when radio signals are blocked between two terrestrial trans-ceiver stations as shown in Figure 11 The satellite then acts as a repeater and the signal beingrepeated must be amplified before being sent backImportant amplifier characteristics are center frequency and span of the pass band gain stabil-

ity input and output matching to the rest of the communication system and noise figure [1]At microwave frequencies a common amplification component that has minimum noise is a

field effect transistor (FET) as shown in Figure 12In this book the FET will be typically modeled as a two-port network where the input is on the

gate and the output is on the drain The source is mainly used for biasing of the transistor

12 Scattering Parameters and Signal Flow Graphs

At high frequencies voltages and currents are difficult to measure directly However scatteringparameters determined from incident and reflected waves can be measured with resistive termin-ations The scattering matrix of a two-port system provides relations between input and outputreflected waves b1 and b2 and input and output incident waves a1 and a2 when the structure is

Earth antenna Earth antenna

Figure 11 High-speed signals must be amplified in a satellite repeater

Ve Vs

Drain

Source

Grille

Figure 12 A FET modeled as a two-port network

2 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

terminated on its characteristic impedance Z0 as shown in Figure 13 Typically the referencesource and load Z0 used in commercial network analyzers is 50ΩIn the case of a two-port system the equations relating incident and reflected waves and the

scattering parameters are given by

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

1 1

b1b2

=S11 S12S21 S22

a1a2

1 2

The incident and reflected waves are related to the voltages and currents in Figure 13

a1 =V1 + Z0I12 Z0

1 3

a2 =V2 + Z0I22 Z0

1 4

b1 =V1minusZ0I12 Z0

1 5

b2 =V2minusZ0I22 Z0

1 6

The parameter S11 is the input reflection coefficient and is the ratio of input reflected waveover input incident wave when the output incident wave is equal to zero The output incidentwave a2 is equal to zero when the output of the system is connected to the characteristicimpedance Z0

S11 =b1a1 a2 = 0

1 7

The parameter S21 is the forward transmission coefficient and is the ratio of the output reflectedwave over the input incident wave when the output incident wave is equal to zero

S21 =b2a1 a2 = 0

1 8

Z0

Z0VG V1

I1

V2

I2

S11

S21 S22

S12a1

b1 a2

b2

Figure 13 Scattering matrix of a two-port system terminated on characteristic impedance Z0

3Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

The parameter S22 is the output reflection coefficient and is the ratio of the output reflected waveover the output incident wave when the input incident wave is equal to zero The input incidentwave a1 is equal to zero when the input of the system is connected to the characteristic imped-ance Z0

S22 =b2a2 a1 = 0

1 9

The parameter S12 is the reverse transmission coefficient and is the ratio of the input reflectedwave over the output incident wave when the input incident wave is equal to zero

S12 =b1a2 a1 = 0

1 10

The two-port network and scattering parameters can be modeled using the signal flow graphrepresentation of Figure 14A useful tool when defining system gains using signal flow graphs is the Mason gain

formula [2] It provides the gain T of a system between a source node and an output node

T =TkΔk

Δ1 11

with

Δ= 1minus Li + LiLjminus LiLjLk

where

Tk is the gain of the kth forward path between the source node and the output node

Li is the sum of all individual loop gains

LiLj is the sum of two loop gain products of any two nontouching loops

LiLjLk is the sum of three loop gain products of any three nontouching loops

Δk is the part of Δ that does not touch the kth forward path

a1

b1

S21

S22

S12

S11

b2

a2

Figure 14 Signal flow graph representation of a two-port network

4 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

13 Reflection Coefficients

As shown in Figure 15 the input reflection coefficient when the output is connected to charac-teristic impedance Z0 can be expressed in terms of the input impedance ZIN =V1 I1 by replacing a1and b1 by their expressions in terms of voltages and currents

S11 =b1a1 a2 = 0

=

V1minusZ0I12 Z0

V1 + Z0I12 Z0

=ZIN minusZ0ZIN + Z0

S11 =ZIN minusZ0ZIN + Z0

1 12

The input impedance can be expressed in terms of the input reflection coefficient by

ZIN =Z01 + S111minusS11

1 13

Figure 16 defines additional reflection coefficients when the two-port is terminated on arbitraryloads ZG and ZLReflection coefficient of the source

ρG =a1b1

=ZGminusZ0ZG + Z0

1 14

Z0

Z0VG V1

I1

V2

I2

S11

S21

Zin

S22

S12a1

b1 a2

b2

Figure 15 Input reflection coefficient and input impedance

ZG

ZLVG

I1

V2

I2

S11

S21

Z0

ρin

ρG

ρout

ρL

S22

S12

V1

b1

a1 a2

b2

Figure 16 Reflection coefficients of a two-port when terminated on arbitrary loads

5Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

Reflection coefficient of the load

ρL =a2b2

=ZLminusZ0ZL +Z0

1 15

Input reflection coefficient of the two-port when output loaded on ρL

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

1 16

Output reflection coefficient of the two-port when input loaded on ρG

ρout =b2a2

= S22 +S12S21ρG1minusS11ρG

1 17

For example the expression of the input reflection coefficient when loaded on ρL is obtained byfirst using the general scattering parameter definition of the two-port

b1 = S11a1 + S12a2

b2 = S21a1 + S22a2

Then using the relation between incident and reflected waves a2 = ρLb2 gives

b1 = S11a1 + S12ρLb2

b2 = S21a1 + S22ρLb2

then from the second equation b2 1minusS22ρL = S21a1 and b2 =S21

1minusS22ρLa1 so that

b1 = S11a1 + S12ρLS21

1minusS22ρLa1 = S11 +

S21S12ρL1minusS22ρL

a1

and

ρin =b1a1

= S11 +S12S21ρL1minusS22ρL

The voltage standing wave ratio (VSWR) is given in terms of a reflection coefficient ρ by

VSWR=1+ ρ

1minus ρ1 18

The input VSWR for the two-port in Figure 16 is therefore

VSWRIN =1 + ρin1minus ρin

1 19

6 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

The output VSWR for the two-port in Figure 16 is therefore

VSWROUT =1+ ρout1minus ρout

1 20

14 Gain Expressions

Figure 17 shows the different reflection coefficients used to define various power gainsThe transducer power gain can be computed using the signal flow graph and the Mason gain

formula as shown in Figure 18There is one forward path from node bG to node b2 The path gain of this path

is T1 = 1 times S21 = S21There are three individual loops ρGS11 ρLS22 and ρGS21ρLS12This gives

Δ = 1minus Li + LiLjminus LiLjLk = 1minus S11ρG + S22ρL + S12S21ρGρL + S11ρGS22ρL minus0

and

T =TkΔk

Δ=

T1 times 1minus01minusS11ρGminusS22ρLminusS12S21ρGρL + S11ρGS22ρL

bG a1

a2

ρG

ρL

b1

b2S21

S11

S12

S22

Figure 18 Signal flow graph representation for defining the gain

ZG I1 I2

a1 a2

b1 b2

ρG

ρin

ρout

ρL

ZL

VG V1 V2

S11

S21 S22

S12

Z0

Figure 17 Gain definitions

7Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

T =S21

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL

The transducer power gain is defined as

GT =power delivered to the load

maximum available power from the source

so that

GT =b2

2 1minus ρL2

bG2

1minus ρG2

= T 2 1minus ρG2 1minus ρL

2

and

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρGminusS22ρL + S11S22ρGρLminusS12S21ρGρL2 1 21

Note that GT can also be written as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG 1minusS22ρL minusS12S21ρGρL2 1 22

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusρGρin2 1minusS22ρL

2 1 23

or as

GT =1minus ρG

2 1minus ρL2 S21

2

1minusS11ρG2 1minusρLρout

2 1 24

Note that when S12 = 0 the transducer power gain reduces to the unilateral transducer power gainGTU given by [3]

GTU =GGG0GL

where

GG =1minus ρG

2

1minusS11ρG2 G0 = S21

2 and GL =1minus ρL

2

1minusS22ρL2

8 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

In this case GG represents the losses in the source G0 is the intrinsic gain and GL represents thelosses in the load and can be modeled as in Figure 19The maximum unilateral gain occurs when there is perfect matching of source and load imped-

ances For maximum unilateral gain one would match the source to the input of the transistor bymaking ρG = Slowast11 and match the load to the output of the transistor by making ρL = S

lowast22

The maximum unilateral gain GTU MAX is then given by

GTU MAX =1

1minus S112 S21

2 1

1minus S222 1 25

The available power gain is defined as

GA =maximum power the amplifier can deliver to the load

maximum available power from the source

and it is given by

GA =1minus ρG

2 S212

1minusS11ρG2 1minus ρout

21 26

Note that when the input is perfectly matched then ZG = Z0 and ρG = 0 and

ρout = S22 +S12S21ρG1minusS11ρG

= S22

so that the available power gain becomes

GA =S21

2

1minus S222 1 27

15 Stability

The stability of the amplifier depends on the scattering parameters of the transistor but also on thematching networks and terminations [4]For the two-port shown in Figure 110 ρin is the input reflection coefficient of the transistor

when output loaded on ρL and ρout is the output reflection coefficient of the transistor when input

ρL

ρG

S11

Z0

Z0VGGG GL

G0

(transistor)

S22

Figure 19 Representation in the case of a unilateral amplifier S12 = 0

9Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

loaded on ρG The system is said to be unconditionally stable if the amplitude of ρin and ρout are lessthan unity for all the real parts of load impedance ZL and source impedance ZG

ZL withRe ZL gt 0 ρin2 lt 1

ZG withRe ZG gt 0 ρout2 lt 1

It can be shown that for unconditional stability one must satisfy three conditions

K gt 1

B1 gt 0

B2 gt 0

1 28

where

K =1 + Δ 2minus S11

2minus S222

2 S12S211 29

B1 = 1minus S222minus S12S21 1 30

B2 = 1minus S112minus S12S21 1 31

It is seen that these conditions only depend on the scattering parameters of the transistor Whenthese three conditions are met the amplifier can be connected to the loads without risk of becomingunstable and producing oscillations

16 Noise

Figure 111 shows an active two-port between input impedance ZG and load impedance ZGThe noise in the amplifier can be characterized by the noise figure F defined by

F =Fmin +Rn

GGYGminusYGmin

2 1 32

where

Fmin is the minimum noise figure obtained when YG = YGmin

Rn is the equivalent noise resistance of the active device

ρL

ρG

ρin

ρout

VG

ZG

ZLS11

S21

S12

Transistor

S22

Figure 110 Stability of a two-port

10 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

YGmin is the source admittance that makes the noise figure minimumYG is the source admittance such that YG =GG + jBG

The Section 17 consists of the rewritten noise figure formula in terms of reflection coefficientsrather than in terms of admittancesReferring to Figure 111 the reflection coefficient ρG from the source admittance is given by

ρG =Y0minusYGY0 + YG

1 33

where Y0 is the characteristic admittance usedThis gives the source admittance in terms of the source reflection coefficient such that

YG =1minusρG1 + ρG

Y0 1 34

In (132) taking YG = YGmin makes the noise figure become minimum This translates to a reflec-tion coefficient ρGmin where the noise figure is at a minimum such that

ρGmin =Y0minusYGmin

Y0 + YGmin1 35

YGmin =1minusρGmin

1 + ρGminY0 1 36

Then replacing YGmin and YG by their expressions in terms of ρG and ρGmin gives us

YGminusYGmin2 = Y0

2 1minusρG1 + ρG

minus1minusρGmin

1 + ρGmin

2

= Y02 1minusρG + ρGminminusρGρGminminus 1minusρGmin + ρGminusρGρGmin

1 + ρG 1 + ρGmin

2

and

YGminusYGmin2 = Y0

2 2 ρGminminusρG1 + ρG 1 + ρGmin

2

= 4Y02 ρGminminusρG

2

1 + ρG2 1 + ρGmin

2

ρL

ρG

ρin

ρout

VG

ZG

ZLActive

device

Figure 111 Active device and source and load impedances

11Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

So that the noise figure can first be expressed as

F =Fmin +4Rn

GGY0

2 ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

Then one expressesGG as the real part of the source admittance in terms of reflection coefficients

GG =Re YG =YG + YGlowast

2=Y02

1minusρG1 + ρG

+1minusρG

lowast

1 + ρGlowast

=Y02

1minusρG + ρGlowastminus ρG

2 + 1 + ρGminusρGlowastminus ρG

2

1 + ρG2

and

GG = Y01minus ρG

2

1 + ρG2

so that

F =Fmin + 4RnY02 1Y0

1 + ρG2

1minus ρG2

ρGminusρGmin2

1 + ρG2 1 + ρGmin

2

and the noise figure is given in terms of the source reflection parameter ρG and the optimum sourcereflection parameter ρGmin by

F =Fmin + 4RnY0ρGminusρGmin

2

1minus ρG2 1 + ρGmin

21 37

Typically the manufacturer provides the three parameters ρGmin Fmin and rn =Rn R0 the nor-malized equivalent noise resistance Note that the reflection coefficient ρGmin is complex andis often given as magnitude and phase Note that these parameters do change with frequencyso they are provided in table formNext we provide the noise figure corresponding to a cascade of active devices In Figure 112 a

first active device is characterized by a gain G1 and noise figure F1 and a second active device ischaracterized by a gain G2 and noise figure F2

Z0

Z0

Active

device

G2 F2

Active

device

G1 F1

Figure 112 Noise figure of the cascade of two active devices

12 Microwave Amplifier and Active Circuit Design Using the Real Frequency Technique

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals

It can be shown [5] that the noise figure F2 corresponding to the cascade of the two systems isgiven by

F2 =F1 +1G1

F2minus1 1 38

and the gain G2 of the cascaded system is given by

G2 =G1G2 1 39

The system is then placed in cascade with a third active device characterized by a gainG3 and noisefigure F3 as shown in Figure 113Then the noise figure F3 corresponding to the cascade of the three systems is given by

F3 =F2 +1G2

F3minus1

and the gain G3 of the cascaded system is given by

G3 =G2G3

We repeat this approach for the case of a cascade of k active devices as shown in Figure 114In this case the noise figure Fk corresponding to the cascade of the k systems is given by

Fk =Fkminus1 +1

Gkminus1

Fk minus1 1 40

and the gain Gk of the cascaded system is given by

Gk =Gkminus1Gk 1 41

Z0

Z0

Active

device

G3 F3

Active

device

G2prime F2prime

Figure 113 Noise figure of the cascade of three active devices

Z0

Z0

Active

device

G2 F2

Active

device

G3 F3

Active

device

Gk Fk

Active

device

G1 F1

Figure 114 Noise figure of the cascade of k active devices

13Microwave Amplifier Fundamentals