Microwave Transmission Line Circuits [1 ed.] 9781608075706, 9781608075690

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MICROWAVE TRANSMISSION LINE CIRCUITS

Microwave Transmission Line Circuits is the first book to explore and develop the interface between lumped-element circuits and distributed-element circuits. Supported with over 580 equations and 100 illustrations, this volume presents the necessary technological underpinnings and all the practical details needed to fully comprehend and work with the material. William T. Joines is a professor in the Department of Electrical and Computer Engineering, Duke University, where his research and teaching focuses on electromagnetic field and wave interactions with materials and structures. He has authored over 200 technical papers on electromagnetic-wave theory and applications, and holds 20 U.S. patents. He is a Life Fellow of the IEEE, and a recipient of the IEEE Outstanding Engineering Educator Award and the Scientific and Technical Achievement Award presented by the Environmental Protection Agency. W. Devereux Palmer is the program manager for electromagnetics, microwaves, and power at the U.S. Army Research Office, responsible for a portfolio of basic research projects focused on creating the innovations that will drive the next generation of DoD systems for radio communications, sensing, and electronic warfare. A Fellow of the IEEE, he received the U.S. Army Research Laboratory Award for Program Management in 2010 and the U.S. Army Superior Civilian Service Medal in 2011. Jennifer T. Bernhard is a professor in the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, where she is active as a researcher and educator in electromagnetics and antennas. She has authored over 200 journal articles, conference papers and presentations, books, and book chapters, and holds 10 patents and patent applications. She is an IEEE Fellow and served as president of the IEEE Antennas and Propagation Society in 2008.

Include bar code ISBN 13: 978-1-60807-569-0 ISBN 10: 1-60807-569-9

BOSTON

LONDON

www.artechhouse.com

Palmer Bernhard

MICROWAVE TRANSMISSION LINE CIRCUITS

This authoritative resource offers professionals and students valuable assistance with their work and studies involving microwave circuit analysis and design. Readers gain a thorough understanding of the properties of planar transmission lines for integrated circuits. Moreover, this practical book presents matrix and computer-aided methods for analysis and design of circuit components. Engineers find in-depth details on input, output, and interstage networks, as well as coverage of stability, noise, and signal distortion.

Joines

MICROWAVE TRANSMISSION LINE CIRCUITS William T. Joines W. Devereux Palmer Jennifer T. Bernhard

Microwave Transmission Line Circuits

Joines_CIP.indd i

10/18/2012 11:03:30 AM

For a complete listing of titles in the Artech House Mirowave Library, turn to the back of this book.

Joines_CIP.indd ii

10/18/2012 11:03:54 AM

Microwave Transmission Line Circuits William T. Joines W. Devereux Palmer Jennifer T. Bernhard

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Cover design by CC Graphic Design, Durham, NC

ISBN 13: 978-1-60807-569-0

© 2013 ARTECH HOUSE 685 Canton Street Norwood, MA 02062

All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.

10 9 8 7 6 5 4 3 2 1

CONTENTS PREFACE

ix

Chapter 1 INTRODUCTION 1.1 Defining Microwaves 1.2 Radar Systems 1.2.1 Pulse Radar 1.2.2 Doppler Radar 1.2.3 Frequency-Modulated Continuous-Wave Radar 1.3 Microwave Communication Systems 1.3.1 Microwave Links and Repeaters 1.3.2 Carrier Modulation 1.4 Signal-to-Noise Requirements 1.4.1 Channel Capacity and Information Content Bibliography

1 1 1 4 6 7 10 11 12 15 17 20

Chapter 2 MICROWAVE TRANSMISSION LINES 2.1 Useful Transmission Line Configurations 2.2 Wave Equation for Voltage and Current 2.3 Incident, Reflected, and Standing Waves 2.3.1 Voltage Reflection Coefficient 2.3.2 Power Flow 2.3.3 Voltage Standing Wave Ratio 2.4 Transmission Lines with Losses 2.5 Transmission Line Parameter Calculations 2.6 Impedance Matching

23 23 25 31 31 33 34 36 40 45

v

vi

CONTENTS

2.7 Impedance Transformations and the Smith Chart Problems Bibliography Chapter 3 TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS 3.1 Lumped-Element Limitations 3.2 Using Transmission Lines as Lumped Elements 3.2.1 Impedance Matching Using Lumped Ls and Cs 3.2.2 Resonant Circuits Problems Bibliography

57 63 66

69 69 70 77 80 92 95

Chapter 4 MATRIX REPRESENTATION OF MICROWAVE NETWORKS 97 4.1 Z, Y , and ABCD Matrices for Connected Networks 97 4.1.1 Two-Port Network Parameters 97 4.1.2 Two-Port Networks Connected in Cascade 101 4.2 Network Gain or Loss in Terms of ABCD Parameters 103 4.2.1 Insertion Gain for Complex ZS and ZL 103 4.2.2 Transducer Gain for Complex ZS and ZL 104 4.3 Scattering Parameters and the Scattering Matrix of a Network 110 4.4 Signal Flow Graphs and Mason’s Gain Rule 116 Problems 124 Bibliography 129 Chapter 5 SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS 131 5.1 Network Synthesis and Design 131 5.1.1 Lowpass Filters 132 5.1.2 Bandpass Filters 141 5.1.3 Highpass and Bandstop Filters 144 5.1.4 Network Design Using Q Tapering of Sections 152 5.2 Tapped-Stub Resonator 155 5.2.1 The Half-Wavelength Tapped Stub as a Tunable Filter 157 5.3 Coupled Line Filters 160 Problems 166 Bibliography 170 Chapter 6 BROADBAND IMPEDANCE-MATCHING NETWORKS

173

CONTENTS

6.1 6.2 6.3

Network Model for Impedance Matching The Q of λ/4 and λ/2 Transformer Sections Multiple Quarter-Wavelength Transformers in Cascade 6.3.1 Two Cascaded Sections 6.3.2 Three Cascaded Sections 6.4 More Compact Impedance-Matching Networks 6.4.1 Lumped-Element Equivalent of the Quarter-Wavelength Transformer 6.4.2 The Eighth-Wavelength Transformer 6.4.3 Impedance Matching a Real Source to a Complex Load Problems Bibliography

Chapter 7 COMBINING, DIVIDING, AND COUPLING CIRCUITS 7.1 Power Dividers and Power Combiners 7.1.1 Two-Way Power Divider/Combiner 7.1.2 N -Way Power Divider/Combiner 7.1.3 Unequal Power Division 7.2 Couplers 7.2.1 Branch-Line, or 90-Degree Hybrid Coupler 7.2.2 180-Degree Hybrid Coupler 7.3 Frequency Diplexers Problems Bibliography Chapter 8 TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS 8.1 Impedance Matching for Maximum Gain 8.1.1 Transistor Equivalent Circuit 8.1.2 Stability Conditions and the Bilateral Amplifier 8.1.3 Conjugately Matching the Bilateral Amplifier 8.2 Stabilizing Potentially Unstable Transistors 8.3 Dynamic Range of a Transistor Amplifier 8.3.1 Lower Input Power Limit (Minimum Detectable or Usable Signal) 8.3.2 Upper Input Power Limit 8.4 The Transistor as an Oscillator

vii

173 175 177 178 180 180 181 182 191 193 195 197 197 197 199 201 204 204 208 212 218 222 225 225 227 230 232 234 235 236 238 243

viii

CONTENTS

8.4.1 Feedback Oscillators 8.4.2 Negative Resistance Oscillator 8.5 Microwave Diodes 8.5.1 Diode Fundamentals 8.5.2 Diodes as Analog Devices 8.5.3 Diodes as Digital Devices Problems Bibliography

243 246 247 250 250 253 260 261

Chapter A NORMALIZED ELEMENT VALUES FOR LOWPASS FILTERS Bibliography

263 275

Chapter B MICROSTRIP CHARACTERISTIC IMPEDANCE Bibliography

277 279

ABOUT THE AUTHORS

295

Index

297

PREFACE This book is an ideal text for a one-semester course in microwave transmission line circuit analysis and design techniques. It includes information on the properties of planar transmission lines for integrated circuits and matrix and computer-aided methods for analysis and design of circuit components. Fundamental theory and practical information on the analysis and design of input, output, and interstage networks for microwave transistor amplifiers and oscillators is included, along with topics on stability, noise, and signal distortion. Distinct from other texts covering the above topics, this book explores and develops the interface between lumpedelement circuits and distributed element circuits. The material is targeted towards the student with a background in basic electrical circuit theory and a beginning course in electromagnetics. These eight chapters appear in an order designed to present the material in the most logical manner. Chapter 1 illustrates where the microwave region fits within the electromagnetic spectrum, together with the nomenclature of different operating bands and allocations by usage. Also presented in Chapter 1 are a number of microwave system architectures that use the microwave design techniques and components developed in the remaining seven chapters. Chapter 2 presents the theory and applications of various types of transmission line, and the concepts of incident, reflected and transmitted voltage, current and power are formalized with defining equations. Impedance matching for maximum power transfer also is introduced in Chapter 2. Chapter 3 shows how transmission segments can serve as network elements in much the same way as lumped-element capacitors and inductors. The equivalence of lumped elements and distributed elements (transmission line segments) is illustrated with a number of examples. In Chapter 4, analytical

ix

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CONTENTS

techniques are presented that will be used extensively in later chapters: matrix representations of a network, insertion gain and transducer gain. Chapter 5 presents the synthesis and design techniques for frequency-selective networks that will be used extensively in Chapters 6, 7 and 8. Chapter 6 addresses a problem often encountered by the microwave engineer — providing an impedance match over a broad range of operating frequencies. Many of the techniques introduced in earlier chapters are used here to great advantage. Combining, dividing, and coupling microwave signals are addressed in Chapter 7, where the combining, dividing, and coupling may be designed based on desired power level or on frequency channelization using frequency diplexers. In Chapter 8, microwave concepts and transmission line techniques are applied to amplifier, oscillator, and phase shifter circuits with the aim of obtaining a more complete understanding of these circuits. A number of concepts in industry and in the published literature are introduced together with numerous example designs.

Chapter 1 INTRODUCTION 1.1 DEFINING MICROWAVES Microwaves are electromagnetic waves that have wavelengths comparable to the size of physical structures with which the waves may interact. Microwaves extend from 0.1 mm to 1 m wavelength in air, or in frequency from 0.3 × 109 cycles per second (Hz), or 0.3 GHz, to 3,000 GHz. The shortest wavelength could justifiably be set at 0.01 mm or 1 mm, with a corresponding upper frequency of 30,000 GHz or 300 GHz. It is the boundary line between microwaves and infrared that has not been well defined, and the range of each is shown approximately on the electromagnetic spectrum chart in Figure 1.1. Microwaves having the shortest wavelength are called submillimeter waves or terahertz waves, then millimeter waves, and so on as shown in Table 1.1. A way in which radio frequency signals are classified is shown in Table 1.2, where microwaves include the UHF, SHF, EHF, and THF waves. The frequencies ranging from UHF to THF are divided into many microwave bands and designated by letters of the alphabet as indicated in Table 1.3. Frequency ranges are also shown in Table 1.4 as they have been allocated according to use. Two very important uses of microwaves, in radar and communication systems, are discussed in the following sections.

1.2 RADAR SYSTEMS The term radar is a contraction of radio direction and ranging. While radar systems were first developed for use at radio frequencies, the term is now taken to mean

1

2

MICROWAVE TRANSMISSION LINE CIRCUITS

3 × 1021

10-13

Cosmic Rays Gamma Rays

3 × 10

18

10-10

X-Rays Ultraviolet

3 × 1015

10-7

Visible Light Infrared 3 × 1012

10-4

Sub-mm Waves THF EHF

3 × 109

SHF UHF VHF

3 × 106

HF

mm Waves Satellite TV Wi-Fi Cellular GPS Broadcast TV FM Radio

cm Waves Microwaves

Shortwave Radio AM Radio

10-1

102

Long Waves 3 × 10

3

Frequency (hertz) Figure 1.1 The electromagnetic spectrum.

105 Wavelength (meters)

3

INTRODUCTION

Table 1.1 Electromagnetic Waves Electromagnetic Waves Long waves Medium waves Short waves Ultra short waves meter waves Microwaves centimeter waves millimeter waves submillimeter or terahertz waves Infrared rays

Wavelength

Frequency

10–1 km 1–0.1 km 100–10 m

30–300 kHz 0.3–3 MHz 3–30 MHz

10–1 m

30–300 MHz

100–1 cm 10–1 mm 1–0.1 mm 0.1 mm–75 nm

0.3–30 GHz 30–300 GHz 300–3,000 GHz 3,000–405,000 GHz

Table 1.2 Radio Frequencies Radio Frequencies High frequency Very high frequency Ultra high frequency Super high frequency Extremely high frequency Tremendously high frequency

HF VHF UHF SHF EHF THF

Frequency

Wavelength

3–30 MHz 30–300 MHz 0.3–3 GHz 3–30 GHz 30–300 GHz 300–3000 GHz

100–10 m 10–1 m 100–10 cm 10–1 cm 10–1 mm 1–0.1 mm

any electromagnetic system that irradiates a region of space and then detects and measures the reflected signals from objects within that space. Radar can not only detect the presence of a reflecting object and its range or distance from the transmitter by measuring the out-and-back time delay of short duration pulses of a microwave signal, but can also detect the exact position of a target (phasedarray radar) and its radial velocity (Doppler radar). There are numerous civilian and military uses of radar, including air traffic control, aircraft and ship navigation, control and guidance of weapons systems, remote sensing of Earth’s surface and environment, astronomy, and automotive collision avoidance. Radar applications can use frequencies from about 1 MHz to over 200 GHz.

4

MICROWAVE TRANSMISSION LINE CIRCUITS

Table 1.3 Microwave Frequency Band Designations Microwave Bands

Frequency (GHz)

Wavelength in Air

L S C X Ku K Ka Q U V W D G

1–2 2–4 4–8 8–12 12–18 18–27 27–40 33–50 40–60 50–75 75–110 110–170 140–220

30.00–15.00 cm 15.00–7.50 cm 7.50–3.75 cm 3.75–2.50 cm 2.50–1.67 cm 1.67–1.11 cm 1.11–0.75 cm 1.11–0.75 cm 7.50–5.00 mm 6.00–4.00 mm 4.00–2.73 mm 2.73–1.76 mm 2.14–1.36 mm

1.2.1 Pulse Radar A radar system that simply detects the range of targets is one that repetitively transmits a pulse of microwaves of width τ seconds every T seconds, and measures the time elapsed before the reflected pulse is received back at the transmitter site. Thus, the distance or range to the target is d=

v t 2

(1.1)

where v = 3 × 108 m/s in free space and t is the time to and from the target. This pulse radar system is shown in block diagram form in Figure 1.2. The circulator, a three-port magnetic ferrite device routes microwave signals from one port to another. Signals moving in the direction of the arrow (port 1 to port 2, port 2 to port 3) are attenuated by about 0.3 dB, and signals moving against the arrow (port 1 to port 3, port 2 to port 1) are attenuated by 20 dB or more. Thus, the circulator serves to separate the transmit and receive signals, permitting the use of one antenna and eliminating the need for a transmit/receive switch. Also, the reverse transmission from ports 1 to 3 through the circulator allows a replica of the transmitted pulse to be displayed on the receiver indicator. For this range detection system to work properly, it should be clear that the pulse reflected from the target

5

INTRODUCTION

Table 1.4 Frequency Ranges Allocated by Use Allocated Use AM Radio Shortwave Radio

Frequency Range 535–1605 kHz 3–30 MHz

FM Radio

88–108 MHz

Broadcast Television VHF (Ch.2–4) VHF (Ch.5–6) UHF (Ch.7–13) UHF (Ch.14–83)

54–72 MHz 76–88 MHz 174–216 MHz 470–890 MHz

Cable Television Cellular Telephony (GSM, CDMA) uplink/downlink uplink/downlink Global Positioning System (GPS) L1 L2

55.25–999 MHz

824–849/869–894 MHz 1.85–1.91/1.93–1.99 GHz

1575.42 MHz 1227.60 MHz

Microwave Ovens

2.45 GHz

Wireless Data Link (Wi-Fi) IEEE 802.11a IEEE 802.11b IEEE 802.11n

5 GHz 2.4 GHz 2.4/5 GHz

Satellite Television uplink/downlink uplink/downlink

5.925–7.075/3.7–4.2 GHz 14.0–14.5/11.7–12.2 GHz

6

MICROWAVE TRANSMISSION LINE CIRCUITS

TRANSMITTER

CIRCULATOR 1

ANTENNA

TARGET

2

3

T

τ

t

RECEIVER

Figure 1.2 A pulse radar range detection system block diagram.

should arrive back at the transmitter/receiver site at an elapsed time t bracketed by τ ≤ t ≤ T. Example 1.1: A pulse radar for range detection transmits a microwave pulse of width τ = 10 µs every T = 1 ms. (a) At what minimum range can a target be detected, and (b) at what maximum range can a target be detected? Solution: (a) From (1.1), with t = τ = 10 µs, d=

v 3 × 108 τ = × 10−5 = 1500 m = 0.93 mile 2 2

(b) From (1.1), with t = T = 1 ms, d=

3 × 108 v T = × 10−3 = 1.5 × 105 m = 93 miles 2 2

1.2.2 Doppler Radar The Doppler effect is a shift in the frequency of a wave caused by the relative motion of the transmitter, the target, or the receiver. Most Doppler radars have the transmitter and receiver at one location and use a common antenna (i.e., police radar). In the same way that a train whistle is higher in frequency when arriving and lower in frequency when departing, the received frequency fr will be higher than

INTRODUCTION

7

the frequency of the transmitted wave ft if the transmitted wave is reflected and received back from a target moving toward the transmitter/receiver location. Thus, for an approaching target (fr > ft ), and for a receding target (fr < ft ), the Doppler frequency shift that is detected and measured at the receiver is, fD = fr − ft = ±

2vmo ft v

(1.2)

with the plus sign for incoming targets and the minus sign for outgoing targets. In (1.2), vmo is the velocity of the moving object (target) in the inward or outward direction from the receiver/transmitter, and v is the velocity of the electromagnetic wave (3 × 108 m/s in free space). Rearranging (1.2), the velocity of the target is, vmo = ±

v fD 2 ft

(1.3)

Example 1.2: The transmitted frequency of a police radar is ft = 10.5 GHz. What is the velocity of an approaching vehicle if fD = 2, 345 Hz? Solution: Substitution into (1.3) yields, vmo =

v fD 3 × 108 2, 345 = = 33.50 m/s ∼ = 75 mi/hr 2 ft 2 10.5 × 109

It is useful to note that the Doppler shift produces low-frequency tones (in this case in the human audio range) and thus only simple electronics are required to process the return signal. 1.2.3 Frequency-Modulated Continuous-Wave (FMCW) Radar With this method of measuring short-range distances to reflecting objects, the transmitter is continuously and linearly swept from a given lower frequency to a given higher frequency, as shown in Figure 1.3. The signal from the swept source is divided into two parts by the power divider, with most of the signal fed through the circulator and into the irradiated region. A small part of the signal (typically 20 dB down from the input signal) is fed to a modulator or mixer. Reflected signals are returned through the circulator and also are fed to the modulator or mixer. There the two signals are combined and the reflected signal is shifted to a lower

8

MICROWAVE TRANSMISSION LINE CIRCUITS

2 CIRCULATOR

1

fH

ANTENNA

t

fL SWEPT SOURCE

3

POWER DIVIDER

MODULATOR (MIXER)

d1

d2 FREQUENCY COUNTER

Figure 1.3 FMCW radar block diagram.

intermediate frequency (IF). The resulting signal is filtered to pass only the IF, then the IF frequency is measured by the counter. To illustrate the FMCW radar detection method, let fL = 100 MHz and fH = 200 MHz. A 100 MHz signal sent out by the transmitter and reflected by the target will arrive back at the transmitter at a later time when the transmitter has swept to, for instance, 105 MHz. The returning signal at 100 MHz and the currently transmitted signal at 105 MHz are applied to the mixer, yielding an intermediate frequency of 5 MHz. This intermediate frequency is directly proportional to the distance to the reflecting object. For a linear sweep of repetition time T , the frequency at an elapsed time t is: f=



fH − fL T



t + fL

(1.4)

or f − fL = RF S t

(1.5)

where RF S = (fH − fL )/T is the frequency sweep rate. Thus, if t is the elapsed time between transmitting a lower frequency fL and a higher frequency f, then the intermediate frequency is, fIF = RF S t (1.6) To continue the illustration, let the signal leaving the antenna travel through air and dry soil of thickness d1 and d2 , respectively, and be reflected from a buried metal object beneath the soil. The elapsed time down and back through the air and

INTRODUCTION

9

soil is,

d1 d2 +2 (1.7) v1 v2 where v is the velocity of propagation through the given material. Substituting this expression for t into fIF = RF S t yields,   d1 d2 + (1.8) fIF 2 = 2RF S v1 v2 t2 = 2

or

d1 d2 fIF 2 + = (1.9) v1 v2 2RF S Likewise, the reflection from the interface between the air and soil yields an elapsed time d1 t1 = 2 (1.10) v1 and the intermediate frequency fIF 1 = 2RF S or

d1 v1

d1 fIF 1 = v1 2RF S

From (1.11) and (1.8)

Thus, measuring fIF 1 as

(1.11)

(1.12)

d2 fIF 2 − fIF 1 = (1.13) v2 2RF S and fIF 2 determines both d1 and d2 , if v1 and v2 are known, fIF 1 v1 2RF S

(1.14)

fIF 2 − fIF 1 v2 2RF S

(1.15)

d1 = and d2 =

Example 1.3: In this example, an FMCW radar is used as a groundpenetrating radar (GPR) for detection of buried objects. The FMCW radar is operated at a lower frequency to increase penetration depth into the soil. Assume v1 = 3 ×108 m/s in air and v2 = 108 m/s in dry, hardpacked soil. Let RF S = 3 × 1011 Hz/sec (i.e., fH − fL = 300 MHz

10

MICROWAVE TRANSMISSION LINE CIRCUITS

and T = 0.001 sec). If the two measured intermediate frequencies are fIF 1 = 2000 Hz and fIF 2 = 2600 Hz, determine the distance d1 from the antenna to the soil surface, and the depth d2 of the buried metal object.

Solution: Substituting the given information into (1.14) yields d1 = 1 m, and from (1.15) d2 = 0.1 m.

1.3 MICROWAVE COMMUNICATION SYSTEMS Human beings have been devising systems for communicating messages from one distant point to another at least as long as history has been recorded. The fundamental elements of any such communication system include at one end an information source that inputs a message to a transmitter. In the block diagram of Figure 1.4, the transmitter couples the message onto a transmission channel in the form of a signal that matches the transfer properties of the channel. The channel is the medium forming the path between the transmitter and the receiver. This can be a material structure for guiding the signal wave, or an unguided atmospheric or space channel. For a transmission line channel as well as an atmospheric channel, signal power of the propagating electromagnetic wave is lost by attenuation (due to absorption and heat generation) and reflection (due to wave impedance changes). A channel is distortionless if all signals within a frequency band of interest are attenuated, reflected, and delayed in arriving by the same amount, thus allowing the transmitted information to be retrieved exactly. However, some distortion always occurs, and the function of the receiver is to extract the weakened and distorted signal from the channel, amplify it, and restore it to its original form before passing it on to the message destination. A vital part of the signal extraction and restoration process is the filtering characteristic of the receiver. The receiver should reject unwanted out-of-band interference and noise and pass all signals in the frequency band of interest without adding distortion. The amplitude and phase delay of signals passing through the receiver may be represented by a transfer function, H(ω) = |H(ω)|e−jφ(ω)

(1.16)

11

INTRODUCTION

TRANSMITTER

TRANSMISSION CHANNEL

RECEIVER

MESSAGE DESTINATION

MESSAGE SOURCE

Figure 1.4 Fundamental elements of a communication system.

where ω = 2πf is the radian frequency. For distortionless transmission the transfer function should be flat in magnitude over frequencies within the passband, or |H(ω)| = H0 e−αl

(1.17)

where α is the attenuation rate through the receiver and l is the length of travel. The design of maximally flat magnitude filters is covered in Chapter 5. Also for distortionless transmission, the phase delay should be a linear function of frequency, as represented by ωl φ(ω) = = ωTd (1.18) v where v is the signal velocity and Td is the time delay in traveling the distance l. Thus, Td = l/v should be flat or constant at all passband frequencies. The design of maximally flat delay filters also is covered in Chapter 5. 1.3.1 Microwave Links and Repeaters Point-to-point microwave links are used as studio-to-transmitter links (STL) for radio and television broadcasting stations, to link cable television head-ends (antenna sites) to their distribution systems, and as part of a communications network involving telephone, data, or television signals. A fiber-optic link has a greater available bandwidth, but a microwave link is easier to install in difficult terrain and a single line-of-sight microwave link can cover a distance of about 40 km (∼ =25 miles) before propagation loss reduces the signal to unacceptable levels. The link can be extended beyond 40 km using a repeater station that receives, amplifies, and retransmits the signal. A communications satellite functions as a distant repeater that receives uplink signals from Earth stations, processes the signals, and then retransmits them on the

12

MICROWAVE TRANSMISSION LINE CIRCUITS

downlink to their intended Earth destinations. The International Telecommunications Union has allocated specific bands for satellite communications (Table 1.4). The bands used by most U.S. commercial satellites for domestic communications are the 4/6 GHz band (3.7 to 4.2 GHz downlink and 5.925 to 7.075 GHz uplink) and the 12/14 GHz band (11.7 to 12.2 GHz downlink and 14.0 to 14.5 GHz uplink). Each uplink and downlink segment has been allocated 500 MHz of bandwidth. By using different frequency bands for the uplink and downlink, the same antenna on the satellite may be used for transmission and reception without interference between signals in the two bands. The downlink is assigned the lower frequency band because lower frequency signals are attenuated less by the Earth’s atmosphere, thus conserving the limited output power of the satellite.

1.3.2 Carrier Modulation

Information-bearing signals which are intended for direct transmission over a line are called baseband signals. For communications systems having bandpass channels, it becomes advantageous to modulate a carrier wave signal with the information-bearing signals prior to transmission. The frequency of the carrier wave may be many times greater than the modulation bandwidth. Thus, the trend in developing new systems has been to increase the carrier-wave frequency to achieve a corresponding increase in bandwidth or information capacity. This process of modulating the carrier wave with the information to be transmitted led to the birth of television, radar, and microwave communication systems. An enormous amount of information (signal bandwidth) may be sent by selecting a high-frequency carrier wave that is best suited to transmit through the available channel (microwaves for transmission lines, waveguides, and free-space links between antennas; and light waves for optical fibers), and then allowing the amplitude or frequency of the carrier to vary in proportion to the information that is sent. This formatting of the information to be sent is amplitude modulation (AM) if the amplitude of the carrier wave is modified by the information signals while the carrier frequency is held constant, and frequency modulation (FM) if the frequency of the carrier wave is modified by the information while the carrier amplitude is held constant. If the amplitude and frequency variations within the received modulated carrier are exact replicas of what was sent, then no distortion was contributed by the channel and the transmitted information may be removed or demodulated from the carrier in a precise manner.

INTRODUCTION

13

Conventional amplitude modulation places the message (baseband signals) on a carrier wave of frequency much greater than any of the signal frequencies contained in the baseband. The resulting waveform has a spectrum that extends above and below the carrier frequency. In essence, AM shifts the baseband to a new region of the electromagnetic spectrum. Each AM radio station broadcasts at a different carrier frequency, so they can be individually received using bandpass filters tuned to the assigned carrier. After reception, the modulated signals are electronically returned (demodulated) to the original baseband frequencies. The modulation waveforms for both AM and FM is illustrated in Figure 1.5(a) for a sinusoidal baseband signal and a high-frequency sinusoidal carrier, and a block diagram showing the modulation and demodulation process is shown in Figure 1.5(b). Amplitude modulation of a sinusoidal carrier of radian frequency ωc by a sinusoidal message signal of frequency ωm is expressed as i = Is (1 + m cos ωm t) cos ωct

(1.19)

where m is the modulation index, and m ≤ 1. If the baseband signals extend in frequency from 0 to ωm , then the frequency spectrum of information in the AM modulated carrier extends from ωc − ωm up to ωc + ωm , or a frequency bandwidth of 2fm hertz. In conventional frequency modulation (FM), the time phase angle of a carrier wave is made to vary in accordance with the baseband information signals. The modulated current may be expressed as i = Is cos [ωct + θ(t)]

(1.20)

where the message resides in the θ(t) term. Frequency modulation of a sinusoidal carrier of radian frequency ωc by a sinusoidal message signal of frequency ωm is expressed as i = Is cos (ωc t + β sin ωm t)

(1.21)

where β = ∆ω/ωm is the modulation index. If the baseband signals extend in frequency from 0 to ωm , then the frequency spectrum of information in the FM modulated carrier extends from frequencies below to above the carrier. The total bandwidth in hertz required for FM may be expressed as B = 2fm (1 + β)

(1.22)

14

MICROWAVE TRANSMISSION LINE CIRCUITS

CARRIER

MODULATING SIGNAL (BASEBAND)

AM

FM

(a) BASEBAND IN

CARRIER OSCILLATOR

MODULATOR

AMPLIFIER

ANTENNA

LOCAL OSCILLATOR

DEMODULATOR

AMPLIFIER

ANTENNA

BASEBAND OUT

(b) Figure 1.5 Modulation and demodulation process for AM and FM, showing (a) waveforms and (b) block diagram.

INTRODUCTION

15

Thus, for β  1 (low information content), the FM bandwidth is 2fm , the same as for AM. For larger values of β, the required FM bandwidth exceeds that of a comparable AM system. Since β can be much greater than one, the FM bandwidth can greatly exceed the bandwidth required for AM. The added FM bandwidth can improve the signal-to-noise ratio of FM over that of AM. For sinusoidal modulation in relatively low noise receivers, and m = 1 for AM, the signal-to-noise ratios for FM and AM are related by     S S 2 = 3β (1.23) N FM N AM Thus, by increasing the modulation index β, the signal-to-noise ratio can be made much higher in FM than in AM, but at the expense of added bandwidth. For example, when β = 5 the FM signal-to-noise ratio is 75 times that of an equivalent AM system, but the required bandwidth is approximately six times larger than for AM.

1.4 SIGNAL-TO-NOISE REQUIREMENTS Internally and externally generated electrical signals that interfere with the information-bearing signal transmission are classified as noise. Communication systems differ in their requirements for an acceptable signal-to-noise ratio (S/N ). For standard common-carrier telephone service the requirement is S/N ≥ 50 dB, and anything less would be detectable as noise in the audio signal. Thus, in the process of transmitting a telephone signal from one location to another, through telephone wires, coaxial cables, waveguides, microwave repeaters, and communication satellites, the desired signal must be at least 50 dB above the noise when the transmission reaches the telephone handset. Signal-to-noise requirements for television transmission are less severe. For analog television broadcasts, a signal-to-noise ratio of 40 dB produces no noticeable degradation of the TV picture. For S/N = 35 dB, the picture is somewhat snowy. For S/N = 30 dB, the interference is objectionable, and for S/N = 25 dB the picture is unacceptable. Digital television broadcasts, which replaced analog television in the United States in 2009, produces an excellent picture with a much lower S/N of only 15.19 dB. The S/N of any system can obviously be improved by increasing the signal and decreasing the noise. The use of low-noise-amplifiers (LNAs) and lowpass or bandpass filters that pass the wanted signals and reject out-of-band noise are

16

MICROWAVE TRANSMISSION LINE CIRCUITS

excellent ways to improve S/N . Another way to improve S/N is to reduce the effects of noise interference by converting the analog information-bearing signals into digital signals through the use of pulse-code modulation (PCM). The accumulation of noise as the signal travels through many links is avoided by regenerating the signal at each repeater. This requires demodulating the signal, decoding the data, then recoding and remodulating onto a new carrier. If the signalto-noise ratio for each single link is high enough to avoid errors, no increase in error rates will occur as the signal progresses through the system. While a zero error rate is not obtainable in practice, the error rates add directly from link to link, rather than multiply as in an analog system. A disadvantage of digital transmission of data is that more bandwidth is required than when analog AM or FM modulation techniques are used. However, as better data compression algorithms are created, bandwidth for digital modulation becomes less of a problem. To be transmitted by microwaves, the digital signal modulates the microwave carrier in a manner similar to that for an analog system. Repeaters, however, must demodulate the signal to baseband in order to achieve the advantages of digital transmission. That is, they must be regenerative repeaters, as shown in Figure 1.6. In pulse-code modulation, an analog signal to be transmitted versus time is quantized into m amplitude levels. At predetermined times the signal is sampled and represented by an N -digit code and the digits are transmitted as a sequence of 1s and 0s. The number of levels available depends on the number of bits used to express the sample value. The number of levels is given by m = 2N

(1.24)

where m is the number of levels and N is the number of binary digits (bits) per sample. Thus, for N = 8 (telephone), m = 256, and for N = 16 (CD player), m = 65, 536. To illustrate PCM, the analog signal in Figure 1.7 is quantized into m = 8 amplitude levels, and each sample is represented by N = 3 bits/sample. The seven sampled values are transmitted serially in binary form as 001010011101110111111. As is evident in Figure 1.7, quantization errors may occur when the signal samples are midway between two assigned levels. The quantization error can be reduced by making the step size between levels smaller, which requires more digits to encode the samples. While the three-digit code required 23 = 8 quantization levels, an eight-digit code would require 28 = 256 quantization levels, and would yield a signal-to-quantization-noise ratio of approximately 50 dB. Since this would meet

17

INTRODUCTION

COMPLETE RECEIVER

BASEBAND

DATA

DECODER

ENCODER

RECEIVE ANTENNA

BASEBAND

COMPLETE TRANSMITTER

TRANSMIT ANTENNA

Figure 1.6 Digital microwave repeater block diagram.

the S/N requirements for telephone transmission, an eight-digit code is used to represent the analog signal in most PCM systems. Example 1.4: For an eight-digit code, let the maximum signal amplitude be 255 units (256, including zero). If quantization error or noise is the only noise present, determine S/N in dB, based on the maximum quantization error. Solution: The maximum quantization error or noise amplitude will be half of one level, or 0.5. Thus, S/N is   S 255 = 20 log = 54 dB N 0.5 1.4.1 Channel Capacity and Information Content The rate at which samples of the analog signal are taken is: fS = 1/T

(1.25)

If fm is the highest frequency in the original analog waveform, the waveform can be reproduced if fS = 1/T > 2fm (1.26) which is the Nyquist condition. Thus, if the spectral bandwidth (B) of the original signal is B = fm −0, then for faithful reproduction of the original signal we require fS > 2B

(1.27)

18

MICROWAVE TRANSMISSION LINE CIRCUITS

Amplitude Levels

111 110 101 100 011 010 001 000 t Sampling Times

(a)

001

010

011

101

110

111

111

(b) Figure 1.7 Sampling and coding for PCM. (a) Quantized samples of continuous analog signal. (b) A binary coded pulse sequence representing the sampled values.

In preparing the signal waveform for transmission, and allocating amplitude levels (m), we recognize that it is difficult and expensive to distinguish between levels that are closer than AN apart, where AN is the root mean square (RMS) electrical noise level. Expressing the maximum signal amplitude as AS , the required number of levels (m) is related to these amplitudes by: s  2 AS m≥ 1+ (1.28) AN Therefore, from (1.24), (1.27), and (1.28), the minimum number of bits per second (b/s) required is: s "  2  2 # AS AS RI = fS × N = 2B log2 1 + = B log2 1 + (1.29) AN AN This equation (Shannon’s equation or law) sets forth the requirements for recovering the original analog signal. This equation may also be used to represent the channel

19

INTRODUCTION

capacity of a communications receiver, or the information content of different types of signal waveforms, by taking RI as the information rate (b/s), B as the channel bandwidth (Hz), and AS /AN as the receiver signal-to-noise ratio.  2  2 AS AS Assuming the usual case that A  1, and expressing in dB, as AN N S = 10 log10 N



or 

AS AN

AS AN

2

2

= 20 log10



AS AN



= 10(S/N)/10

(1.30)

(1.31)

equation (1.29) may be simplified to

or

h i B(S/N ) log2 10 RI = B log2 10(S/N)/10 = 10

(1.32)

RI = 0.332B(S/N )

(1.33)

Thus, a receiver bandwidth of 10 MHz and a signal-to-noise ratio of 16 dB yields an information rate or bit rate RI = 53.12 Mb/s. Example 1.5: Calculate the channel capacity needed to transmit the following familiar forms of information, and the amount of information contained in each form: (a) a speech channel with B ∼ = 3.5 kHz (fS = 8 > 2B kHz is adequate) and S/N ≥ 30 dB (or AN /AS ≥ 31.6), (b) a 50-minute audio cassette played on a stereo having a receiver-amplifier with B ∼ = 20 kHz and S/N ∼ = 80 dB, (c) a 100-minute VCR tape of a movie played on a system with B = 5.5 MHz and S/N ≥ 50 dB, and (d) the book on which the movie in (c) is based, which contains 100,000 words and 250 pages. Solution: (a) RI = 0.332×3.5 kHz×30 = 34.9 kb/s is the information rate or the minimum channel capacity required. (In practice, RI = 56 kb/s is used.) The information content of a 50-minute speech is: 56 kb/s × 50 min × 60 s/min = 168 Mb = 0.168 Gb. (b) RI = 0.332 × 20 kHz × 80 = 531 kb/s is the information rate, and the information content is: 531 kb/s × 50 min × 60 s/min = 1.6 Gb. (c) The information rate is RI = 0.332×5.5 MHz×50 = 91 Mb/s, and the

20

MICROWAVE TRANSMISSION LINE CIRCUITS

information content is: 91 Mb/s × 100 min × 60 s/min = 546 Gb. (d) With an average of five letters/word, the book contains 500,000 letters. At seven bits/letter (the ASCII code for converting text to digital form) the information content is 3.5 Mb. Thus, as expected, more information is contained in the movie than in the book. (In this example, a picture is worth 546 Gb/3.5 Mb = 156,000 words.) BIBLIOGRAPHY R. Blake, Comprehensive Electronic Communications. Berkeley, CA: West Group, 1997. Federal Communications Commission. “FCC Online Table of Frequency Allocations.” http://transition.fcc.gov/oet/spectrum/table/fcctable.pdf. International Telecommunications Union. “The Radio Regulations, Edition of 2008.” http://www.itu.int/pub/R-REG-RR. G. Keiser, Optical Fiber Communications, 4th ed. New York: McGraw-Hill, 2008. J. A. Kosinski, W. D. Palmer, and M. B. Steer, “Unified Understanding of RF Remote Probing,” IEEE Sensors Journal, vol. 11, no. 12, pp. 3055–3063, December 2011. J. C. Palais, Fiber Optic Communications, 5th ed. Upper Saddle River, NJ: PrenticeHall, 2005. D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley and Sons, 2011. P. A. Rizzi, Microwave Engineering: Passive Circuits. Upper Saddle River, NJ: Prentice-Hall, 1988. S. E. Schwarz, Electromagnetics for Engineers. New York: Oxford University Press USA, 1995. A. W. Scott, Understanding Microwaves, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2005.

INTRODUCTION

21

F. G. Stremler, Introduction to Communication Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1990. F. T. Ulaby, E. Michielssen, and U. Ravaioli, Fundamentals of Applied Electromagnetics, 6th ed. Upper Saddle River, NJ: Prentice-Hall, 2010.

22

MICROWAVE TRANSMISSION LINE CIRCUITS

Chapter 2 MICROWAVE TRANSMISSION LINES 2.1 USEFUL TRANSMISSION LINE CONFIGURATIONS

Any configuration of two or more conductors will serve as a transmission line. The configurations most often used consist of just two conductors of generally uniform cross-section along the direction of wave propagation. Energy propagates within the medium between conductors (dielectric or air) as transverse electromagnetic (TEM) waves. This means that the vector electric field intensity E between conductors (proportional to voltage) and the vector magnetic field intensity H surrounding conductors (proportional to current) are perpendicular to each other and to the direction of wave propagation. Thus, the pattern of E and H fields will be the same at any cross-section perpendicular to the direction of propagation on a transmission line of uniform dimensions. Shown in Figure 2.1 are cross-sections of some two-conductor transmission lines that are used extensively in modern high-frequency and microwave applications. The two-wire line or twin-lead shown in Figure 2.1(a) also would represent a twisted-pair transmission line. The planar transmission lines shown in Figure 2.1(b, c, and d) may be constructed from sheets of low-loss dielectric (e.g., polyethylene) that have a thin layer of copper cladding on each surface. The thickness of the copper cladding is usually 0.0014 inch (1 ounce per square foot) or 0.0028 inch (2 ounces per square foot). Selected areas of the copper cladding are etched away to form the desired transmission line circuit pattern. The planar lines also may be constructed by depositing layers of metal onto the dielectric substrate until the desired conductor thickness is achieved.

23

24

MICROWAVE TRANSMISSION LINE CIRCUITS

a w

w

h

2h

h

(a)

(b)

(c)

b w

a

b

(d)

(e)

Figure 2.1 Cross-sections of some two-conductor transmission lines: (a) Two-wire line or twin-lead; (b) Parallel-plate line; (c) Microstrip; (d) Stripline; (e) Coaxial line. The conductors extend infinitely into and out of the plane of the page, maintaining a constant geometry.

MICROWAVE TRANSMISSION LINES

25

2.2 WAVE EQUATION FOR VOLTAGE AND CURRENT The equations describing the propagation of voltage and current waves (to be derived) will be valid for any two-conductor transmission line. In the derivation, certain parameters will be identified that are functions of the line geometry and the materials used. As a model for the derivations, let a time-varying wave of voltage and current be injected from the source onto the two-conductor line shown in Figure 2.2(a). The waves will propagate from left to right along the distance l between source and load. There may also be reflections from the load (at x = 0), so that total voltage and current at some position and time will be represented by v(x, t) and i(x, t), respectively. The current flowing toward the load in one conductor and toward the source in the other causes a voltage drop along conductors and a magnetic flux between conductors. For an incremental length ∆x, the voltage drop divided by current may be represented by a series resistance R and the flux density divided by current may be represented by a series inductance L. Also, the voltage difference between conductors over an incremental length ∆x causes a conduction current and a displacement current through the dielectric. This leakage current gives rise to a shunt conductance G and capacitance C. The lumped-element equivalent circuit representing the transmission line over the length ∆x is shown in Figure 2.2b. In Figure 2.2(a and b), consider a differential element of length ∆x, so small that the time delay between x and x + ∆x is negligible. The voltage drop across ∆x is di v(x + ∆x, t) − v(x, t) = (Ri + L )∆x (2.1) dt Taking the limit as ∆x approaches zero yields: lim

∆x→0

dv di v(x + ∆x, t) − v(x, t) = = Ri + L ∆x dx dt

(2.2)

Again, considering a length ∆x in Figure 2.2(a and b), the current through capacitance C and conductance G is: i(x + ∆x, t) − i(x, t) = (Gv + C

dv )∆x dt

(2.3)

and i(x + ∆x, t) − i(x, t) di dv = = Gv + C ∆x→0 ∆x dx dt lim

(2.4)

26

MICROWAVE TRANSMISSION LINE CIRCUITS

l i(x, t)

ZS

iL

v(x, t)

VS

v(x + ∆ x, t)

L

(a)

0

x i(x + ∆ x, t)

ZL

vL

R

i(x, t) C

G

v(x, t)

(b)

∆x Figure 2.2 (a) Model of two-conductor transmission line indicating propagating waves of voltage and current. (b) Equivalent circuit for a length ∆x.

Assuming that voltage and current vary sinusoidally with time, we may separate the variables in terms of x and t dependence as: v(x, t) = V (x)ejωt

(2.5)

i(x, t) = I(x)ejωt

(2.6)

and where ω = 2πf is the radian frequency and f is the frequency in hertz (Hz). More accurately, for future calculations it is understood that we use the real part of the above equations, or v(x, t) = Re[V (x)ejωt ] (2.7) and i(x, t) = Re[I(x)ejωt ]

(2.8)

Substituting (2.5) and (2.6) into (2.2) and (2.4), and canceling the common ejωt term, yields: dV = (R + jωL)I (2.9) dx

MICROWAVE TRANSMISSION LINES

27

and

dI = (G + jωC)V (2.10) dx Differentiating (2.9) and (2.10) with respect to x and substituting (2.9) and (2.10) into the results yield the desired second-order differential equations as:

and

d2 V = (R + jωL)(G + jωC)V dx2

(2.11)

d2 I = (R + jωL)(G + jωC)I dx2

(2.12)

Let γ=

p

(R + jωL)(G + jωC) = α + jβ

(2.13)

where γ, α, and β are defined as the propagation, attenuation, and phase constants respectively, then (2.11) and (2.12) become: d2 V = γ2 V dx2

(2.14)

and

d2 I = γ2 I (2.15) dx2 These equations are known as the Telegrapher’s Equations and were first developed by Oliver Heaviside in the 1880s and applied to trans-Atlantic telegraphy. Since V and I satisfy the same differential equation, we will first express the solution for V (x), and then determine I(x) from previous equations. The general solution for the voltage wave V (x) in (2.14) is: V (x) = Aeγx + Be−γx

(2.16)

Differentiating (2.16) and using (2.9) gives: dV = γAeγx − γBe−γx = (R + jωL)I(x) dx

(2.17)

Letting R + jωL = γ

s

R + jωL = Z0 G + jωC

(2.18)

28

MICROWAVE TRANSMISSION LINE CIRCUITS

we may express the current wave I(x) as: I(x) =

A γx B −γx e − e Z0 Z0

(2.19)

where Z0 , as defined in (2.18), is the characteristic impedance (in ohms) of the transmission line having the parameters R, L, C, and G at the frequency f = ω/2π. The voltage wave V (x) in (2.16) and the current wave I(x) in (2.19) satisfy the differential equations (2.14) and (2.15), respectively. They must also satisfy the boundary conditions imposed at x = 0 (the load in Figure 2.1(f)). Thus, at x = 0 we have: V (0) = VL = A + B A−B I(0) = IL = Z0

(2.20) (2.21)

Solving these two equations for A and B yields: VL + Z0 IL VL = (1 + 2 2 VL − Z0 IL VL B= = (1 − 2 2 A=

Z0 ) ZL Z0 ) ZL

(2.22) (2.23)

Substituting the solutions for A and B into the expressions for V (x) and I(x) yields: Z0 IL γx VL γx (e + e−γx ) + (e − e−γx ) (2.24) V (x) = 2 2 and IL γx VL γx I(x) = (e + e−γx ) + (e − e−γx ) (2.25) 2 2Z0 or V (x) = VL cosh γx + Z0 IL sinh γx (2.26) and I(x) = IL cosh γx +

VL sinh γx Z0

(2.27)

Putting (2.26) and (2.27) in matrix form gives 

V (x) I(x)



=



cosh γx 1 Z0 sinh γx

Z0 sinh γx cosh γx



VL IL



(2.28)

MICROWAVE TRANSMISSION LINES

29

This two-by-two matrix fully describes the relationship between the voltage and current on the transmission line, including both propagation and attenuation characteristics. It is completely general for transmission lines with constant cross-section and uniform material properties. This matrix is known as the ABCD matrix and will be discussed in depth in Chapter 4. Since V (x) and I(x) represent the total voltage and current at any point x distance from the load, the impedance presented to waves at any point x is: Z(x) =

V (x) ZL cosh γx + Z0 sinh γx ZL + Z0 tanh γx = Z0 = Z0 I(x) Z0 cosh γx + ZL sinh γx Z0 + ZL tanh γx

(2.29)

To express v(x, t) and i(x, t), we reinsert the ejωt term in (2.16) and (2.19) to obtain, v(x, t) = Aejωt+γx + Bejωt−γx = Aeαx ej(ωt+βx) + Be−αx ej(ωt−βx) and i(x, t) =

A αx j(ωt+βx) B −αx j(ωt−βx) e e − e e Z0 Z0

(2.30)

(2.31)

where γ, α, and β are the propagation, attenuation, and phase constants, respectively. The two terms in the v(x, t) and i(x, t) equations represent waves incident upon the load (from the source) and waves reflected from the load (toward the source). To test this statement let the phase angle of the first term in (2.30) and (2.31) be θi = ωt + βx (2.32) If we change position with time in such a way that θi stays constant, the position is a fixed point on a moving wave. Thus, with θi constant: dθi dx =ω+β =0 dt dt or −

dx ω =v= dt β

(2.33)

(2.34)

and this wave moves in the −x direction (from source to load) at a velocity v = ω/β. To test the remaining phase term in (2.30) and (2.31), let θr = ωt − βx

(2.35)

30

MICROWAVE TRANSMISSION LINE CIRCUITS

and if we move with the wave to maintain a constant phase point

or

dθr dx =ω−β =0 dt dt

(2.36)

dx ω =v= dt β

(2.37)

Hence, the waves containing the θr term in (2.30) and (2.31) travel in the +x direction (from load to source) at a velocity v = ω/β. The waves of voltage and current will be similar in that they travel at the same velocity (ω/β), are attenuated at the same rate (α) with distance, and they oscillate at the same frequency. Since v(x, t) and i(x, t) are related by Z0 , if Z0 is complex, as Z0 = |Z0 |ejφ, then voltage and current will oscillate out of time phase by φ degrees. In the expressions for v(x, t) and i(x, t), A and B are complex (phasor) voltages representing the incident voltage wave and the reflected voltage wave, respectively, at x = 0. The distance that a sinusoidal wave travels in one time period (T = 1/f) is related to the phase constant β by λ = vT =

v 2πv 2π = = f ω β

(2.38)

Hence, if λ is expressed in meters, then β is in radians/meter. As one observes from the development leading up to the expressions for v(x, t) and i(x, t), the nature of the propagating waves ultimately depends upon the distributed parameters R, L, C, and G. Unless we specifically wish to calculate the attenuation constant α (considered later), we may assume for most practical transmission lines that ωL  R and ωC  G. (Of course, if the conductors and dielectrics are perfect and have no resistive or conductive losses, R = 0 and G = 0.) Let us assume for now that R and G are small enough to be neglected, and we will refer to this as the lossless case. The propagation constant becomes p √ γ = (R + jωL)(G + jωC) = jω LC = jβ (2.39) So, the phase constant is

√ β = ω LC

(2.40)

and the velocity of wave propagation is related to L and C by: v=

ω 1 = √ β LC

(2.41)

MICROWAVE TRANSMISSION LINES

31

The characteristic impedance for this lossless case is:

Z0 =

s

R + jωL = G + jωC

r

L 1 = C vC

(2.42)

Since α = 0 for this lossless case, the expressions for V (x), I(x), and Z(x) change accordingly. From (2.16) and (2.19): V (x) = Aejβx + Be−jβx = Vi (x) + Vr (x) A jβx B −jβx I(x) = e − e = Ii (x) + Ir (x) Z0 Z0

(2.43) (2.44)

Where Vi (x), Vr (x), Ii (x), and Ir (x) are the incident and reflected voltage and current waves at the point x on the transmission line. From (2.26) and (2.27): V (x) = VL cos βx + jZ0 IL sin βx VL I(x) = IL cos βx + j sinβx Z0

(2.45) (2.46)

where VL and IL are the voltage and current waves at the transmission line load, and V (x) ZL + jZ0 tan βx Z(x) = = Z0 (2.47) I(x) Z0 + jZL tan βx

2.3 INCIDENT, REFLECTED, AND STANDING WAVES 2.3.1 Voltage Reflection Coefficient The voltage reflection coefficient at any point x on the transmission line is defined as the ratio of reflected voltage to incident voltage at x, and is expressed as Γ(x) =

Vr (x) B = e−2αx e−j2βx = ΓLe−2αx e−j2βx Vi (x) A

where ΓL =

B ZL − Z0 = A ZL + Z0

(2.48)

(2.49)

32

MICROWAVE TRANSMISSION LINE CIRCUITS

is the reflection coefficient at the load (x = 0), which may be rearranged to solve for ZL as, 1 + ΓL ZL = Z0 (2.50) 1 − ΓL Using Γ(x) in the equations for V (x) and I(x), we have V (x) = Vi (x) [1 + Γ(x)] Vi (x) [1 − Γ(x)] I(x) = Z0

(2.51) (2.52)

and the wave impedance becomes Z(x) =

V (x) 1 + Γ(x) = Z0 I(x) 1 − Γ(x)

(2.53)

The last equation may be solved for Γ(x) to obtain: Γ(x) =

Z(x) − Z0 = ΓL e−2αxe−j2βx Z(x) + Z0

(2.54)

For a lossless line, Z0 is real, but ZL , Z(x), ΓL , and Γ(x) generally are complex. Example 2.1: At x = 0.965 cm from the load on a lossless transmission line of characteristic impedance Z0 = 50 Ω, the measured reflection ◦ coefficient is Γ(x) = 0.817ej0 , and λ = 10 cm. Determine the load impedance. Solution: From (2.48), with α = 0, ΓL

=

Γ(x)ej2βx

=

0.817ej0 ej2π×2×0.0965

= =

0.817ej69.44 0.287 + j0.765

◦

◦

and from (2.50), ZL = Z0

1 + ΓL 1.287 + j0.765 = 50 = 71.586 77.74◦ ∼ = 15+j70 Ω 1 − ΓL 0.713 − j0.765

MICROWAVE TRANSMISSION LINES

33

2.3.2 Power Flow The time-averaged power transmitted along the line at any point x is P (x) =

1 Re[V (x)I(x)∗ ] = Pi (x) − Pr (x) 2

(2.55)

where the asterisk denotes the complex conjugate, Pi (x) is the incident power flowing from source to load, and Pr (x) is the reflected power flowing in the opposite direction. Substituting (2.51) and (2.52), P (x) becomes,   Vi (x)∗ 1 ∗ P (x) = Re Vi (x)[1 + Γ(x)] [1 − Γ(x) ] (2.56) 2 Z0∗ or P (x) =

  1 |Vi (x)|2 ∗ 2 Re [1 + Γ(x) − Γ(x) − |Γ(x)| ] 2 Z0∗

Expressing Γ(x) = |Γ(x)|ejφ, we observe that purely imaginary. Thus,   1 1 2 2 P (x) = |Vi (x)| (1 − |Γ(x)| )Re = 2 Z0∗

(2.57)

Γ(x) − Γ(x)∗ = j2|Γ(x)| sin φ is 1 Re [Z0 ] |Vi (x)|2 (1 − |Γ(x)|2) 2 |Z0 |2 (2.58)

Of course, if Z0 is real, P (x) =

1 |Vi (x)|2 (1 − |Γ(x)|2 ) = Pi (x) − Pr (x) 2 Z0

(2.59)

Dividing P (x) by Pi (x) = 12 |Vi (x)|2 /Z0 , the fractional power transmitted past the point x is, P (x) Pr (x) = 1 − |Γ(x)|2 = 1 − (2.60) Pi (x) Pi (x) When ZL = Z0 in (2.49), ΓL = 0, and all incident energy is totally transmitted to the load. When ZL = 0 (short circuit) or ZL = ∞ (open circuit), the magnitude of Γ is 1, and all incident energy is totally reflected. For total reflection, the reflected wave combines with the incident wave to form a purely standing wave. For the more general case of partial reflection (0 < ZL < ∞, including complex values), the smaller reflected wave combines with an equal amplitude of the incident wave to form a standing wave. The remainder of the incident wave is a traveling wave transmitting energy from source to load.

34

MICROWAVE TRANSMISSION LINE CIRCUITS

Example 2.2: For a lossless transmission line, find V (x), I(x), Z(x), and the average power flow toward the load, P (x) = 21 Re[V I ∗ ], if (a) ZL = 0, (b) ZL = ∞, and (c) ZL = Z0 . Solution: (a) For ZL = 0, since VL = 0, then V (x) I(x) Z(x)

= jZ0 IL sin βx = IL cos βx = jZ0 tan βx 1 P (x) = Re[jZ0 IL2 sin βx cos βx] = 0 2

(b) For ZL = ∞, since IL = 0, then V (x)

= VL cos βx VL sin βx I(x) = j Z0 Z(x) = −jZ0 cot βx 1 V2 P (x) = Re[−j L sin βx cos βx] = 0 2 Z0

(c) For ZL = Z0 , since B = 0 and A = VL , then V (x)

=

I(x)

=

Z(x)

=

P (x) =

VL ejβx VL jβx e Z0 Z0 1 |VL |2 2 Z0

2.3.3 Voltage Standing Wave Ratio The voltage standing wave ratio, typically referred to as S or VSWR, is defined as the ratio of the maximum-to-minimum value of the standing wave. The largest value of V (x) on the line is |V (x)|M AX = |A| + |B|, x = xM

(2.61)

MICROWAVE TRANSMISSION LINES

35

and the smallest value is |V (x)|M IN = |A| − |B|, x = xm

(2.62)

where the distance from the load to a voltage-maximum point is xM , and to a voltage-minimum point it is xm . Thus, the standing wave ratio is ZL−Z0 B 1 + 1+ A ZL +Z0 |A| + |B| B = S= (2.63) = ZL−Z0 |A| − |B| 1 − A 1 − Z L +Z0

For a lossless line, |Γ(x)| = |ΓL | = |B/A| does not change with distance x, so that S also may be expressed as 1 + |Γ| S= (2.64) 1 − |Γ| or S−1 |Γ| = (2.65) S+1 and in (2.60), the fractional power transmitted past x becomes, P (x) 4S = Pi (x) (S + 1)2

(2.66)

Since 0 ≤ |Γ| ≤ 1, the range of S is from 1 to ∞. Note from (2.63) that if ZL and Z0 are both real, then S = ZL /Z0 when ZL ≥ Z0 , and S = Z0 /ZL when ZL ≤ Z0 . Just as there will be a standing wave of voltage on the line when ZL 6= Z0 , there also will be a standing wave of current. From (2.19) and (2.62), |I(x)|M IN =

1 |V (x)|M IN (|A| − |B|) = Z0 Z0

(2.67)

But this occurs at x = xM , a voltage-maximum point. Hence, the wave impedance at a voltage-maximum point on the line is, Z(xM ) =

|V (x)|M AX |V (x)|M AX = Z0 = SZ0 |I(x)|M IN |V (x)|M IN

(2.68)

In a similar manner, the impedance at a voltage-minimum point (current maximum) is, |V (x)|M IN |V (x)|M IN Z0 Z(xm ) = = Z0 = (2.69) |I(x)|M AX |V (x)|M AX S

36

MICROWAVE TRANSMISSION LINE CIRCUITS

From (2.68) and (2.47) at x = xM , ZL may be expressed as, ZL = Z0

S − j tan βxM 1 − jS tan βxM

(2.70)

Alternatively, using (2.69) and (2.47) at x = xm , ZL = Z0

1 − jS tan βxm S − j tan βxm

(2.71)

Thus, since β = 2π/λ, if S, λ, Z0 , and xM or xm are known, the load impedance is easily calculated using (2.70) or (2.71). In practice, S, λ, xM , and xm may be measured and a voltage-minimum point may be more accurately determined. Thus, a calculation using (2.71) may be the method of choice. Example 2.3: On a lossless transmission line of characteristic impedance Z0 = 50 Ω, the measured standing wave ratio is S = 10. The distance from the load to the first voltage-maximum point is xM = 0.0965λ, and λ = 10 cm. Determine the load impedance. Solution: From (2.70), with βxM = 2π × 0.0965 = 0.606 = 34.74◦: ZL = 50

10 − j tan 34.74◦ = 71.586 77.74◦ ∼ = 15 + j70 Ω 1 − j10 tan 34.74◦

The distance from the load to the first voltage-minimum point will be xm = 0.0965λ + 0.25λ = 0.3465λ, or βxm = 34.74◦ + 90◦ = 124.74◦. From (2.71), ZL = 50

1 − j10 tan 124.74◦ ∼ = 15 + j70 Ω 10 − j tan 124.74◦

2.4 TRANSMISSION LINES WITH LOSSES For voltage and current waves on a transmission line, the propagation constant (γ = α + jβ) and the characteristic impedance (Z0 ) will be altered by resistive and conductive losses. All real transmission lines have losses, that is R 6= 0 and G 6= 0, and these losses tend to increase with increasing frequency. The construction materials and operating frequency of most transmission lines are such that R  ωL

MICROWAVE TRANSMISSION LINES

37

and G  ωC, and thus the effects of non-zero R and G are negligible for many practical applications. Also, contributions due to losses in conductors and losses in the dielectric tend to cancel in the equations that are used to calculate β and Z0 . Thus, the lossless line equations are often a good approximation even when the total attenuation per unit length (α) is appreciable. We determine why and when the lossless line equations apply by examining the general expressions for γ and Z0 . From (2.13), γ may be expressed as,   12   12 √ R G γ = α + jβ = jω LC 1 + 1+ jωL jωC

(2.72)

For the negligible loss case where R  ωL and G  ωC, we can use the binomial series expansion 1 1 1 (2.73) (1 + a) 2 = 1 + a − a2 + ..., a < 1 2 8 and approximate the two terms in γ as   21 R R R2 1+ + =1+ jωL j2ωL 8ω2 L2

(2.74)

and

 12 G G G2 1+ = 1+ + (2.75) jωC j2ωC 8ω2 C 2 Substituting the above approximations into the general expression for γ and collecting terms, yields " r r  2 # √ G L 1 R G R C + + jω LC 1 + − (2.76) γ = α + jβ = 2 L 2 C 2 2ωL 2ωC 

where the smallest imaginary term has been neglected. The attenuation constant α, the real part of the above expression, is r r R C G L α= + = αc + αd (2.77) 2 L 2 C where αc is the attenuation due to the metallic conductors of the line, and αd is the attenuation due to the dielectric material between conductors, or r R C αc = (2.78) 2 L

38

MICROWAVE TRANSMISSION LINE CIRCUITS

and

r G L αd = (2.79) 2 C Again, R is the per-unit-length resistance in Ohms per meter (Ω/m) to current flowing in the conductors, and G is the per-unit-length conductance in inverse Ohms, or Siemens, per meter (S/m) for leakage current through the dielectric. The units of α may be expressed in nepers per-unit-length (Np/m), or in decibels perunit-length (dB/m) by multiplying α in Np/m by 8.686. The parameters R and G in the equations for α will be examined in more detail in a later section. The imaginary part of γ in (2.76) is the phase constant β, or "  2 # √ 1 R G β = ω LC 1 + − (2.80) 2 2ωL 2ωC The last two terms in (2.80) may also be expressed as r R R C 1 αc √ = √ = 2ωL 2 L ω LC ω LC and G G = 2ωC 2 and the phase constant β becomes,

r

1 αd L √ = √ C ω LC ω LC

  √ (αc − αd )2 β = ω LC 1 + 2ω2 LC

(2.81)

(2.82)

(2.83)

The phase constant β is seen to increase with losses contributed by R and G. Consequently, line losses will cause wavelength (λ = 2π/β) and phase velocity (v = ω/β) to decrease. The general expression for characteristic impedance as defined in (2.18) may also be expressed as r   12  − 12 L R G Z0 = 1+ 1+ (2.84) C jωL jωC Using the first two terms of the binomial series expansion, the characteristic impedance with losses becomes r    L R G Z0 = 1+ 1− (2.85) C j2ωL j2ωC

MICROWAVE TRANSMISSION LINES

39

or, performing the indicated multiplication and neglecting the smallest term: r     r  L L R G (αc − αd) Z0 = 1−j − = 1−j √ (2.86) C 2ωL 2ωC C ω LC In the final expressions for β and Z0 , αc and αd are usually small enough to be neglected for most practical transmission lines. But in any event, the two terms are opposite in sign and tend to cancel, √ so that even when plosses are present it is often a good approximation to let β = ω LC and Z0 = L/C. Note that the squarebracketed terms in (2.83) and (2.86) are functions of frequency. For appreciable and noncanceling losses this would make the phase velocity vary with frequency (a dispersive transmission line), and the characteristic impedance would not only vary with frequency but would have an imaginary component as well. A special situation occurs if αc = αd , or equivalently, if R/L = p G/C. In this √ situation, even if αc and αd are quite large, β = ω LC and Z0 = L/C. This would eliminate frequency dispersion in that signals at different frequencies would travel at the same velocity. Note also that the total attenuation for this so-called distortionless line becomes, with R/L = G/C, r r √ G L 1√ 1√ R C α= + = RG + RG = RG (2.87) 2 L 2 C 2 2 Example 2.4: The parameters of a certain transmission line are adjusted to yield R/L = G/C = π × 106

where L = 10−6 H/m, C = 0.4 × 10−9 F/m, and R = π Ω/m. Determine the characteristic impedance and attenuation of this line. Solution: From (2.86), r Z0 =

L = C

r

R √ = 2500 = 50 Ω G

and from (2.87), the attenuation is, α=

√ 1√ 1√ R π RG + RG = RG = = = 0.0628 Np/m 2 2 Z0 50

or, in dB/m the attenuation is 8.686 × 0.0628 = 0.545 dB/m, where the conversion factor comes from dB= 20 log eαl = 20αl log e = 8.686αl.

40

MICROWAVE TRANSMISSION LINE CIRCUITS

2.5 TRANSMISSION LINE PARAMETER CALCULATIONS The per-unit-length parameters R, L, C, and G in the equivalent circuit of the transmission line determine the nature of wave propagation in that these parameters, along with frequency, make up the propagation constant (γ) and the characteristic impedance (Z0 ). These four parameters are all functions of the geometry of the particular transmission line as well as the electrical properties of the materials. As we shall see, L, C, and G share a common function of geometry that is quite different from the function used to determine R. Consider the coaxial line in Figure 2.3, where are shown within the dielectric between conductors, the radial electric field intensity (E in V/m) and the circumferential magnetic field intensity (H in A/m). Also shown are the current (I) flowing in the inner and outer conductors, and the voltage drop per-unit-length (Exo , in V/m) along the conductor surface caused by I flowing through a finite conductivity (σc , in S/m). The conductivity and permittivity of the dielectric are symbolized by σd and , respectively. The field quantities E and H for a typical cross-section of the coaxial line are, ql dV (r) E= =− (2.88) 2πr dr and I (2.89) H= 2πr where ql is the charge-per-unit length on the inner conductor, r is the radial distance from the center of the inner conductor, and V (r) the potential or voltage at any point r between a and b. Since E is the negative gradient of potential as indicated in (2.88), then the implied integration yields V (r) = −

ql ln r + const. 2π

(2.90)

The potential difference or voltage between the inner and outer conductors obtained using (2.90) is ql b V = ln (2.91) 2π a Thus, combining (2.88) and (2.91), E may be expressed as, E=

V r ln ab

(2.92)

MICROWAVE TRANSMISSION LINES

ε, σd

41

I EXO ql

I

a σC b

E

r H

Figure 2.3 Cross section of coaxial line showing field quantities, material properties, and geometry used to calculate the parameters R, L, C, and G.

The power flow from source to load along the line is carried by the field quantities E and H within the dielectric. The time-averaged power flow per unit area (W/m2 ) is 1 p(r) = Re[EH ∗ ] (2.93) 2 where ∗ denotes the complex conjugate. Substituting (2.92) and (2.89) into (2.93), p(r) =

1 V I∗ Re[ ] 2 2πr 2 ln ab

(2.94)

Integrating p(r) over the cross-sectional area of the dielectric would yield the total time-averaged power transmitted along the line as, P =

1 Re[V I ∗ ] 2

(2.95)

a familiar result. In a similar manner, the power flow into the surface of the inner conductor (and therefore lost) may be determined from the surface field quantities Exo and

42

MICROWAVE TRANSMISSION LINE CIRCUITS

H(a) as

1 Re[ExoH(a)∗ ] (2.96) 2 Since psa is the power flow per-unit-area through any point on the inner conductor surface, the power flow per-unit-length is obtained by multiplying psa by 2πa, or psa =

P = psa2πa

(2.97)

Exo = ηcH(a)

(2.98)

The surface fields are related by

where ηc is the intrinsic impedance of the metallic inner conductor, given by r ωµ ηc = (1 + j) (2.99) 2σc Substituting yields 1 P = Re[ηcH(a)H(a)∗ ](2πa) = πa 2 where

r

1 Ri = 2πa

ωµ 2σc

r



I 2πa

2

=

ωµ 2σc

1 2 |I| Ri 2

(2.100)

(2.101)

is the resistance per-unit-length due to the inner conductor. In a similar manner, the power flow into the inside surface of the outer conductor results in a resistance per-unit-length due to the outer conductor of, r 1 ωµ Ro = (2.102) 2πb 2σc Thus, for the coaxial line, the total resistance per-unit-length is   r 1 ωµ 1 1 R = Ri + Ro = + 2π 2σc a b

(2.103)

The inductance per-unit-length, or magnetic flux-linkage per ampere, is given by, µo Ψ L= = I

R

S

H · dS I

(2.104)

43

MICROWAVE TRANSMISSION LINES

where Ψ is the magnetic flux through the surface S of unit length and of width b − a in Figure 2.3. Substituting H from (2.89), the inductance is, L=

µo 2π

Z

b

a

dr(1) µo b = ln r 2π a

(2.105)

which is the external inductance per-unit-length, since only the flux between or external to the metallic conductors has been used.The capacitance per-unit-length is the charge per-unit-length ql on the inner conductor divided by the potential difference V between conductors, as C=

ql 2π = V ln ab

(2.106)

where V was substituted from (2.91). Finally, the conductance per-unit-length is the leakage current Id through the dielectric divided by the potential difference V . The current density in A/m2 flowing between conductors is Jd = σd E. Thus, the leakage current is σd q l (2.107) Id = σd E(a) × 2πa × 1 =  and Id 2πσd G= = (2.108) V ln ab Note that L, C, and G all contain the same function of the conductor geometry, F (g) =

1 b ln 2π a

(2.109)

such that, L = µo F (g) C = /F (g) G = σd /F (g)

(2.110) (2.111) (2.112)

Substituting (2.110), (2.111), and (2.112) into the general expressions for propagation constant (γ) and characteristic impedance (Z0 ), we have, s p R γ = (R + jωL)(G + jωC) = ( + jωµo )(σd + jω) (2.113) F (g)

44

MICROWAVE TRANSMISSION LINE CIRCUITS

and Z0 =

s

R + jωL = G + jωC

s

R F (g)

+ jωµo

σd + jω

F (g)

(2.114)

It is instructive to determine the altered forms of γ and Z0 in (2.113) and (2.114) when the distortionless line requirement (R/L = C/G) is imposed. Using (2.110), (2.111), and (2.112) in R/L = G/C yields: R µ0 = σd F (g) 

(2.115)

and when this expression is substituted into (2.113) and (2.114), γ and Z0 become: r r µ0 µ0 √ γ = α + jβ = (2.116) (σd + jω) = σd + jω µ0    and

r

µ0 F (g) (2.117)  Thus, as required, v = ω/β is independent of frequency and Z0 is real. As implied, r σd µ0 αc = αd = (2.118) 2  Z0 =

Example 2.5: In a previous example the parameters of a transmission line were adjusted to yield the distortionless condition, R/L = G/C = π × 106 where L = 10−6 H/m and C = 0.4 × 10−9 F/m. Determine the conductivity and relative permittivity of the dielectric material between the metallic conductors of this line. Solution: From the problem statement, (2.110), (2.111), and (2.112), L = µ0 F (g) = 10−6 −→ F (g) = C=

L 10−6 2.5 = = −7 µ0 4π × 10 π

  0.4 × 10−9 F (g) = 0.4 × 10−9 −→ = = 36 F (g) 0 10−9 /(36π)

R/L = G/C = σd / = π × 106 −→ σd = π × 106 = 0.001 S/m

45

MICROWAVE TRANSMISSION LINES

For most practical transmission lines operating at 1 MHz and above, ωµo  R/F (g) and ω  σd . These are the conditions for a low-loss or lossless line, and γ and Z0 become √ √ γ = jω LC = jω µo  = jβ (2.119) and Z0 =

r

L = C

r

µo F (g) 

(2.120)

The functional relationship common to L, C, G, and Z0 is true for all two-conductor transmission lines. While the function F (g) and R are given for the coaxial line by (2.109) and (2.103), respectively, these two parameters are different for different types of line. The parameters under discussion are displayed in Tables 2.1 for coaxial line, 2.2 for two-wire line, and 2.3 for parallel-plate line, in Tables 2.4 for stripline and 2.5 for microstrip, and in Tables 2.6 for coplanar waveguide and 2.7 for coplanar stripline. These all are widely used forms of twoconductor transmission line. Also shown in these tables are the attenuations αc and αd expressed as, R R = p µo αc = (2.121) 2Z0 2  F (g)

and

1 σd GZ0 = αd = 2 2 F (g)

r

µo σd F (g) =  2

r

µo 

(2.122)

2.6 IMPEDANCE MATCHING Consider the network shown in Figure 2.4, where the subscript S refers to source parameters and the subscript 1 refers to load parameters or to the input port of a more elaborate network. The current flow through both ZS and Z1 is, I1 =

VS VS = ZS + Z1 (RS + R1 ) + j(XS + X1 )

(2.123)

and the time-averaged power delivered to the load Z1 is, P1 =

1 1 Re[V1 I1∗ ] = |I1 |2 R1 2 2

(2.124)

46

MICROWAVE TRANSMISSION LINE CIRCUITS

b a ε

Coaxial line Table 2.1 Parameters for Coaxial Transmission Lines

Parameter

Coaxial Line

F (g)

1 2π

L (H/m)

µo F (g)

C (F/m)

/F (g)

G (S/m)

σd /F (g)

Z0 (Ω)

ηF (g), η =

αd (Np/m)

σd 2 η

αc =

R 2Z0

(Np/m) β = 2π λ (radians/m)

ln ab

Rs 4πZ0

Rs =

1 + 1b a q ωµo 2σc

√ ω µo 

p µo 




MICROWAVE TRANSMISSION LINES

2a

2h

ε

Two-wire line Table 2.2 Parameters for Two-Wire Transmission Lines

Parameter

Two-Wire Line

F (g)

1 π

L (H/m)

µo F (g)

C (F/m)

/F (g)

G (S/m)

σd /F (g)

Z0 (Ω)

ηF (g), η =

αd (Np/m)

σd 2 η

ln 2h a

R 2Z0

Rs 2πaZ0

(Np/m)

Rs =

αc =

β = 2π λ (radians/m)

p µo

p 1 2 a q 1−( h )

√ ω µo 

ωµo 2σc



47

48

MICROWAVE TRANSMISSION LINE CIRCUITS

ε0

w 2h

εd

Parallel-plate line Table 2.3 Parameters for Parallel-Plate Transmission Lines

Parameter

Parallel-Plate Line

F (g)

1 π

ln

8h w

+

w 4h 2

w h


,

w h +1.393+0.667 ln

≤1

( wh +1.444)

,w h ≥1

L (H/m)

µo F (g)

C (F/m)

o /F (g), 2 = d + o + √d −12h

1+

G (S/m)

σd /F (g)

Z0 (Ω)

ηF (g), η =

αd (Np/m)

σd η 2

αc =

R 2Z0

(Np/m) β = 2π λ (radians/m)

Rs Z0 πη 2 h

Rs =



p µo 

(2+ πw h ) 1+ 1+ πw ( q h)

√ ω µo 

ωµo 2σc

w h




w

49

MICROWAVE TRANSMISSION LINES

w b εd , σd

t

Stripline Table 2.4 Parameters for Stripline Transmission Lines

Parameter

Stripline

F (g)

1 2π

ln



8 πw t b + b (1+ln

4πw b

)



,

w b−t

1 4w 4 b−t + π



b b−t

L (H/m)

µo F (g)

C (F/m)

/F (g) o +  = d + 2

G (S/m)

σd /F (g)

Z0 (Ω)

ηF (g), η =

αd (Np/m)

p µo σd

αc =

R 2Z0

(Np/m) β = 2π λ (radians/m)

2

4Rs Z0 η2 b

Rs =

t(2b−t) 2b−t t +ln (b−t)2

ln

d −o 2

√

,

≤ 0.35 w b−t

≥ 0.35

1 1+ 12h w

p µo 



h



b 2w qb−t b−t ωµo 2σc

√ ω µo 

 +1 +

b(b+t) π(b−t)2

ln

2b−t t


i

50

MICROWAVE TRANSMISSION LINE CIRCUITS

ε0

w t=0

h

εd , σd Microstrip Table 2.5

Parameters for Microstrip Transmission Lines

Parameter

Microstrip

F (g)

1 2π

8h w

ln

+

w 4h 1


,

w h +1.393+0.667 ln

L (H/m)

µo F (g)

C (F/m)

/F (g) o  = d + + 2

G (S/m)

σd /F (g)

Z0 (Ω)

ηF (g) p η = µo

αd (Np/m) αc =

R 2Z0

(Np/m) β = 2π λ (radians/m)

σd 2

p µo

Rs Z0 η2 h

Rs =

w h

≤1

( wh +1.444)

d −o 2

1 1+ 12h w

√





(2+ πw 2h ) 1+ 1+ πw ( q 2h )

√ ω µo 

ωµo 2σc

,

w h




w h

≥1

51

MICROWAVE TRANSMISSION LINES

ε0

s

w

s h

εd , σd Coplanar waveguide Table 2.6 Parameters for Coplanar Waveguide

Parameter

Coplanar Waveguide

F (g)

π/4 , ws ln ( 4w s +2) 1 2π

8s w

ln

≥3


+4 ,

w s

L (H/m)

µo F (g)

C (F/m)

/F (g) o  = d + ,hs 2

G (S/m)

σd /F (g)

Z0 (Ω)

ηF (g) p η = µo

αd (Np/m) αc =

R 2Z0

σd 2

p µo 

8Rs Z0 (w+s) w πη 2 s(2w+s) , s

≥ 3, Rs =

q

ωµo 2σc

Rs (w+s) w πZ0 w(w+2s) , s

≤ 3, Rs =

q

ωµo 2σc

(Np/m)

β = 2π λ (radians/m)

≤3

√ ω µo 

52

MICROWAVE TRANSMISSION LINE CIRCUITS

ε0

w

s

w

εd , σd

h

Coplanar stripline Table 2.7 Parameters for Coplanar Stripline

Parameter

Coplanar Stripline

F (g)

π/2 , ws ln ( 8w s +4) 1 π

ln

4s w

≥ 1/3


+2 ,

w s

≤ 1/3

L (H/m)

µo F (g)

C (F/m)

/F (g) o  = d + 2 ,hs

G (S/m)

σd /F (g)

Z0 (Ω)

ηF (g) p η = µo

αd (Np/m) αc =

R 2Z0

σd 2

p µo 

4Rs Z0 (w+s) w πη 2 s(2w+s) , s

≥ 1/3, Rs =

q

ωµo 2σc

2Rs (w+s) w πZ0 w(w+2s) , s

≤ 1/3, Rs =

q

ωµo 2σc

(Np/m)

β = 2π λ (radians/m)

√ ω µo 

MICROWAVE TRANSMISSION LINES

ZS = RS + jXS VS

53

i1 Z1 = R1 + jX1

Figure 2.4 Network for which the load is to be matched to the source.

or, P1 =

1 |VS |2 R1 1 |VS |2 R1 = 2 2 |ZS + Z1 | 2 (RS + R1 )2 + (XS + X1 )2

(2.125)

If we are free to choose the load resistance (R1 ) and reactance (X1 ) such that maximum power will be extracted from the source and delivered to the load, then at the maximum value of P1 we require, ∂P1 =0 ∂X1 ∂P1 =0 ∂R1

(2.126) (2.127)

The first condition yields X1 = −XS , and the second condition yields R1 = RS (since R1 cannot be negative). Thus, the load impedance must be the complex conjugate of the source impedance, or Z1 = ZS∗ . Using this requirement in (2.125), the maximum available power from the source, or the maximum that P1 can be is: P1 (max) = P0 =

1 |VS |2 2 4RS

(2.128)

It is convenient to normalize the actual value of P1 in (2.125) to the maximum value P0 in (2.128), to obtain, P1 4RS R1 4RS R1 = = 2 P0 |ZS + Z1 | (RS + R1 )2 + (XS + X1 )2

(2.129)

Thus, P1 /P0 in (2.129) is the fractional power transmitted to the load for any choice of load impedance Z1 . Conjugate impedance matching where Z1 = ZS∗ , yields P1 /P0 = 1 as indicated above, or from (2.129). If the complex impedances are simply equalized

54

MICROWAVE TRANSMISSION LINE CIRCUITS

as Z1 = ZS , then P1 /P0 in (2.129) would be P1 1 = X2 P0 1 + R2S S

Thus, the fractional power delivered to the load would be diminished from unity by the fractional power stored term in the denominator, which is just the ratio of the power stored in reactance to the power delivered to the load or dissipated in the source. Clearly then, it is the conjugate impedance match that yields maximum power transfer between source and load. The situation most often encountered in practice is where ZS = RS is real, and both types of impedance match become the same, requiring R1 = RS and X1 = 0, and yielding P1 /PS = 1. An extension of the input network of impedance Z1 might be as shown in Figure 2.5. The transmission line is assumed to be lossless, and the input impedance is ZL + jZ0 tan βl = R1 + jX1 (2.130) Z1 = Z0 Z0 + jZL tan βl where Z0 is real and the phase length βl may be expressed in various ways as βl = θ =

ωl √ 2π ω √ l = l = ωl µo  = r λ v c

(2.131)

Notice that when βl = nπ, n = 0, 1, 2, ..., or l = nλ/2, then from (2.130) Z1 = ZL . The load impedance is repeated at multiples of a half wavelength along the line, no matter what the value of Z0 . Also note from (2.130) and Figure 2.5 that when βl = (2n + 1) π2 , n = 0, 1, 2, ..., or l = (2n + 1) λ4 , then 1/ tan βl = 0 in (2.130) and Z1 = Z02 /ZL . This quarter-wavelength transformer also is known as an impedance inverter and is commonly used for impedance matching. For a conjugate impedance match we set Z1 = ZS∗ =

Z02 ZL

(2.132)

or, since Z0 is real: Z02 = Re[ZS∗ ZL ] = Re[RS RL + j(RS XL − RL XS ) + XS XL ]

(2.133)

Thus, for Z0 to be real requires XS XL = RS RL

(2.134)

55

MICROWAVE TRANSMISSION LINES

ZS = RS + jXS

l

VS

Z0

ZL = RL + jXL

Z1 = R1 + jX1 Figure 2.5 A lossless transmission line between source and load.

yielding, p Z0 = RS RL

s

1+



XS RS

2

(2.135)

Equation (2.134), requiring that ZS and ZL must have the same ratio of reactance to resistance, is usually too restrictive to be practical. However, if ZS and ZL are both real (XS = XL = 0), then from (2.135), Z0 =

p

RS RL

(2.136)

and an impedance match is always practical if a quarter-wavelength section with this value of Z0 is available. Example 2.6: As in Figure 2.5, a lossless transmission line of characteristic impedance Z0 and length l = λ/4 connects a real source impedance ZS = RS = 50 Ω to a real load impedance ZL = RL = 200 Ω. (a) Determine the Z0 that yields an impedance match and maximum power transfer between the source and load. (b) Show that the value of Z0 determined in (a) yields an impedance match and maximum power transfer at both ends of the lossless line. Solution: (a) Using RS and RL in (2.136), Z0 =

√ 50 × 200 = 100 Ω

(b) At the source end of the line, Z1 from (2.132) is, Z1 = (100)2 /200 = 50 Ω

56

MICROWAVE TRANSMISSION LINE CIRCUITS

which matches the source impedance, and it is obvious from (2.129) that P1 /P0 = 1. At the load end of the line, but looking back toward the source, an impedance Z2 = Z02 /ZS = (100)2 /50 = 200 Ω is obtained from (2.132) with a slight change in subscript notation. Thus, Z2 = ZL at the load end of the line. The fractional power transmitted from the end of the line and into the load is obtained by modifying (2.129) as P2 4R2RL = =1 P0 (R2 + RL)2 Thus, all of the source power entering the input of the lossless line is delivered to the load. If θ = βl in (2.130) is allowed to become one of the unknown variables along with Z0 , a conjugate-impedance match is possible when both ZS and ZL are complex. Setting Z1 = ZS∗ in (2.130) and equating real terms to real terms and imaginary terms to imaginary terms, yields two equations in the two unknowns Z0 and θ. The two equations have the solutions

Z0 =

s

RL|ZS |2 − RS |ZL|2 RS − RL

and tan θ = Z0

RS − RL RS XL − RL XS

(2.137)

(2.138)

The characteristic impedance in (2.137) must be real (lossless line), and this restriction leads to the following conditions that must be met: If RS ≥ RL then |XS | ≥ |XL|, or if RS ≤ RL then |XS | ≤ |XL |. Example 2.7: Again referring to Figure 2.5 as a model, determine the values of Z0 and l (in wavelengths) that make Z1 = ZS∗ if (a) ZS = 500 + j60 Ω and ZL = 5 + j5 Ω, (b) ZS = 500 + j5 Ω and ZL = 5 + j60 Ω, and (c) ZS = 500 + j60 Ω and ZL = 5 − j5 Ω.

57

MICROWAVE TRANSMISSION LINES

Solution: (a) These impedances meet the condition |XS | ≥ |XL | for RS ≥ RL. From (2.137) and (2.138), we obtain Z0 = 50.11 Ω and θ = 84.93◦ (or l = 0.236λ). (b) These impedances do not meet the conditions |XS | ≥ |XL| for RS ≥ RL or |XS | ≤ |XL | for RS ≤ RL, and Z0 determined from (2.137) is purely imaginary. (c) These impedances also meet the condition |XS | ≥ |XL | for RS ≥ RL. From (2.137) and (2.138), we obtain Z0 = 50.11 Ω and the electrical length θ = −83.56◦ or 96.44◦. Of course, θ = (2π/λ)l = (360◦ /λ)l = 96.44◦ (or l = 0.268λ) is the correct length, since l can not be negative. The impedance-matching transformer considered here for matching complex ZS and ZL is sometimes referred to as a short transformer since the phase length is often less than 90◦ as in part (a) of the previous example. 2.7 IMPEDANCE TRANSFORMATIONS AND THE SMITH CHART The impedance Z1 in (2.130) and in Figure 2.5 is a transformation of the load impedance ZL through the transmission line of characteristic impedance Z0 and length l. It is convenient to normalize all impedances with respect to Z0 , so that (2.130) becomes Z1 zL + j tan βl = z1 = = r1 + jx1 Z0 1 + jzL tan βl

(2.139)

where r1 and x1 are the real and imaginary parts, respectively, of the normalized impedance z1 . From (2.53) and (2.54), z1 is expressed in terms of the input reflection coefficient Γ1 as z1 = or

1 + Γ1 1 + ΓL e−j2βl = = r1 + jx1 1 − Γ1 1 − ΓL e−j2βl

(2.140)

z1 − 1 r1 − 1 + jx1 = (2.141) z1 + 1 r1 + 1 + jx1 Also, with zL real, the standing-wave ratio S = zL for zL ≥ 1, and S = 1/zL for zL ≤ 1, yielding Γ1 =

S + j tan βl 1 + jS tan βl 1 + jS tan βl z1 = S + j tan βl z1 =

zL ≥ 1

(2.142)

zL ≤ 1

(2.143)

58

MICROWAVE TRANSMISSION LINE CIRCUITS

Thus, a plot of z1 = r1 + jx1 in any coordinate system would involve βl, Γ1 , and S. On a rectangular impedance plot, lines of constant βl, |Γ1 |, and S form families of non-concentric circles. On a polar plot of Γ1 = |Γ1 |ejφ , where |Γ1 | ≤ 1 for a passive load, the center of the circular plot corresponds to |Γ1 | = 0 and z1 = 1 + j0, and the outermost circle corresponds to Γ1 = 1ejφ and z1 = 0 + jx1 . Such a circular impedance plot is the Smith chart shown in Figure 2.6. This chart is very convenient for solving transmission line problems because constant βl is a straight radial line from the center, and constant |Γ1 | and S are circles concentric with the center of the plot. These lines and circles are not shown, but they may be sketched onto the chart as needed. Movement between points on a lossless transmission line is represented by movement along a concentric circle. Movement toward the load is counterclockwise on the chart and movement toward the source is clockwise, with the distance of movement given in wavelengths on the periphery of the chart. The usefulness of the Smith chart will be made clear by several examples. Example 2.8: Find the magnitude and phase angle of the reflection coefficient corresponding to the Smith chart impedances (a) z = 0, (b) z = ∞, (c) z = 1, (d) z = +j1, (e) z = −j1, (f) z = 3, and (g) z = 1/3. Solution: Reading the reflection coefficient magnitude as a linear variation from 0 to 1 in moving from the center to the perimeter of the chart, and reading the angle directly from the scale on the perimeter, we find for each impedance, (a) Γ = 16 180◦ , (b) Γ = 16 0◦ , (c) Γ = 0, (d) Γ = 16 +90◦ , (e) Γ = 16 −90◦ , (f) Γ = 0.56 0◦ , and (g) Γ = 0.56 180◦ . Example 2.9: A lossless transmission of characteristic impedance Z0 = 50 Ω is terminated in ZL = 15 + j70 Ω. Use the Smith chart to determine (a) the reflection coefficient at the load, (b) the standing-waveratio on the line, and (c) the distance from the load to the nearest voltage maximum point, if λ = 10 cm. Solution: With Figure 2.5 as a model, normalize all impedances to Z0 , so that z0 = 1 and zL = 0.3 + j1.4. (a) Enter the Smith chart (Figure 2.6) with the value of zL and read ΓL = 0.8176 69.44◦. This

59

MICROWAVE TRANSMISSION LINES

0.0

5

0.4

0.4

45

1.2

1.0

50

0.9

0.8

1.6

1.8 2.0

65 0.5

0.1 0.3

50

8

2

25 0.4

20

3.0

0.6

0.3

0.8

4.0

15

0.28

1.0

5.0

0.22

1.0

IND UCT IVE

80

9 0.2

REA 0.0 75 CT 6 14 AN 0.4 0 CE 4 CO MP ON EN 70 T( +j X/ Zo

0.2

30

0.0 4

7

30

1 0.2

15 0

o) jB/Y E (+

3

0.2

0.4 6

0.1 0.3

60

0.3

20

0.2

85

CI PA CA

4

40

R ), O

C AN PT CE US ES TIV

6

0.3

35

1 0.3 10

0.8

0.27

0.24

0.6

0.25 0.24 0.26 COEFFICIENT IN DE OF REFLECTION GREES ANGLE ISSION COEFFICIENT IN D EGREES OF TRANSM ANGLE

10

0.23

0.1

0.4

20 0.2

—> WAVEL ENGTH S TOW ARD 0.49 GEN ERA 0.48 TOR —> 170 0.47 160 90

3 0.4 0 13

0.1

70

40

9 0.1

5

0

0.35

80

1.4

0 12

.07

0.15

0.36

90

0.7

2 0.4

55

1 0.4

8

0.6 60

0.0

110

0.14

0.37

0.38

0.39 100

0.4

0.13

0.12

0.11

0.1

9 0.0

50

20

10

5.0

4.0

3.0

2.0

1.8

1.6

1.4

1.2

1.0

0.9

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.49 0.48

10 7

1 1 30 ∞ 0

0.1

0 0

1.1

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1

0.99

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

CENTER 1

1.1

5

4

1.1

1.2

1.3 1.4

0.2

0.4

0.6

1.2

1.3 0.95

1.2

1.4

0.8

1.5

0.9

1.3

3 1.6 1

1.5

2 2

1.6 1.7 1.8 1.9 2 0.8

1.4

2 1.8

0.7

1.5

3

4

3

4

5

0.5

1.7

10

5

2.5 0.6

1.6

αd . Determine the frequency at which αc = αd , for (a), (b), and (c) of the coaxial line in problem 5. 7. Determine the reflection coefficient Γ(x) and the standing-wave-ratio S at (a) x = 0, (b) x = λ/8, and (c) x = λ/4 on a Z0 = 50 Ω line terminated in ZL = 20 + j50 Ω at x = 0. 8. With reference to Figure 2.7, determine Z01 and the shortest length l1 (in wavelengths) to make Zin = 50 Ω. 9. With reference to Figure 2.8, determine the shortest lengths l1 and l2 (in wavelengths) to make Zin = 50 Ω. 10. The optimum load impedance for a transistor is ZL = 25 + j25 Ω. If the actual load is 50 Ω, use sections of transmission line of 50 Ω characteristic impedance in conjunction with the 50 Ω load to present the optimum load impedance to the output of the transistor. 11. For maximum power to be delivered from the ZS = 50 Ω source to the ZL = (1 + j0.23)50 Ω load of the network in Figure 2.9, (a) what must be the value of characteristic impedance Z01 ? (b) What is the standing-waveratio S on the 0.366λ section? 12. Referring to Figure 2.10, (a) find the location and length of the shorted stub such that no reflections occur to the left of the stub. (b) What is the standingwave-ratio S on the lines l1 , l2 , and to the left of the shorted stub?

ZS = 50 Ω VS

Z0 = 50 Ω Zin

λ/4

l1

Z01

Z0 = 50 Ω ZL = 20 + j50 Ω

Figure 2.7 Determine Z01 and the shortest length l1 (in wavelengths) to make Zin = 50 Ω.

65

MICROWAVE TRANSMISSION LINES

Zin

ZS = 50 Ω

l1

Z0 = 50 Ω

Z0 = 50 Ω ZL = 20 + j50 Ω

0

l 2

Z

=

50

Ω

VS

Figure 2.8 Determine the shortest lengths l1 and l2 (in wavelengths) to make Zin = 50 Ω.

50 Ω VS

50 Ω

0.250 λ

0.366 λ

Z01

50 Ω

ZL

ZL = (1 + j0.23) 50 Ω Figure 2.9 Determine the value of the characteristic impedance Z01 so that maximum power is delivered from the source to the load using the values in Problem 11. What is the standing-wave-ratio S on the 0.366λ section?

66

MICROWAVE TRANSMISSION LINE CIRCUITS

ZS = Z0

l1

VS

ZL = 2Z0

l

0

Z0

2

Z

Z0

Figure 2.10 Find the location and length of the shorted stub such that no reflections occur to the left of the stub. What is the standing-wave-ratio S on the lines l1 , l2 , and to the left of the shorted stub?

BIBLIOGRAPHY R. L. Coren, Basic Engineering Electromagnetics: An Applied Approach. Upper Saddle River, NJ: Prentice-Hall, 1989. T. C. Edwards and M. B. Steer, Foundations of Interconnect and Microstrip Design, 3rd ed. Hoboken, NJ: John Wiley and Sons, 2001. V. F. Fusco, Microwave Circuits: Analysis and Computer-Aided Design. Upper Saddle River, NJ: Prentice-Hall, 1987. H. Howe, Stripline Circuit Design. Dedham, MA: Artech House, 1974. J. D. Kraus and D. A. Fleisch, Electromagnetics: with Applications, 5th ed. New York: McGraw-Hill, 1999. G. F. Miner, Lines and Electromagnetic Fields for Engineers. New York: Oxford University Press USA, 1996. M. A. Plonus, Applied Electromagnetics, 2nd ed. New York: McGraw-Hill, 1978. D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley and Sons, 2011.

MICROWAVE TRANSMISSION LINES

67

P. A. Rizzi, Microwave Engineering: Passive Circuits. Upper Saddle River, NJ: Prentice-Hall, 1988. P. H. Smith, Electronic Applications of the Smith Chart: in Waveguide, Circuit, and Component Analysis. Malabar, FL: Krieger, 1983.

68

MICROWAVE TRANSMISSION LINE CIRCUITS

Chapter 3 TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS 3.1 LUMPED-ELEMENT LIMITATIONS As a general rule, the voltage-current relationship of lumped or discrete elements such as resistors, inductors, and capacitors (R, L, and C) is described by Ohm’s law at all frequencies where the electrical length of the lumped element is much smaller than a wavelength. Due to the time delay through the element of length l, the t l ejωt time-dependence becomes ej(ωt−βl) = ej2π( T − λ ) . The choice may depend upon the application, but an electrical length l ≤ λ/36 is usually adequate, since this makes the phase delay through the element βl ≤ 10◦ . Components with leads that are mounted through holes in a printed circuit board generally are larger and thus may be limited to operating frequencies below about 100 MHz. Surface-mount components generally are much smaller and therefore can be used at frequencies well into the GHz range. For example, a chip capacitor of length 0.5 mm= λ/36 would be usable up to about 16.7 GHz. At low frequencies, interconnections between devices are treated as lumped or discrete connections that transfer signals with no appreciable phase delay or reflection loss. At high frequencies, the same length limitation (l ≤ λ/36) applies to the interconnection, as illustrated in Example 3.1. Example 3.1. Two Z0 = 50 Ω systems are interconnected by a Z01 = 100 Ω transmission line that has an electrical length βl = 10◦ (or l = λ/36) at the highest operating frequency. At this frequency, determine the input impedance at the interconnection, the return loss (RL) in dB (RL = −10 log |Γ|2 ), and the transmission loss (TL) in dB 69

70

MICROWAVE TRANSMISSION LINE CIRCUITS

(TL = −10 log [1 − |Γ|2 ]) through the interconnection. Solution. The input impedance is, Z = Z01

1 + j ZZ010 tan βl Z0 + jZ01 tan βl = Z0 Z01 + jZ0 tan βl 1 + j ZZ010 tan βl

(3.1)

or,

1 + j0.352 = 51.15 + j13.10 Ω (3.2) 1 + j0.088 Using the calculated value of Z, the input reflection coefficient is, Z = 50

Γ=

1.15 + j13.10 Z − Z0 = = 0.1296 77.6◦ Z + Z0 101.15 + j13.10

(3.3)

and RL = −10 log |0.129|2 = 17.79 dB

(3.4)

The transmission loss is, TL = −10 log [1 − |0.129|2] = 0.07 dB

(3.5)

In most applications, the calculated values of Z, RL, and TL in this example would be within acceptable limits.

3.2 USING TRANSMISSION LINES AS LUMPED ELEMENTS The lumped-element concept, and all of the network analysis and synthesis techniques often used at lower frequencies, may be retained at microwave frequencies by appropriately replacing the lumped elements with transmission line segments. To begin this analogous replacement, compare the admittance looking into the terminals of a capacitor, and an open-circuited transmission line of length l and characteristic admittance Y01 , as shown in Figure 3.1. The admittance at the input terminals of the capacitor is Y = jωC (3.6) and at the input terminals of the open-circuited stub the admittance is Y = jY01 tan

ωl v

(3.7)

71

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

l Y01

C

YL = 0

 ωl  Y = jY01 tan    v 

Y = jωC Y

Y

ω

πv

ω

2l Figure 3.1 Input admittance versus frequency of a lumped-element capacitor and an open-circuited section of transmission line.

Equating the terminal admittance yields C=

Y01 ωl tan ω v

(3.8)

Thus, the two elements are analogous, and the capacitance is determined precisely by (3.8) over the frequency range where ωl v < π/2 or l < λ/4. In a similar manner, the input impedance at the terminals of an inductance and a short-circuited section of transmission line may be compared in Figure 3.2 to yield the impedances and the equivalent inductance as Z = jωL = jZ01 tan and

ωl v

(3.9)

Z01 ωl tan (3.10) ω v Again, the two elements are analogous, and the inductance is determined precisely by (3.10) over the frequency range where ωl < π/2 or l < λ/4. v L=

72

MICROWAVE TRANSMISSION LINE CIRCUITS

Table 3.1 Lumped-Element and Distributed-Element (Transmission-Line-Segment) Equivalent Transformations

C

Y0

Y0

Design Equation

Y0

l

Y

Y0

Distributed-Element

01

Lumped-Element

Y01 ω

tan

ωl , v

l < λ/4

C=

Y01 ω

tan

ωl , v

l < λ/4

L=

Z01 ω

tan

ωl , v

l < λ/4

L=

Z01 ω

tan

ωl , v

l < λ/4

tan ωl , v

Z0 Z01

tan

Z0

Y0

Z0

Z0

Z

L Z0

Z0

Z0

L Z0

01

l

l

Z

L

Y0

01

Y0

Y0

01

l Y

C

Z0

C=

Z0

l

Z0

Z0

Z01

Z0 L=

Z01 ω

ωl v

1

73

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

l Z01

L

ZL = 0

 ωl  Z = jZ 01 tan    v 

Z = jωL Z

Z

ω

πv

ω

2l Figure 3.2 Input impedance versus frequency of a lumped-element inductor and a short-circuited section of transmission line.

Lumped-element Cs and Ls, connected in parallel or in series with the rest of the circuit, are replaced by transmission line segments as illustrated in Table 3.1, where the input and output lines are represented by a characteristic impedance Z0 . Also represented in Table 3.1 is an additional convenient method for forming a series inductance from a series-cascade section of line. To illustrate, in the last row and central column of Table 3.1, the impedance looking into the line of characteristic impedance Z01 , length l, and terminated in Z0 is,

Z = Z01

Z0 + jZ01 tan ωl v Z01 + jZ0 tan ωl v

In (3.11), if the condition is met that

Z0 Z01

=

Z0 + jZ01 tan ωl v 0 1 + j ZZ01 tan ωl v

(3.11)

tan ωl  1, then the input impedance is, v

Z = Z0 + jZ01 tan

ωl = Z0 + jωL v

(3.12)

74

MICROWAVE TRANSMISSION LINE CIRCUITS

Thus, under the condition stated, the cascaded line of electrical length θ = ωl/v and characteristic impedance Z01 forms an inductance, L=

Z01 ωl tan ω v

(3.13)

Note in (3.8), (3.10), and (3.13) that two parameters may be varied (Z01 and l or ωl/v) to determine C and L. The usual case is to pick one parameter for convenience and calculate the remaining one to yield the desired value of L or C. From Chapter 2, recall that Z0 is a function of the line geometry and materials. For some extreme (large or small) values of Z0 the dimensions of the line may make it difficult to fabricate. Therefore, it is important to pick realistic values early in the design process. Example 3.2: The lowpass filter in Figure 3.3(a) has lumped elements C1 = C3 = 3.18 pF and L2 = 15.92 nH, with RS = RL = Z0 = 50 Ω. For the analogous transmission line circuit in Figure 3.3(b), determine the characteristic impedances (Z01 , Z02 ) and the electrical lengths (θ1 = ωl1 /v, θ2 = ωl2 /v) at f = 1 GHz. Solution: In (3.8), choose θ1 = 30◦ and calculate, Z01 =

tan θ1 0.577 = = 28.87 Ω ωC1 0.02

(3.14)

In (3.13), choose Z02 = 150 Ω and calculate, θ2 = tan−1 (ωL2 /Z02 ) = tan−1 (100/150) = 33.69◦

(3.15)

Example 3.3: The lowpass filter in Example 3.2 and Figure 3.3(b) is to be constructed in a microstrip format on a 1/16 inch thick substrate of epoxy-glass. Using the tabulation in Appendix B of characteristic impedance (Z0 ) and effective relative permittivity (epsilon effective or EPSE) versus line width (w), determine (in both inches and mm) the widths w1 and w2 corresponding to Z01 and Z02 , and the lengths l1 and l2 corresponding to θ1 and θ2 . Solution: From the tabulation in Appendix B, Z01 = 28.87 Ω corresponds to w1 = 0.256 inch (in the next-to-last column), or 6.50 mm.

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

RS

L2 C1

VS

C3

RL

(a)

RS

θ2 Z02

1

θ

01

RL

Z

01 1

θ

Z

VS

(b) Figure 3.3 (a) A lumped-element lowpass filter and (b) the analogous transmission line circuit.

Likewise, Z02 = 150 Ω corresponds to w2 = 0.006 inch, or 0.152 mm. To determine the line lengths, the effective velocity √of propagation (v) to be used is the free-space value divided by EPSE. Thus, Z01 = 28.87 Ω yields EPSE= 3.86, and θ1 = ωll /v yields, 3 × 108 30◦ l1 = vθ1 /ω = ( √ )( ) = 12.72 mm 3.86 360◦ × 109

(3.16)

or 0.501 inch. Likewise, Z02 = 150 Ω yields EPSE= 3.069, and θ2 = ωl2 /v yields, 3 × 108 33.69◦ l2 = vθ2 /ω = ( √ )( ) = 16.03 mm 3.069 360◦ × 109 or 0.631 inch.

(3.17)

75

76

MICROWAVE TRANSMISSION LINE CIRCUITS

Note that the calculated line widths, 0.256 inch (6.50 mm) and 0.006 inch (0.152 mm), cover a wide range, which may complicate fabrication of the circuit. The input impedances of the lumped-element lowpass filter in Figure 3.3(a) and the analogous transmission line circuit in Figure 3.3(b) are determined as Z1a =

1 jωC1 +

1

jωL2 +

=

1 jωC1 + 1 50

(1 − ω2 L2 C1 )50 + jωL2 (1 − ω2 L2 C1 ) + jωC1 (2 − ω2 L2 C1 )50

and Z1b =

(3.18) 1

jY01 tan θ1 +

1 jZ02 tan θ2 +

(3.19)

1 jY01 tan θ1 + 1 50

or, combining terms, (1 − Y01 Z02 tan θ1 tan θ2 )50 + jZ02 tan θ2 (1 − Y01 Z02 tan θ1 tan θ2 ) + jY01 tan θ1 (2 − Y01 Z02 tan θ1 tan θ2 )50 (3.20) From the input impedances expressed in the foregoing equations the input reflection coefficient is determined as, Z1 − 50 Γ= (3.21) Z1 + 50 Z1b =

Recall that the fractional power reflected from the input is Pr /Pi = |Γ|2 , and the fractional power transmitted through the network to the load is Pt /Pi = 1 − |Γ|2 . These fractional powers are expressed in dB as, Γ(dB) = 10 log |Γ|2 = 20 log |Γ|

(3.22)

T (dB) = 10 log (1 − |Γ|2 )

(3.23)

and

Using these equations and the parameters given in Examples 3.2 and 3.3, the fractional powers reflected and transmitter are determined and plotted versus frequency in Figure 3.4. Note in Figure 3.4 that the two networks are truly analogous in operation, and the transmission line network is even slightly better in that Γb drops more sharply in the passband and Tb drops more sharply in the stopband.

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

77

Ta Γa

Γb Tb

Figure 3.4 For the equivalent lowpass filters in Figure 3.3, the fractional powers reflected (Γa and Γb ) and transmitted (Ta and Tb ) are expressed in dB versus frequency.

3.2.1 Impedance Matching Using Lumped Ls and Cs An impedance match to a complex load impedance may be achieved by using two lumped elements, an L and a C, or by using their transmission line equivalents from Table 3.1. A real source (ZS ) is matched to any complex load (ZL ) by properly choosing one of the four LC networks shown in Figure 3.5(a), where all impedances and admittances are normalized to Z0 and Y0 , respectively. The impedance-matching circuits numbered (1) through (4) correspond to the particular connection of L and C required to match the normalized load impedance zL = rL + jxL as (1)rL > 1, xL > 0 (2)rL < 1, xL > 0 (3)rL < 1, xL < 0 (4)rL > 1, xL < 0 The numbers also correspond to each quadrant of the Smith chart where zL may be located, as indicated in Figure 3.5(b). The procedure used to make Zin = Z0 , or zin = 1 in the normalized network of Figure 3.5(a), is illustrated in Figure 3.6 for zL in each quadrant of the Smith chart. For example, in Figure 3.6(a), if zL = 2 + j2 (the first quadrant of the Smith chart): zL is inverted on the chart to yield yL = 0.25 − j0.25, shunt admittance jωC/Y0 = j0.683 is added to yL to make y = 0.25 + j0.433 the inverse of z = 1 − j1.732, and series impedance jωL/Z0 = j1.732 is added to make

78

MICROWAVE TRANSMISSION LINE CIRCUITS

jωL/Z0 jωC/Y0

(1) ZS =1 Z0

Y0 /jωC (2)

Z0 /jωL VS

ZL = zL Z0

jωL/Z0 Z in = zin = 1 Z0

(3)

jωC/Y0

(2)

Y0 /jωC (4)

Z0 /jωL

(a)

(1) (4)

(3) (b)

Figure 3.5 (a) Lumped LC impedance-matching networks for zL = ZL /Z0 in each quadrant of the Smith chart. (b) The numbers give the quadrant location of zL and the required matching circuit in (a).

zin = 1. The real parts of y and z in this example must be maintained at 0.25 and 1, respectively. The imaginary parts of y and z are just the values required to make y and z reciprocals of each other, or the straight-line distances from y and z to the center of the Smith chart are equal. Specific values of load impedance occuring in the remaining three quadrants of the Smith chart are illustrated in Figure 3.6 as (b): zL = 0.2 + j0.7, z = 0.2 − j0.4, y = 1 + j2, (c): zL = 0.1 − j0.6, z = 0.1 + j0.3, y = 1 − j3, and (d): zL = 2.5 − j2.5, yL = 0.2 + j0.2, y = 0.2 − j0.4, z = 1 + j2. Example 3.4: A complex load ZL = 5 − j30 Ω is to be impedancematched to a source ZS = Z0 = 50 Ω using an LC network. If the operating frequency is 100 MHz, determine the L and C values of the

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

zL

y

ADD SHUNT L

79

y

zL ADD SHUNT C

ADD SERIES C

yL

ADD SERIES L

z

z

(a)

(b)

z

ADD SERIES L

zL

(c)

ADD SERIES C

yL

z

ADD SHUNT L

ADD SHUNT C

zL

y

y

(d)

Figure 3.6 Impedance matching procedures for zL in each quadrant of the Smith chart using the matching networks in Figure 3.5. (a) Use series L, shunt C network (1). (b) Use shunt L, series C network (2). (c) Use shunt C, series L network (3). (d) Use series C, shunt L network (4).

80

MICROWAVE TRANSMISSION LINE CIRCUITS

appropriate network. Solution: Normalizing, zL = ZL /Z0 = 0.1 − j0.6

(3.24)

which is in quadrant three of the Smith chart. Thus, LC network (3) in Figure 3.5(a) is selected, and the Smith chart solution for this particular load is already plotted in Figure 3.6(c) using the following procedure: Add jωL/Z0 = j0.9 in series with zL = 0.1 − j0.6 to yield z = 0.1 + j0.3, which is the reciprocal of y = 1 − j3. Add jωC/Y0 = j3 in parallel with y = 1 − j3 to obtain yin = 1. From this procedure the element values are: L=

0.9Z0 0.9 × 50 = = 71.62 nH ω 2π × 108

(3.25)

3Y0 3 × 0.02 = = 95.49 pF ω 2π × 108

(3.26)

and C= 3.2.2 Resonant Circuits

Certainly, the lumped Ls and Cs, or their transmission line equivalents in Table 3.1, may be connected in parallel or in series to form resonant circuits in the conventional manner. A parallel or series resonant circuit may also be formed by using a single shorted or opened segment of transmission line, if the line is a quarter wavelength at the resonant frequency. The lumped element circuit and the transmission line circuit will have equivalent frequency response at the resonance frequency if they have the same impedance or admittance value and the same impedance or admittance slope (derivative) with respect to the radian frequency ω. Consider the admittance at the input terminals of the parallel LC circuit in Figure 3.7(a) and the shorted section of transmission line in Figure 3.7(b). The admittance of the LC circuit in Figure 3.7(a) is   1 Y = j ωC − = jB (3.27) ωL Taking the derivative of the susceptance B with respect to ω yields dB 1 =C+ 2 dω ω L

(3.28)

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

81

√ At resonance, where B = 0 and ω = ω0 = 1/ LC, dB 1 |ω=ω0 = C + 2 = C + C = 2C dω ω0 L

(3.29)

The input admittance of the shorted transmission line in Figure 3.7(b) is Y = −jY01 cot

ωl = jB v

(3.30)

The derivative of the susceptance B yields, dB Y01 l 1 = dω v sin2 ωl v At resonance, where B = 0 and ω = ω0 =

πv 2l ,

or l =

(3.31) πv 2ω0

=

dB Y01 l |ω=ω0 = dω v

πv 4πf0

= λ0 /4, (3.32)

Thus, the two circuits have the same input admittance at resonance and the same admittance slope as the frequency passes through resonance if C=

Y01 l Y01 = 2v 8f0

(3.33)

Equivalence of the lumped series LC circuit in Figure 3.8(a) and the openended line segment in Figure 3.8(b) is obtained by equating input impedances and impedance slopes versus frequency at resonance. The impedance of the series LC circuit in Figure 3.8(a) is   1 Z = j ωL − = jX (3.34) ωC Taking the derivative of the reactance X yields, dX 1 =L+ 2 dω ω C √ At resonance, where X = 0 and ω = ω0 = 1/ LC, dX 1 |ω=ω0 = L + 2 = L + L = 2L dω ω0 C

(3.35)

(3.36)

82

MICROWAVE TRANSMISSION LINE CIRCUITS

l L

Y = − jY01 cot

Y = j(ωC - 1/ωL) = jB B

0

dB = 2C dω

ω0

Y01

C

B

ω

ωl v

= jB

dB l = Y01 dω v

0

ω0

(a)

ω

(b)

Figure 3.7 (a) Parallel LC circuit and (b) analogous transmission line circuit.

l L Z01

C Z = − jZ 01 cot

Z = j (ω L − 1 ωC ) = jX

X

0

dX = 2L dω

ω0

(a)

X

ω

0

ωL v

= jX

dX l = Z 01 dω v

ω0

(b)

Figure 3.8 (a) A series LC circuit and (b) the analogous transmission line circuit.

ω

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

83

The input impedance of the open-ended transmission line in Figure 3.8(b) is Z = −jZ01 cot

ωl = jX v

(3.37)

The derivative of the reactance X yields, Z01 l 1 dX = dω v sin2 ωl v At resonance, where X = 0 and ω = ω0 =

πv 2l ,

or l =

dX Z01 l |ω=ω0 = dω v

(3.38) πv 2ω0

=

πv 4πf0

= λ0 /4, (3.39)

At resonance, the two circuits have the same input impedance and the same impedance slope versus frequency if L=

Z01 l Z01 = 2v 8f0

(3.40)

In a typical design where a transmission line segment is to replace an LC circuit, set l = λ0 /4 at f = f0 and solve for Y01 or Z01 from known values of C or L in the lumped-element circuit. For convenient reference, the resonantcircuit equivalences for lumped elements and transmission line segments are given in Table 3.2, along with the appropriate design equations. Figure 3.9 illustrates the transmission-line replacement of cascaded sections of lumped-element resonant circuits, where (a) and (b) represent a bandpass filter, and (c) and (d) represent a bandstop filter. However, problems occur when the transmission line networks in Figure 3.9(b) and Figure 3.9(d) are constructed using the transmission line formats discussed in Chapter 2. One problem is that the series stubs (length l2 ) are more difficult to realize in stripline and microstrip than the shunt stubs (lengths l1 and l3 ). Another problem is that all of the stubs are physically so close together that mutual coupling of their fields may prevent the proper operation of the network. Both of these problems may be resolved through the use of series-to-parallel and parallel-to-series transformations with the quarterwave impedance inverters introduced in Chapter 2. To derive the parallel-to-series transformation equation, refer to Figure 3.10. The parallel admittance Y 0 in (a) is connected on each side through a line of characteristic impedance Z0 and length λ0 /4. With this circuit terminated in Z0

84

MICROWAVE TRANSMISSION LINE CIRCUITS

Table 3.2 Lumped-Element Resonant Circuits and Their Transmission Line Equivalents

L

C

Y0

Y0

Design Equation

Y0

l

Y

Y0

Transmission Line Circuit

01

Lumped-Element Circuit

C=

Y01 l 2v

=

Y01 , 8f0

l = λ0 /4

C=

Y01 l 2v

=

Y01 , 8f0

l = λ0 /4

L=

Z01 l 2v

=

Z01 , 8f0

l = λ0 /4

L=

Z01 l 2v

=

Z01 , 8f0

l = λ0 /4

Z0

Y0

C

L

Z0

Y0

Y0

Z0

Z0

Z

L C Z0

Z0

01

l

l

Z

C

Z0 01

Y0

01

Y

l

L

Z0

85

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

L2 C2

Z0 L 1

C1

L3

C3 Z 0

l2

Z02

Z0 03

Z

l

(b)

3

01

l

Z

Z0

1

(a)

L2

Z0

l2

C3

Z02

Z0 03

l

(d)

3

l

Z

Z0

Z

(c)

L3

01

C1

C2

1

L1

Z0

Figure 3.9 Transmission line replacement of cascaded sections of lumped-element resonant circuits. (a) Lumped-element bandpass filter, and (b) transmission line equivalent, where l1 = l2 = l3 = λ0/4, Y01 = 8f0 C1 , Z02 = 8f0 L2 , and Y03 = 8f0 C3 . (c) Lumped-element bandstop filter, and (d) transmission-line equivalent, where l1 = l2 = l3 = λ0 /4, Z01 = 8f0 L1 , Y02 = 8f0 C2 , and Z03 = 8f0 L3 .

86

MICROWAVE TRANSMISSION LINE CIRCUITS

λ0 /4

λ0 /4 Y'

Z0

Z0

Z0

Y1 = Y0 Y2 = Y0 + Y ' Z in' = Z 0 + Z 02Y ' (a)

Z Z0

Z in = Z 0 + Z (b) Figure 3.10 Parallel-to-series transformation of elements from network (a) to (b).

the input impedance is 0 Zin = Z0 + Z02 Y 0

(3.41)

The impedance Z in series with the Z0 termination in (b) has the input impedance Zin = Z0 + Z

(3.42)

Clearly, the input impedances in (a) and (b) of Figure 3.10 are equal if Z = Z02 Y 0

(3.43)

Thus, if Y 0 = jωC 0 due to a parallel capacitance, then Z = jωZ02 C 0 = jωL is due to a series inductance L = Z02 C 0 . In a similar manner, if Y 0 = 1/(jωL0 ) due to a parallel inductance, then Z = 1/(jωL0 /Z02 ) = 1/(jωC) is due to a series capacitance C = L0 /Z02 . If L0 and C 0 are connected in parallel as Y 0 , they transform the same way to become series connected L and C in Z.

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

87

Using a similar analysis, the two networks in Figure 3.11(a) and (b) have the same input admittance if Yin = Y0 + Y = Y0 + Y02 Z 0

(3.44)

Y = Y02 Z 0

(3.45)

or This means that series elements L0 and C 0 in Z 0 transform into parallel elements L and C in Y as L = Z02 C 0 and C = L0 /Z02 . For convenient reference, all of the transformations implied in Figure 3.10 and Figure 3.11 are illustrated in Table 3.3. To illustrate the series-to-parallel and parallel-to-series transformations, the equivalent circuits in Table 3.3 are used to transform the bandpass and bandstop filters considered earlier so that all resonant sections are in shunt with the main transmission line and spaced λ0 /4 apart to prevent coupling. The resulting bandpass filter is shown in Figure 3.12, and the bandstop filter is shown in Figure 3.13. The example that follows may also add clarity to the transformation process. Example 3.5: The lumped-element bandpass filter in Figure 3.9(a) has the element values C1 = C3 = 5 pF, L1 = L3 = 5.07 nH, L2 = 25 nH, and C2 = 1.01 pF. Determine the characteristic impedances of the transmission line equivalent circuits in Figs. 3.9(b) and 3.12(c), where all line lengths are λ0 /4. Solution: From Figure 3.9(b) and the equations in the caption, Y01 = 8f0 C1 = 8 × 109 × 5 × 10−12 = 0.04 S = Y03

(3.46)

or Z01 = 1/Y01 = Z03 = 25 Ω

(3.47)

and Z02 = 8f0 L2 = 8 × 109 × 25 × 10−9 = 200 Ω

(3.48)

From Figure 3.12(c), Z01 = Z03 = 1/(8f0 C1 ) = 25 Ω

(3.49)

88

MICROWAVE TRANSMISSION LINE CIRCUITS

Z'

λ0 /4 Z0

λ0 /4 Z0

Z0

Z1 = Z0 Z2 = Z0 + Z ' Yin = Y0 + Y02 Z ' (a)

Y

Z0

Yin = Y0 + Y (b) Figure 3.11 Series-to-parallel transformation of elements from network (a) to (b).

as before, and Y02 = 8f0 C20 =

8f0 L2 8 × 109 × 25 × 10−9 = = 0.08 S Z02 502

(3.50)

or Z02 = 1/Y02 = 12.5 Ω

(3.51)

89

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

Table 3.3 Parallel-to-Series and Series-to-Parallel Transformations

Element Between λ/4 Sections

λ0 /4

λ0 /4

λ0 /4

C

λ0 /4 Z0

λ0 /4

C ' Z0

λ0 /4

Z0

C

'

L' C '

λ0 /4

Z0

C

Z0

Z0

L

Z0

C

λ0 /4

λ0 /4 Z0

L'

λ0 /4 L'

L'

Equivalent Circuit Element

λ0 /4

Z0

L λ0 /4

C'

Z0

L C

λ0 /4

Z0 L'

λ0 /4 Z0

Z0

L'

Z0

Element Between λ/4 Sections

L

λ0 /4 C'

Z0

Equivalent Circuit Element

C'

L

C

λ0 /4 Z0

L C

In each network pair the equivalent circuit elements are related by L = Z02 C 0 and C = L0 /Z02 .

90

MICROWAVE TRANSMISSION LINE CIRCUITS

L 2 C2

C1

λ0 /4

λ0 /4

C1 L'2

C2' L3

l

(a)

C3 Z 0

(b)

Z0

(c)

03

02

Z

Z0

l

l

01

Z0

Z

Z0

C3 Z 0

l

Z

Z 0 L1

L3

l

Z 0 L1

Figure 3.12 (a) Lumped-element bandpass filter, (b) with central section transformed using L02 = Z02 C2 and C20 = L2 /Z02 from Table 3.3, and (c) with each parallel LC replaced by a shorted, λ0 /4 transmission line, where, Y01 = 1/Z01 = 8f0 C1 , Y02 = 1/Z02 = 8f0 C20 = 8f0 L2 /Z02 , Y03 = 1/Z03 = 8f0 C3 , and l = λ0 /4 throughout.

91

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

L2

L1

L3

C2

C1

λ0 /4 L1

L'2

L3

C1

C2'

C3

l

(b)

Z0

(c)

03

02

Z

Z0

l

01

l

Z0

l

Z0

Z

Z0

(a)

λ0 /4

Z

Z0

Z0

C3

l

Z0

Figure 3.13 (a) Lumped-element bandstop filter, (b) with central section transformed using L02 = Z02 C2 and C20 = L2 /Z02 from Table 3.3, and (c) with each series LC replaced by an open, λ0 /4 transmission line, where, Z01 = 8f0 L1 , Z02 = 8f0L02 = 8f0 Z02 /C2 , Z03 = 8f0 L3 , and l = λ0 /4 throughout.

92

MICROWAVE TRANSMISSION LINE CIRCUITS

ZS = 50 Ω VS

L C

C

ZL

Z, Γ Figure 3.14 Lumped-element prototype for the microwave circuit described in Problem 1.

PROBLEMS 1. The lumped-element circuit in Figure 3.14 has ZS = ZL = 50 Ω, C = 3.183 pF, and L = 15.90 nH. (a) Design the equivalent microstrip circuit at f = 1 GHz, showing all line lengths and widths. The input and output lines are Z0 = 50 Ω. Use 1/16 inch thick epoxy-glass substrate (r = /0 = 4.8), and assume that the metalization thickness is negligible. (b) On the input 50 Ω line just before the microstrip circuit begins, calculate the input impedance (Z) and reflection coefficient (Γ) at the three frequencies, f = 0.5 GHz, f = 1 GHz, and f = 2 GHz. (c) Using metal-foil tape and a 7.5 cm by 15 cm epoxy-glass board (1/16 inch thick), construct the circuit designed in (a), and measure the transmission and reflection losses in dB over the 0 to 1.5 GHz range. 2. The input impedance of the networks on the left in Figure 3.15 is to be the same as that of the networks on the right for any termination. (a) Determine C for each network in terms of Z0 and L. (b) If L = 50 nH and Z0 = 50 Ω, what is C (in pF) for each network? 3. A parallel-resonant circuit is to be formed by connecting an open-circuited stub in parallel with a short-circuited stub as indicated in Figure 3.16. If l1 = l2 = 2.54 cm, Z01 = Z02 = 100 Ω, and the dielectric medium is air, determine (a) the lowest resonant frequency, and (b) the value of L and C at resonance. 4. The three resonant circuits in Figure 3.17 are to be equivalent at the terminals a − a0 . Determine Z01 and Z02 if C = 2.5 pF and f0 = 1 GHz at resonance. 5. Determine Z01 , Z02 , and Z03 to make the two circuits in Figure 3.18 equivalent as bandpass filters with f0 = 1 GHz.

93

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

λ0 /4 Z0

C

Z0

λ0 /4 Z0

L

λ0 /4

⇔ C

λ0 /4

L

Z0

⇔

Figure 3.15 Determine the component values required to make the input impedances equivalent using the values given in Problem 2.

C

L

l1

l2

Z01

Z02

Figure 3.16 Find the lowest resonant frequency for the distributed circuit using the transmission line lengths and characteristic impedances given in Problem 3. Find the value of L and C in the lumped element circuit at that frequency.

a

C

a

L

a' Figure 3.17 Problem 4.

a'

a λ0 /4

λ0 /8

λ0 /8

Z01

Z02

Z02

a'

Determine Z01 and Z02 to make these circuits equivalent using the values given in

94

MICROWAVE TRANSMISSION LINE CIRCUITS

L 2 C2

ZS = 50 Ω L1

VS

C1

L3

ZS = 50 Ω VS

l Z0

Z0

ZL = 50 Ω C1 = C3 = 5 pF L2 = 25 nH

C3

l Z0

Z0

ZL = 50 Ω

Z

03

l

Z

02

l

01

l

Z

Z0 = 50 Ω

Figure 3.18 Determine Z01 , Z02 , and Z03 to make these circuits equivalent as bandpass filters with f0 = 1 GHz.

TRANSMISSION LINE SEGMENTS AS NETWORK ELEMENTS

95

BIBLIOGRAPHY I. J. Bahl and P. Bhartia, Microwave Solid State Circuit Design, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2003. T. C. Edwards and M. B. Steer, Foundations of Interconnect and Microstrip Design, 3rd ed. Hoboken, NJ: John Wiley and Sons, 2001. V. F. Fusco, Microwave Circuits: Analysis and Computer-Aided Design. Upper Saddle River, NJ: Prentice-Hall, 1987. H. Howe, Stripline Circuit Design. Dedham, MA: Artech House, 1974. S. Y. Liao, Microwave Circuit Analysis and Amplifier Design. Upper Saddle River, NJ: Prentice-Hall, 1987. M. A. Plonus, Applied Electromagnetics, 2nd ed. New York: McGraw-Hill, 1978. D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley and Sons, 2011. P. A. Rizzi, Microwave Engineering: Passive Circuits. Upper Saddle River, NJ: Prentice-Hall, 1988. G. D. Vendelin, A. M. Pavio, and U. L. Rohde, Microwave Circuit Design, Using Linear and Nonlinear Techniques, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2005. E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications. New York: Wiley-Interscience, 1988.

96

MICROWAVE TRANSMISSION LINE CIRCUITS

Chapter 4 MATRIX REPRESENTATION OF MICROWAVE NETWORKS 4.1 Z, Y , AND ABCD MATRICES FOR CONNECTED NETWORKS 4.1.1 Two-Port Network Parameters An arbitrary, two-port microwave network is represented as shown in Figure 4.1. A linear relationship between terminal voltages and currents may be expressed as:

or in matrix form:

V1 = Z11 I1 − Z12 I2

(4.1)

V2 = Z21 I1 − Z22 I2

(4.2)



    V1 Z11 Z12 I1 = (4.3) V2 Z21 Z22 −I2 Note that while it is common to define currents as entering the network, in this case making the definition of I2 as the current leaving the network will be useful in later matrix formulations, where the negative sign on I2 correctly reflects the concept of impedance. The coefficients Z11 , Z12 , Z21 , and Z22 are referred to as the open-circuit impedances of the network since they are defined and measured in the following way: Z11 = VI11 Z12 = − VI21 I2 =0

I1 =0

(4.4)

Z21

= VI12

I2 =0

Z22

97

= − VI22

I1 =0

98

MICROWAVE TRANSMISSION LINE CIRCUITS

ZS

I1

I2 MICROWAVE

V1

VS

V2

NETWORK

ZL

Z1 Figure 4.1 Arbitrary two-port microwave network.

For individual networks connected in series at the input and output, as in Figure 4.2, the open-circuit impedance parameters of each network add directly to form a composite Z matrix. A linear relationship between terminal currents and voltages may be expressed as: I1 = Y11 V1 + Y12 V2 (4.5) −I2 = Y21 V1 + Y22 V2

(4.6)

or in matrix form: 

I1 −I2



=



Y11 Y21

Y12 Y22



V1 V2



(4.7)

The coefficients Y11 , Y12 , Y21 , and Y22 are referred to as the short-circuit admittances of the network since they are defined and measured in the following way: Y11 = VI11 Y12 = VI12 V2 =0

V1 =0

(4.8)

Y21

= − VI21

V2 =0

Y22 =



− VI22

V1 =0

For individual networks connected in parallel at the input and output, as in Figure 4.3, the short-circuit admittance parameters of each network add directly to form a composite Y matrix. It should be clear from the foregoing developments that the Y matrix is the inverse of the Z matrix and of course, vice versa. The input voltage and current may be related to the output voltage and current of the arbitrary two-port network in Figure 4.1 as: V1 = AV2 + BI2

(4.9)

99

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

Z11'

Z12'

' Z 21

' Z 22

'' 11

'' 12

Z11

Z12

Z21

Z22

Z11' + Z11'' Z12' + Z12''

=

I1

' '' ' '' Z 22 + Z 22 Z 21 + Z 21

I2

V1

Z

Z

'' Z 21

'' Z 22

V2

Figure 4.2 Two-port networks with inputs connected in series and the outputs in series.

I1

I2

V1

Y11'

Y12'

Y21'

Y22'

V2

Y11''

Y12''

Y11

Y12

Y21''

Y22''

Y21

Y22

Y11' + Y11''

Y12' + Y12''

Y21' + Y21''

Y22' + Y22''

=

Figure 4.3 Two-port networks with inputs and outputs in parallel.

I1 = CV2 + DI2

(4.10)

or in matrix form; 

V1 I1



=



A C

B D



V2 I2



(4.11)

100

MICROWAVE TRANSMISSION LINE CIRCUITS

I1

I2 Z

V1

A=

C=

V1 V2 I1 V2

=1

V2

V1 =Z I2 V =0

B=

I2 = 0

2

=0

D=

I2 = 0

A

I1 =1 I2 V =0 2

B

1

Z

0

1

= C

D

Figure 4.4 The ABCD parameters for a series impedance Z.

where the ABCD parameters may be defined and measured as: A = VV21 B = VI21 I2 =0

C=



I1 V2 I =0 2

V2 =0

D=



I1 I2 V =0 2

(4.12)

The elements of the ABCD matrix may be converted to the elements of the Z matrix or the Y matrix, and vice versa, as:     1 A (AD − BC) Z11 Z12 (4.13) = Z21 Z22 1 D C     1 Y11 Y12 D −(AD − BC) = (4.14) Y21 Y22 A B −1     1 A B Z11 (Z11 Z22 − Z12 Z21 ) = (4.15) C D 1 Z22 Z21

101

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

I1

I2

I3

NETWORK

V1

NETWORK

V2

1

V3

2

Figure 4.5 Cascaded two-port networks.



A C



B D

−1 = Y21



Y22 1 (Y11 Y22 − Y12 Y21 ) Y11



(4.16)

For reciprocal or linear, passive networks a simplification may be made since Z12 = Z21 , Y12 = Y21 , and AD − BC = 1. An example determination of A, B, C, and D for a network consisting of a series impedance Z is shown in Figure 4.4. The ABCD matrix elements for some additional networks are given in Table 4.1. 4.1.2 Two-Port Networks Connected in Cascade Of the three matrices defined thus far (Z, Y , and ABCD), the ABCD matrix is the most useful for analyzing microwave networks, since signals are most often passed from one network to another in cascade. To illustrate the utility of the ABCD matrix, consider the cascade connection of two-port networks shown in Figure 4.5. The input and output variables of network 1 are related by      V1 A1 B1 V2 = (4.17) I1 C1 D1 I2 and for network 2, 

V2 I2



=



A2 C2

B2 D2



Substituting (4.18) in (4.17) yields     V1 A1 B1 A2 = I1 C1 D1 C2 or



V1 I1



=



A C

B D



V3 I3



B2 D2



V3 I3



(4.18)

V3 I3



(4.19)

(4.20)

102

MICROWAVE TRANSMISSION LINE CIRCUITS

Table 4.1 The ABCD Matrix of Some Useful Networks Network

Schematic Diagram

ABCD Matrix

θ

θ = ωl v

Z0 Section of lossless line

cosθ

jZ 0 sin θ

jY0 sin θ

cosθ

Z 1

Z

0

1

1

0

Y

1

Series impedance

Y Shunt impedance

Z

Z 1 + YZ

Z(2 + YZ)

Y

1 + YZ

Y Tee network

Z Y

1 + YZ

Z

Y(2 + YZ)

1 + YZ

−1 g m R2 −1 g m R1 R2

−1 gm −1 g m R1

Y

Pi network

V1 Voltage amplifier

R1

gmV1

gm = transconductance (S)

R2

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

103

Thus, the ABCD matrix of two networks in cascade is simply the product of the individual network matrices, as      A B A1 B1 A2 B2 = (4.21) C D C1 D1 C2 D2 This result can be generalized for the case of any number of networks in cascade. This provides a mathematically straightforward way to determine the overall ABCD matrix of any network composed of cascaded two-ports if the ABCD matrix of each individual network is known.

4.2 NETWORK GAIN OR LOSS IN TERMS OF ABCD PARAMETERS 4.2.1 Insertion Gain for Complex ZS and ZL Let the power delivered to the load with the two-port network in Figure 4.1 removed be: VS P00 = |IS |2 RL = | |2 RL (4.22) ZS + ZL and the power delivered to the load with the network inserted be: PL = |I2 |2 RL

(4.23)

Hence, the insertion gain Gi = PL/P00 (or insertion loss Li = P00 /PL) is 1 PL |I2 |2 = 0 = |ZS + ZL |2 Li P0 |VS |2

(4.24)

I1 = CV2 + DI2 = C(I2 ZL ) + DI2

(4.25)

I1 = (CZL + D)I2

(4.26)

Gi = But or and

I2 =

I1 CZL + D

(4.27)

I1 =

VS ZS + Z1

(4.28)

Also,

104

MICROWAVE TRANSMISSION LINE CIRCUITS

where Z1 =

V1 AZL + B = I1 CZL + D

(4.29)

Hence, the load current I2 may be expressed as I2 =

VS ZS +

AZL +B CZL +D

1 VS = CZL + D AZL + B + CZS ZL + DZS

(4.30)

Substitution into the expression for Gi yields, 2 ZS + ZL Gi = AZL + B + CZS ZL + DZS

(4.31)

4.2.2 Transducer Gain for Complex ZS and ZL

The maximum available power from the source connected to the network input is P0 =

|VS |2 4RS

(4.32)

As before, the power delivered to the load ZL with the network in place is PL = |I2 |2 RL

(4.33)

Again, I2 =

VS AZL + B + CZS ZL + DZS

(4.34)

and the transducer gain Gt = PL/P0 (or transducer loss Lt = P0 /PL) is 1 PL |I2 |2 4RS RL = = Lt P0 |VS |2

(4.35)

1 4RS RL = Lt |AZL + B + CZS ZL + DZS |2

(4.36)

Gt = or Gt =

While under certain conditions, Gt = Gi , the expression for Gt is more often used as it is based on maximum available power which is also the incident power at the network input.

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

105

Next, we examine some special cases that may simplify calculations. First, note that if ZS and ZL are real and equal (ZS = ZL = Z0 ), then Gi = Gt , and the gain may be expressed as 2 1 2 G= = (4.37) L A + BY0 + CZ0 + D If source and load impedances are real but unequal (ZS = the transducer gain (or loss) may be expressed as 1 2 q Gt = = q √ Lt R B L A +√ + C R R + D RS RS

RS RL

S

L

RL

RS , ZL = RL ),

2

(4.38)

If source and load are real and the two-port network contains only reactive elements (inductance, capacitance, sections of lossless transmission line or waveguide), there is of course no power dissipated in the network. Thus, the fractional power transmitted to the load (PL/P0 ) is related to the input reflection coefficient (Γ) and voltage-standing-wave-ratio (S) by PL 4S = 1 − |Γ|2 = P0 (S + 1)2

(4.39)

since

S−1 (4.40) S+1 For the nondissipative or lossless network, it can be shown that the matrix elements A and D are real and the elements B and C are imaginary. Using this information and the fact that AD − BC = 1, the transducer loss (or gain) in (4.38) may be expressed as P0 1 = Lt = PL Gt or   r !2  r 2 p P0 1 RL RS B =1+ A− D − √ − RS RL C  (4.41) PL 4 RS RL RS RL |Γ| =

This form is convenient to use in the case of a symmetrical √ network (RS = RL , A = D) and for certain impedance-matching networks ( RS RL = Z0 , B/Z0 = Z0 C).

106

MICROWAVE TRANSMISSION LINE CIRCUITS

Z0 = 50 Ω

Z0 = 50 Ω

θ

θ

θ Z0 = 25 Ω

SHORT

Figure 4.6 Microstrip bandpass filter with two shorted stubs.

Example 4.1: A microstrip bandpass filter consists of two shorted stubs of characteristic impedance Z01 = 25 Ω and electrical length and spacing θ = βl = π2 ff0 , as shown in Figure 4.6. The main line is of characteristic impedance Z0 = 50 Ω. Express P0 /PL as a function of f/f0 , where f0 is the center of the passband. Solution: The composite network consists of three individual sections; the shorted stubs form shunt admittances on each end of the transmission line separating them. Thus, we multiply the individual ABCD matrices together to obtain  =



1 −jY01 cot θ

0 1



A C

B D

cos θ jY0 sin θ



=

jZ0 sin θ cos θ



1 −jY01 cot θ

0 1



107

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

=

"

θ j sin Z0

0 (1 + ZZ01 ) cos θ 2 j θ − Z01 (2 + ZZ010 ) cos sin θ

jZ0 sin θ 0 (1 + ZZ01 ) cos θ

Substitution into (4.41) with RS = RL = Z0 , yields   P0 1 B 2 2 =1+ (A − D) − ( − Z0 C) PL 4 Z0

#

(4.42)

(4.43)

Noting from (4.42) that A = D and thus (A − D) = 0, this equation can be simplified as    2 P0 1 Z0 Z0 cos2 θ =1− j sin θ − j sin θ + j 2+ (4.44) PL 4 Z01 Z01 sin θ Further noting that Z0 /Z01 = 50/25 = 2 gives P0 cos4 θ = 1 + 16 2 PL sin θ

(4.45)

At f = f0 , θ = π/2 = 90◦ , and L = P0 /PL = 1. Setting L = P0 /PL = 2 in (4.45) and solving yields the 3 dB frequencies above and below f0 as: f2 = 1.311f0 , and f1 = 0.689f0. Hence, the Q of this network is f0 /(f2 − f1 ) = 1.61. Note that in Figure 4.6, the transmission line lengths θ are defined exactly. Upon closer examination of the figure, it becomes clear that the locations of the junctions between transmission lines in fact are not well-defined because the transmission lines have finite width. The choice of whether the junction occurs at the edge or at the center of the transmission lines will affect the overall operation of the circuit, and may require several design iterations to achieve the desired circuit performance. Example 4.2: A lossless transmission line of electrical length θ = (π/2)(f/f0 ) and characteristic impedance Z0 is connected between RS and RL to make Z1 = RS at f = f0 where θ = π/2, as in Figure 4.7. Express P0 /PL as a function of f/f0 and the ratio of RL /RS . Solution: The ABCD matrix of the transmission line is     A B cos θ jZ0 sin θ = C D jY0 sin θ cos θ

(4.46)

108

MICROWAVE TRANSMISSION LINE CIRCUITS

RS

θ

VS

Z0

RL

Z1 Figure 4.7 Lossless transmission line of length θ between RS and RL .

and from (4.29), with θ = π/2, Z1 = RS = Z02 /RL, or Z0 = √ RS RL. Substitution into (4.41) yields P0 PL

=

Lt

=

1 1+ 4

=

1+

1 4

!2 r RL RS A− D RS RL !2 r r RL RS − cos2 θ RS RL r

(4.47)

At f = f0 , θ = π/2 = 90◦ , and Lt = P0 /PL = 1. The loss ratio at other frequencies depends on the ratio RL/RS . Note that at f = 0 and f = 2f0 , cos2 θ = 1, and Lt ≤ 2 if RL /RS ≤ 5.828.

Example 4.3: A lossless transmission line of electrical √ length θ = (π/2)(f/f0 ) and characteristic impedance Z02 = RS RL is connected between RS and RL to make Z1 = RS at f = f0 where θ = π/2, as in Figure 4.8. The shorted stub of characteristic impedance Z01 in parallel with the input is invisible at f = f0 . Express P0 /PL as a function of θ, RS , RL, and Z01 . Solution: The ABCD matrix of the network between RS and RL is 

A C

B D



=



1 −jY01 cot θ

0 1



cos θ jY02 sin θ

jZ02 sin θ cos θ



109

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

RS

θ

VS

01

RL

θ

Z

Z02

Figure 4.8 Shorted stub and lossless transmission line of lengths θ between RS and RL .

=

"

cos θ  cot2 θ −j Z01 − Z102 sin θ 

#  jZ02 sin θ (4.48) 02 1+ Z cos θ Z01

Substitution into (4.41) yields P0 1 = Lt = 1+ PL 4 or, with Z02 =

√

"

Z02 RS RS − − RS Z02 Z01

2

2

cos θ +



Z02 Z01

2

# cos4 θ sin2 θ (4.49)

RS RL

P0 = Lt PL  r  !2 r √ 2 4 1 RL RS RS RS RL cos θ  = 1+  − − cos2 θ + 4 RS RL Z01 Z01 sin2 θ (4.50)

The characteristic impedance of the shorted stub, Z01 , may be chosen to flatten the frequency response of Lt , thus broadening the impedance match at f = f0 . Since θ contains the frequency and cos2 θ is a suitable frequency variable for this example, we let dLt θ=π/2 = 0 d(cos2 θ)

(4.51)

110

MICROWAVE TRANSMISSION LINE CIRCUITS

to obtain r

RL − RS

r

RS RS − RL Z01

!

=0

(4.52) √

as the condition to be met. Thus, if RL = 3RS , then Z01 = 23 RS , 4 θ in (4.50). If RL = 9RS , then Z01 = 38 RS , and and Lt = 1 + cos sin2 θ 4 θ Lt = 1 + 16 cos in (4.50). Also, note that (4.50) reduces to (4.47) if sin2 θ Z01 = ∞ (the shorted stub vanishes). 4.3 SCATTERING PARAMETERS AND THE SCATTERING MATRIX OF A NETWORK Another useful network representation is the scattering matrix. The elements of the scattering matrix specify the ratio of reflected to incident waves at each port of the network. These ratios are just the reflection and transmission coefficients of the network that are conveniently measured in the laboratory. Measurement and calculation of the scattering parameters require that the network be terminated in some reference impedance (usually Z0 of the connecting lines), rather than a short or an open as for the Z, Y , and ABCD parameters. To develop the scattering parameters, refer again to the arbitrary two-port network shown in Figure 4.1. The total voltages V1 and V2 at the input and output ports may be separated into incident and reflected components as: V1 = V1i + V1r

(4.53)

V2 = V2i + V2r

(4.54)

Since we have a linear network describable by linear equations, the reflected waves may be expressed in terms of the incident waves as: V1r = S11 V1i + S12 V2i

(4.55)

V2r = S21 V1i + S22 V2i (4.56) √ Dividing both sides of the above equations by Z0 (the reference impedance) adds little mathematical complexity, and allows calculation of the incident or reflected power at a particular point in the circuit simply by squaring the result, as V1r V1i V2i √ = S11 √ + S12 √ Z0 Z0 Z0

(4.57)

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

111

V2r V1i V2i √ = S21 √ + S22 √ Z0 Z0 Z0

(4.58)

b1 = S11 a1 + S12 a2

(4.59)

b2 = S21 a1 + S22 a2

(4.60)

or

and in matrix form:



    b1 S11 S12 a1 = (4.61) b2 S21 S22 a2 The coefficients S11 , S12 , S21 , and S22 are the scattering parameters that make up the scattering matrix as indicated. Using the above equations, the S parameters may be calculated or measured as follows: S11 = ab11 S12 = ab12 a2 =0

a1 =0

(4.62)

S21

= ab21

a2 =0

S22

= ab22

a1 =0

Note that S11 is the input reflection coefficient with ZL = Z0 , S12 is the reverse transmission coefficient with ZS = Z0 , S21 is the forward transmission coefficient with ZL = Z0 , and S22 is the output reflection coefficient with ZS = Z0 . Again, the S parameters describe the behavior of the network for any ZS and ZL , but they must be measured and calculated with ZS = ZL = Z0 , where Z0 is the characteristic impedance of the transmission line connecting the input to ZS and the output to ZL . S parameters easily can be measured, but are not particularly useful for calculating the characteristics of cascaded networks. ABCD parameters are mathematically expedient for calculating the characteristics of cascaded networks, but are difficult to measure. The S parameters may be related to the ABCD parameters under the condition that ZS = ZL = Z0 . Thus, since (4.37) and |S21 |2 both represent the forward gain of the network, we have 2 2 2 (4.63) G = |S21 | = A + BY0 + CZ0 + D or

2 A + BY0 + CZ0 + D Also, the input reflection coefficient is S21 =

S11 = Γ11 =

Z1 − Z0 = Z1 + Z0

AZ0 +B CZ0 +D AZ0 +B CZ0 +D

(4.64)

− Z0 + Z0

(4.65)

112

MICROWAVE TRANSMISSION LINE CIRCUITS

or

A + BY0 − CZ0 − D (4.66) A + BY0 + CZ0 + D The complete correspondence between the S matrix and the ABCD parameters is     1 A + BY0 − CZ0 − D S11 S12 2(AD − BC) = S21 S22 2 −A + BY0 − CZ0 + D ∆ (4.67) or   A B = C D   1 (1 + S11 )(1 − S22 ) + S12 S21 Z0 [(1 + S11 )(1 + S22 ) − S12 S21 ] = (1 − S11 )(1 + S22 ) − S12 S21 2S21 Y0 [(1 − S11 )(1 − S22 ) − S12 S21 ] (4.68) where ∆ = A+BY0 +CZ0 +D. If the network is linear and passive, AD−BC = 1, and S21 = S12 . If the network is nonlinear or active, then S21 and S12 may have quite different values. For a transistor amplifier, for example, it is desirable to have S21  1 and S12  1. S11 =

Example 4.4: Use (4.67) and Table 4.1 to determine the S parameters of a network consisting of (a) a series impedance Z, (b) a shunt admittance Y , and (c) a section of lossless transmission line of electrical length θ and characteristic impedance Z0 , where for the source and load impedances, ZS = ZL = Z0 . Solution: (a) For the series Z, the ABCD parameters in Table 4.1 are substituted into (4.67) to obtain:     1 S11 S12 ZY0 2 = (4.69) S21 S22 2 ZY0 ZY0 + 2 (b) For the shunt admittance Y , Table 4.1 and (4.67) yield:     1 S11 S12 −Y Z0 2 = S21 S22 2 −Y Z0 Y Z0 + 2

(4.70)

(c) Again, from Table 4.1 and (4.67), the S parameters of the lossless line are:       1 S11 S12 0 2 0 e−jθ = = S21 S22 e−jθ 0 2(cos θ + j sin θ) 2 0 (4.71)

113

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

ZS

b1

VS

a1 Z0

ΓS

a2 s11

s12

s21

s22

Γ1

b2 Z0

Γ2

ZL

ΓL

Figure 4.9 Two-port network with arbitrary ZS and ZL .

If ZS and ZL are not equal to Z0 , as indicated in Figure 4.9, then the input reflection coefficient Γ1 is not S11 and the transducer gain Gt is not |S21 |2 . The network is described by (4.59) and (4.60), and the reflection coefficient of the load is ZL − Z0 a2 = (4.72) ΓL = b2 ZL + Z0 Using this equation and (4.59) yields

or

b1 − S11 a1 = S12 a2 = S12 ΓL b2

(4.73)

b1 b2 = S11 + S12 ΓL a1 a1

(4.74)

S22 ΓLb2 − b2 = −S21 a1

(4.75)

From (4.60) and (4.72) or

b2 −S21 = a1 S22 ΓL − 1 Substituting (4.76) into (4.74) gives the input reflection coefficient as Γ1 =

S12 S21 ΓL b1 = S11 − a1 S22 ΓL − 1

(4.76)

(4.77)

In (4.77), if ZL = Z0 , then ΓL = 0 and Γ1 = S11 as before. In terms of reflection coefficients and scattering parameters the transducer gain for any ZS and ZL is Gt =

(1 − |ΓS |2 )|S21 |2 (1 − |ΓL|2 ) |(1 − S11 ΓS )(1 − S22 ΓL ) − S12 S21 ΓS ΓL |2

(4.78)

114

MICROWAVE TRANSMISSION LINE CIRCUITS

where ΓS = (ZS − Z0 )/(ZS + Z0 )

(4.79)

is the source reflection coefficient. Substituting (4.79), (4.59), and (4.67) into (4.78) would yield the expression for Gt derived earlier and expressed as (4.36). Note that, as expected, when ZS = ZL = Z0 in (4.78), ΓS = ΓL = 0, and Gt = |S21 |2 . For a nonreciprocal network like an amplifier, S12 may be zero or negligible, and (4.78) reduces to Gt =

(1 − |ΓS |2 )|S21 |2 (1 − |ΓL|2 ) |(1 − S11 ΓS )(1 − S22 ΓL )|2

(4.80)

If the input and output impedances are conjugately matched to the source and load impedances, respectively, as Z1 = ZS∗ and Z2 = ZL∗ , then S11 = Γ∗S and S22 = Γ∗L . Substituting these results into (4.80) yields: Gt =

(1 − |ΓS |2 )|S21 |2 (1 − |ΓL|2 ) |S21 |2 = |(1 − |ΓS |2 )(1 − |ΓL |2 )|2 (1 − |ΓS |2 )(1 − |ΓL|2 )

(4.81)

Thus, impedance matching at the input and output can make Gt > |S21 |2 , since |ΓS | ≤ 1 and |ΓL| ≤ 1. Example 4.5: At 1 GHz the S parameters of a transistor amplifier are: S11 = 0.996 −25◦ , S21 = 3.806 160◦, S12 = 0.016 76◦ , and S22 = 0.706 −16◦ . (a) If ZS = ZL = Z0 , in which case, ΓS = ΓL = 0, determine the transducer gain in magnitude and in dB. (b) If ZS = Z1∗ and ∗ ∗ ZL = Z2∗ , in which case (taking S12 = 0), ΓS = S11 and ΓL = S22 , determine the transducer gain in magnitude and in dB. Solution: (a) Using (4.78), (4.80), or (4.81), Gt = |S21 |2 = (3.80)2 = 14.44

(4.82)

Gt = 10 log |S21 |2 = 11.60 dB

(4.83)

or (b) Using (4.81), Gt

=

|S21 |2 (1 − |ΓS |2 )(1 − |ΓL|2 )

115

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

= = = =

3.802 (1 − 0.992 )(1 − 0.702) 50.25 × 14.44 × 1.96 1422.76 or 17.01 + 11.60 + 2.92 31.53 dB

(4.84)

Thus, we note that conjugately matching the input and output of the amplifier produces a highly significant increase in gain. Methods of producing the conjugate match for several different transistor amplifiers will be presented in Chapter 8. The scattering matrix may be used to describe a network with any number of ports. Generalizing from the two-port equations in (4.59) and (4.60) we have n ports: b1 = S11 a1 + S12 a2 + ... + S1n an b2 = S21 a1 + S22 a2 + ... + S2n an ... = ......................................... bn = Sn1 a1 + Sn2 a2 + ... + Snn an or



(4.85)

    b1 S11 S12 . . . S1n a1  b2   S21 S22 . . . S2n   a2        .   .   . .   =  .  (4.86)  .   .  .  . .       .   . . .  .  bn Sn1 Sn2 . . . Snn an The S parameters for a network are often specified in matrix form to characterize a device at a single frequency. For example, an ideal three-port circulator (a ferrite device) passes a signal from port 1 to port 2, port 2 to port 3, and port 3 to port 1 with a phase delay θ = −βl and no signal attenuation or reflection. The circulator blocks the signal in the reverse direction, so the signal transmission from port 1 to port 3, port 3 to port 2, and port 2 to port 1 is zero. All ports are terminated with transmission lines of characteristic impedance Z0 , so there is no self-reflection at any port. Thus, the scattering matrix for the circulator is:     S11 S12 S13 0 0 ejθ  S21 S22 S23  =  ejθ 0 0  (4.87) jθ S31 S32 S33 0 e 0

116

MICROWAVE TRANSMISSION LINE CIRCUITS

Example 4.6: A three-port, Y-junction network is formed by connecting three identical transmission lines together at a single point so that all inputs and outputs are in parallel. Each line is of characteristic admittance Y0 . Determine the scattering matrix of this three-port network. Solution: The input reflection coefficient at each port will be the same, or S11

=

S22 = S33

= =

(Zin − Z0 )/(Zin + Z0 ) (Z0 /2 − Z0 )/(Z0 /2 + Z0 ) = −1/3

(4.88)

The forward and reverse transmission between any two ports with the third port terminated in Z0 will also be the same, or S12

= = =

S21 = S13 = S31 = S23 = S32 p Gt 2 = 2/3 A + BY0 + CZ0 + D

(4.89)

where the ABCD matrix is that of a shunt admittance Y0 . Thus, the complete scattering matrix is, 

S11  S21 S31

S12 S22 S32

   S13 −1/3 2/3 2/3 S23  =  2/3 −1/3 2/3  S33 2/3 2/3 −1/3

(4.90)

4.4 SIGNAL FLOW GRAPHS AND MASON’S GAIN RULE Signal flow graphs are useful tools with which to describe the operation of complex networks, both graphically and analytically. Mason’s gain rule, introduced by S. J. Mason and applied to describe microwave networks by J. K. Hunton, is now a powerful tool with which to derive network power and voltage gains, especially those with more than two ports or intricate coupling. Indeed, over the years, signal flow graphs and Mason’s gain rule have been used to analyze and design microwave amplifiers, a variety of microwave passive circuits, transmission line circuits with

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

117

nonlinear loads, and even passive and active microstrip patch antennas. They have also been used as an undergraduate teaching tool in microwave courses. Fundamentally, signal flow graphs are composed of nodes and branches, which represent waves at ports and paths between ports, respectively. Each port of a network has two nodes, designated as a and b. Node ai is the wave entering port i and node b is the wave exiting port i. Each branch has an associated S parameter or reflection coefficient that captures the interaction between the nodes, and the branches are directional. A branch is always labeled a directed path from an a-node to a b-node (with the exception of a isolated generator, as described later). For this reason, the a nodes are often referred to as incident waves or independent variables, and the b nodes are referred to as reflected waves or dependent variables. Consider the simple example of a signal flow graph shown in Figure 4.10 of a two-port network. Here the signals entering and exiting port 1 are represented by nodes a1 and b1 , respectively, while the comparable signals for port 2 are represented by nodes a2 and b2 . The branch labeled S21 captures one of the directional interactions between ports 1 and 2, with the incident signal originating at port 1 (node a1 ), while the branch labeled S12 captures the reverse interaction, with the incident signal originating at port 2 (node a2 ). The branches labeled S11 and S22 capture the interactions at the single ports between the incident and reflected signals. To connect this network to a generator and a load, signal flow graphs for those elements are also required. Here, we implement the models for the generator and the load described by Gonzalez. The generator depicted in Figure 4.11(a) has a source voltage, VS , and a source impedance, ZS , with the current and voltage seen by a connected network designated as Ig and Vg , respectively. Figure 4.11(a) also indicates the entering and exiting signals from the generator as ag and bg as shown. Since the generator is fundamentally a one-port device, the a and b nodes may at first seem to be mislabeled, but remember that all a nodes represent signals entering a port and b nodes represent signals exiting a port. To develop the signal flow graph for the generator, one needs to develop an equation in terms of the traveling waves (ag and bg ) related to the circuit properties of the generator. The voltage equation at the terminals of the generator can be expressed as: VS + Ig ZS = Vg

(4.91)

Using the conventions of Section 4.3, the traveling waves represented by ag and bg can be expressed as: Vg i ag = √ Z0

(4.92)

118

MICROWAVE TRANSMISSION LINE CIRCUITS

a1

S21

S11

b2

S22

b1

S12

a2

Figure 4.10 Signal flow graph for a simple two-port network.

ZS

bs

Ig +

ag

1

Vg

VS

−

bg

Γs bg ag

(a) Figure 4.11 Generator model (a) and equivalent signal flow graph (b).

(b)

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

Vg r bg = √ Z0

119

(4.93)

with Vg = Vgi + Vgr . Recognizing that Vgi Vgr − Z0 Z0

(4.94)

b g = b S + ΓS ag

(4.95)

Ig = one can solve for bg : √

with bS = (VS Z0 )/(Z0 + ZS ) and ΓS = (ZS − Z0 )/(ZS + Z0 ). Translating (4.95) into signal flow graph format (keeping the nature of the a and b nodes in mind) results in the graph shown in Figure 4.11(b). The signal flow graph for a load depicted in Figure 4.12(a) can be derived in a similar fashion. The voltage across the load, VL , is given by IL ZL . Again, we express the traveling waves represented by aL and bL in terms of the incident and reflected components of VL : VL i aL = √ Z0 VL r bL = √ Z0

(4.96) (4.97)

with VL = VLi +VLr . Substituting into the Ohm’s law equation for the total voltage and current with the traveling wave components separated, we arrive at the equation that provides the design for the signal flow graph of the load: b L = ΓL aL

(4.98)

with ΓL = (ZL − Z0 )/(ZL + Z0 ) as usual. The signal flow graph of the load indicated by (4.98) is given in Figure 4.12(b). To combine the flow graphs for the generator and load with that of the simple two-port network, we recognize the following equivalencies: b g → a1 ag → b 1

b L → a2 aL → b 2

(4.99) (4.100)

The signal flow graph for the complete circuit (generator, network, load) is shown in Figure 4.13.

120

MICROWAVE TRANSMISSION LINE CIRCUITS

IL

aL

ZL

ΓL

+

aL

VL bL

− bL (a)

(b)

Figure 4.12 Load model (a) and equivalent signal flow graph (b).

bS

1

a1

S21

S11

ΓS

b1

b2

S22

S12

ΓL

a2

Figure 4.13 Signal flow graph for a complete two-port system with generator and load.

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

121

Mason’s gain rule is applied to signal flow graphs to determine transfer functions, that is, ratios of dependent (b nodes) to independent variables (a nodes). The Rule requires the definition and specification of certain traits of the signal flow graph, namely paths (labeled with Pp ) and loops of order n (labeled as L(n)). Paths are routes following the directional arrows that connect the independent and dependent variables, moving from independent to dependent nodes and encountering any node only once. The value of any path is the product of all branch coefficients along that path. A first order loop, indicated by L(1), is the product of the branch coefficients encountered in a round trip from one node back to itself, again following the directionality of all branches. A second order loop, indicated by L(2), is the product of any two non-touching first order loops. Similarly, a third order loop, indicated by L(3), is the product of any three non-touching first order loops, and so on. In the transfer function of Mason’s gain rule, these loops appear in sums, such that ΣL(1) is the sum of all first order loops, ΣL(2) is the sum of all second order loops, etc. Additionally, there are special sums of loops that appear in the numerator of the transfer function, which are defined generally as ΣL(n)(p) , which is the sum of all loops order n that do not touch path Pp . With the paths and loops so defined, the basic format for a transfer function defined using Mason’s gain rule is given as: P1 [1 − ΣL(1)(1) + ΣL(2)(1) − · · ·] + P2 [1 − ΣL(1)(2) + · · ·] + · · · 1 − ΣL(1) + ΣL(2) − ΣL(3) + · · · (4.101) As an example, let us calculate the input reflection coefficient at port 1 of the circuit depicted in Figure 4.13 for the case of an unmatched load that has a load reflection coefficient given by ΓL . We use a slightly modified version of the signal flow graph, shown in Figure 4.14, that does not include the generator in this case. The input reflection coefficient, ΓI N is given by the ratio of b1 (dependent variable) to a1 (independent variable). These two nodes define the possible paths in the graph. Applying the definition of a path, this graph has the following paths: T =

P1 = S11

(4.102)

P2 = S21 ΓL S12

(4.103)

This particular graph has only one first order loop, so that ΣL(1) = S22 ΓL S12

(4.104)

122

MICROWAVE TRANSMISSION LINE CIRCUITS

a1

b2

S11

Γ IN =

b1 a1 b =0

b1

S22

S12

ΓL

a2

s

Figure 4.14 Signal flow graph used to calculate the input reflection coefficient for two-port network with a load.

and ΣL(1)(1) = S22 ΓL

(4.105)

Substituting these values for the paths and loop variables into the general expression for the transfer function (4.101) gives ΓIN

S11 (1 − S22 ΓL ) + S12 ΓLS21 1 − S22 ΓL S12 S21 ΓL = S11 + 1 − S22 ΓL

=

(4.106) (4.107)

Note that if the load is matched, ΓL is zero and the input reflection coefficient simplifies to ΓIN = S11 as expected. The output reflection coefficient can also be calculated in a parallel fashion, referencing the signal flow graph shown in Figure 4.15. The expression for the output reflection coefficient, ΓOU T is given by: Γout = S22 +

S22 S21 Γs . 1 − S11 Γs

(4.108)

While reflection coefficients are important for network analysis, perhaps the more important application of signal flow graphs and Mason’s gain rule come with the determination of power gains. As an example, here we derive the transducer gain of the simple two-port represented by the signal flow graph shown in Figure 4.13. As defined in Section 4.2.2, transducer gain is given as the ratio between the power delivered to a load with the network in place and the maximum available power from the source connected to the network, PL /P0 . To determine power

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

a1

S21

S11

ΓS

b1

123

b2

S22

S12

a2

ΓOUT =

b2 a2

bs = 0

Figure 4.15 Signal flow graph used to calculate the output reflection coefficient for two-port network terminated with an inactive source (bS = 0).

gains, we must remember that the squared magnitudes of the incident and reflected (voltage) waves indicate relative power flow. The maximum power available from the source occurs under the condition when ΓIN = Γ∗S . Under this condition, referencing the signal flow graph of Figure 4.11, the equation that captures this relationship is given by P0

= =

1 1 | b g | 2 − | ag | 2 2 2 1 2 | b | 2 2 1− | ΓS |2

(4.109) (4.110)

with bg = bS + bg ΓS Γ∗S

(4.111)

ag = bg Γ∗S

(4.112)

Likewise, the power delivered to the load is given by PL =

1 1 1 | b2 |2 − | a2 |2 = | b2 |2 (1− | ΓL |2 ) 2 2 2

(4.113)

Combining (4.110) and (4.113), we arrive at an equation that helps us determine what transfer function is necessary to calculate the transducer gain, namely b2 /bS : PL | b2 | 2 Gt = = (1− | ΓL |2 )(1− || Γs |2 ) (4.114) P0 | bs | 2

124

MICROWAVE TRANSMISSION LINE CIRCUITS

Using the same techniques outlined earlier for the determination of paths and loops applied to the signal flow graph in Figure 4.13, we can arrive at the expression for b2 /bS : b2 bs

= =

S21 (4.115) 1 − (S11 Γs + S22 ΓL + S21 ΓLS12 Γs ) + S11 Γs S22 ΓL S21 (4.116) (1 − S11 Γs )(1 − S22 ΓL)S21 S12 ΓL Γs

Substituting this transfer function into (4.114) gives the expression for the network’s transducer gain: Gt =

| S21 |2 (1− | ΓS |2 )(1 − (| ΓL |2 )) | (1 − S11 ΓS )(1 − S22 ΓL ) − S21 S12 ΓL ΓS |2

(4.117)

Note that this expression is the same given by (4.78) and that other power gains of interest can be calculated using a parallel procedure. For straightforward signal flow graphs, the method described above to attain transfer functions is perfectly reasonable, but one can easily imagine more complicated graphs that would make this procedure somewhat tedious. Engineers have several options to simplify this process. First, Pozar provides decomposition rules for signal flow graphs that can be helpful in simplifying the representation of complex circuits before Mason’s gain rule is applied. Additionally, Walton developed a MATLABTM script that helps to apply Mason’s gain rule for extremely complicated signal flow graphs. PROBLEMS 1. Referring to Figure 4.16, determine (a) Z0 for maximum power extraction from the source (ZS = 40 Ω), and (b) the transducer gain Gt . 2. For the network in Figure 4.17, determine Z0 as a function of ZS and ZL to make Z1 = ZS at θ0 = 45◦ . Assume that ZS and ZL are real but unequal. 3. In Figure 4.18, ZS = ZL = Z0 = 50 Ω and θ = π2 ff0 . (a) At f = f0 , find R1 as a function of R2 to make Γ = 0. (b) Using the result in (a), find the transducer gain at f = 0, f = f0 , and f = 2f0 , if R2 = 50 Ω. 4. By equating ABCD matrices, determine L and C as functions of M , f0 , and Z0 to make the discrete-element network in Figure 4.19(a) equivalent to the distributed-element network (quarter wavelength line) in Figure 4.19(b), at f = f0 , where M is a positive number between zero and infinity.

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

ZS = 40 Ω VS

125

λ/4 3 × 10−4 36

Z0

ZL = 80 kΩ

5 × 10−7 0.02

ABCD MATRIX OF LINEAR AMPLIFIER

Figure 4.16 Determine Z0 for maximum power extraction from the source. Determine the transducer gain Gt .

ZS

θ

VS

ZL 0

Z

θ

0

θ

Z

Z0

Figure 4.17 Determine Z0 as a function of ZS and ZL to make Z1 = ZS at θ0 = 45◦ . Assume that ZS and ZL are real but unequal.

5. By equating ABCD matrices, determine L and C, as functions of M , Z0 , and f0 , to make the two networks in Figure 4.20 equivalent at f = f0 , where M is a positive number between zero and infinity. 6. In Figure 4.21, (a) determine Z01 and Z02 to make the two networks equivalent at f = f0 . (b) Determine Zin at f = f0 for the√ transmission line network. (c) Determine the transmission coefficient (t = Gt ) at f = f0 for the transmission line network. 7. A simple equivalent circuit for an amplifier is shown in Figure 4.22. (a) Express the ABCD matrix of the amplifier in terms of the equivalent circuit parameters. (b) Express the transducer gain Gt in terms of the network

126

MICROWAVE TRANSMISSION LINE CIRCUITS

ZS = 50 Ω

R1

R2

θ

VS

Z0 = 50 Ω

ZL = 50 Ω

Figure 4.18 Use the parameters given in Problmem 3 to find R1 as a function of R2 to make Γ = 0 at f = f0 . Find the transducer gain.

θ=

M Z0 VS

π f 2 f0

M Z0

Z0

(a) M Z0 VS

L C

C

Z0

(b) Figure 4.19 Determine component values to make these two networks equivalent by equating ABCD matrices.

parameters. (c) If RS = R1 and RL = R2 , express Gt in terms of the network parameters. 8. The ABCD parameters of an amplifier are A = −0.025, B = −12.5, C = −0.005, and D = −2.5. (a) Determine the corresponding S parameters with reference to Z0 = 50 Ω. (b) Determine the transducer gain Gt if RS = RL = 50 Ω. (c) Determine the transducer gain Gt if RS = 5 Ω and RL = 500 Ω.

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

127

λ0/4

M Z0 VS

Z0

M Z0

(a) M Z0

L

VS

L Z0

C

(b) Figure 4.20 Determine component values to make these two networks equivalent by equating ABCD matrices.

RS = 4 Z0 VS

L C

C

RS = 4 Z0 VS

l RL = Z0 01

Z

l

01

Z

Z02

l

RL = Z0

Figure 4.21 Determine Z01 and Z02 to make these two networks equivalent as described in Problem 6. Determine Zin at f = f0 for the transmission line network. Determine the transmission coefficient √ (t = Gt ) at f = f0 for the transmission line network.

128

MICROWAVE TRANSMISSION LINE CIRCUITS

I1

RS VS

V1

I2 R1

gmV1

A

B

C

D

Figure 4.22 A simple equivalent circuit for an amplifier.

R2 V2

RL

MATRIX REPRESENTATION OF MICROWAVE NETWORKS

129

BIBLIOGRAPHY I. J. Bahl and P. Bhartia, Microwave Solid State Circuit Design, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2003. R. Clark, G. H. Huff, and J. T. Bernhard, “An integrated active microstrip reflectarray element with an internal amplifier,” IEEE Transactions on Antennas and Propagation, vol. 51, no. 5, pp. 993–999, May 2003. R. M. Fano and A. W. Lawson, “The Theory of Microwave Filters,” in Microwave Transmission Circuits, vol. 9, G. L. Ragan, Ed. New York: McGraw-Hill, 1948. V. F. Fusco, Microwave Circuits: Analysis and Computer-Aided Design. Upper Saddle River, NJ: Prentice-Hall, 1987. G. Gonzalez, Microwave Transistor Amplifiers, Analysis and Design, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. J. K. Hunton, “Analysis of microwave measurement techniques by means of signal flow graphs,” I.R.E. Trans. on Microwave Theory and Techniques, vol. MTT-8 , pp. 206–212, Mar. 1960. S. J. Mason, “Feedback Theory — Some properties of signal flow graphs,” Proceedings of the I.R.E., vol. 41, pp. 1144–1156, Sep. 1953. S. J. Mason, “Feedback Theory — Further properties of signal flow graphs,” Proceedings of the I.R.E., vol. 44, pp. 920–926, Jul. 1956. D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley and Sons, 2011. P. A. Rizzi, Microwave Engineering: Passive Circuits. Upper Saddle River, NJ: Prentice-Hall, 1988. J. E. Ruyle and J. T. Bernhard, “Signal flow graph for a probe-fed microstrip patch antenna,” IEEE Antennas and Wireless Propagation Letters, vol. 8, pp. 935–938, 2009.

130

MICROWAVE TRANSMISSION LINE CIRCUITS

J. E. Schutt-Aine and R. Mittra, “Scattering parameter transient analysis of transmission line loaded with nonlinear terminations,” IEEE Transactions on Microwave Theory and Techniques, vol. 36, pp. 529–536, Mar. 1988. G. D. Vendelin, A. M. Pavio, and U. L. Rohde, Microwave Circuit Design, Using Linear and Nonlinear Techniques, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2005. R. Walton. “Mason.m.” http://www.mathworks.com S. W. Wedge, “Wave computations for microwave education,” IEEE Transactions on Education, vol. 36, pp. 127–131, Feb. 1993. E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications. New York: Wiley-Interscience, 1988.

Chapter 5 SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS 5.1 NETWORK SYNTHESIS AND DESIGN Consider the linear, lossless, passive network connected between a source of resistance R ohms and a normalized load impedance of 1 Ω, as shown in Figure 5.1. For a specified ratio of power delivered to the load (PL) to maximum available source power (P0 ), and taking s = σ + jω as the complex frequency, the process for determining Z1 (s), the input impedance, for the lossless case (σ = 0 or s = jω) is as follows PL = |T (s)|2 = T (s)T (−s) ≤ 1 (5.1) P0 where T is the transmission coefficient. Since the power incident upon a lossless network must either be transmitted or reflected, the squared magnitude of the reflection coefficient, Γ, must be |Γ|2 = 1 − |T |2

(5.2)

Γ(s)Γ(−s) = 1 − T (s)T (−s)

(5.3)

or To satisfy the limiting condition in (5.1), Γ(s) is formed such that the poles of Γ(s) occur in the left-half of the s-plane, otherwise Γ and thus T (s) would approach infinity. It is convenient but not necessary to choose the zeros of Γ(s) to lie in the left-half of the s-plane, as zeros do not cause the same mathematical complications

131

132

MICROWAVE TRANSMISSION LINE CIRCUITS

I

N

R

1

Ζ1 Figure 5.1 Linear, lossless, passive network terminated at both ends.

as poles. It is easily shown that Γ(s) is also given by Z1 (s) − R Z1 (s) + R

(5.4)

Z1 (s) 1 + Γ(s) = R 1 − Γ(s)

(5.5)

Γ(s) = which is solved for Z1 (s) as

and the elements of the network required to produce the specified response are determined from Z1 (s). 5.1.1 Lowpass Filters Now suppose that the network of Figure 5.1 is used to approximate the ideal or lossless (σ = 0) lowpass filter characteristic in Figure 5.2(a), where the squared magnitude of the normalized transmission coefficient T (s)/T0 is plotted versus normalized real frequency ω/ω0 . If we specify the approximating function to be T (s)T (−s)|s=jω =

T02 1 + Fn (ω2 )

(5.6)

where Fn (ω2 ) is a rational function of ω2 (that is, a finite number of poles where Fn (ω2 ) → ∞ and no other singularities), it is necessary that Fn (ω2 )  1, 0 ≤ Fn (ω2 )  1,

ω ω0
ω0

1

1

(5.7) (5.8)

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

T T0

2

T T0

1.0

133

2

1.0

ε

CUTOFF 0.5

0.5 PASSBAND

0

0.5

STOPBAND

1.0

1.5

(a) Figure 5.2 istics.

ω ω0

0

0.5

1.0

1.5

(b)

ω ω0

(a) Ideal lowpass filter transmission, and (b) approximations to the ideal lowpass character-

The scale factor T02 is T02 = |T (jω)|2ω=0 =

4R (R + 1)2

(5.9)

Expressing |T |2 as the ratio of two polynomials, T (s)T (−s) = T02

Nm (s)Nm (−s) Dn (s)Dn (−s)

(5.10)

or T (s)T (−s)|s=jω = T02

1 + a1 ω2 + a2 ω4 + ... + am ω2m 1 + b1 ω2 + b2 ω4 + ... + bn ω2n

(5.11)

and m < n, since |T |2 = 0 at ω = ∞. The passband flatness can be approximated by setting n − 1 derivatives of (5.11) with respect to ω2 equal to zero. This makes the denominator of (5.11) equal to the numerator plus the single term bn ω2n and results in a maximally flat magnitude (MFM) approximation to the ideal lowpass filter characteristic which can be written from (5.11) as T (s)T (−s)|s=jω =

T02 1+

ω 2n

bn 1+a1 ω 2 +...+am ω 2m

=

T02 1 + Fn (ω2 )

(5.12)

134

MICROWAVE TRANSMISSION LINE CIRCUITS

If in addition to maximal flatness we require that all the zeros of |T |2 lie at ω = ∞ and that |T |2 = (1/2)T02 at ω/ω0 = 1, then |T |2 becomes |T |2 =

T02 T02 = 2n 1+ω Bn (s)Bn (−s)

(5.13)

T0 Bn (s)

(5.14)

or T (s) =

where Bn (s) are Butterworth polynomials of degree n as given in Table 5.1. Note from (5.13) that the Butterworth response is down by 1/2 (3 dB) at ω/ω0 = 1. Another approximation to the ideal lowpass filter which has all zeros of transmission at infinity but distributes the passband error in an oscillatory manner and has a stopband response which decreases monotonically at a faster rate than a Butterworth filter of the same order is the equal-ripple or Chebyshev filter. For the Chebyshev filter T (s)T (−s)|s=jω =

T02 T02 = 1 + 2 Tn2 (ω) Cn (s/)Cn (−s/)

(5.15)

where  < 1 is a real constant which determines the ripple amplitude, and Tn (ω) is an nth -degree Chebyshev polynomial defined by Tn (ω) = cos (n cos−1 ω), Tn (ω) = cosh (n cosh

−1

ω),

|ω| ≤ 1

|ω| > 1

(5.16) (5.17)

From (5.15) the transmission coefficient is T (s) =

T0 Cn (s/)

(5.18)

where, for a specified  (ripple), Cn (s/) is an nth -degree polynomial for Chebyshev filters. More extensive tables are available in Rizzi, but the first five polynomials for Chebyshev filters for a ripple of 0.5 dB ( = 0.349) and 1 dB ( = 0.509) are given in Table 5.1. In some applications it is important to know that in the limiting case of  = 0, the Chebyshev filter becomes a Butterworth filter. If the ideal lowpass characteristic in Figure 5.2(a) is relabeled as time delay Td instead of |T |2 , the requirements for a lowpass filter approximating the ideal time delay response are determined by considering the transmission coefficient expressed

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

135

Table 5.1 Polynomials for Butterworth, Chebyshev, and Bessel Filters

Butterworth Polynomials n 1 2 3 4 5

s + 1√ s2 + 2s + 1 s3 + 2s2 + 2s + 1 = (s + 1)(s2 + s + 1) s4 + 2.613s3 + 3.414s2 + 2.613s + 1 = (s2 + 0.765s + 1)(s2 + 1.848s + 1) s5 + 3.236s4 + 5.236s3 + 5.236s2 + 3.236s + 1 = (s + 1)(s2 + 0.618s + 1)(s2 + 1.618s + 1)

Chebyshev Polynomials for Filters with 0.5 dB Ripple n 1 2 3 4 5

s + 2.863 s2 + 1.425s + 1.516 s3 + 1.253s2 + 1.535s + 0.716 = (s + 0.626)(s2 + 0.626s + 1.142) s4 + 1.197s3 + 1.717s2 + 1.025s + 0.379 = (s2 + 0.351s + 1.064)(s2 + 0.845s + 0.356) s5 + 1.172s4 + 1.937s3 + 1.309s2 + 0.753s + 0.179 = (s + 0.362)(s2 + 0.224s + 1.036)(s2 + 0.586s + 0.477)

Chebyshev Polynomials for Filters with 1 dB Ripple n 1 2 3 4 5

s + 1.965 s2 + 1.098s + 1.103 s3 + 0.988s2 + 1.238s + 0.491 = (s + 0.494)(s2 + 0.494s + 0.994) s4 + 0.953s3 + 1.454s2 + 0.743s + 0.276 = (s2 + 0.279s + 0.987)(s2 + 0.674s + 0.279) s5 + 0.937s4 + 1.689s3 + 0.974s2 + 0.581s + 0.123 = (s + 0.289)(s2 + 0.179s + 0.988)(s2 + 0.468s + 0.429)

Bessel Polynomials n 1 2 3 4 5

s+1 s2 + 3s + 3 s3 + 6s2 + 15s + 15 = (s + 2.322)(s2 + 3.678s + 6.460) s4 + 10s3 + 45s2 + 105s + 105 = (s2 + 5.792s + 9.140)(s2 + 4.208s + 11.488) s5 + 15s4 + 105s3 + 420s2 + 945s + 945 = (s + 3.647)(s2 + 6.704s + 14.272)(s2 + 4.649s + 18.156)

136

MICROWAVE TRANSMISSION LINE CIRCUITS

as T (s)|s=jω = T0

a(ω2 ) + jωb(ω2 ) c(ω2 ) + jωd(ω2 )

(5.19)

where a, b, c, and d are rational functions of ω2 . The phase of the transmission coefficient in (5.19) is φ(ω) = tan−1

ωb ωd − tan−1 a c

(5.20)

and the delay Td (also called the group delay) is defined as Td = −

dφ(ω) dω

(5.21)

Expressing Td as a ratio of polynomials Td (ω2 ) = T0

1 + a1 ω2 + a2 ω4 + ... + am ω2m 1 + b1 ω2 + b2 ω4 + ... + bn ω2n

(5.22)

where m < n, a maximally flat delay (MFD) or constant phase response is obtained by setting n − 1 derivatives of Td with respect to ω2 equal to zero. The resulting Td is 1 Td (ω2 ) = T0 (5.23) 2n 1 + 1+a1 ω2 +ab2nωω4 +...+am ω2m Requiring the transmission coefficient which produces the maximally flat delay to have all its zeros at infinity generates a special class of MFD filters called Thomson filters. The transmission coefficient of a Thomson filter is T (s) =

T0 Pn (s)

(5.24)

where Pn (s) are nth -degree Bessel polynomials as given in Table 5.1. Thus, we see from the foregoing development that the transmission coefficient of a Butterworth, Chebyshev, or Thomson filter is expressed as T (s) =

T0 Dn (s)

(5.25)

and we simply choose the appropriate polynomial for Dn (s) from Table 5.1, where n will be the number of reactive elements (Ls and Cs) in the filter. For a given n, the

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

137

Butterworth filter has the flattest possible magnitude of transmission in the passband which is important for analog signal quality, the Chebyshev filter has the fastest possible rate of cutoff in the stopband which reduces potential interference with out-of-band signals, and the Thomson filter has the flattest possible time delay in the passband which minimizes distortion in digital signals. In practice one chooses the filter type according to the factor most important in a particular design application. An example will illustrate how the foregoing procedure is used to synthesize a network: Example 5.1: Determine the number of elements n, and the normalized element values for a Butterworth filter where R = RS /RL = 1, and the magnitude response at ω = 1.5 is to be down at least 10 dB from the magnitude at ω = 0. Solution: From (5.13) at ω = 1.5 10 log

T02 = 10 log [1 + ω2n ] |T (jω)|2

(5.26)

or 10 dB = 10 log [1 + (1.5)2n] and n=

log 9 = 2.71 2 log (1.5)

(5.27)

(5.28)

Hence a Butterworth filter with n = 3 elements is required. Now with n = 3 and T0 = 1 (since R = 1), in (5.13) PL 1 = |T |2 = P0 1 + ω6 and |Γ|2 = 1 − |T |2 =

ω6 1 + ω6

(5.29)

(5.30)

or Γ(s)Γ(−s) =

−s6 (s3 )(−s3 ) = 3 6 2 1−s (s + 2s + 2s + 1)(−s3 + 2s2 − 2s + 1) (5.31)

138

MICROWAVE TRANSMISSION LINE CIRCUITS

Taking s3 s3 + 2s2 + 2s + 1

(5.32)

1 + Γ(s) 2s3 + 2s2 + 2s + 1 = 1 − Γ(s) 2s2 + 2s + 1

(5.33)

Γ(s) = with R = 1 in (5.5) yields Z1 (s) =

or, dividing numerator by denominator Z1 (s) = s +

1 1 2s + s+1

(5.34)

which is recognized as the input impedance of the network in Figure 5.3(a). Had we chosen Γ(s) =

s3

−s3 + 2s2 + 2s + 1

(5.35)

the equally valid network in Figure 5.3(b), which is the dual of the one in Figure 5.3(a), would have been obtained. The synthesis procedure illustrated, which determines the form and element values of the network, becomes more complicated if R 6= 1. An often simpler method of finding the element values, once n is determined, is to compare the transmission coefficient of the desired network with the polynomial specified in (5.25). In terms of the ABCD matrix, the transmission coefficient of the network in Figure 5.1 is (with RS = R, RL = 1) √ √ 2 RS RL 2 R T = = ARL + B + CRS RL + DRS A + B + CR + DR

(5.36)

which must be the same as (5.25) for Butterworth, Chebyshev or Thomson filters. Example 5.2: For n = 3, take the desired lowpass form to be as shown in Figure 5.4 (an extension of Figure 5.3(b)) and determine the normalized element values.

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

1 I

139

1 2

1

1

(a) 2 I

1

1

1

1

(b) Figure 5.3 (a) Realization of third-degree Butterworth filter for R = RS /RL = 1, and (b) the dual, equally valid network.

Solution: The ABCD matrix of the network with n = 3 is       A B 1 0 1 sL2 1 0 = C D sC1 1 0 1 sC3 1   2 L2 C3 s + 1 L2 s = C1 L2 C3 s3 + (C1 + C3 )s C1 L2 s2 + 1 (5.37) and substituting into (5.36) √ 2 R/(R + 1) T = RC1 L2 C3 2 +L2 C3 3 +L2 ( R+1 )s3 + ( RC1 LR+1 )s2 + ( RC1 +RC )s + 1 R+1 (5.38) √ From (5.25), since T0 = 2 R/(R + 1), T =

√ 2 R/(R + 1) Dn (s)

(5.39)

and Dn (s), which can be any of the third-degree polynomials in Table 5.1, must equal the denominator of (5.25). If we again select

140

MICROWAVE TRANSMISSION LINE CIRCUITS

L2 I

R

C1

Ln C3

1 n EVEN

Cn

1

n ODD

Figure 5.4 General form of lowpass filter with n even or odd.

a Butterworth response with R = 1, then from Table 5.1 and (5.25) D3 (s)

= =

s3 + 2s2 + 2s + 1 C1 L2 C3 3 C1 L2 + L2 C3 2 C1 + L2 + C3 s +( )s + ( )s + 1 2 2 2 (5.40)

and component values C1 = 1 F L2 = 2 H

(5.41) (5.42)

C3 = 1 F

(5.43)

as in the previous example. For most applications the preceding calculations are not necessary because the element values for many types of lowpass filters have been extensively tabulated in the literature. Such a tabulation is included here for Butterworth, Chebyshev and Bessel (Thomson) filters in Appendix A. The element values in Appendix A have of course been normalized with respect to the load resistance RL and the filter cutoff frequency ωc. Letting the subscript a denote the actual value, the actual reactance of a series branch is Xa = ωa La (5.44) Normalizing with respect to RL , multiplying by ωc /ωc, and rearranging terms yields Xa ωaLa ωc ωa ωc La = = ( ) = ωL (5.45) RL R L ωc ωc R L

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

141

Hence, the actual inductance for any ωc and RL is La =

RL L ωc

(5.46)

where L is a normalized inductance from the tables in Appendix A. By a similar process of scaling, the actual capacitance is Ca =

C R L ωc

(5.47)

where C is a normalized capacitance from the tables in Appendix A. 5.1.2 Bandpass Filters Bandpass filter specifications typically include the center frequency ω0 and the bandwidth QT defined as ω0 (5.48) QT = ω2 − ω1 where ω1 and ω2 are the lower and upper cutoff frequencies, respectively. The frequency transformation ω ←→ Ω (5.49) where Ω=

ω0 ω ω0 ω ω0 ( − ) = QT ( − ) ω2 − ω1 ω0 ω ω0 ω

(5.50)

transforms the elements of a lowpass filter into the corresponding elements of a bandpass filter, as shown in Figure 5.5. The transformation shifts a lowpass response (including negative frequencies) upward along the frequency scale where it becomes an identical bandpass response. For example, a lowpass Butterworth response P0 1 + ω2n (R + 1)2 |LP = = (1 + ω2n ) PL T02 4R

(5.51)

becomes a bandpass Butterworth response   P0 1 + Ω2n (R + 1)2 ω0 2n 2n ω |BP = = 1 + Q ( − ) T PL T02 4R ω0 ω

(5.52)

142

MICROWAVE TRANSMISSION LINE CIRCUITS

L'2 C2'

L2 C1

C3

ω

LOWPASS

ω

1 L2

1/ω

1 C1

1 C3

Ω

L1'

C1'

L'3

C3'

BANDPASS L'2

ω

Ω

HIGHPASS

L1' C1'

C2'

L'3 C3'

BANDSTOP

Figure 5.5 Frequency transformations that convert a lowpass prototype filter to a highpass, bandpass, or bandstop filter with the same frequency response.

As can be determined from (5.49), the frequency points (−1, 0, 1) in the lowpass response become (ω1 , ω0 , ω2 ) in the bandpass response, and ω0 =

√

ω1 ω2

(5.53)

where ω0 is the resonant midband frequency, and ω2 and ω1 are the upper and lower cutoff frequencies (3 dB points for Butterworth filters and top-of-the-ripple points for Chebyshev filters). Applying (5.49) to the lowpass filter elements of Figure 5.4, the susceptance of the k th shunt capacitance becomes (with k odd) ωCk ←→ ΩCk =

QT ω0 QT C k 1 ωCk − = ωCk0 − ω0 ω ωL0k

(5.54)

and the reactance of the k th series inductance becomes (with k even) ωLk ←→ ΩLk =

ω0 QT Lk 1 QT ωLk − = ωL0k − ω0 ω ωCk0

(5.55)

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

143

Hence, the transformation replaces each Ck in the lowpass filter by QT Ck ω0

(5.56)

1 ω0 QT C k

(5.57)

Ck0 = and L0k =

in parallel, and each Lk in the lowpass filter by QT Lk ω0

(5.58)

1 ω0 QT Lk

(5.59)

L0k = and Ck0 =

in series. Figure 5.6 shows the resulting bandpass filter transformed from the lowpass prototype in Figure 5.4. For a given bandpass filter design application, ω0 and QT are specified, then a lowpass filter prototype with the desired response and number of elements is selected, and the bandpass filter elements are computed using (5.56) through (5.59). Each section of the bandpass filter has the same resonant or midband frequency but, in general, a different Q or selectivity. The Qs of the individual sections and how they relate to the total Q of the filter determine the response of a bandpass filter. The individual Q of a bandpass section is computed as if it were the only section connected between source and load. Thus, the Q of the k th parallel LC section in Figure 5.6 is Qk =

ω0 Ck0 Ck = QT G+1 G+1

k odd

(5.60)

where G = 1/R, and the Q of the k th series LC section is Qk =

ω0 L0k Lk = QT R+1 R+1

k even

(5.61)

Thus, the bandpass filter may be designed using (5.56)-(5.59) or (5.60) and (5.61). For filters at microwave frequencies, (5.60) and (5.61) often are more convenient to use because they are simpler to calculate and thus speed up the design process.

144

MICROWAVE TRANSMISSION LINE CIRCUITS

L'2 C2'

R L1'

I

C1'

L'3

L'n Cn' C3'

Cn'

1 L'n

n EVEN

1

n ODD

Figure 5.6 Bandpass filter obtained from the lowpass filter of Figure 5.4.

5.1.3 Highpass and Bandstop Filters A lowpass prototype filter that has a desired response may also be used to design a highpass filter or a bandstop filter with the same response. To transform network elements and response functions from lowpass to highpass, replace the lowpass ω by 1/ω. This converts each lowpass C to a highpass L, and each lowpass L to a highpass C, as illustrated in Figure 5.5. As also illustrated in Figure 5.5, a bandstop filter is obtained by replacing the highpass ω by Ω. Table 5.2 summarizes the network transformations illustrated in Figure 5.5 and gives scaled element values in accordance with (5.46) and (5.47). A number of examples will illustrate how the preceding transformations, tables, and design equations may be used. Example 5.3: Determine the element values of a lowpass filter with fc = 0.75 GHz, RS = 50 Ω, RL = 500 Ω, that has a response at 1.5 GHz that is at least 30 dB below the DC level. Solution: Testing an n = 5 Butterworth response in (5.13), yields 2   t = T = −10 log 1 + ( f )2n T0 fc

(5.62)

dB

or

  1.5 10 T = −10 log 1 + ( ) = −30.11 dB 0.75

(5.63)

From Appendix A, with R = RS /RL = 0.1 and n = 5, we read C1 = 3.1522, L2 = 0.0912, C3 = 14.0945, L4 = 0.1727, and

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

Table 5.2 Summary of Network Transformations with Expressions for Finding the Element Values kth element

Network

k odd

k even

L2 Cka

C1

C3

Ck

ωc RL RL Lk

Lka

ωc

LOWPASS PROTOTYPE

L'2 C2' L1'

L'3

C1'

Cka'

QT Ck ω0 RL

1 ω0QT RL Lk

L'ka

RL ω0QT Ck

QT RL Lk

C3'

BANDPASS FILTER

ω0

C2' = 1 L2 1

Cka' L1' =

1 C1

L'3 =

1 C3

L'ka

ωc RL Lk RL

ωc C k

HIGHPASS FILTER

L'2 ' 1

L

' 1

C2'

C

BANDSTOP FILTER

' 3

Cka'

L

C3'

L'ka

Ck

QT

ω0QT RL

ω0 RL Lk

QT RL ω0 C k

RL Lk ω0QT

145

146

MICROWAVE TRANSMISSION LINE CIRCUITS

C5 = 15.7103. Thus, from (5.46) and (5.47), or Table 5.2: C1a =

C1 RL ωc

L2a =

RL ωc L2

C3a =

C3 RL ωc

L4a =

RL L ωc 4

C5a =

C5 RL ωc

= 1.34 pF = 9.68 nH = 5.98 pF = 18.32 nH = 6.67 pF

Example 5.4: Use a 0.5 dB ripple Chebyshev filter with n = 4 to meet the specifications in the previous example. Solution: We have from (5.15) and (5.17):

or

2   t = T = −10 log 1 + 2 cosh2 (n cosh−1 f ) T0 fc dB 

2

2

T = −10 log 1 + (0.349) cosh (4 cosh

−1

(5.64)

 1.5 ) = −30.6 dB 0.75 (5.65)

From Appendix A with 1/R = RL/RS = 10, n = 4, and 0.5 dB ripple, we read L1 = 0.0975, C2 = 15.352, L3 = 0.194, C4 = 14.2616. Thus, from (5.46) and (5.47): L1a =

RL L1 ωc

= 10.34 nH

C2a =

C2 RL ωc

= 6.52 pF

L3a =

RL L3 ωc

= 20.58 nH

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

C4a =

C4 RL ωc

= 6.05 pF

Example 5.5: Determine the element values of an n = 3 (a) lowpass filter and (b) highpass filter that have a Butterworth response for R = RS /RL = 1, RL = 50 Ω, and fc = 1 GHz. Solution: From Appendix A, the lowpass prototype elements are: C1 = 1, L2 = 2, and C3 = 1. (a) Thus, the actual element values of the lowpass filter are: C1a =

C1 ωc RL

= 3.183 pF = C3a

RL L2 ωc

= 15.92 nH

and L2a =

(b) The element values of the highpass filter are: L1a =

RL ωc C 1

= 7.96 nH = L3a

and C2a =

1 ωc RL L2

= 1.592 pF

Example 5.6: Determine the element values of a three-section bandpass filter that has a Butterworth response for R = RS /RL = 1, RL = 50 Ω, f0 = 1 GHz, and QT = π/2. Solution: From Appendix A, the lowpass prototype elements are: C1 = 1, L2 = 2, and C3 = 1. Thus, from (5.46), (5.47), and (5.56)-(5.59), or Table 5.2: L01a =

RL ω0 QT C1

= 5.066 nH = L03a

147

148

MICROWAVE TRANSMISSION LINE CIRCUITS

0 C1a =

QT C1 ω0RL

L02a =

QT RL L2 ω0

0 C2a =

1 ω0 QT RL L2

0 = 5 pF = C3a

= 25 nH = 1.013 pF

Example 5.7: Determine the element values of a three-section bandstop filter that has a Butterworth response for R = RS /RL = 1, RL = 50 Ω, f0 = 1 GHz, and QT = π/2. Solution: From Appendix A, the lowpass prototype elements are: C1 = 1, L2 = 2, and C3 = 1. Thus, from Figure 5.6, and scaling with respect to RL as in (5.46) and (5.47): L01a =

QT RL ω0 C 1

= 12.5 nH = L03a

0 C1a =

1 ω02 L01a

=

C1 ω0 QT RL

L02a =

1 0 ω02 C2a

=

RL L2 ω0 QT

0 C2a =

QT ω0 RL L2

0 = 2.03 pF = C3a

= 10.13 nH

= 2.5 pF

The design of a bandpass filter using transmission line sections in place of the lumped Ls and Cs follows the procedure outlined in Figure 5.7. The lowpass prototype elements are transformed to bandpass sections, and for ease of fabrication, series LC sections may be converted to shunt LC sections with a λ0 /4 section of line on each side. Each shunt LC section is replaced by a shunt λ0 /4 shorted stub of characteristic impedance, Z0 4 = QT C k , Z0k π

k odd

(5.66)

Z0 4 = QT Lk , Z0k π

k even

(5.67)

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

149

L2 I

C1

R

C3

1

LOWPASS PROTOTYPE

ω⇔Ω

L'2 C2'

I

R L1'

L'3

C3'

QT

   k odd 1 1  L'k = 2 ' = ω0 Ck ω0QT Lk  Ck' =

1

ω0

Ck

L'k Ck' (k even) L''k Ck'' + λ 4 SECTIONS

REPLACE SERIES WITH PARALLEL

λ0/4

' 1

'' 2

C

R L1'

C

L''2

z0 = 1

L'3

ZS

l

02

z

1

λ0/4

λ0/4

Z0

REPLACE PARALLEL LCs WITH

λ0/4 SHORTED SECTIONS 1

1

l

01

1

l I

C

l

z

R

z

I

' 3

z0 = 1 l

 L'k Q → Ck'' = L'k = T Lk  ω0 z02   k even 1  '' ' 2 '' ' Lk = Ck z0 → Lk = Ck = ω0QT Lk  Ck'' =

03

I

λ0/4

 ' y0 k l y0 k QT  Ck = 2v = 8 f = ω Ck (k odd) 0 0   4 y0 k = QT Ck (k odd)  π   y l '' C = 0 k = y0 k = QT L (k even) k  k 2v 8 f 0 ω0  4  y0 k = QT Lk (k even)  π

Z0=RL

Z0 03

Z

/4

USUALLY ZS

0

λ

02

Z

/4 0

λ

01 0

λ

/4

Z

SCALE IMPEDANCES

Figure 5.7 Evolution of bandpass filters from the lowpass prototype.

= RL = Z0

150

MICROWAVE TRANSMISSION LINE CIRCUITS

where Ck and Lk are the lowpass prototype elements tabulated in Appendix A, and the characteristic impedance of the main line is assumed to be Z0 = RL. The design of a bandstop filter using transmission line sections in place of the lumped Ls and Cs proceeds in a similar fashion as outlined in Figure 5.8. The lowpass prototype elements are transformed to highpass elements and then to bandstop sections. Again, for ease of fabrication in a transmission line format, series connected sections are converted to shunt connected sections with a λ0 /4 length of line on each side. Each shunt LC section is replaced by a shunt λ/4 open stub of characteristic impedance, Z0k 4 1 = QT , Z0 π Ck

k odd

(5.68)

Z0k 4 1 = QT , k even (5.69) Z0 π Lk where again, Ck and Lk are the lowpass prototype elements tabulated in Appendix A, and the characteristic impedance of the main line is assumed to be Z0 = RL. Example 5.8: For R = 1, n = 3, Z0 = 50 Ω, and QT = f0 /(f2 −f1 ) = π/2, design a bandpass filter with three shorted stubs as depicted in Figure 5.7 for (a) a Butterworth response, and (b) a 0.1 dB-ripple Chebyshev response. (c) Repeat the designs in (a) and (b) for a bandstop filter with three open stubs as depicted in Figure 5.8. Solution: (a) From Appendix A, with n = 3 and R = 1, C1 = 1, L2 = 2, and C3 = 1 for a Butterworth response. Hence, from (5.66) and (5.67), the stub characteristic impedances are: Z01 = Z0 /( π4 QT C1 ) =

Z0 2C1

= 25 Ω = Z03

Z0 2L2

= 12.5 Ω

and Z02 = Z0 /( π4 QT L2 ) =

(b) From Appendix A, with n = 3 and R = 1, C1 = 1.4328, L2 = 1.5937, and C3 = 1.4328 for a 0.1 dB-ripple Chebyshev response. Hence, from (5.66) and (5.67), the stub characteristic impedances are:

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

151

L2 I

C1

R

C3

1

ω↔

C2h

I

h 1

R

h 3

L

1

ω

1

L

ω

L'2

ω ↔ Ω = QT 

 ω0

−

ω0  ω 

 h 1  Lk = C k  Ckh = 1  Lk

k odd k even

QT 1  ω0 Ck   k odd C 1 ' = k  Ck = h ω0QT Lk ω0QT  L'k =

R

L1'

C

C1'

I

R

L'3

' 2

1

C3'

' 3 ' 3

L

'' 2

L

L

C

C2''

C

' 1 ' 1

λ0/4

λ0/4

z0 = 1

z0 = 1

I

ZS

1

l

02

l

z

1

l

01

1

z

R

z

I

λ0/4

λ0/4

Z0

1

l

l

03

Z

/4

USUALLY ZS

0

λ

02

Z

/4 0

λ

01

Z

/4 0

Lhk =

 ' z0 k l z0 k QT 1  Lk = 2v = 8 f = ω C (k odd) 0 0 k   4 1 z0 k = QT (k odd)  π Ck    L'' = z0 k l = z0 k = QT 1 (k even)  k 2v 8 f 0 ω0 Lk  4 1  z0 k = QT (k even)  π Lk 

SCALE IMPEDANCES

λ

ω0

L'k L  1 → Ck'' = L'k = = k ω0QT Ckh ω0QT  z02  k even Q Q L''k = Ck' z02 → L''k = Ck' = T Ckh = T  ω0 ω0 Lk 

Z0=RL

Z0

QT

Ck'' =

03

I

Figure 5.8 Evolution of bandstop filters from the lowpass prototype.

= RL = Z0

152

MICROWAVE TRANSMISSION LINE CIRCUITS

Z01 = Z0 /( π4 QT C1 ) =

Z0 2C1

= 17.45 Ω = Z03

Z0 2L2

= 15.69 Ω

and Z02 = Z0 /( π4 QT L2 ) =

(c) From (5.68) and (5.69), the open stub characteristic impedances for a maximally-flat bandstop filter are: Z01 = Z0 ( π4 QT C11 ) =

2Z0 C1

= 100 Ω = Z03

2Z0 L2

= 50 Ω,

and Z02 = Z0 ( π4 QT L12 ) =

and for a 0.1 dB-ripple Chebyshev bandstop filter the open stub characteristic impedances are: Z01 = Z0 ( π4 QT C11 ) =

2Z0 C1

= 69.79 Ω = Z03

2Z0 L2

= 62.75 Ω

and Z02 = Z0 ( π4 QT L12 ) =

5.1.4 Network Design Using Q Tapering of Sections With R = 1 in Figure 5.7, the Q of a section with k odd is: Qk =

ω0 Ck0 ω0 QT C k Ck = ( )= QT 2 2 ω0 2

(5.70)

But equivalence of the shunt L0k Ck0 circuit and the λ/4 shorted stub requires: ω0 Ck0 =

π Z0 4 Z0k

(5.71)

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

Thus, from (5.70)

153

π Z0 8 Z0k

(5.72)

ω0 Ck00 ω0 QT Lk Lk = ( )= QT 2 2 ω0 2

(5.73)

Qk = Likewise, the Q of a k-even section is: Qk =

Again, for equivalence of the shunt L00k Ck00 circuit and the λ0 /4 shorted stub: ω0 Ck00 =

π Z0 4 Z0k

(5.74)

Thus, Qk for k even is:

π Z0 (5.75) 8 Z0k To design a bandpass network using Q tapering, the desired Qk of each section of the network is determined from (5.70) and (5.73), with QT specified and Ck and Lk selected from Appendix A. Then, Z0k for each λ0 /4 shorted stub is determined from (5.72) or (5.75). This somewhat different approach will of course yield the same results obtained earlier, since equating (5.70) and (5.72), and (5.73) and (5.75) reproduces (5.66) and (5.67). Following a similar development for bandstop networks with R = 1 in Figure 5.8, results in 2 QT , k odd (5.76) Qk = Ck and 2 Qk = QT , k even (5.77) Lk Qk =

for the lumped-element network, and Qk =

π Z0k 2 Z0

(5.78)

for k odd or even. Again, equating (5.76) and (5.77) with (5.78) yields (5.68) and (5.69). Summarizing, with R = 1, Qk = g2k QT for bandpass networks, and Qk = g2k QT for bandstop networks, where gk , with k = 1, 2, 3, ..., is C1 , L2 , C3 , ... or L1 , C2 , L3 , ... from Appendix A. Once the Qk s are adjusted in this way to

154

MICROWAVE TRANSMISSION LINE CIRCUITS

achieve the desired response, the required characteristic impedances of λ/4 stubs are determined using (5.75) or (5.78). Since we know how the Q tapering relates to the gk s in Appendix A, this design method may be extended to other frequency selective networks (other than λ/4 shorted and open stubs) if we had an expression relating Qk to the important properties of the network. A universal definition for Q is Q = ω0

peak energy stored average power lost

(5.79)

For a network describable in terms of admittance Y = G + jB, applying (5.79) yields   ω ∂B Q= (5.80) 2G ∂ω ω=ω0 where G is the total shunt conductance. Also, for a network describable in terms of impedance Z = R + jX, (5.79) yields   ω ∂X (5.81) Q= 2R ∂ω ω=ω0 where R is the total series resistance. Example 5.9: A shorted stub of characteristic admittance Y01 and length l = nλ0 /4, n = 1, 3, 5, 7..., is connected in parallel with a lossless transmission line of characteristic admittance Y0 that is also terminated in ZS = ZL = Z0 . Determine the Q of this stub on the transmission line. Solution: The admittance at the common node connection is, Y = 2Y0 − jY01 cot From (5.80), Q=

nπω 2ω0

ω0 nπ Y01 π Y01 =n 2 nπ 4Y0 2ω0 sin 2 8 Y0

(5.82)

(5.83)

Thus, with n = 1, the Q is the same as determined earlier for a λ0 /4 shorted stub, higher values of Q are obtained by odd-multiple increases in the stub length.

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

155

01

Z

2

θ

01

Z0

θ

−

2

Z

Z0

θ

θ = βl = θ2 = k

ωl v

=

π f 2 f0

π f 2 f0

Figure 5.9 Tapped-stub resonator on Z0 line.

5.2 TAPPED-STUB RESONATOR The simplest form of a tapped-stub resonator is obtained by changing the connection point of a shorted stub on the main transmission line. This is illustrated in Figure 5.9, where the total stub length is l = λ0 /4, and the range of the tapping factor k is 0 ≤ k < 1. Note that k = 0 represents the original untapped shorted stub. The Q of the tapped-stub resonator on the line is determined by using previous methods. At the common connection node, the total admittance is: Y = 2Y0 + jY01 (tan θ2 − cot θ1 ) = G + jB

(5.84)

Since θ1 = (1 − k)ωl/v, and θ2 = kωl/v, the susceptance may be expressed as:   πf πf − cot (1 − k) B = Y01 (tan kωl/v − cot (1 − k)ωl/v) = Y01 tan k 2f0 2f0 (5.85) Note that at f = f0 in (5.85), B = 0, as required for a bandpass characteristic. Taking the partial derivative of B with respect to ω and evaluating at ω = ω0 yields: ∂B πY01 h π πi |ω=ω0 = k sec 2 k + (1 − k) csc 2 (1 − k) (5.86) ∂ω 2ω0 2 2

156

MICROWAVE TRANSMISSION LINE CIRCUITS

Using (5.86) the Q of the tapped-stub resonator is:   ω0 ∂B π Y01 h π πi Q= = k sec2 k + (1 − k) csc2 (1 − k) 2G ∂ω ω=ω0 8 Y0 2 2 or

π Y01 Q= 8 Y0

"

k cos2 k π2

1−k + 2 sin (1 − k) π2

#

(5.87)

(5.88)

But sin2 (1 − k) π2 = cos2 k π2 , and Q is compactly expressed as: Q=

π Y01 1 8 Y0 cos2 k π2

(5.89)

For k = 0, Q = (π/8)(Y01 /Y0 ), as given earlier for an untapped stub. As k approaches unity, the shorted end of the stub moves close to the main transmission line, and the Q of the tapped stub approaches infinity. The increase in Q of the tapped stub may be interpreted as an effective increase in the characteristic admittance of the untapped stub. To show this, let (5.89) be expressed as 0 π Y01 (5.90) Q= 8 Y0 where Y01 0 Y01 = (5.91) cos2 k π2 is the effective characteristic admittance one would require for an untapped stub. To 0 0 0 illustrate the increase in Y01 : for k = 0.5, Y01 = 2Y01 , for k = 0.667, Y01 = 4Y01 , 0 0 for k = 0.77, Y01 = 8Y01 , and for k = 0.839, Y01 = 16Y01 . The placement of the short circuit becomes very critical for high Q requirements where k approaches unity. In such cases, it may be more convenient, and more accurate values of Q may be obtained if the short-circuited stub is replaced by an open-circuited stub that is λ0 /4 longer. The total stub length from open-end to open-end is now 2l = λ0 /2, as illustrated in Figure 5.10, and the node admittance is: Y = 2Y0 + jY01 (tan θ2 + tan θ1 ) = G + jB (5.92) Since θ1 = (2 − k)ωl/v, θ2 = kωl/v, and l = λ0 /4, as before, the susceptance becomes:   πf πf + tan (2 − k) B = Y01 (tan kωl/v + tan (2 − k)ωl/v) = Y01 tan k 2f0 2f0 (5.93)

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

157

01

Z

2

θ

01

Z0

θ

−

2

Z

Z0

θ = βl =

θ π f θ1 = (2 − k )

2 f0

ωl v

=π

θ2 = k

f f0

π f 2 f0

Figure 5.10 Tapped-stub resonator with open stubs on Z0 line.


and at f = f0 , B = Y01 tan k π2 + tan (2 − k) π2 = 0, as required for a bandpass structure. From (5.93), the partial derivative of B with respect to ω evaluated at ω0 is: ∂B πY01 h π πi |ω=ω0 = k sec 2 k + (2 − k) sec 2 (2 − k) ∂ω 2ω0 2 2

(5.94)

Using (5.94) the Q of the tapped-stub resonator with open circuits only is: 

ω0 ∂B Q= 2G ∂ω



ω=ω0

=

π Y01 π π Y01 1 sec2 k = 4 Y0 2 4 Y0 cos2 k π2

(5.95)

since sec 2 (2 − k) π2 = sec2 k π2 . Note from (5.95) that the Q is further increased by a factor of two when the tapped λ0 /2 open stub is used instead of the tapped λ0 /4 shorted stub. 5.2.1 The Half-Wavelength Tapped Stub as a Tunable Filter A transformation of the half-wavelength tapped stub makes it tunable in bandwidth and operating frequency. As a starting point, Figure 5.11(a) shows a half-wavelength tapped-stub on a transmission line between source and load (Z0 is usually, but not

158

MICROWAVE TRANSMISSION LINE CIRCUITS

50 Ω

Z01 θ2 Z0

C2

50 Ω Z0

Z0 Z01

50 Ω

Z01 θ1

Z0 90o

50 Ω

C1

(a)

(b) VC2 C2'

50 Ω Z0

Z0 L0

50 Ω

IN

OUT

0.5”

VC1

' 1

C

(c)

(d)

0.5”

Figure 5.11 Transforming the tapped-stub resonator to a compact tunable filter: (a) The halfwavelength tapped stub on uniform line; (b) Open ends are replaced by capacitors; (c) The remaining 90-degree section is replaced by capacitors and an inductor; (d) The inductance is simulated by a highimpedance short-length line, and the capacitors are replaced by varactor diodes.

necessarily, 50 Ω), where θ1 +θ2 = 180 electrical degrees at the resonant frequency. The admittance at the common junction to ground is Y

=

2Y0 + jY01 [tan θ1 + tan θ2 ]

= = =

2Y0 + jY01 [tan (180◦ − θ2 ) + tan θ2 ] 2Y0 + jY01 [− tan θ2 + tan θ2 ] 0

(5.96)

Thus, excluding the case where θ1 = θ2 , the tapped stub is resonant at all θ1 + θ2 = 180◦ , or where the total length is λ/2. Assuming that θ1 > θ2 , move on to Figure 5.11(b) and replace the open-ended line of length θ2 by a capacitor C2

159

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

where, jωC2 = jY01 tan θ2

(5.97) ◦

◦

◦

On the lower end of the tapped stub, since θ1 = 180 − θ2 = 90 + 90 − θ2 , replace the open end of θ1 by, jωC1 = jY01 tan (90◦ − θ2 )

(5.98)

which leaves the 90◦ or λ/4 section shown in Figure 5.11(b). From the main line the impedance looking into the λ/4 line terminated in the impedance 1/jωC1 is an inductive reactance given by, 2 jωL1 = Z01 jωC1

(5.99)

or 2 L1 = Z01 C1

(5.100)

Hence, the original λ/2 tapped stub is now a parallel LC circuit of resonant frequency, 1 1 √ √ f0 = = (5.101) 2π L1 C2 2πZ01 C1 C2 and quality factor (Q), C2 ω0 C 2 1 Z0 √ Q= = = 2Y0 2Y 2Z Z01 C1 C2 0 01

r

C2 C1

(5.102)

Thus it is seen from the two equations and Figure 5.11(b) that resonant frequency and Q (or bandwidth) may be adjusted independently by adjusting the product or the ratio of C1 and C2 , respectively. At frequencies below approximately 3 GHz the λ/4 section of line may be prohibitively long, so often it is useful to replace the λ/4 line with something equivalent but smaller. In Figure 5.11(c) the λ/4 line of characteristic impedance Z01 is exactly equivalent to a shunt C0 series L0 shunt C0 in the neighborhood of resonance, as will be shown in Example 6.4 and Figure 6.8. The two capacitors are equal and these elements are related to Z01 by, ω0 L0 = Z01 =

1 ω0 C 0

where ω0 = 2πf0 is the radian resonant frequency.

(5.103)

160

MICROWAVE TRANSMISSION LINE CIRCUITS

In Figure 5.11(c), the two capacitors denoted as C0 may be added in parallel to C1 and C2 , so that C10 = C1 +C0 and C20 = C2 +C0 . The inductance L0 could be left as a lumped inductor at frequencies below approximately 200 MHz. Above that frequency the inductor could be replaced with a short section of transmission line of impedance Z0t and length l0 , as shown in Table 3.1. This equivalent replacement is made by making Z0t as large as possible, and the inductive reactance of this line will be, 2π jω0 L0 = jZ0t tan l0 (5.104) λ But ω0 L0 = Z01 , as determined previously, and this yields, l0 =

λ Z01 tan−1 2π Z0t

(5.105)

If, for example, Z01 = 50 Ω and Z0t = 150 Ω, l0 = 0.051λ, rather than the 0.25λ length in Figure 5.11(b). In Figure 5.11(d), VC1 is the adjustable bias voltage that would be applied to C1 when replaced by a varactor diode, and VC2 is the bias on C2 replaced by a varactor diode. The junction capacitance of a varactor diode may be expressed as C= 

C0 1−

V V0

2

(5.106)

Since C0 is the value of capacitance at V = 0, if V is varied from 0.5V0 to −9V0 (where V0 is the maximum forward bias voltage), then C can be varied approximately from 4C0 down to 0.01C0 or less. In other words, The resonant frequency can be varied by a factor of 2 upward or downward, and the Q (or bandwidth) can also be increased or decreased by a factor of 2. Both variations are independent of each other.

5.3 COUPLED LINE FILTERS While the methods for filter design described in the previous sections can deliver fairly wideband filters, coupled line filters are more appropriate for narrowband selectivity. Here, we discuss first the basics of coupled lines in general, and then apply this information to the design of coupled line filters. Coupled lines are typically used in microstrip and stripline transmission line topologies for the usual reasons of ease of fabrication and compactness. A diagram of a generic coupled line in microstrip is shown in Figure 5.12.

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

w

161

w s

εr

h

Figure 5.12 Coupled lines in microstrip, with dimensions as indicated.

Coupled lines operate in two modes, even and odd. The relative strength of the modes propagating in the lines depends on the nature of excitation. For pure even mode excitation, the voltages on each of the lines are identical and there is no coupling between the lines. One can imagine then, a plane of constant, zero magnetic field between the two conductors, called an H-wall, as depicted in Figure 5.13. The equivalent circuit for the even mode is shown to the right in Figure 5.13. Here the even mode excitation results in two identical capacitances per unit length to ground (assuming that the two lines are identical) , given by C11 and C22 . This capacitance per unit length of a single line to ground is often termed the evenmode capacitance, Ce , and this capacitance translates to an effective even-mode characteristic impedance for the line, Z0e , given by r L 1 Z0e = = (5.107) Ce vCe with L being the usual inductance per unit length of the transmission line and v being the phase velocity. Conversely, the odd mode is the exact opposite of the even mode, with the voltages on each of the lines completely out of phase with each other, and coupling between the lines is maximum. In this case, one can imagine a plane of constant, zero electric field between the two conductors, called an E-wall, as depicted in Figure 5.14. The equivalent circuit for the even mode is shown to the right in Figure 5.14. Here the odd mode excitation results in the maintenance of the two identical capacitances per unit length to ground (assuming that the two lines are identical), still given by C11 and C22. Additionally, to account for the strong coupling between lines, the circuit model of Figure 5.14 shows two capacitances per unit length between the lines and the zero-potential E-wall, termed 2C12. Taking into account these coupling capacitances, we can find the odd-mode capacitance per unit length, which is given by the total capacitance of each line to ground operating in the odd

162

MICROWAVE TRANSMISSION LINE CIRCUITS

H-wall

+V

+V

C11

C22

s Figure 5.13 Coupled lines operating in a pure even mode in microstrip, with the equivalent circuit as shown to the right (all values per unit length).

E-wall 2C12 +V

−V

C11

2C12 C22

s Figure 5.14 Coupled lines operating in a pure odd mode in microstrip, with the equivalent circuit as shown to the right (all values per unit length).

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

163

mode, Co : Co = C11 + 2C12 = C22 + 2C12 .

(5.108)

This capacitance translates to an effective odd-mode characteristic impedance for the line, Z0o , given by r L 1 Z0o = = (5.109) Co vCo with L again being the usual inductance per unit length of the transmission line and v being the phase velocity. In practice, the excitation of coupled lines is a superposition of even and odd modes, and designs for filters here take this superposition into account. For stripline designs, one can use the results of conformal mapping techniques to obtain excellent approximations for all of these coupled line parameters. For microstrip, closed form design equations are only approximations, since the phase velocities of the even and odd modes differ because of differences between the nature of the electromagnetic fields in the substrate and air for the two modes. Today, most designers, leveraging the development of numerous computational tools, specify substrate parameters and desired even and odd mode impedances, and the tools provide line dimensions. Here, we assume that such tools are readily available and concentrate instead on device design using these structures as building blocks. In general, the coupled line section (viewed from above) is a four-port device (shown in Figure 5.15). Each port has a total current as indicated, but these total currents can be decomposed into even- and odd-mode currents in a line of electrical length θ as shown in Figure 5.16. In this case, i1 is the even-mode current source into the left hand side of the structure, while i3 is the even-mode current source into the right hand side of the structure. The odd-mode current source at the left hand port is given by i2 , while the odd-mode current source at the right hand port is given by i4 . The total currents at each of the ports can then be expressed as sums and differences of the even- and odd-mode currents: I1 I2 I3 I4

= = = =

i1 + i2 i1 − i2 i3 − i4 i3 + i4

(5.110)

164

MICROWAVE TRANSMISSION LINE CIRCUITS

I2

I3 Port 2

Port 3

Port 1

Port 4

I1

I4

Figure 5.15 General four-port model for a coupled line segment, viewed from above.

Using the methods of Chapter 4, the elements of the first row of the impedance matrix for the coupled line section can be determined: Z11

=

Z12

=

Z13

=

Z14

=

−j 2 −j 2 −j 2 −j 2

(Z0e + Z0o ) cot θ

(5.111)

(Z0e − Z0o ) cot θ (Z0e − Z0o ) csc θ (Z0e + Z0o ) csc θ

Using the symmetry of the circuit, we can fill out the entire impedance matrix with Z11 Z12

= =

Z22 = Z33 = Z44 Z21 = Z34 = Z43

Z13 Z14

= =

Z31 = Z24 = Z42 Z41 = Z23 = Z32

(5.112)

There are ten possible combinations of two-port connections for this fourport device if the other two ports are terminated in either open or short circuits. For implementation in bandpass filters, we use the configuration shown in Figure 5.17, with ports 2 and 4 left open (I2 = I4 = 0), and ports 1 and 3 serving as input and output ports, respectively. To implement multiple coupled line sections in a bandpass filter, a circuit model for each section is required. For the bandpass arrangement of Figure 5.17, the equivalent circuit is shown in Figure 5.18, with the J network defined as an admittance inverter. Specifically, the admittance inverter is a quarter wavelength of transmission line at the center design frequency, of characteristic impedance 1/J. Using the ABCD parameters of both circuits, one can arrive at the design equations for the coupled line segment (nominally a quarter wavelength, θ = π/2)

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

θ

i1

165

i3 i4

i2 i1

i3

Figure 5.16 Decomposed even- and odd-mode currents on a coupled line segment of electrical length θ, viewed from above.

θ

Port 3

Port 1 Figure 5.17

Port connections for use of a coupled line section in a bandpass filter, viewed from above.

in terms of the impedance inverter value, J: Z0e Z0o

= =

Z0 [1 + JZ0 + (JZ0 )2 ] Z0 [1 − JZ0 + (JZ0 )2 ]

(5.113)

A bandpass filter constructed out of coupled line segments is shown in Figure 5.19. Note that the sections are cascaded in a stagger, so that the output of one line serves as the input to the next section. The following equations relate these even- and odd-mode characteristic impedances to the normalized element values in Appendix A to allow us to again specify the frequency response of the bandpass filters designed with these coupled line segments. In particular, we use these equations to solve for the J value for each coupled line segment, and then use these values to determine the even- and odd-mode characteristic impedances of each section using (5.114). r π Z0 J1 = (5.114) 2QT C1 π Z0 Jn = for n − 2, 3, · · · N √ 2QT en−1 en r π Z0 JN+1 = 2QT eN eN+1

166

MICROWAVE TRANSMISSION LINE CIRCUITS

θ

θ J −90o

Z0

Z0

Figure 5.18 Equivalent circuit of the coupled line segment shown in Figure 5.17.

Z0

Z 0e , Z 0o Z 0e , Z 0o

Z0

3

{

2

{

1

{

{

Z 0e , Z 0o

N+1 Figure 5.19 Bandpass filter composed of N+1 sections.

with en representing either capacitive or inductive element values provided in Appendix A, as dictated by the index and the total number of segments in the filter. For all of the filters designed using this method and the tables in Appendix A, eN+1 = 1. PROBLEMS √ √ 1. In Figure 5.20, (a) if ω0 C1 = 2, and ω0 L2 = 0.5, with C1 = C3 , use the Smith chart to determine Z1 at ω0 . (b) If ω0 C1 = 1.1811, ω0 L2 = 0.7789, and ω0 C3 = 3.2612 (from Butterworth lowpass element tables, n = 3, R = 0.5), use the Smith chart to determine Z1 at ω0 . 2. If f0 = 800 MHz, f1 = 640 MHz, f2 = 1000 MHz, ZS = ZL = 50 Ω, and a 0.1 dB ripple Chebyshev response is required with n = 5, (a) design a bandpass filter and (b) a bandstop filter using transmission line sections. Sketch each filter and label all characteristic impedances and line lengths. 3. For each of the bandstop filters shown in Figure 5.21 (a), (b), and (c), determine the type of amplitude response versus frequency (Butterworth,

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

167

L2 I

R

C1

C3

1

R = 0.5 Figure 5.20 Use the Smith chart to determine Z1 with the component values given in Problem 1.

Chebyshev, Thomson) that is produced and the total selectivity (QT ). The Z0 of each θ-length section is as indicated, where θ = π2 ff0 . 4. For the tapped-stub filter in Figure 5.22, determine Z01 and Z02 for a Butterworth amplitude response, where θ = π2 ff0 , θ0 = π2 , QT = π2 , and Z0 = 50 Ω.

168

MICROWAVE TRANSMISSION LINE CIRCUITS

θ

100 Ω

θ

89.9 Ω

θ

100 Ω (a)

50 Ω 50

θ

θ

θ

50

50

143.82 Ω

θ

50 Ω

50

θ

50 Ω

22.02 Ω (b)

50 Ω 50

θ

θ

θ

50

50

100 Ω

θ

50 Ω

50

θ

50 Ω

100 Ω (c)

50 Ω 50

θ

θ

50

50

50

50 Ω

Figure 5.21 Determine the type of amplitude response versus frequency (Butterworth, Chebyshev, Thomson) that is produced and the total selectivity (QT ) for each bandstop filter.

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

Z01 Z0

1 θ0 2

Z0

Z02

2 θ0 3

Z0

θ0 Z01

1 θ0 2

Z01 Z0

Z02

1 θ0 3

1 θ0 2 Z0

θ0 Z01

Figure 5.22 Determine Z01 and Z02 for a Butterworth amplitude response.

1 θ0 2

50 Ω

169

170

MICROWAVE TRANSMISSION LINE CIRCUITS

BIBLIOGRAPHY I. J. Bahl and P. Bhartia, Microwave Solid State Circuit Design, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2003. J. T. Bernhard and W. T. Joines, “Microwave Bandpass Filters Using SeriesCascaded Sections of Microstrip Line,” Microwave and Optical Technology Letters, vol. 5, pp. 177–181, Apr. 1992. S. B. Cohn, “Shielded Coupled-Strip Transmission Line,” IRE Transactions on Microwave Theory and Techniques, vol. 3, no. 5, pp. 29–38, Oct. 1955. R. M. Cottee and W. T. Joines, “Synthesis of Lumped and Distributed Networks for Impedance Matching of Complex Loads,” IEEE Trans. Circuits Syst., vol. 26, no. 5, pp. 316–329, May 1979. J. M. Drozd and W. T. Joines, “Determining Q Using S Parameter Data,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 11, pp. 2123–2127, Nov. 1996. T. C. Edwards and M. B. Steer, Foundations of Interconnect and Microstrip Design, 3rd ed. Hoboken, NJ: John Wiley and Sons, 2001. M. S. Ghausi, Principles and Design of Linear Active Circuits. New York: McGraw-Hill, 1965. J. R. Griffin and W. T. Joines, “The General Transfer Matrix for Networks with Open-Circuited or Short-Circuited Stubs,” IEEE Trans. Circuit Theory, vol. 16, no. 3, pp. 373–376, Aug. 1969. W. T. Joines and J. R. Griffin, “On Using the Q of Transmission Lines,” IEEE Trans. Microw. Theory Tech., vol. 16, no 4, pp. 258–260, Apr. 1968. G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Artech House, Dedham, MA: 1980. W. W. Mumford, “Maximally-Flat Filters in Waveguide,” Bell Syst. Tech. J., vol. 277, pp. 684–713, Oct. 1948. D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley and Sons, 2011.

SYNTHESIS AND DESIGN OF FREQUENCY-FILTERING NETWORKS

171

P. A. Rizzi, Microwave Engineering: Passive Circuits. Upper Saddle River, NJ: Prentice-Hall, 1988. D. Vatis and W. T. Joines, “An Exact, Time-Saving Synthesis Procedure for UHF and Microwave Networks,” in Proc. IEEE Southeast Region 3 Conf., Williamsburg, VA, 1997, pp. 479–502. G. D. Vendelin, A. M. Pavio, and U. L. Rohde, Microwave Circuit Design, Using Linear and Nonlinear Techniques, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2005. L. Weinberg, Network Analysis and Synthesis. Malabar, FL: Krieger Publishing Co., 1975. E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications. New York: Wiley-Interscience, 1988. A. I. Zverev, Handbook of Filter Synthesis. New York: Wiley-Interscience, 1967.

172

MICROWAVE TRANSMISSION LINE CIRCUITS

Chapter 6 BROADBAND IMPEDANCE-MATCHING NETWORKS 6.1 NETWORK MODEL FOR IMPEDANCE MATCHING As a two-port model an impedance-matching network (IMN) between a real source (ZS = RS ) and a real load (ZL = RL) is represented in Figure 6.1, where Z1 = R1 (ω) + jX1 (ω)

(6.1)

Z2 = R2 (ω) + jX2 (ω)

(6.2)

at the input, and at the output. From the input side of the network, the load RL is contained as part of Z1 , and the real part of Z1 may be taken as the impedance to which power actually is delivered from the source. Hence, from the input side, the impedance-matching network may be represented as shown in Figure 6.2, where all impedances have been divided by R1 (ω) to normalize with respect to the real part of Z1 . For a given frequency ω0 , an impedance match is achieved when the source impedance RS is equal to the input impedance of the matching network R1 . This requires that X1 (ω0 ) = 0

(6.3)

R1 (ω0 ) = RS

(6.4)

and or z1 =

Z1 (ω0 ) =R=1 R1 (ω0 ) 173

(6.5)

174

MICROWAVE TRANSMISSION LINE CIRCUITS

ZS = RS VS

IMN

ZL = RL

Z 2 = R2 (ω ) + jX 2 (ω )

Z1 = R1 (ω ) + jX 1 (ω )

Figure 6.1 Two-port model of an impedance-matching network showing the impedance at the input (Z1 ) and at the output (Z2 ) of the network.

RS =R R1 (ω )

j

X 1 (ω ) R1 (ω ) RL (ω ) =1 R1 (ω )

VS Z1 X (ω ) = z1 = 1 + j 1 R1 (ω ) R1 (ω )

Figure 6.2 Model of the impedance-matching network from port 1, the input side. The matching network is chosen such that R1 (ω) = RL (ω).

The frequency-selective network represented by jX1 (ω)/R1 (ω) in Figure 6.2 can be composed of lumped or distributed elements, or any combination of the two, which preferably are lossless and hence purely reactive. There can be few or many such elements, which then can be grouped in sections. Since the network is terminated in unity source and load resistances, the sections within the network may be designed by using the Q-tapering techniques presented in Chapter 5. Thus, frequency-selective sections within the impedance-matching network may be related to lowpass prototype elements in such a way that a specified response versus frequency is achieved along with the desired impedance match at f = f0 . The method of Q-tapering is very useful for designing impedance-matching networks, and since R = 1 at f = f0 in Figure 6.2, the Q of the k th section is, as in Chapter 5, related to the total Q by Qk =

gk QT 2

(6.6)

BROADBAND IMPEDANCE-MATCHING NETWORKS

175

where gk is a lowpass prototype element chosen to yield a desired frequency response. The exact design procedure will be made clear through example solutions. However, for later use, we need to determine the frequency selectivity (Qk ) of an impedance-transforming section of transmission line.

6.2 THE Q OF λ/4 AND λ/2 TRANSFORMER SECTIONS The impedance-matching network in Figure 6.1 can be as simple as a length of transmission line. Recalling from Chapter 2 the expression for the input impedance of a loaded transmission line, we can write the input impedance of the transformation section between the load RL and the source RS in Figure 6.3 as Z1 = Z0t

RL + jZ0t tan θt Z0t + jRL tan θt

At resonant frequencies we require an impedance match such that:  2 Z0t /RL for θt = π2 (2p − 1) Z1 = RS = RL for θt = pπ

(6.7)

(6.8)

where p is a positive integer. Note that for p = 1, the upper equation represents a quarter-wave line and the lower equation represents a half-wave line. Solving for the real and imaginary parts of (6.7) yields, Z1 =

2 2 Z0t RL(1 + tan2 θt ) + jZ0t (Z0t − R2L ) tan θt = R1 + jX1 2 + R2 tan2 θ Z0t t L

The frequency selectivity or Q of the circuit in Figure 6.3 at ω = ω0 is,   ω ∂X1 Q= R ∂ω ω=ω0 where R = RS + R1 . Carrying out the operation indicated yields, θt Z0t RL π Q= − , θt = p 4 RL Z0t 2

(6.9)

(6.10)

(6.11)

Note that (6.11) is valid for both quarter-wavelength and half-wavelength transformers of integer length p. For the shortest quarter-wavelength transformer (p =

176

MICROWAVE TRANSMISSION LINE CIRCUITS

1, θt = π/2): π Q= 8

r r Z0t π RS π RS R Z R L 0t L RL − Z0t = 8 Z0t − RS = 8 RL − RS

(6.12)

2 where Z0t /RL = RS , or Z0t /RL = RS /Z0t , has been used to express the Q of the λ/4 transformer in the three equivalent ways. Applying the Q-tapering concept to the λ/4 transformer/shorted-stub impedance-matching network considered earlier, √ as shown in Figure 6.4, a Butterworth response requires that Q1 = Q2 = QT / 2. Thus,

Q1 =

π RS 1 = √ QT 8 Z01 2

(6.13)

and π Q2 = 8 or, equating Q1 and Q2 ,

Z02 RS 1 RS − Z02 = √2 QT

Z02 RS RS RS − Z02 − Z01 = 0

(6.14)

(6.15)

Also, an impedance match at θ = θ0 = π/2 requires, 2 RS = Z02 /RL

(6.16)

From (6.15) and (6.16), Z01 =

RS p RS RL RL − RS

(6.17)

and Z02 =

p RS RL

(6.18)

These solutions for Z01 and Z02 yield a Butterworth response for any RS and RL , and as shown in an earlier example, the shorted stub serves to broaden the impedance match over what it would be with the transformer alone.

BROADBAND IMPEDANCE-MATCHING NETWORKS

RS VS

177

θt Z0t

RL

Ζ1 Figure 6.3 Impedance-transforming section of transmission line.

RS

Ζ1 θ RL>RS

θ

01

Z02

Z

VS

Figure 6.4 Impedance-matching network using a shorted stub and λ/4 transformer.

6.3 MULTIPLE QUARTER-WAVELENGTH TRANSFORMERS IN CASCADE

To broaden the frequency band of an impedance match and also achieve a specified response versus frequency, multiple quarter-wavelength transformers may be used. At the center-band frequency (f = f0 ) the impedance at each junction of the cascaded transformer is as shown in Figure 6.5, where Z1 = RS for the impedance match. Using the impedances in Figure 6.5, the Q of each cascaded section is

178

MICROWAVE TRANSMISSION LINE CIRCUITS

RS VS

Z1 =

Z 012 Z2

λ0/4

λ0/4

λ0/4

Z01

Z02

Z0n

Z2 =

Z 022 Z3

Z3

Zn =

RL

Z 02n RL

Figure 6.5 Impedance-matching network using cascaded λ/4 transformers.

Q1

=

Q2 . . .

= = = =

Qn

=

π 8

RS π Z01 = Z01 − ZR01 8 Z2 − 2 S π Z01 /RS 02 − Z 2Z/R 8 Z02 S 01 . . . RL π Z0n 8 RL − Z0n



Z2 Z01

(6.19)

The design proceeds by setting Z1 = RS for a match at f = f0 , and tapering Q1 , Q2 , ..., Qn as gk Qk = QT , k = 1, 2, ..., n (6.20) 2 for a desired frequency response using any number of sections. Note that each section will have a different input impedance and so there will be multiple internal reflections within the network. Rigorous mathematical analysis shows that these internal reflections actually cancel at the input, yielding the desired impedance match. 6.3.1 Two Cascaded Sections Two cascaded quarter-wavelength line sections between RS and RL are shown in Figure 6.6, and Q1 and Q2 are π Q1 = 8

RS Z01 π Z01 Z2 Z01 − RS = 8 Z2 − Z01

(6.21)

BROADBAND IMPEDANCE-MATCHING NETWORKS

RS VS

λ0/4

λ0/4

Z01

Z02

.

Z2 = 2

179

RL

Z 022 RL

Z2  Z  Z1 = 01 =  01  RL = RS Z 2  Z 02  Figure 6.6 Two cascaded λ/4 transformers as an impedance-matching network.

and

Z02 RL − (6.22) RL Z02 √ √ If a Butterworth response is required then (g1 = 2, g2 = 2), and Q1 = Q2 = √1 QT . Equating (6.21) and (6.22) yields 2 Q2 =

π 8

RS Z01 Z02 RL − = − Z01 RS RL Z02

(6.23)

From Figure 6.6 the input impedance is Z2 Z1 = 01 = Z2



Z01 Z02

2

RL = RS

(6.24)

Solving (6.23) and (6.24) for Z01 and Z02 : 3

1

Z01 = RS4 RL4 and

1

(6.25)

3

Z02 = RS4 RL4

(6.26)

Thus, if RS = 300 Ω and RL = 50 Ω, then Z01 = 191.68√ Ω and Z02 = 78.25 Ω. If RS = 200 Ω and R = 50 Ω, then Z = 100 2 = 141.42 Ω and L 01 √ Z02 = 100/ 2 = 70.71 Ω.

180

MICROWAVE TRANSMISSION LINE CIRCUITS

6.3.2 Three Cascaded Sections With n = 3, or three cascaded λ/4 transformers, the input impedance for a match at f = f0 is,  2  2 Z01 Z03 Z1 = RS = RL (6.27) Z02 RL as indicated in Figure 6.7. Again, if a Butterworth response is required (Q1 = QT /2, Q2 = QT , Q3 = QT /2), the section Qs are tapered accordingly, as: π RS Z01 1 Q1 = − = QT (6.28) 8 Z01 RS 2 2 π Z02 RL Z03 = QT Q2 = − (6.29) 2 8 Z03 Z02 RL π Z03 RL 1 Q3 = − = QT (6.30) 8 RL Z03 2 Using (6.27) through (6.30) yields: r  2 r  2   RS Z01 RL RS Z01 RS − +2 − =0 RL RS RS Z01 RS Z01 p Z02 = RS RL

(6.31) (6.32)

and

RS RL (6.33) Z01 For given values of RS and RL, Z01 , Z02 , and Z03 are determined from (6.31), (6.32), and (6.33). Clearly, the solutions to (6.32) and (6.33) for Z02 and Z03 follow quite easily once (6.31) is solved for Z01 . Finding Z01 from (6.31) can be done numerically or by trial and error. For example, if RS = 75 Ω and RL = 300 Ω: Z01 = 89.25 Ω, Z02 = 150 Ω, and Z03 = 252 Ω. Also, if RS = 50 Ω and RL = 200 Ω: Z01 = 59.5 Ω, Z02 = 100 Ω, and Z03 = 168 Ω. Z03 =

6.4 MORE COMPACT IMPEDANCE-MATCHING NETWORKS As an impedance-matching element, a single λ/4 transformer generally is broader in bandwidth than the stub tuners and the two-element (LC) matching networks

181

BROADBAND IMPEDANCE-MATCHING NETWORKS

RS VS 2

λ0/4

λ0/4

λ0/4

Z01

Z02

Z03

RL

2

Z Z Z1 =  01   03  RL = RS  Z 02   RL  Figure 6.7 Three cascaded λ/4 transformers as an impedance-matching network.

considered earlier. Multiple λ/4 transformers extend the impedance match over even greater bandwidths. However, cascaded λ/4 transformers, or even a single λ/4 transformer, may become excessively long unless the operating frequency is quite high. We might ask: Can we simulate the properties of the λ/4 transformer by using more compact structures? To find the answer, we will compare the ABCD matrix of the λ/4 transformer with the ABCD matrices of candidate equivalent circuits consisting of lumped elements and transmission line segments. 6.4.1 Lumped-Element Equivalent of the Quarter-Wavelength Transformer √ With θt = π2 ff0 and Z0t = RS RL, the ABCD matrix of the λ/4 transformer in Figure 6.8(a) is     A B cos θt jZ0t sin θt = (6.34) C D jY0t sin θt cos θt and the ABCD matrix of the LC network in Figure 6.8(b) is     A B 1 − ω2 LC jωL = C D jωC(2 − ω2 LC) 1 − ω2 LC

(6.35)

√ These two matrices are equal term by term at f = f0 if f0 = 1/(2π LC), thus yielding the design equation ω0 L = Z0t =

1 ω0 C

(6.36)

This same design equation relates the other three networks in (c), (d), and (e) of Figure 6.8 to the λ/4 transformer by making all of the ABCD matrices equal at f = f0 .

182

MICROWAVE TRANSMISSION LINE CIRCUITS

The LC networks in Figure 6.8 may be used as direct replacements for the λ/4 transformer if the operating frequency is not so high that the component size becomes comparable to a wavelength, as mentioned in Chapter 3. Alternatively, the LC elements may be replaced by appropriate transmission line segments that are less than λ/4 in length. 6.4.2 The Eighth-Wavelength Transformer To explore another method that replaces the λ/4 transformer with shorter segments of transmission line, consider the relatively simple networks in Figure 6.9. Each segment in Figure 6.9(a) has the same characteristic impedance and phase length, symbolized by Z0 and θ, respectively. These parameters have the same symbols in Figure 6.9(b), but the actual values of Z0 may differ from those in Figure 6.9(a). The ABCD matrix of the network in Figure 6.9(a) is     A B (1 − tan2 θ) cos θ jZ0 sin θ = (6.37) C D jY0 (3 − tan2 θ) sin θ (1 − tan2 θ) cos θ This matrix is equal to the matrix of the λ/4 transformer at f = f0 , if θ= and Z0 =

π f 4 f0

(6.38)

√ 2Z0t

(6.39)

Thus, the three line-segment lengths in Figure 6.9(a) are λ/8 at f = f0 , and this network will be referred to as an λ/8 transformer. From the ABCD√matrix of the λ/4 transformer in (6.34), the transmission coefficient with Z0t = RS RL is √ 2 RS RL 2 q  t= = q ARL + B + CRS RL + DRS RL RS RS + RL cos θt + j2 sin θt (6.40) and t = −j = 16 −90◦ at f = f0 , indicating perfect transmission and a 90◦ phase delay, as expected. For comparison, using the ABCD parameters of √ √ the λ/8 transformer in (6.37), the transmission coefficient with Z0 = 2Z0t = 2RS RL is √ 2 RS RL t = ARL + B + CRS RL + DRS

183

BROADBAND IMPEDANCE-MATCHING NETWORKS

RS

θt

VS

Z0t

RL

(a)

L C

C

L

(b)

L (c)

C

C L

L

C

(d)

C L

(e)

Figure 6.8 Comparison of the λ/4 transformer in (a) with LC equivalents in (b), (c), (d), and (e).

MICROWAVE TRANSMISSION LINE CIRCUITS

RS

λ0/8

VS

(a)

RL Z 0 = 2 Z 0t

/8 0

λ

0

/8 0

λ

0

0

λ

Z

0

λ

/8

0

Z

0

/8

Z

Z0

Z

184

RS VS

Z0 λ0/8

Figure 6.9 λ/8 transformers that are equivalent to the λ/4 transformer.

RL Z0 =

(b) Z 0t 2

185

BROADBAND IMPEDANCE-MATCHING NETWORKS

=

2 q

RL RS

+

q

RS RL



(1 − tan2 θ) cos θ + j √12 (5 − tan2 θ) sin θ (6.41)

and at f = f0 , where θ = 45◦ , t = −j = 16 −90◦ . The ABCD matrix of the network in Figure 6.9(b) is 

A C

B D



=



(1 − tan2 θ) cos θ jY0 sin θ

j(3 − tan2 θ)Z0 sin θ (1 − tan2 θ) cos θ



(6.42)

This matrix is equal to the matrix of the λ/4 transformer at f = f0 , if θ= and

π f 4 f0

√ Z0 = Z0t / 2

(6.43)

(6.44)

Thus again, the three line-segment lengths in Figure 6.9(b) are λ/8 at f = f0 , and this network, which has the same transmission coefficient as given in (6.41), will also be referred to as a λ/8 transformer. The λ/8 transformer in Figure 6.9(a) may be constructed conveniently using stripline or microstrip, whereas the transformer in Figure 6.9(b) is well suited for coplanar waveguide (CPW) or twin lead construction. Example 6.1: For RS = 50 Ω and RL = 100 Ω, use two cascaded λ/4 transformers to form an input impedance match that has a Butterworth response function, and then use lumped-element networks of the type shown in Figure 6.8(b) to replace the λ/4 transformers at f = 500 MHz. Solution: From (6.25) and (6.26), the λ/4 transformer impedances are Z01 = 500.751000.25 = 59.46 Ω and Z02 = 500.251000.75 = 84.09 Ω. In the replacement network there will be five lumped elements. If these are numbered with subscripts 1 through 5, the central element is the sum of two adjacent elements as C3 = C1 + C5 . The calculations proceed using (6.36) to obtain:

186

MICROWAVE TRANSMISSION LINE CIRCUITS

C1 =

1 = 5.35 pF 59.46ω0

L2 =

59.46 = 18.93 nH ω0

C3 = C1 + C5 = 9.14 pF L4 =

84.09 = 26.77 nH ω0

C5 =

1 = 3.79 pF 84.09ω0

and

The resulting networks are shown in Figure 6.10(a) and (b). The input reflection coefficient (S11 in dB) is plotted versus frequency in Figure 6.11 for the transmission line network in Figure 6.10(a) and for the lumpedelement network in Figure 6.10(b). As illustrated, both networks are perfectly matched (S11 ∼ = −∞ dB) at 500 MHz (the design frequency). The bandwidth for S11 = −15 dB (an acceptable impedance match) is 507 MHz for the transmission line circuit while the bandwidth for the lumped-element circuit is considerably less at 246 MHz. The reduced bandwidth would not be a problem unless an extremely wide bandwidth is required. Notice particularly in this example and the plot of S11 versus frequency that impedance matching networks are also frequency filters. In many applications this is a desirable feature that may be used to prevent out-ofband signals from being transmitted to the load. Example 6.2: (a) Replace the two cascaded λ/4 transformers in the previous example with λ/8 transformers of the type shown in Figure 6.9(a). (b) Repeat using λ/8 transformers of the type shown in Figure 6.9(b). Solution: In both parts (a) and (b), the result will be five transmission line segments each of length λ/8, and the central segments will be the combination of two adjacent segments. In part (a), before combining the central segments, the√characteristic impedance of each of the first three segments is 59.46 2 = 84.09 Ω and for the last three

BROADBAND IMPEDANCE-MATCHING NETWORKS

ZS = 50 Ω VS

λ0/4

λ0/4

Z01 = 59.46 Ω

Z02 = 84.09 Ω

ZS = 50 Ω VS

L2 C1

187

ZL = 100 Ω

(a)

ZL = 100 Ω

(b)

L4 C3

C5

Figure 6.10 Lumped-element replacement of cascaded λ/4 transformers. (a) Cascaded transformers yielding a Butterworth response. (b) The lumped-element replacement network where: C1 = 5.35 pF, L2 = 18.93 nH, C3 = C1 + C5 = 9.14 pF, L4 = 26.77 nH, and C5 = 3.79 pF.

Figure 6.11 Input reflection coefficient (S11 in dB) versus frequency for the two equivalent networks in Figure 6.10(a) and (b).

MICROWAVE TRANSMISSION LINE CIRCUITS

λ0/8

92

Ω

VS

59.46 Ω

λ0/8

λ0/8

.4 6

/8 0

λ

0

42.04 Ω

59

/8

50 1. 10

/8 0

Ω

Ω

0

0

λ

λ

42

.0 4

Ω

0

λ

/8

/8

49 .4

6

Ω 9 /8

84 .0 50 Ω

100 Ω (a)

118.92 Ω

Ω

84.09 Ω

λ

VS

λ0/8

λ

50 Ω

11 8.

188

100 Ω (b)

Figure 6.12 Cascaded λ/8 transformers that are equivalent to the cascaded λ/4 transformers in Figure 6.10(a).

√ it is 84.09 2 = 118.92 Ω. The completed network is shown in Figure 6.12(a), where the 49.26 Ω characteristic impedance is the parallel combination of 84.09 Ω and 118.92 Ω. In part (b), before combining the central segments, the characteristic impedance of each of the √ first three√segments is 59.46/ 2 = 42.04 Ω and for the last three it is 84.09/ 2 = 59.46 Ω. The completed network is shown in Figure 6.12(b), where the 101.50 Ω characteristic impedance is the series combination of 42.04 Ω and 59.46 Ω. Example 6.3: A network consisting of a lossless transmission line of characteristic impedance Z0 = 10 Ω and electrical length θ = 90◦ transmits power from a 50 Ω source to a 2 Ω load. (a) Find the transducer gain (Gt ) at the frequency where θ = 90◦ . (b) If VS = 200 V (RMS), find I1 , V1 , I2 , and V2 . (c) Find P1 (the power extracted from

189

BROADBAND IMPEDANCE-MATCHING NETWORKS

the source) and PL (the power delivered to the 2 Ω load). Solution: (a) From (4.36), Gt at f = f0 is Gt

= = =

4RS RL |ARL + B + CRS RL + DRS |2 400 |0 + j10 + j100/10 + 0|2 1

(6.45)

(b) The input voltage and current are related to the output (load) voltage and current by V1 = AV2 + BI2 (6.46) I1 = CV2 + DI2

(6.47)

With I1 = VS /100 = 2 A and V1 = 2 × 50 = 100 V, then, I2 = 100/(j10) = −j10 A and V2 = −j10 × 2 = −j20 V. (c) The power input to the transmission line from the source is, P1 = |V1 I1 | = 100 W, and the power delivered to the load is, P2 = |V2 I2 | = 100 W. Example 6.4: The transmission line of characteristic impedance Z0 = 10 Ω and electrical length θ = 90◦ in Example 6.3 is replaced by the LCL network in Figure 6.8 (c). (a) Find the numerical values of ω0 L and ω0 C as functions of Z0 = 10 Ω that make this replacement valid. (b) Find Gt for the LCL circuit. (c) The LCL network is now replaced by smaller sections of transmission lines as shown in Table 3.1 in Chapter 3. If Z01 = 100 Ω for the inductive lines and Y02 = 0.1 S = 101 Ω for the capacitive line, find the associated lengths l1 and l2 in fractional wavelengths. Solution: (a) For the θ = 90◦ , Z0 = 10 Ω line: 

A C

B D



= =

 

cos θ jY0 sin θ 0 jY0

jZ0 0

jZ0 sin θ cos θ 

 (6.48)

190

MICROWAVE TRANSMISSION LINE CIRCUITS

and the ABCD matrix of the LCL network is     A B 1 − ω2 LC jωL(2 − ω2 LC) = C D jωC 1 − ω2 LC   0 jω0 L = (6.49) jω0 C 0 These two matrices are equal at f = f0 if ω02 LC = 1, to yield ω0 L = Z0 = 1/ω0 C. (b) As in Example 6.3 Gt

= = =

4RS RL |ARL + B + CRS RL + DRS |2 400 |0 + j10 + j100/10 + 0|2 1

(6.50)

Note that since Gt = |S21 |2 , then √ ◦ 20 2 RS RL S21 = = = −j1 = 1 × e−j90 ARL + B + CRS RL + DRS j20 (6.51) (c) The inductances are replaced by series sections of transmission line using 2πl1 jω0 L = jZ01 tan , or l1 = 0.0159λ (6.52) λ and the shunt capacitance is replaced by an open-ended section of line using 2πl2 jω0 C = jY02 tan , or l2 = 0.125λ (6.53) λ Example 6.5: If in Example 6.4, ZL = RL + jXL = 2 + j10 Ω, match the 50 Ω source to this ZL by using the same LCL network. Solution: When the j10 Ω inductive reactance of the load was absent, the match to RL = 2 Ω was performed using series jω0 L = j10 Ω, shunt jω10 C = −j10 Ω and series jω0 L = j10 Ω. Therefore, all we need do here is to omit the last series jω0 L in the network, since the exact value of series inductive reactance is already supplied by the load ZL .

BROADBAND IMPEDANCE-MATCHING NETWORKS

191

6.4.3 Impedance Matching a Real Source to a Complex Load The procedure introduced in Example 6.5 for impedance matching a real source (RS ) to a complex load that has series elements (ZL = RL + jXL ) or that has parallel elements (YL = GL + jBL ) may be made more generally useful by the following steps. First, since only the real part of the load impedance absorbs power, √ we can ignore the imaginary part and use a λ/4 transformer of impedance Z0t = RS RL to match RS to RL. For series load elements (ZL = RL + jXL ), if jXL = jω0 LL , use the network in Figure 6.8(c) and let the reactance of the third element of the network be jX3 = jω0 L − jω0 LL

(6.54)

If jXL = −j ω01CL , use the network in Figure 6.8(e) and let the reactance of the third element of the network be   1 1 −j ω0 C jX3 = −j +j = 1− (6.55) ω0 C ω0 C L ω0 C ω0 C L For load elements connected in parallel (YL = GL + jBL ), if jBL = 1/jXL = jω0 CL, use the network in Figure 6.8(b) and let the susceptance of the third element of the network be   ω0 C L jB3 = jω0 C − jω0 CL = jω0 C 1 − (6.56) ω0 C If jBL = 1/jXL = −j ω01LL , use the network in Figure 6.8(d) and let the susceptance of the third element of the network be jB3 =

−j j −j + = ω0 L ω0 LL ω0 L



1−

ω0 L ω0 LL



(6.57)

Note that the third element in each matching network (jX3 or jB3 ) decreases in value as the imaginary part of the load increases from zero in (6.54), (6.55), (6.56) and (6.57), so that the third element is diminished to zero when ω0 LL = ω0 L or ω0 CL = ω0 C. Also note that if the imaginary part of the load is greater than the original value of the third element in (6.54), (6.55), (6.56) and (6.57), these equations correctly yield the value and sign of the new impedance or admittance that becomes jX3 or jB3 of the third element.

192

MICROWAVE TRANSMISSION LINE CIRCUITS

Example 6.6: As in Example 6.5, matching a 50 Ω source to a 2 Ω load requires √ a quarter-wavelength transformer of characteristic impedance Z0t = 50 × 2 = 10 Ω, and each lumped element in Figure 6.8 will have the impedance jω0 L = jZ0t = j10 Ω, or j ω−1 = 0C −j10 Ω (jω0 C = j0.1 S).

(a) If the load impedance is ZL = 2 + j5 Ω, choose the appropriate network in Figure 6.8 and determine jX3 or jB3 for an impedance match to the 50 Ω source. (b) Repeat part (a) if ZL = 2 + j15 Ω.

(c) Repeat part (a) if the load is 2 Ω in parallel with −j5 Ω.

(d) Repeat part (a) if the load is 2 Ω in parallel with −j15 Ω. Solution: (a) Since the load has a series inductance we choose Figure 6.8(c), and use (6.54) to find the new value of the third element as jX3 = j10 − j5 = j5 Ω.

(b) We again use Figure 6.8(c) and (6.54) to find the new value of the third element as jX3 = j10 − j15 = −j5 Ω. Thus, the original j10 Ω inductive reactance of the third element must be replaced by −j5 Ω of capacitive reactance. This capacitive reactance of −j5 Ω resonates with the extra j5 Ω in the load to maintain the impedance match, and the overall bandwidth will be decreased by a small amount.

(c) Here we want to combine parallel capacitive elements, so we choose Figure 6.8(b) and use (6.56) to find the new value of the third element as jB3 = j0.1−j0.2 = −j0.1 S. Thus, the third element is changed from j0.1 S of capacitive susceptance to −j0.1 S of inductive susceptance. Viewed another way, the new value of jX3 = 1/(−j0.1) = j10 Ω is combined in parallel with −j5 Ω of the load to yield −j10 Ω which was the original value of jX3 . (d) Here again, the new value of jX3 must be combined in parallel with −j15 Ω of capacitive reactance in the load to yield the original value of jX3 = −j10 Ω. We again choose Figure 6.8(b) and use (6.56) to find the new value of jB3 = j0.1 − j0.0667 = j0.0333 S. This is the same as adding the new value of jX3 = 1/(j0.0333) = −j30 Ω in parallel with −j15 Ω in the load to yield the original value of jX3 = −j10 Ω.

BROADBAND IMPEDANCE-MATCHING NETWORKS

M Z0

θ=

VS

VS

π f 2 f0

Z0

M Z0

M Z0

193

L C

C

Z0

Figure 6.13 Show that the lumped-element circuit is equivalent to the λ/4 transformer by equating ABCD matrices. Determine the transducer gain of each circuit.

PROBLEMS 1. (a) In Figure 6.13, by equating ABCD matrices, show that the √ lumpedelement circuit is equivalent to the λ/4 transformer if ω L = M Z0 and 0 √ ω0 C = 1/( M Z0 ), where the multiplier M is some positive number and Z0 √ is a real load impedance. (b) If Z0 = 50 Ω, M = 36 or M Z0 = 300 Ω, determine the transducer gain (in dB) of each circuit at f = f0 and f = 2f0 . 2. The equivalent networks shown in Figure 6.14 will provide a broadband impedance match of a 300 Ω TV set to a 75 Ω antenna at a center frequency of 166 MHz. For the lumped-element equivalent circuit shown, determine the values of L and C in µH and pF, respectively. 3. The network shown in Figure 6.14 is to provide a broadband impedance match of a 300 Ω TV set to a 75 Ω antenna at a center frequency of 166 MHz. If all transmission lines are twin-lead in air, (a) determine the total length l = λ0 /2 in cm. (b) If the physical length is to be halved by replacing each λ/4 transformer by a λ/8 transformer, sketch the replacement network and determine all characteristic impedances. 4. The broadband impedance-matching network between RS = 200 Ω and RL = 50 Ω in Figure 6.15 consists of the transmission line segments of characteristic impedance Z01 , Z02 , and Z03 , all λ0 /4 in length at f0 . (a) If the network is to have a Butterworth response versus frequency, determine

194

MICROWAVE TRANSMISSION LINE CIRCUITS

75 Ω VS

λ0/4

λ0/4

Z01 = 106.066 Ω

Z02 = 212.132 Ω

300 Ω

ANTENNA

TV SET

75 Ω VS

L2 C1

L4 C3

C5

300 Ω

Figure 6.14 Transmission line and lumped-element matching networks for Problem 2 and Problem 3.

Z01 , Z02 , and Z03 . (b) Three cascaded λ0 /4 transformers could have been used instead of the shorted stub and two transformers. Which choice would yield the lowest QT , and therefore the broadest bandwidth?

λ0/4

Z01

Z02

RL = 50 Ω

0

λ

/4

Z

VS

λ0/4

03

RS = 200 Ω

Figure 6.15 Broadband impedance-matching network for Problem 4.

BROADBAND IMPEDANCE-MATCHING NETWORKS

195

BIBLIOGRAPHY R. E. Collin, Foundations for Microwave Engineering, 2nd ed. Hoboken, NJ: Wiley-IEEE Press, 2001. R. M. Cottee and W. T. Joines, “Synthesis of Lumped and Distributed Networks for Impedance Matching of Complex Loads,” IEEE Trans. Circuits Syst., vol. 26, no. 5, pp. 316–329, May 1979. T. C. Edwards and M. B. Steer, Foundations of Interconnect and Microstrip Design, 3rd ed. Hoboken, NJ: John Wiley and Sons, 2001. R. S. Elliot, Introduction to Guided Waves and Microwave Circuits. Upper Saddle River, NJ: Prentice-Hall, 1993. V. F. Fusco, Microwave Circuits: Analysis and Computer-Aided Design. Upper Saddle River, NJ: Prentice-Hall, 1987. S. Y. Liao, Microwave Circuit Analysis and Amplifier Design. Upper Saddle River, NJ: Prentice-Hall, 1987. G. Owyang and T. Wu, “The approximate parameters of slot lines and their complement,” IRE Transactions on Antennas and Propagation, vol. 6, no. 1, pp. 49–55, January 1958. D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley and Sons, 2011. P. A. Rizzi, Microwave Engineering: Passive Circuits. Upper Saddle River, NJ: Prentice-Hall, 1988. S. D. Robertson, “The Ultra-Bandwidth Finline Coupler,” IRE Transactions on Microwave Theory and Techniques, vol. 3, no. 6, pp. 45–48, December 1955. G. D. Vendelin, A. M. Pavio, and U. L. Rohde, Microwave Circuit Design, Using Linear and Nonlinear Techniques, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2005. E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications. New York: Wiley-Interscience, 1988.

196

MICROWAVE TRANSMISSION LINE CIRCUITS

Chapter 7 COMBINING, DIVIDING, AND COUPLING CIRCUITS 7.1 POWER DIVIDERS AND POWER COMBINERS When microwave signals are divided, combined, or coupled, the usual requirements are that each input or output port be impedance matched to its source or load, and that certain ports be isolated from each other. While these requirements may be met at a selected operating frequency f = f0 , the performance of the circuit over some bandwidth containing f0 depends (as usual) upon the design of the circuit. 7.1.1 Two-Way Power Divider/Combiner For the two-way power divider in Figure 7.1, power from port 1 is divided equally between ports 2 and 3, with an impedance match at port 1, if Z10 =

2 Z01 = Z0 Z0

or Z01 =

√

2Z0

(7.1)

(7.2)

The S parameter frequency response of the two-way power divider in Figure 7.1 is shown in Figure 7.2. Note, however, that ports 2 and 3 are not impedance matched to Z0 , and that signals entering ports 2 and 3 will be transmitted to each other, as well as to port 1. An impedance match may be achieved at ports 2 and 3, and transmission between

197

198

MICROWAVE TRANSMISSION LINE CIRCUITS

λ0/4 ZS = Z 0 VS

2

Z0

ZL2 = Z0

3

Z0

ZL3 = Z0

Z01 Z0

1 Z01 λ0/4

Figure 7.1 Two-way power divider.

S21

S11

S42

S22

Figure 7.2 S parameter frequency response of the two-way power divider in Figure 7.1.

COMBINING, DIVIDING, AND COUPLING CIRCUITS

199

these ports may be eliminated by connecting a resistor R between ports 2 and 3, as in Figure 7.3. The electrical length of the resistor, including the leads, should be less than a few degrees, and the resistance value should be R = 2Z0 , or 100 Ω if Z0 = 50 Ω. The required value for R in Figure 7.3 is determined as follows: With ports 1 and 3 terminated in Z0 , let a signal input at port 2 be denoted by Vi2 = |Vi2 |6 0◦ . A portion of this signal arrives at port 3 by traveling through the resistor R, denoted by V3R = |V3R |6 0◦ if the resistor path is electrically small. The rest of the signal input at port 2 takes the path from port 2 to port 1, where some of this signal exits at port 1, and the remaining portion arrives at port 3, shifted in phase by 180◦ and denoted by V32 = |V32 |6 180◦. If it happens that |V32 | = |V3R|, then the equal-amplitude opposite-phase signals cancel each other exactly at port 3, thereby creating a null or an effective short-circuit to ground across port 3. To insure that a null does occur under the conditions stated, the value of R is determined by assuming a null or short-circuit at port 3. With port 2 as the input, the impedance looking from port 2 2 toward port 1 through the Z01 , λ0 /4 line is Z12 = Z01 /Z0 = 2Z0 , and the total input impedance at port 2 is Z2 = 2Z0 ||R = 2Z0 R/(2Z0 + R). Thus, for no reflection when a signal is input at port 2 we let Z2 = Z0 , which yields R = 2Z0 . Due to symmetry, an input at port 3 would lead to the same results. Also due to symmetry, the resistor R is effectively invisible to an input at port 1, since the signals at port 1 divide equally and remain in phase at ports 2 and 3, so that no current flows through R. The end result is that the circuit in Figure 7.3 is impedance-matched at all ports and serves as a power divider for an input at port 1, or as a power combiner when equal-amplitude in-phase signals are input at ports 2 and 3. This type of twoway power divider/combiner where all of the ports are impedance matched is known as the Wilkinson power divider/combiner, as the design was first published in 1960 by Ernest Wilkinson. 7.1.2 N -Way Power Divider/Combiner The two-way power divider/combiner is easily extended to become an N -way power divider/combiner, also known as a corporate combiner, by repeating the same two-way network at each output. For example, a four-way power divider/combiner is shown in Figure 7.4, where the underlying, common-ground line is not shown. Since the input and output ports of the basic two-way networks are matched to Z0 , the input and output ports of the four-way network are also matched to Z0 . Likewise, each of the four output ports are isolated from each other by the R = 2Z0 resistor.

200

MICROWAVE TRANSMISSION LINE CIRCUITS

λ0/4 ZS = Z0

2

Z0

Z0

Z0

Z0

Z01 R

1

Z0

VS

Z01 3 λ0/4

Figure 7.3 Two-way power divider/combiner, also known as the Wilkinson power divider/combiner.

Z0

λ0/4 Z01

Z01 RS = Z0 Z01 VS

Z0 = RL

R Z0

R

λ0/4

Z0 Z01

R

Z0 Z0

Figure 7.4 Four-way power divider/combiner.

201

COMBINING, DIVIDING, AND COUPLING CIRCUITS

2Z0

2 2 Z0

λ0/4

VS

λ0/4 Z01

R

Z0

2 Z0

Z0

2Z0 VS

2 2 Z0

λ0/4

2 Z0

λ0/4

Z0

Figure 7.5 Bisection of the four-way power divider/combiner.

Does the four-way power divider/combiner work over a narrower or wider bandwidth of frequencies than the two-way power divider/combiner? To answer this question, assume an input at port 1, and taking advantage of symmetry, bisect the four-way network two times as shown in Figure 7.5. After the second bisection, √ label the characteristic impedances as RS = 4Z0 , RL = Z0 , Z01 = 2 2Z0 = √ 3/4 1/4 1/4 3/4 RS RL , and Z02 = 2Z0 = RS RL . Thus, we observe that each of the four symmetrical sections exhibit a maximally-flat magnitude or Butterworth response between input and output. The four-way network response will be flatter over a given operating band than the two-way network because of the broadbanding effect of the two λ0 /4 transformers in cascade. The S parameter frequency response of the four-way power divider in Figure 7.4 is shown in Figure 7.6. 7.1.3 Unequal Power Division Returning to the basic two-way power divider, an impedance match and an unequal division of power at port 1 may be obtained by using λ0 /4 transmission line sections of unequal characteristic impedance, as indicated in Figure 7.7. With the input impedance matched to Z0 at port 1, all of the incident power (P1 ) enters the junction

202

MICROWAVE TRANSMISSION LINE CIRCUITS

S41

S11 S42 S44 Figure 7.6 S parameter frequency response of the four-way power divider in Figure 7.4.

of Z01 and Z02 and is delivered to the Z0 loads at ports 2 and 3, or P1 = P2 + P3

(7.3)

Due to the phase delays of input voltage and current discussed in section 7.1.1, ports 2 and 3 are isolated from each other, and all of the impedances looking into the three connected lines at port 1 are real at the central operating frequency. At 2 2 port 1 in Figure 7.6, the impedances Z1 = Z0 , Z2 = Z01 /Z0 , and Z3 = Z02 /Z0 are in parallel. Taking the common voltage across the parallel combination as V , then (7.3) becomes, |V |2 |V |2 |V |2 = + (7.4) Z1 Z2 Z3 If we now require that P2 = kP1 and P3 = (1 − k)P1 , where 0 < k < 1, then P2 = k or

|V |2 |V |2 |V |2 Z0 = = 2 Z0 Z2 Z01

(7.5)

Z0 Z01 = √ k

(7.6)

|V |2 |V |2 |V |2 Z0 = = 2 Z0 Z3 Z02

(7.7)

and P3 = (1 − k)

203

COMBINING, DIVIDING, AND COUPLING CIRCUITS

λ0/4 Z01

Z0 = 50 Ω VS

1

2 Z0 R

Z0 Z02 λ0/4

PL2 ZL2 = Z0

3 Z0

PL3 ZL3 = Z0

Figure 7.7 Two-way power divider/combiner with unequal power division.

or Z0 Z02 = √ 1−k

(7.8)

An impedance match to Z0 is desired at ports 2 and 3 if the power divider in Figure 7.7 is to function also as a power combiner. At port 2, impedances are matched if (again assuming a null exists on the opposite end of resistor R), 1 1 Z0 = + 2 Z0 R Z01

(7.9)

or R=

Z0 1−k

(7.10)

Likewise, a match at port 3 requires, 1 1 Z0 = + 2 Z0 R Z02

(7.11)

or R=

Z0 k

(7.12)

Obviously, both ports 2 and 3 can be matched only if k = 0.5. A design choice must be made on which port to match or to use an intermediate value for R.

204

MICROWAVE TRANSMISSION LINE CIRCUITS

7.2 COUPLERS 7.2.1 Branch-Line, or 90-Degree Hybrid Coupler 7.2.1.1 Equal Power Division A four-port network is formed by two side-by-side transmission lines of characteristic impedance Z0 coupled at two points by segments of another line of characteristic impedance Z01 , as shown in Figure 7.8, where each segment is λ0 /4 at some frequency f = f0 . For a signal at f = f0 entering port 1, with all other ports terminated in Z0 , part of that signal arrives at port 2 directly from port 1, but delayed by 90◦ in phase, and another part arrives via the longer path around the rectangle and is delayed by 270◦. These two signals if equal in amplitude would cancel exactly and create a null at port 2 with respect to an input at port 1. Assuming that this cancellation does occur, Z01 is determined as a function of Z0 . With a null or short to ground at port 2, an open circuit is created at λ0 /4 back along the paths taken by the signal injected at port 1, as illustrated in Figure 7.9. Thus, the input impedance at port 1 is, Z1 =

2 Z01 Z0 /2

(7.13)

and setting this equal to Z0 for an impedance match at port 1 yields, Z0 Z01 = √ 2

(7.14)

which completes the design of the branch-line coupler. Ports 1 and 2 in Figure 7.8 are isolated from each other, and power entering port 1 is divided equally between ports 3 and 4. Due to symmetry, any of the ports could serve as an input and the isolation, power division and phase relationship would follow in a similar manner. The output signals at ports 3 and 4 differ in phase by 90◦, so that this coupler also is called a 90◦ hybrid coupler. This type of coupler also can be used to generate the I/Q (in-phase and quadrature) local oscillator signal essential to many modern digital communications systems. 7.2.1.2 Unequal Power Division The design of a 90-degree hybrid coupler with unequal power division follows directly from Figure 7.8 by changing Z0 of the segments between ports 1 and 2

205

COMBINING, DIVIDING, AND COUPLING CIRCUITS

1 P1

Z0

Z0

Z0

Z0

Z0 3 P3 = P1 2

Z0

λ0/4

01

λ0/4

01

Z Z0

2 P2 = 0

Z0

Z0

Z

VS

λ0/4

Z0

4 P4 = P1 2

Figure 7.8 90-degree hybrid coupler with equal power division.

VIRTUAL SHORT OR NULL

Z =∞

1

P1

VS

λ0/4

Z0

Z

01

λ0/4

Z0

Z0

Z0 3

Z0

Z0

Z Z0

2

λ0/4

01

Z0

λ0/4 Z0

4

Figure 7.9 90-degree hybrid coupler with null at port 2 due to an input at port 1.

Z0

206

MICROWAVE TRANSMISSION LINE CIRCUITS

and ports 3 and 4 to Z02 , as shown in Figure 7.10. As before, for a signal at f = f0 entering port 1, with all other ports terminated in Z0 , part of that signal arrives at port 2 delayed by 90◦ in phase, and another part arrives via the longer path delayed by 270◦. Once again, signal cancellation is assumed to occur and create a null at port 2 with respect to an input at port 1, and Z01 and Z02 are determined as a function of Z0 . With signal power P1 injected at port 1 and port 2 grounded, an open circuit is created at λ0 /4 back along the signal paths, as illustrated in Figure 7.11. For no reflections, all of the input power P1 arrives at the node near port 3 and divides between ports 3 and 4 as P3 = kP1 and P4 = (1 − k)P1 . Thus, at the node near port 3: |V |2 P1 = 2 (7.15) Z01 /Z0 |V |2 = kP1 Z0

(7.16)

|V |2 2 /Z = (1 − k)P1 Z02 0

(7.17)

P3 = and P4 =

Solving these three equations yields, √ Z01 = Z0 k and

(7.18)

r Z01 k Z02 = √ = Z0 (7.19) 1 − k 1−k As before, any of the ports could serve as an input and the isolation, power division and phase relationships would follow in a similar way. Example 7.1: In Figure 7.10, the null can be placed at port 2 (as was done in Figure 7.11) or at port 3, thus making P3 = 0 at the central operating frequency. (a) Place the null at port 3, and determine the new design equations that are similar to (7.18) and (7.19) for any 0 < k < 1. (b). Determine Z01 and Z02 for equal power division (k = 0.5) from an input at port 1 to outputs at port 2 and port 4. (c). Repeat part (b) for P2 = 0.9P1 and P4 = 0.1P1. Solution: (a) Referring to Figure 7.10 with a null at port 3, the connection from port 1 to port 3 and from port 4 to port 3 can be ignored

COMBINING, DIVIDING, AND COUPLING CIRCUITS

Z0

1 P1

2 Z02

VS

207

λ0/4 Z01

Z0 P2 = 0 Z01

Z02 P3 = kP1 Z0

3

λ0/4

4

Z0 P4 = (1 − k ) P1

Figure 7.10 90-degree hybrid coupler with unequal power division, where k is the fraction of the input power delivered to port 3 and (1 − k) is the fraction of the input power delivered to port 4.

Z0

1

2

VS

Z0 λ0/4 Z01 Z02 Z0

3

λ0/4

4

Z0

Figure 7.11 90-degree hybrid coupler with unequal power division. A null is generated at port 2 due to an input at port 1.

208

MICROWAVE TRANSMISSION LINE CIRCUITS

in determining the design equations. Taking the common node to be at port 2, the equations for P1 , P2 , and P4 are: |V2 |2 2 Z02 /Z0

(7.20)

|V2 |2 = kP1 Z0

(7.21)

|V2 |2 2 /Z = (1 − k)P1 Z01 0

(7.22)

P1 = P2 = and P4 =

Solving these three equations yields, √ Z02 = Z0 k and Z01

Z02 = √ = Z0 1−k

r

(7.23)

k 1−k

(7.24)

(b) For equal power out at ports 2 and 4, we set k = 0.5 andq determine √ 0.5 Z02 = Z0 0.5 = 0.7071Z0 = 35.355 Ω, and Z01 = Z0 1−0.5 = Z0 = 50 Ω, with reference to Figure 7.10 and Z0 = 50 Ω. (c). For P2 = 0.9P1 and P4 =√0.1P1 in Figure 7.10, we let k = 0.9 and determine q Z02 = Z0 0.9 = 0.9487Z0 = 47.434 Ω, and 0.9 Z01 = Z0 1−0.9 = 3Z0 = 150 Ω, with reference to Figure 7.10 and Z0 = 50 Ω. Following the designs in (b) and (c) of Example 7.1, the S parameter frequency responses for equal and unequal power division are shown in Figure 7.12 and Figure 7.13, respectively. The design is in microstrip format, and the central frequency is 2 GHz. 7.2.2 180-Degree Hybrid Coupler A representation of the 180-degree hybrid coupler, also known as a ring or “ratrace” coupler, is obtained by increasing the length of the line between ports 3 and 4 in Figure 7.8 from λ0 /4 to 3λ0 /4, and letting Z01 be the common characteristic impedance of all line segments connecting the ports, as shown in Figure 7.14. In

COMBINING, DIVIDING, AND COUPLING CIRCUITS

S11

209

S41 S21 S31

Figure 7.12 S parameter frequency response of 90-degree hybrid coupler with equal power out at ports 2 and 4 and no power out at port 3 for a central frequency f0 = 2 GHz. A top view of the design in microstrip format also is shown, with a = 50 Ω, 90◦ and b = 35.355 Ω, 90◦ .

210

MICROWAVE TRANSMISSION LINE CIRCUITS

S21 S41 S31 S11

Figure 7.13 S parameter frequency response of 90-degree hybrid coupler with unequal power out at ports 2 (P2 = 0.9P1 ) and 4 (P4 = 0.1P1) and no power out at port 3 for a central frequency f0 = 2 GHz. A top view of the design in microstrip format also is shown, with a = 150 Ω, 90◦ and b = 47.434 Ω, 90◦ .

COMBINING, DIVIDING, AND COUPLING CIRCUITS

1

Z0

211

2 Z02

VS λ0/4 Z01

Z0 Z01

3

4

Z0

Z0 λ0/4 Z02

Z02 Z02 λ0/4

Figure 7.14 180-degree hybrid coupler, also known as a “rat-race” coupler.

this arrangement, an f = f0 signal input at port 1 would produce a null at port 4. The signal power entering port 1 would be divided equally to ports 2 and 3, which are terminated in Z0 loads. As observed in 7.14, ports 2 and 3 are spaced electrically 180◦ apart, thus the name, 180◦ hybrid coupler. 7.2.2.1 Equal and Unequal Power Division The null at port 4 (with respect to an input at port 1) effectively open-circuits the connections between ports 2 and 4 and ports 3 and 4 in Figure 7.14. Thus the design equations for equal or unequal power division at ports 2 and 3 are determined in a manner similar to that used in Example 7.1. Taking the common node to be at port 1, the equations for P1 , P2 and P3 are: |V1 |2 Z0

(7.25)

|V1 |2 = kP1 2 /Z Z02 0

(7.26)

P1 = P2 = and P3 =

|V|2 2 Z01 /Z0

= (1 − k)P1

(7.27)

212

MICROWAVE TRANSMISSION LINE CIRCUITS

Solving these three equations yields, Z0 Z02 = √ k

(7.28)

Z0 Z01 = √ 1−k

(7.29)

and

Example 7.2: (a) Determine Z01 and Z02 for equal power division from an input at port 1 to ports 2 and 3. (b) Determine Z01 and Z02 for a power division of P2 = 0.9P1 at port 2 and P3 = 0.1P1 at port 3, with P3 = 0. Solution: (a) For equal power out √ at ports 2 and 3, we set k = 0.5 and determine Z01 = Z02 = Z0 0.5 = 70.711 Ω, with reference to Figure 7.14 and Z0 = 50 Ω. (b) For P2 = 0.9P1 and P3 = 0.1P1 in 0 Figure 7.14, we let k = 0.9 and determine Z02 = √Z0.9 = 52.705 Ω, and Z01 =

√ Z0 1−0.9

= 158.114 Ω, for Z0 = 50 Ω.

The S parameter frequency responses of the designs in (a) and (b) of Example 7.2 are shown in Figure 7.15 and Figure 7.16, respectively. The design is in microstrip format, and the central frequency is 2 GHz.

7.3 FREQUENCY DIPLEXERS A division of signal power may be made on the basis of frequency within a given operating band. A frequency diplexer sends all signals below a given fc (cut-off or cross-over frequency) to one output port and all signals above fc to another output port. To accomplish this, a lowpass (LP) and a highpass (HP) filter are connected with their inputs in parallel or in series at port 1 and their outputs going to port 2 and port 3. These two arrangements are shown in Figure 7.17. This type of circuit commonly is used to separate the uplink and downlink frequencies for the communications systems described in Chapter 1. An input impedance match is obtained at all frequencies for the series connection if, zin = zLP + zHP = 1 (7.30)

COMBINING, DIVIDING, AND COUPLING CIRCUITS

213

S11 S31 S21 S41

Figure 7.15 S parameter frequency response of 180-degree hybrid coupler with equal power out at ports 2 and 3 and no power out at port 4 for a central frequency f0 = 2 GHz. A top view of the design in microstrip format also is shown, with a = 70.711 Ω, 90◦ .

214

MICROWAVE TRANSMISSION LINE CIRCUITS

S11

S21 S31

S41

Figure 7.16 S parameter frequency response of 180-degree hybrid coupler with unequal power out at ports 2 (P2 = 0.9P1 ) and 3 (P3 = 0.1P1) and no power out at port 4 for a central frequency f0 = 2 GHz. A top view of the design in microstrip format also is shown, with a = 52.705 Ω, 90◦ and b = 158.114 Ω, 90◦ .

COMBINING, DIVIDING, AND COUPLING CIRCUITS

215

zLP LP

1

HP

1

LP

1

HP

1

zin zHP (a) yLP

yin yHP (b) Figure 7.17 Frequency diplexers using lowpass (LP) and highpass (HP) network combinations with (a) network inputs connected in series and (b) network inputs connected in parallel.

Likewise, an input match is obtained at all frequencies for the parallel connection if, yin = yLP + yHP = 1 (7.31) For these relationships to be true at all frequencies, the denominator polynomials must be common, and the numerator polynomials must add and be equal to the denominator polynomial. The only LP and HP networks meeting these requirements are the Butterworth networks in Appendix A, and HP must be the compliment of LP obtained by replacing ω by 1/ω. For the series connection using impedances, the ideal current source prototype must be used, where R = ∞ in the network at the top of the page. For the parallel connection using admittances, the ideal voltage source prototype must be used, where R = 0 in the network at the bottom of the page.

216

MICROWAVE TRANSMISSION LINE CIRCUITS

Example 7.3: From Appendix A, a Butterworth LP√network with R = 0 √ (or 1/R = ∞) and n = 2 has L1 = 2, C2 = 1/ 2 and is terminated in √ zL = yL = 1. The complimentary HP network has C10 = 1/L √ 1 = 1/ 2 in series with the parallel combination of L02 = 1/C2 = 2 and a unity load. (a) To function as a diplexer with a perfect source match at all frequencies, should these networks be connected with their inputs in series or in parallel? (b) Taking RS = RL = 50 Ω, de-normalize the network elements for ω = ωc (the cut-off or cross-over frequency) as in Table 5.2. (c) Connect the networks to source and loads, and show that the input impedances ZLP and ZHP combine in such a way that yields Zin = 50 Ω. Solution: (a) As indicated earlier, when the ideal voltage source prototype is used, the lowpass and highpass networks must be connected in parallel. (b) From Table 5.2, all inductors convert to their actual impedance in ohms as, jωc La = j50L

(7.32)

and all capacitors convert to their actual impedance in ohms as, 1 50 = −j jωc Ca C

(7.33)

where L and C√ are the normalized elements in √ each circuit. Thus, jωc L1a = j50 2 = j70.71 Ω, jωc1C2a = −j50 2 = −j70.71 Ω, √ √ 1 = −j50 2 = −j70.71 Ω and jωc L02a = j50 2 = j70.71 Ω. 0 jωcC1a (c) The LP and HP networks are connected in parallel as shown in Figure 7.18, where it is seen that, ZLP = j70.71 +

50(−j70.71) 100 √ Ω = 50 − j70.71 1−j 2

(7.34)

and ZHP = −j70.71 +

50(j70.71) 100 √ Ω = 50 + j70.71 1+j 2

(7.35)

These two impedances added in parallel yield Zin = ZLP ||ZHP = 50 Ω at all frequencies.

217

COMBINING, DIVIDING, AND COUPLING CIRCUITS

LP

j 70.71 Ω

50 Ω

1

− j 70.71 Ω

VS

50 Ω

3

− j 70.71 Ω j 70.71 Ω

Zin = 50 Ω

2

50 Ω

HP Figure 7.18 The n = 2 frequency diplexer using LP and HP networks with their individual input impedances combined in parallel to yield a 50 Ω input impedance at the source.

S21

S31 S22

S33

Figure 7.19 Computer simulated S parameter response versus frequency of the diplexer in Figure 7.18.

218

MICROWAVE TRANSMISSION LINE CIRCUITS

The computer simulated S parameter response versus frequency of the diplexer in Figure 7.18 is shown in Figure 7.19 over the frequency range from 0 to 2 GHz, where the cross-over or cut-off frequency is 1 GHz. The input reflection, S11 in dB, does not appear in Figure 7.19 because it is off-scale and below −100 dB at all frequencies, as predicted by theory. It should be clear from Figures 7.18 and 7.19 that since ports 2 and 3 are terminated in 50 Ω loads, these ports can be the inputs to two additional diplexers that divide the signal frequencies below 1 GHz into LP and HP channels, and that divide the signal frequencies above 1 GHz into LP and HP channels. Thus, four different output channels of signal frequencies between 0 and 2 GHz would be created. This process can be repeated at the outputs any number of times, resulting in an important application called frequency-division multiplexing (or wavelength-division multiplexing). Another important application for a frequency diplexer is for improving the operation of modulators and demodulators used in communication systems. This application is illustrated by modifying Figure 1.6 from Chapter 1 and shown here as Figure 7.20. At the output of the modulator in Figure 7.20, the higher frequency carrier wave and the lower frequency baseband signals form sidebands of signal information spaced above and below the carrier wave frequency. Either sideband contains all of the signal information, but the upper sideband should be transmitted and received, so the antennas can be more efficient and smaller in size. Thus, in the upper part of the block diagram all signals below the carrier wave are terminated in a matched load (often 50 Ω), and all signals above the carrier wave are passed on to the amplifier and the transmitting antenna. The received upper sideband is passed through the amplifier and demodulator where on the output the same upper and lower sidebands are once again created. Here we want to select the lower sideband, which is the baseband of information signals. If we did not use diplexer filters in the transmit and receive legs as indicated, the rejected sideband would be reflected back into the modulator or demodulator and much distortion of the information would occur.

PROBLEMS 1. For the two-way power divider in Figure 7.21, determine Z01 and Z02 such that Γ = 0 at port 1, PL2 = (1/4)P1 , and PL3 = (3/4)P1 . 2. In Problem 1, determine the value of an isolation resistor placed between ports 2 and 3 so that all ports are matched to 50 Ω. If the condition is impossible to realize, find a compromise solution.

219

COMBINING, DIVIDING, AND COUPLING CIRCUITS

LOAD

LP BASEBAND IN

CARRIER OSCILLATOR

HP

MODULATOR

AMPLIFIER

ANTENNA

DEMODULATOR

AMPLIFIER

ANTENNA

LOAD HP LP

BASEBAND OUT

Figure 7.20 Modification of Figure 1.6 to show how the modulation and demodulation processes may be greatly improved (less distortion) by using frequency diplexer filters.

2 50 Ω VS

1 Z0 = 50 Ω

λ0/4 Z01 Z02 λ0/4

PL2 R

ZL2 = Z0

3 PL3 ZL3 = Z0

Figure 7.21 Determine Z01 and Z02 such that Γ = 0 at port 1.

220

MICROWAVE TRANSMISSION LINE CIRCUITS

Z0 θ Z0 VS

Z0

Z01

Z02

θ

Z02 θ

Z0 = 50 Ω

R Z0

Z0 = 50 Ω

Figure 7.22 Determine Z01 and Z02 for a maximally flat magnitude response for the power divider circuit. Determine R for no reflections from any port when the circuit is used as a power combiner.

3. (a) If Z0 = 50 Ω and θ = π2 ff0 , determine Z01 and Z02 for a maximally flat magnitude response for the power divider circuit in Figure 7.22. (b) Determine R for no reflections from any port when the circuit is used as a power combiner. 4. For the branch-line coupler in Figure 7.23, where relevant line lengths are λ0 /4 at f = f0 , (a) determine Z01 and Z02 such that ports 1 and 2 are isolated from each other, and, of the power P1 incident upon port 1, 1/3 of P1 will exit at port 4 and the other 2/3 of P1 will exit at port 3. (b) Write the scattering matrix which describes the coupler operation at f = f0 . 5. For the branch-line coupler in Figure 7.24, determine Z01 and Z02 such that ports 1 and 2 are isolated from each other, Γ = 0 at port 1, PL4 = (3/4)P1 , and PL3 = (1/4)P1 . 6. The substrate of the microstrip line circuit in Figure 7.25 is epoxy-glass (r = 4.8). If Z0 = 50 Ω and all line segments have an electrical length of 45◦, determine all lengths and all widths in inches for f0 = 1 GHz.

COMBINING, DIVIDING, AND COUPLING CIRCUITS

λ0/4

1

Z0 VS

2 Z0 = 50 Ω λ0/4

01

Z

01

Z

Z02

Z0

221

Z0 = 50 Ω

Z02 3

4

Figure 7.23 Determine Z01 and Z02 such that ports 1 and 2 are isolated from each other, and, of the power P1 incident upon port 1, 1/3 of P1 will exit at port 4 and the other 2/3 of P1 will exit at port 3.

Z0

Z0

1

Z02

2

Z0

λ0/4

VS

Z01 λ0/4

ZL2 = Z0 = 50 Ω

Z01 Z02

ZL4 = Z0

Z0

3

4

Z0

ZL3 = Z0 = 50 Ω

Figure 7.24 Determine Z01 and Z02 such that ports 1 and 2 are isolated from each other, Γ = 0 at port 1, PL4 = (3/4)P1, and PL3 = (1/4)P1.

MICROWAVE TRANSMISSION LINE CIRCUITS

Z0

R=

2Z

Z0

0

222

1 INCH 16

2Z0

2Z0 EPOXY-GLASS

tB0

Z0

w1

w0

Figure 7.25 For this microstrip circuit, determine all lengths and all widths in inches using the parameters from Problem 6.

BIBLIOGRAPHY R. E. Collin, Foundations for Microwave Engineering, 2nd ed. Hoboken, NJ: Wiley-IEEE Press, 2001. T. C. Edwards and M. B. Steer, Foundations of Interconnect and Microstrip Design, 3rd ed. Hoboken, NJ: John Wiley and Sons, 2001. R. S. Elliot, Introduction to Guided Waves and Microwave Circuits. Upper Saddle River, NJ: Prentice-Hall, 1993. V. F. Fusco, Microwave Circuits: Analysis and Computer-Aided Design. Upper Saddle River, NJ: Prentice-Hall, 1987. H. Howe, Stripline Circuit Design. Dedham, MA: Artech House, 1974. S. Y. Liao, Microwave Circuit Analysis and Amplifier Design. Upper Saddle River, NJ: Prentice-Hall, 1987. D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley and Sons, 2011. P. A. Rizzi, Microwave Engineering: Passive Circuits. Upper Saddle River, NJ: Prentice-Hall, 1988.

COMBINING, DIVIDING, AND COUPLING CIRCUITS

223

G. D. Vendelin, A. M. Pavio, and U. L. Rohde, Microwave Circuit Design, Using Linear and Nonlinear Techniques, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2005. E. J. Wilkinson, “An N-Way Hybrid Power Divider,” IRE Transactions on Microwave Theory and Techniques, vol. 8, no. 1, pp. 116–118, January 1960. E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications. New York: Wiley-Interscience, 1988.

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MICROWAVE TRANSMISSION LINE CIRCUITS

Chapter 8 TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS 8.1 IMPEDANCE MATCHING FOR MAXIMUM GAIN A transistor amplifier, with passive input and output signal processing networks N1 and N2 , is represented in Figure 8.1. The characteristic impedance of connecting lines, and the impedance to the left of N1 and to the right of N2 , is Z0 (usually 50 Ω). The transducer gain of the amplifier may be expressed in terms of the ABCD parameters or the scattering parameters as: 4RS RL |AZL + B + CZS ZL + DZS |2

(8.1)

(1 − |ΓS |2 )|S21 |2 (1 − |ΓL|2 ) |(1 − S11 ΓS )(1 − S22 ΓL ) − S21 S12 ΓS ΓL |2

(8.2)

Gt = or Gt =

where the source and load reflection coefficients are: ΓS =

ZS − Z0 ZS + Z0

(8.3)

ΓL =

ZL − Z0 ZL + Z0

(8.4)

and

225

226

MICROWAVE TRANSMISSION LINE CIRCUITS

Z0 VS

Z0 TRANSISTOR Z0 AMPLIFIER

N1 ZS

Z2

Z2

N2

Z0

ZL

Figure 8.1 Transistor amplifier with passive input and output signal processing networks N1 and N2 .

If the reverse transmission coefficient S12 = 0, the amplifier is unilateral, and (8.2) becomes 2 1 − |ΓS |2 2 1 − |ΓL | Gtu = |S | (8.5) 21 |1 − S11 ΓS |2 |1 − S22 ΓL |2

In this product of three terms, the first term is controlled by the input network N1 , the second is due to the amplifier alone, and the last term is controlled by the output network N2 . If N1 is chosen such that ZS∗ = Z1 or Γ∗S = S11 , and N2 is chosen such that ZL∗ = Z2 or Γ∗L = S22 , then the maximum gain is Gtu (MAX) =

2 1 − |S11 |2 2 1 − |S22 | |S | 21 |1 − |S11 |2 |2 |1 − |S22 |2 |2

(8.6)

1 1 |S21 |2 |1 − |S11 |2 | |1 − |S22 |2 |

(8.7)

or Gtu(MAX) =

Example 8.1: For the NE 21889 gallium arsenide metal semiconductor field effect transistor (GaAs MESFET), the S parameters at 4 GHz are: S11

=

S12 S21 S22

= = =

0.756 −117◦

0.126 15◦ 2.566 79◦ 0.546 −78◦

with VDS = 3 V and IDS = 10 mA. Making the unilateral assumption (S12 = 0), determine Gtu when the transistor is (a) conjugately matched at input and output, (b) connected to Z0 ohms at the input and output.

227

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

Solution: (a) For S12 = 0, ZS∗ = Z1 , and ZL∗ = Z2 , substitution of the remaining S parameters into (8.7) yields: Gtu (MAX) = 2.29 × 6.55 × 1.41 = 21.15

(8.8)

Gtu (MAX) = 3.60 + 8.16 + 1.49 = 13.25 dB

(8.9)

or, in dB:

(b) In a Z0 Ω unmatched system, ZS = ZL = Z0 , ΓS = 0, and ΓL = 0. Thus, from (8.2) or (8.5), Gt = |S21 |2 = 6.55

(8.10)

or 8.16 dB. Note that conjugately matching the input added 3.60 dB to the total gain, and matching the output added 1.49 dB. Maximum gain and minimum noise figure generally require different impedance conditions. For example, from the manufacturer’s specifications on the NE 21889, a minimum noise figure (of 1 dB) at 4 GHz requires ΓS = Γopt = ∗ 0.566 103◦ 6= S11 . While the imput impedance is purposely mismatched to achieve minimum noise figure, the output may still be conjugately matched for maximum power transfer to the load. 8.1.1 Transistor Equivalent Circuit The transistor may be modeled by an equivalent circuit, and a simple one is illustrated in Figure 8.2. From circuit theory, the input and output impedances of the equivalent circuit are:   1 Z1 = R1 + j ωL1 − (8.11) ωC1 and Z2 =

  R2 ωR22 C2 + j ωL − 2 1 + (ωR2 C2 )2 1 + (ωR2 C2 )2

(8.12)

For an unpackaged chip, the lead inductances L1 and L2 may be negligible; but for a packaged device, these inductances must be included in the equivalent circuit.

228

MICROWAVE TRANSMISSION LINE CIRCUITS

L1

R1

L1

vi

C1

gmvi

C2

R2

Figure 8.2 Simplified transistor equivalent circuit.

In the unilateral case (S12 = 0), Z1 and Z2 versus frequency are easily determined from S11 and S22 since   1 + S11 Z1 = Z0 (8.13) 1 − S11 and



 1 + S22 Z2 = Z0 (8.14) 1 − S22 If S11 and S22 are known at two frequencies, all of the components in the equivalent circuit of Figure 8.2 may be determined from (8.11) through (8.14). Example 8.2: The S parameters of the NE 21889 GaAs MESFET at 4 GHz are: S11

=

S12 S21 S22

= = =

S11 S12

= =

S21 S22

= =

0.756 −117◦ 0.126 15◦ 2.566 79◦ 0.546 −78◦

and at 6 GHz they are: 0.676 −160◦ 0.116 −6◦ 2.026 43◦ 0.506 −110◦

Making the unilateral assumption (S12 = 0), and taking Z0 = 50 Ω, determine all of the component values in the equivalent circuit of Figure 8.2.

229

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

Solution: From (8.13), Z1 at 4 GHz is: Z1 = 9.80 − j29.42 Ω

(8.15)

Z1 = 10.20 − j8.48 Ω

(8.16)

and at 6 GHz: Since R1 is the real part of Z1 , R1 = 10 Ω is a reasonable choice for operation over the 4 to 6 GHz band. Setting the imaginary part of (8.11) equal to −29.42 at 4 GHz, and equal to −8.48 at 6 GHz, the two equations are solved to obtain: L1 = 0.532 nH and C1 = 0.930 pF. From (8.14), Z2 at 4 GHz is: Z2 = 33.19 − j49.50 Ω

(8.17)

Z2 = 23.55 − j29.52 Ω

(8.18)

and at 6 GHz: Equating the real and imaginary parts of (8.12) to the calculated values at 4 GHz and at 6 GHz yields: R2 = 530.57 Ω, L2 = 2.12 nH, and C2 = 0.232 pF. Example 8.3: (a) Determine the input and output networks (N1 and N2 ) to conjugately match the NE 21889 field effect transistor (FET) for maximum gain at 4 GHz, again assuming that S12 = 0. (b) Using a 1/32 inch dielectric board with r = 2.20, find the dimensions of the microstrip circuit to realize N1 and N2 . Solution: (a) Since the transistor input impedance at 4 GHz is Z1 = 9.8 − j29.42 Ω, the imaginary part is cancelled by adding a series inductive reactance ωLn1 = 29.42 Ω. Looking through this series inductance toward the transistor input, the impedance is R1 = 9.8 Ω. To match R1 to the signal source of impedance Z0 = √ 50 Ω requires a λ/4 transformer of characteristic impedance Z01 = 50 × 9.8 = 22.14 Ω. In a similar manner, the transistor output impedance at 4 GHz is Z2 = 33.19 − j49.50 Ω, and the reactive term is cancelled by adding a series inductive reactance ωLn2 = 49.50 Ω. The remaining real part of Z2 is matched to the Z0 = 50 Ω√termination by a λ/4 transformer of characteristic impedance Z02 = 33.19 × 50 = 40.74 Ω.

230

MICROWAVE TRANSMISSION LINE CIRCUITS

(b) The gate and drain leads of the NE 21889 stripline package are 0.1 mm thick and 0.51 mm wide. On the microstrip substrate specified, these leads form a characteristic impedance of 116 Ω and may be used to simulate the required series inductive reactances ωLn1 and ωLn2 as, ωLn1 = 116 tan

2π ln1 = 29.42 Ω λ

(8.19)

and

2π ln2 = 49.50 Ω (8.20) λ With r = 2.20, an impedance of 116 Ω yields eff = 1.74, and λ at 4 GHz is 56.86 mm. Solving for ln1 and ln2 yields, ωLn2 = 116 tan

ln1 =

29.42 λ tan−1 = 2.25 mm 2π 116

(8.21)

and

λ 49.50 tan−1 = 3.65 mm (8.22) 2π 116 For the input λ/4 transformer, 22.14 Ω corresponds to a width of 7.21 mm and eff = 1.99, so λ/4 = 13.29 mm at 4 GHz. Similarly, the output λ/4 transformer of 40.74 Ω has a width of 3.30 mm and eff = 1.90, so λ/4 = 13.60 mm at 4 GHz. ln2 =

The solution resulting from Example 8.3 is diagrammed in Figure 8.3, together with the bias circuitry and extra capacitors (C0 ) that must be added to isolate the DC bias from the microwave signals to be amplified. The capacitor C0 is made large enough so that 1/(ωC0 ) ≤ 0.1 Ω. This impedance easily will pass all the microwave signals from source to load, while blocking the DC bias from source and load. C0 at the bias feed point produces an effective short circuit to ground, which appears as an open circuit at the other end of the 150 Ω, λ/4 line and so does not load the impedance matching network. Caution must be taken to select a capacitor of the appropriate size and characteristics such that it does maintain the desired low impedance at and well above the operating frequency of the amplifier. 8.1.2 Stability Conditions and the Bilateral Amplifier If S12 6= 0 (the amplifier is bilateral), the question of stability becomes an important factor. With reference to Figure 8.1, the amplifier is unconditionally stable at a given

231

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

C0

C0

−VGG

+VDD

R2

R1

R3

150 Ω, λ/4

C0 (CHIP) 50 Ω

G

50 Ω

22.14 Ω λ/4

R4 150 Ω, λ/4

C0 (CHIP)

D

S

40.74 Ω λ/4

116 Ω 2.25 mm

50 Ω

50 Ω

116 Ω 3.65 mm FET

Figure 8.3 Diagram of the solution resulting from Example 8.3, showing the bias circuitry and extra blocking capacitors (C0 ).

frequency if Re[ZS + Z1 ] > 0 and Re[ZL + Z2 ] > 0, so that input and output loop currents cannot circulate in the absence of voltage. The amplifier is potentially unstable if some passive ZS and ZL cause Re[ZS + Z1 ] < 0 and Re[ZL + Z2 ] < 0. Since ZS and ZL are always passive for the amplifier considered in Figure 8.1, the requirements for unconditional stability may be expressed as |ΓS | < 1

Z1 − Z0 = S11 + |Γ1 | = Z1 + Z0 Z2 − Z0 = S22 + |Γ2 | = Z2 + Z0

|ΓL| < 1 S12 S21 ΓL 0 and Re[ZL + Z2 ] > 0 to be true. This can be accomplished by resistively loading the transistor or by using negative feedback. Several examples using resistive loading are given. Example 8.5: The S parameters of a bipolar junction transistor (BJT) at 800 MHz are: S11 = 0.656 −95◦, S12 = 0.0356 40◦ , S21 = 56 115◦ , and S22 = 0.86 −35◦ . If the transistor is potentially unstable, make it unconditionally stable by adding a resistor on the output from collector to emitter, and use (8.7) to determine Gtu (MAX) before and after the resistor is included. Solution: Using the S parameters given, K = 0.547 and ∆ = 0.5046 249.6◦. Since K < 1, the transistor is potentially unstable for some passive values of ZS and ZL . Before resistive loading (using for convenience the unilateral gain), Gtu (MAX) = G1 +G0 +G2 = 2.38+13.98+4.44 = 20.8 dB (8.43) As mentioned above, resistive loading can stabilize a previously unstable transistor. Choosing a resistance much larger than the load impedance, say 500 Ω, so as not to affect the impedance match, a resistor is added in parallel with the output. For the combination the new S parameters are: S11 = 0.656 −94◦, S12 = 0.0326 41.2◦, S21 = 4.626 116.2◦, and S22 = 0.666 −36◦ . (These are obtained by converting the original S parameters to ABCD parameters, determining the ABCD matrix for the transistor-resistor combination, and then converting the ABCD parameters back to S parameters). From the new S parameters, K = 1.04 and ∆ = 0.4096 250.13◦. Thus, the transistor is

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

235

now unconditionally stable. Also, from the new S parameters, Gtu (MAX) = G1 + G0 + G2 = 2.38 + 13.29 + 2.48 = 18.15 dB (8.44) Example 8.6: The S parameters of a field effect transistor at 4 GHz are: S11 = 0.886 −79◦ , S12 = 0.066 35◦ , S21 = 2.856 107◦, and S22 = 0.616 −58◦. If the transistor is potentially unstable, make it unconditionally stable by adding a resistor on the output from drain to source, and determine Gtu (MAX) before and after the resistor is included. Solution: Using the S parameters given, K = 0.46 and |∆| = 0.55. Since K < 1, the transistor is potentially unstable for some passive values of ZS and ZL . Before resistive loading, Gtu (MAX) = G1 +G0 +G2 = 6.47+9.10+2.02 = 17.6 dB (8.45) A 50 Ω resistor is added in parallel with the output, and for the combination the new S parameters are: S11 = 0.916 −77◦ , S12 = 0.046 44◦, S21 = 1.76 116◦ , and S22 = 0.256 −134◦ . (These are obtained by converting the original S parameters to ABCD parameters, determining the ABCD parameters for the transistor-resistor combination, and then converting back to S parameters). From the new S parameters, K = 1.10 and |∆| < 1. Thus, the transistor is now unconditionally stable. Also, from the new S parameters, Gtu (MAX) = G1 +G0 +G2 = 7.65+4.61+0.28 = 12.54 dB (8.46) 8.3 DYNAMIC RANGE OF A TRANSISTOR AMPLIFIER The dynamic range of an electrical device (in this case a transistor amplifier) may be defined as: Dynamic Range (dB)

= Upper input power limit (dBm) − Lower input power limit (dBm)

(8.47)

or DR (dB) = PinMAX (dBm) − PinMIN (dBm)

(8.48)

236

MICROWAVE TRANSMISSION LINE CIRCUITS

The lower input power limit for useful gain is set by the background noise level of the amplifier. The upper input power limit is set by saturation effects (gain compression) and resulting distortion caused by intermodulation products. 8.3.1 Lower Input Power Limit (Minimum Detectable or Usable Signal) For a transistor amplifier to be useful, the output signal must be detectable in the presence of noise. Let us define PinMIN as twice the total noise power referred to the amplifier input as: PinMIN = 2Nout /G (8.49) or PinMIN (dBm) = Nout (dBm) − G (dB) + 3 dB

(8.50)

where G is amplifier gain and Nout is the total noise power at the output of the amplifier. This quantity sometimes is referred to as the minimum detectable signal. The above equation is put into a more useful form by introducing the noise figure (F ) of the amplifier as: F =

Pin /Nin Nout Na = =1+ Pout/Nout GNin GNin

(8.51)

where Nout = GNin + Na , Nin is the noise power at the amplifier input, and Na is the output noise power contributed by the amplifier. In terms of amplifier parameters at a given operating frequency, a simple expression for the minimum noise figure (Fmin ) is s R1 (8.52) Fmin = 1 + Kf ωC1 gm where R1 and C1 are the series input resistance and capacitance, respectively, of the transistor circuit model in Figure 8.2, ω = 2πf, and Kf is a frequency-independent fitting factor. Thus, if Fmin , C1 , R1 , and gm are known at one frequency, Kf may be determined for use with the same amplifier at other frequencies. For example, a certain amplifier at 4 GHz has R1 = 10 Ω, C1 = 0.93 pF, gm = 0.04 S, and Fmin = 1.20 (or 0.79 dB). From (8.52) this yields Kf = 0.54. Using this value of Kf at 5 GHz, where gm = 0.03 and R1 and C1 remain the same, yields Fmin = 1.29 (or 1.10 dB). When Nout = F GNin is substituted into the PinMIN equation, we have PinMIN = 2F Nin

(8.53)

237

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

or, in dB PinMIN (dBm) = Nin (dBm) + F (dB) + 3 dB

(8.54)

Statistical analysis shows that the input noise power may be expressed as Nin = kT B

(8.55)

where k is Boltzmann’s constant (1.38 × 10−23 J/K), T is temperature in degrees Kelvin (T = T0 = 290 K is room temperature), and B is the system bandwidth (usually expressed in MHz). Expressing kT B in dBm at room temperature with B in MHz yields Nin (dBm) = kT BMHz (dBm) = −114 dBm + BMHz (dB)

(8.56)

where BMHz(dB) = 10 log(BMHz ). Thus, PinMIN becomes PinMIN (dBm) = −111 dBm + BMHz (dB) + F (dB)

(8.57)

Example 8.7: (a) Determine the minimum detectable input power of an amplifier with noise figure F (dB) = 1 dB (F = 1.26) and B = 10 MHz. (b) Repeat part (a) for a 1000 MHz bandwidth. Solution: (a) Substituting into (8.57), PinMIN (dBm) = −111 dBm + 10 + 1 = −100 dBm

(8.58)

or PinMIN = 10−10 mW = 10−13 W = 0.1 pW

(8.59)

(b) Again, substituting into (8.57), PinMIN (dBm) = −111 dBm + 30 + 1 = −80 dBm

(8.60)

or PinMIN = 10−8 mW = 10−11 W = 10 pW

(8.61)

It should be noted that the calculation for minimum detectable power in Example 8.7 is valid only at a noise temperature of 290 K. If the input noise were higher, as it usually is in real applications, then the noise floor is higher and the minimum detectable power also increases.

238

MICROWAVE TRANSMISSION LINE CIRCUITS

8.3.2 Upper Input Power Limit The input power must not exceed a value that makes the output power too large. The output power is related to the input power by the gain of the amplifier as, Pout = GPin

(8.62)

Pout(dBm) = GdB + Pin (dBm)

(8.63)

or If GdB is constant for all Pin , then dPout(dBm) =1 dPin (dBm)

(8.64)

or, a plot of Pout(dBm) versus Pin (dBm) is a straight line of unity slope. This is illustrated in Figure 8.4 for a transistor with a gain of 20 dB (dashed line). The actual power out and gain versus power in are shown by the solid curves. The upper input power limit is often chosen as the input power where the actual output power is 1 dB below the straight-line or ideal power out. This level of input power is called the 1 dB gain compression point, and on this basis: PinMAX (dBm) = Pin (dBm) at 1 dB gain compression point

(8.65)

or approximately −20 dBm for the transistor in Figure 8.4. This sets the upper input power limit for single-frequency operation, and typically is taken as the upper boundary of the region where small-signal analysis using S parameters is valid. For multiple-frequency operation, let f1 and f2 be two primary signals in the band to be transmitted simultaneously. Then, expanding Pout in a power series and retaining the first three terms: 2 3 = P1 + P2 + P3 Pout = K1 Pin + K2 Pin + K3 Pin

(8.66)

Due to the first term in the expansion, P1 = K1 Pin , input signals at f1 and f2 will yield output signals at f1 and f2 that constitute the ideal linear response. The 2 second term, P2 = K2 Pin , will cause inputs at f1 and f2 to produce output signals at 2f1 , 2f2 , f1 +f2 , and f2 −f1 . These signals usually lie well outside of the frequency 3 band between f1 and f2 . The third term in the power series, P3 = K3 Pin , will cause inputs at f1 and f2 to produce output signals at 2f1 + f2 , 2f1 − f2 , 2f2 + f1 , and 2f2 − f1 , known as intermodulation products. The intermodulation products at

239

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

20 IDEAL RESPONSE

10

POWER OUT

-10

1 dB GAIN COMPRESSION POINT

ACTUAL RESPONSE

-20

20

-30

15

GAIN (dB)

POWER OUT (dBm)

0

GAIN

-40

10

-50

5

-60 -60

-50

-40

-30 -20 -10 0 POWER IN (dBm)

10

20

Figure 8.4 Ideal and actual power out and gain versus power in for a typical transistor amplifier with 20 dB of gain showing 1 dB compression point.

240

MICROWAVE TRANSMISSION LINE CIRCUITS

2f1 − f2 and 2f2 − f1 usually will fall on either side of f1 and f2 (in-band), and are the most troublesome. For example, if f1 = 100 MHz and f2 = 110 MHz, 2f1 − f2 = 90 MHz and 2f2 − f1 = 120 MHz. One usually tries to make these components at least 20 or 30 dB down from the primary power P1 . Expressing P1 , P2 , and P3 of the power series in dB yields: P1 (dBm) = K1 (dB) + Pin (dBm)

(8.67)

P2 (dBm) = K2 (dB) + 2Pin (dBm)

(8.68)

P3 (dBm) = K3 (dB) + 3Pin (dBm)

(8.69)

Since the Ks are constant, the slopes of P1 , P2 , and P3 versus Pin are 1, 2, and 3, respectively, as illustrated in Figure 8.5. For low values of Pin , P2 and P3 are well below P1 (K2 and K3  K1 ). However, because of the increased slopes of P2 and P3 versus Pin , the projected lines of P1 , P2 , and P3 versus Pin will intersect at some value of Pin (usually well above any normal value of Pin ). Specifying a common intercept point for P1 , P2 , and P3 is the most convenient way to characterize the transistor amplifier for intermodulation distortion effects. The primary power out P1 versus Pin is the same straight line of unity slope plotted in Figure 8.4. The 1 dB gain compression output power is usually given in the transistor data sheets. For most transistors, the third-order intercept point is around 11 dB above the 1 dB gain compression point. The second-order intercept point is usually higher by 5 to 10 dB, but as a worst case approximation, we let all three lines intersect at 11 dB above the 1 dB gain compression point, as shown in Figure 8.6. Example 8.8: A transistor amplifier has the following specifications: G = 10 dB, F = 1 dB, the 1 dB gain compression point occurs at Pin = −30 dBm, the amplifier operates at room temperature (T = 290 K) with a bandwidth B = 100 MHz, and the primary, second-order, and third-order intermodulation signals intersect at 11 dB above the 1 dB gain compression point. If third-order signals are to be at least 35 dB below primary power signals, what is the dynamic range in dB? Solution: At the intersect point, the input and output powers are: Pin = −30 dBm + 11 = −19 dBm

(8.70)

241

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

P1

P2

P3 3

2 1 1

1

Pin

1

Pin

Pin

Figure 8.5 Slopes of P1 , P2 , and P3 versus Pin .

and Pout = −19 dBm + 10 = −9 dBm

(8.71)

From (8.69), with P3 = −9 dBm and Pin = −19 dBm, K3 = 48 dB, or P3 = 48 + 3Pin (dBm) (8.72) Since from (8.67), P1 = 10 + Pin (dBm)

(8.73)

the 35 dB difference requires, P1 − P3 = 10 + Pin (dBm) − 48 − 3Pin (dBm) = 35

(8.74)

10 − 48 − 35 = −36.5 dBm = PinMAX 2

(8.75)

or Pin (dBm) =

From (8.57) the minimum power input must be PinMIN = −111 dBm + 20 + 1 = −90 dBm

(8.76)

Hence, the dynamic range of this amplifier is, DR = −36.5 − (−90) = 53.5 dB

(8.77)

242

MICROWAVE TRANSMISSION LINE CIRCUITS

20 10 11 dB

POWER OUT (dBm)

0 -10 -20

1 dB GAIN COMPRESSION POINT

INTERCEPT POINT

PRIMARY SIGNAL THIRD ORDER INTERMODULATION SIGNALS

-30 -40

SECOND ORDER INTERMODULATION SIGNALS

-50 -60 -60

-50

-40 -30 -20 -10 0 PRIMARY POWER (dBm)

10

20

Figure 8.6 Output power of primary, second-order intermodulation, and third-order intermodulation signals versus primary input power showing intercept point.

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

243

8.4 THE TRANSISTOR AS AN OSCILLATOR 8.4.1 Feedback Oscillators Many oscillators use positive feedback, and that technique is appropriate for the present application. A block diagram of a feedback oscillator is shown in Figure 8.7a, where the gain block A is the transistor indicated by the simple equivalent circuit, and B is the feedback circuit to be selected from a number of possibilities most applicable to microwave oscillators. The DC bias connections and other networks required for small-signal operation of the transistor are not shown in the figure. For the connection A and B, the gain with feedback vout /vin = AF is A 1 − AB

(8.78)

AB = |AB|ejφ

(8.79)

AF = Expressing AB as

if φ = 0◦ , 360◦, ..., and |AB| ≥ 1, the conditions are met for oscillations to occur. This equation is known as the Barkhausen criterion. A phase shift of 180◦ occurs from input to output of a common-source FET or a common-emitter BJT. Hence, the feedback block B must provide exactly 180◦ of phase shift at the oscillation frequency. When the circuit in Figure 8.7(b) is used in the feedback block B, a Colpitts oscillator is formed with an output frequency of f0 =

1 r   C3 2π L2 CC11+C 3

(8.80)

The voltage transfer function from output to input (B) is: B=

C3 j180◦ e C1

(8.81)

and the transistor must have enough gain to satisfy the requirement: |A| = gm RT ≥

C1 C3

(8.82)

244

MICROWAVE TRANSMISSION LINE CIRCUITS

A vin

vout

Ri

gmvin

R2

B (a)

L2 C1

C3

(b)

Z02 , λ/8 Z01

Z03

λ/8

λ/8

Z0 , λ/2

(c)

(d)

Figure 8.7 Possible feedback connections to make a microwave transistor oscillator.

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

245

where gm is the transconductance and RT is the terminating resistance. At high microwave frequencies a lumped element circuit may not be feasible, so that C1 , L2 , and C3 would be replaced by their transmission line equivalents. Thus, if eighth-wavelength sections are used to replace the lumped elements of the Colpitts oscillator, the resulting circuit is shown in Figure 8.7(c), where: ω0 C1 = 1/Z01

(8.83)

ω0 L2 = Z02

(8.84)

ω0 C3 = 1/Z03

(8.85)

and |B| =

C3 Z01 = C1 Z03

(8.86)

Substituting (8.83), (8.84), and (8.85) into (8.80) yields the simple requirement: Z02 = Z01 + Z03

(8.87)

Thus, with λ/8 line sections, (8.86) and (8.87) are the only design equations needed. For example, Z01 = 40 Ω, Z02 = 100 Ω, and Z03 = 60 Ω satisfy (8.87) and yield |B| = 2/3 in (8.86). A much simpler oscillator design is to provide the required 180◦ feedback from output to input with a section of transmission line that is 180◦ in electrical length or λ/2 in physical length at the desired frequency of oscillation. This choice for the feedback block B is shown in Figure 8.7(d). The frequency selectivity or Q of this feedback circuit is approximately: Z0 RT π Q = − RT Z0 4

(8.88)

Thus, for a well-defined frequency of oscillation (high Q circuit), Z0 should be chosen to be either much greater than RT or much less than RT . A possible transistor oscillator is diagrammed in Figure 8.8 using the λ/2 feedback circuit of Figure 8.7(d). All of the feedback circuit components in Figure 8.8 would need to be adjusted in value to compensate for extra line lengths, lead inductances, and interelectrode capacitances. In addition, the active device requires DC bias, and so usually the feedback networks are capacitively coupled to the active device, as in Figure 8.8.

246

MICROWAVE TRANSMISSION LINE CIRCUITS

PACKAGED TRANSISTOR

CHIP CAPACITOR C1

Z0, λ/2

+VGG

CHIP CAPACITOR C2

SMA OUT

+VCC ε

Figure 8.8 Transistor oscillator using λ/2 feedback.

8.4.2 Negative Resistance Oscillator Before oscillation can occur in an active circuit, an unstable condition must exist. The feedback circuits in Figure 8.8 force the transistor into an unstable state, since the output signal is fed back in phase with the input signal. If the transistor or active device is inherently unstable at a given frequency, then a negative resistance oscillator may be constructed. The active device is stable and cannot be used as a negative resistance oscillator if the following conditions prevail in terms of the device scattering parameters at a given frequency: K=

1 − |S11 |2 − |S22 |2 + |∆|2 > 1 2|S12 S21 |

(8.89)

and |∆| < 1

(8.90)

|∆| = |S11 S22 − S12 S21 |.

(8.91)

where

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

247

Conversely, the active device is potentially unstable and may be used as a negative resistance oscillator for K < 1, if |∆| > 1 at a given frequency where the S parameters of the device are known. To produce oscillations, the input reactance is tuned out (resonated with an external element), thus leaving a negative resistance at the input so that current may circulate in the absence of voltage. To complete the design and increase the output power, the output of the transistor or active device is conjugately matched to the load. As an example, a computer-aided design is easily accomplished using the Fujitsu FSC10 FET. The completed design, and a plot of the output reflection coefficient (|Γ2 |), is shown in Figure 8.9. Note that |Γ2 | = 13.6 dB at f = 4 GHz, the peak frequency of oscillation. The FSC10 is unstable at 4 GHz (S11 = 0.7636 − 99.8◦, S21 = 3.5266 89.5◦, S22 = 0.4996 − 62.2◦, S12 = 0.0746 33.0◦), so the negative resistance condition was readily accomplished. The FSC10 is stable at 5 GHz (S11 = 0.6896 − 120.3◦, S21 = 3.1306 71.4◦, S22 = 4716 − 75.4◦, S12 = 0.0816 23.4◦), thus a negative resistance oscillator could not be operated at this frequency. However, positive feedback could be used at 5 GHz to force oscillations to occur. An example using the FSC10 with positive feedback to produce oscillations at 5 GHz is shown in Figure 8.10. Note that |Γ2 | has a peak value of 8.2 dB at 5.2 GHz. Oscillator circuits similar to the ones shown in Figures 8.7(c), 8.7(d) (or 8.8), 8.9, and 8.10 may be designed and constructed for use at frequencies up to about 40 GHz. Since a spectrum analyzer displays the specific amplitude and frequency of the output from an oscillator, one is able to measure the output power, the frequency of oscillation, and the stability of output power and frequency versus time.

8.5 MICROWAVE DIODES

Diodes are critical and versatile components in microwave systems. Part of this versatility stems from the use of diodes for both analog and digital applications. In the analog domain, solid-state diodes’ nonlinearity makes them ideal candidates for use in rectifiers, detectors, and mixers. Digital applications implement microwave diodes as switches/relays for high frequency control circuits, including phase shifters. Here we outline some basic diode characteristics for a variety of microwave applications.

248

MICROWAVE TRANSMISSION LINE CIRCUITS

FSC10 FET 50 Ω, 45°

115 Ω, 29° 33 Ω, 90°

Γ2

20

Γ2

dB

-20 3.0

f GHz

5.0

Figure 8.9 Negative resistance oscillator using FSC10 FET at 4 GHz where the FET is unstable.

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

115 Ω, 90° 70 Ω, 30°

33 Ω, 90°

115 Ω, 12°

Γ2

115 Ω, 17°

20

Γ2

dB

-20 3.0

f GHz

7.0

Figure 8.10 Feedback oscillator using FSC10 at 5 GHz where the FET is stable.

249

250

MICROWAVE TRANSMISSION LINE CIRCUITS

8.5.1 Diode Fundamentals The basic operational DC current characteristic of a diode, shown in Figure 8.11, is given by:     qV I(V ) = Isat exp −1 (8.92) nkT with Isat equal to the reverse saturation current (typically on the order of 10−9 to 10−12 amps, q equal to the electron charge, V equal to the voltage across the diode terminals, n equal to the ideality factor, k equal to Boltzmann’s constant (1.380 × 10−23 J/K), and T equal to temperature in K. The ideality factor varies depending on the physical structure and material properties of the diode. Here, we confine our discussion to diodes that are most appropriate for microwave applications, namely Schottky diodes (best for analog applications) and PIN diodes (usually used for digital/switched applications). Schottky diodes have an abrupt metalsemiconductor junction instead of a traditional pn junction, which allows them to have the fast response times necessary to operate at microwave frequencies (ideality factor around 1.2). PIN diodes have their p and n regions separated by a region of intrinsic material (hence, the “I” in the name) that slows their instantaneous responses at microwave frequencies, allowing them to appear predominantly as either high- or a low-value resistor (or relay) (ideality factor around 2.0). Note that this basic model does not include any parasitics that arise from packaging and other geometric and material factors, but these factors can affect the overall performance of the diode. 8.5.2 Diodes as Analog Devices If we consider the behavior of the diode at microwave frequencies, a three-term small signal model is most appropriate. In this case, we assume a DC bias with a smaller microwave sinusoidal signal superimposed across the diode terminals, such that Vapplied = VDC + vRF cos ω0 t with ω0 = 2πf0 . If we expand (8.92) using a Taylor series around the DC operating point, we arrive at the following: I(V ) = IDC

dI 1 2 d2 I +v + v +··· dV VDC 2 dV 2 VDC

(8.93)

The first voltage derivative of the current can be interpreted as the junction conductance of the diode, Gj = 1/Rj . The second voltage derivative of the current is

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

251

I

V −Isat I + V − Figure 8.11 Generalized diode I-V characteristic.

given by d2 I q Gj = G0j = 2 dV VDC nkT

(8.94)

Rewriting the expression for the voltage-dependent current and substituting for the derivatives with our assumed applied voltage gives: I(V ) = IDC + i = IDC + vRF Gj cos ω0 t +

2 vRF G0j cos2 ω0 t 2

(8.95)

If we apply some common trigonometric identities, (8.95) becomes: I(V ) = IDC +

2 v2 vRF G0j + vRF Gj cos ω0 t + RF G0j cos2 2ω0 t 4 2

(8.96)

The expression in (8.96) lends itself to straightforward interpretation of a collection of signals that we can now exploit for a variety of different applications. The first term of (8.96) is simply the DC bias current, but the second term is also a DC term, and represents that DC rectified current. The third and fourth terms are the fundamental and the second harmonic. In this way, (8.96) gives us options for what kind of information we can obtain by having a microwave signal pass through the diode. The first device is a rectifier, which converts a small percentage of the RF signal into a DC component, as indicated in (8.96). In this application the non-zero

252

MICROWAVE TRANSMISSION LINE CIRCUITS

frequency components of the signal are typically filtered out using a lowpass filter. Rectifiers are essential components in everyday wireless devices. Some examples include a cellular phone, which uses rectifiers as components of power monitors (translated to the signal strength bars on a cell phone display) and automatic gain control circuits (that help to ensure that cell phones transmit only enough power to reach their own base station, thus conserving battery power). Another application for rectifiers that has emerged over the past decade is in the area of RF energy harvesting. Given the small values of signal converted to DC with a single diode, however, it is very important to calculate the incident RF power density on the diode to determine whether or not this kind of energy harvesting is feasible for a given application. The second common application for microwave diodes is as a detector, where the nonlinearity of the diode is used to demodulate an AM modulated RF carrier. In this case, the input signal to the diode can be expressed as: Vapplied = VDC + vRF (1 + m cos ωm t)cosωc t, with m being the modulation index (0 ≤ m ≤ 1), ωm being the modulation radian frequency, and ωc = 2πfc as the radian frequency of the RF carrier, which is much higher than ωm . Applying this voltage across the diode results in the following current output: 2 vRF G0j (1+m cos ωm t)2 cos2 ωc t 2 (8.97) If we again rearrange this expression using trigonometric identities for the squared cosine terms, the output contains components at the following radian frequencies from lowest to highest: 0, ωm , 2ωm , ωc − ωm , ωc , ωc + ωm , 2(ωc − ωm ), 2ωc − ωm , 2ωc, 2ωc + ωm , and 2(ωc + ωm ). For the detection operation, the component of interest is the signal at the fundamental modulation radian frequency, ωm , which can be separated from the other signals with a lowpass filter. For this particular application, it is important to provide a DC operating point such that the diode operates in its “square-law” region, where the ωm component arises from the second derivative term in the fundamental diode equation. Specifically, the input power should be large enough to overcome the noise floor of the device, but small enough to maintain small-signal conditions (below saturation of the diode). More details about the operation of the role of a detector in a microwave system were provided in Section 1.3.2. The third common application of microwave diodes is in devices called mixers, first discussed in Section 1.2.3. These devices use the nonlinearity of diodes to generate an output spectrum that contains the sum and difference frequency of two signals. Mixers are designed for both up-conversion (for transmitters) and

I(V ) = IDC +vRF Gj (1+m cos ωm t) cos ωc t+

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

253

down-conversion, (for receivers) both of which are illustrated in Figure 8.12. The inputs to the mixer for up-conversion are the local oscillator signal, indicated by ωLO , and the intermediate frequency signal, indicated by ωIF . In a singleended mixer, these two signals are combined using a simple T-junction combiner or a directional coupler and fed into a diode. Depending upon the specific diode properties, a matching circuit may be used before the diode input, and the diode is biased at a DC bias above the noise floor but below a level that would cause saturation. The local oscillator frequency is typically generated using either crystal oscillators or tunable transistor oscillators. The output signal, indicated as ωRF is specified as either the sum or difference of the local oscillator and intermediate frequencies. The sum and difference signals both exist in the pure mixer output, typically centered around the local oscillator frequency as shown in Figure 8.12, and a sideband filter is required to select either the sum or difference to go on through the signal chain. Often, this sideband filter is included in packaged microwave mixers to restrict the output to either the sum or difference of the input frequencies as indicated. Down-conversion operates in the same general process, but produces the intermediate frequency as the output, which is the difference between the two input signals – the RF signal at ωRF and the local oscillator frequency ωLO . Mixers are typically rated according to their conversion loss, which can be expressed as: Lc [dB] = 10 log

available RF input power IF output power

(8.98)

8.5.3 Diodes as Digital Devices PIN diode control circuits are useful not only to adjust signal pathways but also to reconfigure switched circuits, such as phase shifters, filters, and antennas. PIN diodes are easily integrated with planar circuitry and have very fast switching speeds, on the order of 10s of nanoseconds. When considering PIN diodes for switches, it is very useful to consider their equivalent circuits for their two states: forward biased and reverse biased. These two circuits are provided in Figure 8.13. When the diode is forward biased, the diode simply appears as a very low real impedance, Rf + jωLi . In this state, the diode is considered to be ON. Li is a small inherent inductance in the structure that does not change with bias state, and is typically between a tenth and five tenths of a nH, with the forward bias resistance usually less than 1 Ω. When the diode is reverse biased, the small series junction capacitance, Cj , presents a large reactive impedance, and the diode is considered to be OFF. This junction capacitance is on the order of 1 pF, and the reverse bias

254

MICROWAVE TRANSMISSION LINE CIRCUITS

IF

ωIF

ωRF = ωLO ± ω IF

SIGNAL MAGNITUDE

LO

MIXER

ωIF

ωLO ωLO − ω IF

ωLO + ω IF ω

(a)

LO

ωLO ωRF ωIF = ωRF ± ωLO

SIGNAL MAGNITUDE

RF

MIXER

ωLO

ωRF

ωRF − ωLO

ωRF + ωLO

(b)

ω

Figure 8.12 Mixer operation for up-conversion (a) and down-conversion (b).

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

Zf

Li

Zr

255

Li Cj

Rf

(a)

Rr (b)

Figure 8.13 Equivalent circuits for a (a) forward-biased and (b) reverse-biased PIN diode, with no packaging parasitics included.

resistance is not much larger than 5 Ω. Since these circuit element values only approximate the ideal “open” and “short” connection of an ideal switch, the amount of loss when ON is non-zero and the amount of isolation when OFF is finite. The insertion loss of the diode in either state can be calculated using the equivalent circuits and the analysis methods provided in Chapter 4. Typical values for DC/control bias are 3-5 V, and the forward bias current ranges from 10 to 35 mA. The fact that these diodes do not latch and need constant bias to remain on (thereby constantly drawing current), is one of the largest drawbacks to their implementation, especially in phase shifters (described later). Although it seems that one could simply insert a diode into a circuit to serve as a switch, at microwave frequencies, the circuit for the switch is necessarily more complicated. Figure 8.14 shows the circuits for shunt and series configured singlepole switches. The two circuits contain several components in common. First, in the RF signal chain, two DC blocking capacitors are necessary, indicated by Cblock . The values of these capacitors are selected to present low RF impedance but nearly infinite DC impedance, to keep the DC bias signal from leaking into the rest of the RF circuit. Also included in both shunt and series configurations are the RF chokes, indicated by Lchoke . These are inductors, or even high impedance quarterwavelength sections of transmission line, that present a high RF impedance and prevent the RF signal from leaking onto the DC bias lines. (Care should be taken when using quarter-wavelength sections of line as chokes, since this creates a frequency-dependent impedance for the choke that can limit the effective frequency bandwidth of the switch.) Finally, both circuits include an isolation capacitor, indicated by Cisol , that isolates the RF short at the end of the choke from the DC bias supply.

256

MICROWAVE TRANSMISSION LINE CIRCUITS

DC/CONTROL BIAS

Cisol

Lchoke Cblock

Cblock

(a)

DC/CONTROL BIAS

Lchoke

Cisol

Cblock

Cblock Lchoke (b)

Figure 8.14 Shunt (a) and series (b) single-pole switches using PIN diodes.

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

257

Note that for the shunt switch configuration, the switch (now the entire circuit) is ON when the diode is reverse biased, and for the series switch configuration, the switch is ON when the diode is forward biased. Determination of which circuit configuration to implement often depends on the anticipated most likely state of the switch that can either minimize power consumption in the ON state or provide the highest isolation in the OFF state. 8.5.3.1 Diode-Based Phase Shifters Many microwave phase shifters feature microwave PIN diodes as switching devices to control the relative phases of signals. The first, and most popular, of these phase shifters is called the switched line phase shifter, which, as the name implies, involves switching between two microwave transmission lines of differing lengths. This kind of phase shifter is also called a “true time delay” phase shifter, since the phase shift achieved by the device is linear with frequency and does not experience distortion over broad frequency bands. An example of a single-bit switched line phase shifter is shown in Figure 8.15. The phase shifter shown in Figure 8.15 uses two single-pole, double-throw switches (indicated by the back-to-back diodes at the top and bottom line junctions). The switches direct the signal either through the leftward path, which experiences a transmission phase shift of φ1 = βl1 , or the rightward path, which experiences a transmission phase shift of φ2 = βl2 , with β being the phase constant of the transmission line. This reciprocal device then produces a differential phase shift in the signal of δφ = β(l2 − l1 ).

(8.99)

Cascaded switched phase shifters are typically designed to provide a digitallyselectable phase shift that quantified in “bits” of phase shift. In these cases, cascaded phase shifters can be designated by the number of binary phase shifts possible (in decreasing order of significance) of 180◦ , 90◦, 45◦ , 22.5◦ , and so on. Some practical issues to consider in the implementation of this kind of diode phase shifter include the signal losses incurred through the two diodes in each path for each bit of phase shift, as well as unwanted resonances or radiation from the unused but coupled paths at certain frequencies. Care with circuit layout can help prevent undesired coupling. Another related phase shifter is the reflection phase shifter, which uses properties of a hybrid 90◦ coupler (or other non-reciprocal device) to produce a designated signal phase shift in a reflected signal. (See Chapter 7 for more details on basic coupler design.) Shown in Figure 8.16, the device relies on two diode switches to

258

MICROWAVE TRANSMISSION LINE CIRCUITS

IN

φ1 = β l1

φ2 = β l2

OUT

Figure 8.15 Single bit switched line phase shifter.

switch in or out (simultaneously) an additional path length in a reflected signal at each of the output ports, that are then re-combined in phase at the coupler output. The total phase shift achieved at the output is simply given by δφ, determined by the line lengths between the diodes and the shorted terminations at each reflection port. This idealized operation assumes that the diodes are perfect shorts or opens in their respective states. Therefore, it is important to choose matched diodes with these characteristics in mind so that the signal at the output experiences as little dispersion as possible due to errors created by differences between the diodes. Finally, one more switched phase shifter that can use PIN diodes as switching elements is the loaded-line phase shifter, shown in Figure 8.17. The concept behind a loaded-line phase shifter is a very simple one — the introduction of a shunt reactive or susceptive element introduces a phase shift in a transmitted signal. The practical phase shifts available using this method are typically smaller than those achievable with the other two methods already discussed, since this particular embodiment creates reflections that create insertion loss for the circuit. The actual circuit element for the reactive element could be a lumped capacitor (or inductor, at very low microwave frequencies) or a transmission line stub of any specified length. The reflection and transmission coefficients for

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

IN

259

OUT

{

∆φ / 2

∆φ / 2

{

90o HYBRID COUPLER

Figure 8.16 A reflection phase shifter implemented with a 90◦ hybrid coupler.

such a phase shifter are given by the following two equations, respectively: Γ=

Y0 − (Y0 + jB) −jB = Y0 + (Y0 + jB) 2Y0 + jB

(8.100)

−jB + 2Y0 + jB 2Y0 = (8.101) 2Y0 + jB 2Y0 + jB Examination of the transmission coefficient indicates that the phase shift achieved with the single shunt susceptance B is given by −B/2Y0 . Of course, depending on the nature of the shunt element, the absolute relative phase shift could be positive or negative. Introduction of two switched reactances at a set quarter-wavelength spacing reduces the amount of reflected signal and also can increase the range of apparent signal phase shifts but also decreases the effective bandwidth of the device. As with the other designs, care should be taken in the selection of the diodes for loss and frequency behavior. Loaded-line phase shifters that implement varactor diodes (or varactors) in place of the PIN diodes and reactive elements/stubs can provide continuous phase shifting. There are several commercially-available designs that implement these phase shifters using ferroelectric varactors as well as solid-state versions. In design or selection of these devices, care should be taken to assess the insertion loss characteristics of each phase shifter, the range of phase shift available, and the required actuation voltages required. Finally, one key point to remember when implementing diodes as digital (switched) components is that even though their fundamental operation is intended to be digital, some of the nonlinear behavior that we leverage in analog applications can still be present, and package parasitics can be important. Without proper care in design (use of back-to-back diodes, for instance), this can result in undesirable frequency harmonics and intermodulation products. T =Γ+1=

260

MICROWAVE TRANSMISSION LINE CIRCUITS

Z0

jB

Z0

Figure 8.17 A loaded-line phase shifter implemented with a single switched shunt susceptance, B, using a notional relay that can take a variety of diode-enabled forms.

PROBLEMS 1. Using discrete elements (lumped Ls and Cs) in networks N1 and N2 of Figure 8.18, provide the optimum source and load impedances for the transistor amplifier at f = 1 GHz. 2. The scattering parameters for the NE 70083 transistor amplifier as measured in a 50 Ω system at 4 GHz are: S11 S21 S12 S22

= = = =

0.886 2.406 0.066 0.666

−76◦ 108◦ 33◦ −56◦

From the scattering parameters at 4 GHz, determine (a) the input impedance Z1 with ZL = 50 Ω, (b) the output impedance Z2 with ZS = 50 Ω, and (c) the unilateral transducer gain (in dB) of the amplifier. 3. An input matching network N1 between ZS and Z1 and an output matching network N2 between Z2 and ZL are added to the NE 70083 transistor amplifier described in the previous problem. (a) If N1 and N2 are properly designed, what is the maximum transducer gain (in dB) that can be achieved at f = 4 GHz? (b) Design the networks N1 and N2 to achieve maximum transducer gain at f = 4 GHz. (c) Sketch a top view of the microstrip layout for N1 and N2 on a 1/32 inch Duroid board, showing all line widths and lengths in inches.

TRANSMISSION LINE APPLICATIONS IN ACTIVE CIRCUITS

261

50 Ω VS

N1

TRANSISTOR AMPLIFIER ZS OPT = 5 + j10 Ω

N2

50 Ω

ZL OPT = 100 - j200 Ω

Figure 8.18 Using discrete elements (lumped Ls and Cs) in networks N1 and N2 , provide the optimum source and load impedances for the transistor amplifier at f = 1 GHz.

BIBLIOGRAPHY I. J. Bahl and P. Bhartia, Microwave Solid State Circuit Design, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2003. R. S. Carson, High-Frequency Amplifiers, 2nd ed. New York: John Wiley and Sons, 1982. H. Fukui, “Design of Microwave GaAs MESFET’s for Broadband, Low-Noise Amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 27, no. 7, pp. 643–650, Jul. 1979. H. Fukui, “Optimal Noise Figure of Microwave GaAs MESFET’s,” IEEE Trans. Electron Devices, vol. 26, no. 7, pp. 1032–1037, Jul. 1979. G. Gonzalez, Microwave Transistor Amplifiers, Analysis and Design, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. T. T. Ha, Solid-State Microwave Amplifier Design. Hoboken, NJ: John Wiley and Sons, 1981. S. Y. Liao, Microwave Circuit Analysis and Amplifier Design. Upper Saddle River, NJ: Prentice-Hall, 1987. S. A. Maas, Microwave Mixers, Artech House, Dedham, MA: 1986. W. D. Palmer and W. T. Joines, “Traveling-Wave Amplifiers with Prescribed Frequency Response,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 6, pp. 1223– 1229, Jun. 1992.

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MICROWAVE TRANSMISSION LINE CIRCUITS

D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley and Sons, 2011. M. B. Steer, Microwave and RF Design: A Systems Approach, Scitech Publishing, 2010. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed., John Wiley and Sons, 2007. G. D. Vendelin, Design of Amplifiers and Oscillators by the S-Parameter Method. New York: Wiley-Interscience, 1982. G. D. Vendelin, A. M. Pavio, and U. L. Rohde, Microwave Circuit Design, Using Linear and Nonlinear Techniques, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2005. E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications. New York: Wiley-Interscience, 1988.

Appendix A NORMALIZED ELEMENT VALUES FOR LOWPASS FILTERS WITH BUTTERWORTH, CHEBYSHEV, AND BESSEL RESPONSES Chapter 5 describes the synthesis and design of frequency-filtering networks using sets of polynomials that generate the required filter response. Common requirements include maximally flat magnitude across the passband, generated by the Butterworth polynomials; equal-ripple response in the passband with faster rolloff in the stopband, generated by the Chebyshev polynomials; and maximally flat delay across the passband, generated by the Bessel polynomials. Direct calculation of the filter element values from these sets of polynomials can be quite complicated. For convenience, element values are presented in the following tables for maximally flat magnitude (Table A.1), equal-ripple (0.01 dB, 0.1 dB, and 0.5 dB in Tables A.2, A.3, and A.4 respectively), and maximally flat delay (Table A.5) for prototype lowpass filters with two to seven elements. The element values have of course been normalized with respect to the load resistance RL and the filter cutoff frequency ωc . Chapter 5 gives detailed procedures for denormalizing the element values and transforming the prototype lowpass filter into a bandpass, bandstop, or highpass filter. The elements of a prototype lowpass filter are series inductors (Ln ) and shunt capacitors (Cn ). The filter network can begin with either type of element. The choice of element dictates how the values in the tables are used. Figure A.1 shows a filter driven by a current source with a shunt capacitor as the first element. For

263

264

L2 I

R

Ln

C1

C3

1 n EVEN

Figure A.1 element.

L1

n ODD

L3 C2

Ln Cn n EVEN

Figure A.2 element.

1

Prototype lowpass filter driven by a current source with a shunt capacitor as the first

R V

Cn

1

1 n ODD

Prototype lowpass filter driven by a voltage source with a series inductor as the first

this configuration, element values are assigned using the labels across the top of the tables. Figure A.2 shows a filter driven by a voltage source with a series inductor as the first element. For this configuration, element values are assigned using the labels across the bottom of the tables. Once the normalized element values for the prototype lowpass filter are determined, the filter design proceeds using the methods from Chapter 5.

NORMALIZED ELEMENT VALUES FOR LOWPASS FILTERS

Table A.1 Butterworth Response: Maximally Flat Magnitude n

R

C1

L2

2

1.0000 1.1111 1.2500 1.4286 1.6667 2.0000 2.500 3.3333 5.0000 10.0000 ∞

1.4142 1.0353 0.8485 0.6971 0.5657 0.4483 0.3419 0.2447 0.1557 0.0743 1.4142

1.4142 1.9352 2.1213 2.4387 2.8284 3.3461 4.0951 5.3126 7.7067 14.8138 0.7671

C3

L4

3

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

1.0000 0.8052 0.9442 0.9152 1.0225 1.1811 1.4254 1.8380 2.6687 5.1672 1.5900

2.0000 1.6332 1.3840 1.1652 0.9650 0.7789 0.6042 0.4396 0.2842 0.1377 1.3333

1.0000 1.5994 1.9259 2.2774 2.7024 3.2612 4.0642 5.3634 7.9102 15.4554 0.5000

4

1.0000 1.1111 1.2500 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.7654 0.4657 0.3882 0.3251 0.2699 0.2175 0.1692 0.1237 0.0304 0.0392 1.5307

1.8478 1.5924 1.6946 1.9619 2.1029 2.4524 2.9858 3.8826 5.6835 11.0942 1.5772

1.8478 1.7439 1.5110 1.2913 1.0824 3.8826 0.6911 0.5072 3.3307 0.1616 1.0824

0.7654 1.4690 1.8109 2.1752 2.6131 3.1868 4.0094 5.3381 7.9397 15.6421 0.3827

n

1/R

L1

C2

L3

C4

265

266

Table A.1 Butterworth Response: Maximally Flat Magnitude (Continued) n

R

C1

L2

C3

L4

C5

5

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

0.6180 0.4416 0.4698 0.5173 0.5860 0.6857 0.8378 1.0937 1.6077 3.1522 1.5451

1.6180 1.0265 0.8660 0.7313 0.6094 0.4955 0.3877 0.2848 0.1861 0.0912 1.6944

2.0000 1.9095 2.0605 2.2849 2.5998 3.0510 3.7357 4.8835 7.1849 14.0945 1.3820

1.6180 1.7562 1.5443 1.3326 1.1255 0.9237 0.7274 0.5367 0.3518 0.1727 0.8944

0.6180 1.3887 1.7380 2.1083 2.5524 3.1331 3.9648 5.3073 7.9345 15.7103 0.3090

L6

C7

6

1.0000 1.1111 1.2500 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.5176 0.2890 0.2445 0.2072 0.1732 0.1412 0.1108 0.0816 0.0535 0.0263 1.5529

1.4142 1.0403 1.1163 1.2363 1.4071 1.6531 2.0275 2.6559 3.9170 7.7053 1.7593

1.9319 1.3217 1.1257 0.9567 0.8011 0.6542 0.5139 0.3788 0.2484 0.1222 1.5529

1.9319 2.0539 2.2389 2.4991 2.8580 3.3687 4.1408 5.4325 8.0201 15.7855 1.2016

1.4142 1.7443 1.5498 1.3464 1.1431 0.9423 0.7450 0.5517 0.3628 0.1788 0.7579

0.5176 1.3347 1.6881 2.0618 2.5092 3.0938 3.9305 5.2804 7.9216 15.7375 0.2588

7

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

0.4450 0.2985 0.3215 0.3571 0.4075 0.4799 0.5899 0.7745 1.1448 2.7571 1.5576

1.2470 0.7111 0.6057 0.5154 0.4322 0.3536 0.2782 0.2055 0.1350 0.0665 1.7988

1.8018 1.4043 1.5174 1.6883 1.9284 2.2726 2.7950 3.6706 5.4267 10.7004 1.6588

2.0000 1.4891 1.2777 1.0910 0.9170 0.7512 0.5917 0.4373 0.2874 0.1417 1.3972

1.8019 2.1249 2.3338 2.6177 3.0050 3.5532 4.3799 5.7612 8.5263 16.8222 1.0550

1.2470 1.7268 1.5461 1.3498 1.1503 0.9513 0.7542 0.5600 0.3692 0.1823 0.6560

0.4450 1.2961 1.6520 2.0277 2.4771 3.0640 3.9037 5.2583 7.9079 15.7480 0.2225

n

1/R

L1

C2

L3

C4

L5

C6

L7

NORMALIZED ELEMENT VALUES FOR LOWPASS FILTERS

Table A.2 Chebyshev Response: Ripple = 0.01 dB n

R

C1

L2

2

1.1007 1.1111 1.2500 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

1.3472 1.2472 0.9434 0.7591 0.6091 0.4791 0.3634 0.2590 0.1642 0.0781 1.4118

1.4829 1.5947 1.9974 2.3442 2.7490 3.2772 4.0328 5.2546 7.6495 14.7492 0.7415

C3

L4

3

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

1.1811 1.0917 1.0969 1.1600 1.2737 1.4521 1.7340 2.2164 3.1934 6.1411 1.5012

1.8214 1.6597 1.4431 1.2283 1.0236 0.8294 0.6452 0.4704 0.3047 0.1479 1.4330

1.1811 1.4802 1.8057 2.1653 2.5984 3.1644 3.9742 5.2800 7.8338 15.3899 0.5905

4

1.0000 1.1111 1.2500 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.9500 0.8539 0.6182 0.4948 0.3983 0.3156 0.2418 0.1744 0.1121 0.0541 1.5287

1.9382 1.9460 2.0749 2.2787 2.5709 2.9943 3.6406 4.7274 6.9102 13.4690 1.6939

1.7608 1.7439 1.5417 1.3336 1.1277 0.9260 0.7293 0.5379 0.8523 0.1729 1.3122

1.0457 1.1647 1.6170 2.0083 2.4611 3.0448 3.4746 5.2085 7.8126 15.5100 0.5229

n

1/R

L1

C2

L3

C4

267

268

Table A.2 Chebyshev Response: Ripple = 0.01 dB (Continued) n

R

C1

L2

C3

L4

C5

5

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

0.9766 0.8798 0.8769 0.9263 1.0191 1.1658 1.3983 1.7966 2.6039 5.0406 1.5466

1.6849 1.4558 1.2350 1.0398 0.8626 0.6985 0.5442 0.3982 0.2592 0.1266 1.7950

2.0366 2.1738 2.3785 2.6582 3.0408 3.5835 4.4027 5.7721 8.5140 16.7406 1.6449

1.6849 l.6412 1.4991 1.3228 1.1345 0.9421 0.7491 0.5573 0.3679 0.1819 1.2365

0.9766 1.2739 1.6066 1.9772 2.4244 3.0092 3.8453 5.1925 7.8257 15.6126 0.4883

L6

C7

6

1.1007 1.1111 1.2500 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.8514 0.7597 0.5445 0.4355 0.3509 0.2786 0.2139 0.1547 0.0997 0.0483 1.5510

1.7956 1.7817 1.8637 2.0383 2.2978 2.6781 3.2614 4.2448 6.2227 12.1707 1.8471

1.8411 1.7752 1.4886 1.2655 1.0607 0.8671 0.6816 0.5028 0.3299 0.1623 1.7897

2.0266 2.0941 2.4025 2.7346 3.1671 3.7683 4.6673 6.1631 9.1507 18.1048 1.5976

1.6312 1.6380 1.5067 1.3318 1.1451 0.9536 0.7606 0.5676 0.3760 0.1865 1.1904

0.9372 1.0533 1.5041 1.8987 2.3568 2.9483 3.7899 5.1430 7.7852 15.5950 0.4686

7

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

0.9127 0.8157 0.8111 0.8567 0.9430 1.0799 1.2971 1.6692 2.4235 4.7006 1.5593

1.5947 1.3619 1.1504 0.9673 0.8025 0.6502 0.5072 0.3716 0.2423 0.1186 1.8671

2.0021 2.0886 2.2618 2.5158 2.8720 3.3822 4.1563 5.4540 8.0565 15.8718 1.8657

1.8704 1.7217 1.5252 1.3234 1.1237 0.9276 0.7350 0.5459 0.3604 0.1784 1.7651

2.0021 2.2017 2.4647 2.8018 3.2496 3.8750 4.8115 6.3703 9.4844 18.8179 1.5633

1.5947 1.5805 1.4644 1.3066 1.1310 0.9468 0.7584 0.5682 0.3776 0.1879 1.1610

0.9127 1.2060 1.5380 1.9096 2.3592 2.9478 3.7900 5.1476 7.8019 15.6523 0.4564

n

1/R

L1

C2

L3

C4

L5

C6

L7

NORMALIZED ELEMENT VALUES FOR LOWPASS FILTERS

Table A.3 Chebyshev Response: Ripple = 0.1 dB n

R

C1

L2

2

1.3554 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

1.2087 0.9771 0.7326 0.5597 0.4169 0.2933 0.1841 0.0868 1.3911

1.6382 1.9824 2.4885 3.0538 3.8265 5.0502 7.4257 14.4332 0.8191

C3

L4

3

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

1.4328 1.4258 1.4511 1.5210 1.6475 1.8530 2.1857 2.7630 3.9418 7.5121 1.5133

1.5937 1.4935 1.3557 1.1927 1.0174 0.8383 0.6603 0.4860 0.3172 0.1549 1.5090

1.4328 1.6219 1.8711 2.1901 2.6026 3.1594 3.9675 5.2788 7.8503 15.4656 0.7164

4

1.3554 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.9924 0.7789 0.5764 0.4398 0.3288 0.2329 0.1475 0.0704 1.5107

2.1476 2.3480 2.7304 3.2269 3.9605 5.1777 7.6072 14.8873 1.7682

1.5845 1.4292 1.1851 0.9672 0.7599 0.5602 0.3670 0.1802 1.4550

1.3451 1.7001 2.2425 2.8563 3.6976 5.0301 7.6143 15.2297 0.6725

n

1/R

L1

C2

L3

C4

269

270

Table A.3 Chebyshev Response: Ripple = 0.1 dB (Continued) n

R

C1

L2

C3

L4

C5

L6

C7

5

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

1.3013 1.2845 1.2998 1.3580 1.4694 1.6535 1.9538 2.4765 3.5457 6.7870 1.5613

1.5559 1.4329 1.2824 1.1170 0.9469 0.7777 0.6119 0.4509 0.2950 0.1447 1.8069

2.2411 2.3794 2.5819 2.8679 3.2688 3.8446 4.7193 6.1861 9.1272 17.9569 1.7659

1.5559 1.4878 1.3815 1.2437 1.0846 0.9126 0.7333 0.5503 0.3659 0.1820 1.4173

1.3013 1.4883 1.7384 2.0621 2.4835 3.0548 3.8861 5.2373 7.8890 15.7447 0.6507

6

1.3554 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.9419 0.7347 0.5422 0.4137 0.3095 0.2195 0.1393 0.0666 1.5339

2.0797 2.2492 2.6003 3.0679 3.7652 4.9266 7.2500 14.2200 1.8838

1.6581 1.4537 1.1830 0.9575 0.7492 0.5514 0.3613 0.1777 1.8306

2.2473 2.5437 3.0641 3.7119 4.6512 6.1947 9.2605 18.4267 1.7485

1.5344 1.4051 1.1850 0.9794 0.7781 0.5795 0.3835 0.1901 1.3937

1.2767 1.6293 2.1739 2.7936 3.6453 4.9962 7.6184 15.3495 0.6383

7

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

1.2615 1.2422 1.2550 1.3100 1.4170 1.5948 1.8853 2.3917 3.4278 6.5695 1.5748

1.5196 1.3946 1.2449 1.0826 0.9169 0.7529 0.5926 0.4369 0.2862 0.1405 1.8577

2.2392 2.3613 2.5481 2.8192 3.2052 3.7642 4.6179 6.0535 8.9371 17.6031 1.9210

1.6804 1.5784 1.4430 1.2833 1.1092 0.9276 0.7423 0.5557 0.3692 0.1838 1.8270

2.2392 2.3966 2.6242 2.9422 3.3841 4.0150 4.9702 6.5685 9.7697 19.3760 1.7340

1.5196 1.4593 1.3619 1.2326 1.0807 0.9142 0.7384 0.5569 0.3723 0.1862 1.3786

1.2615 1.4472 1.6967 2.0207 2.4437 3.0182 3.8552 5.2167 7.8901 15.8127 0.6307

n

1/R

L1

C2

L3

C4

L5

C6

L7

NORMALIZED ELEMENT VALUES FOR LOWPASS FILTERS

Table A.4 Chebyshev Response: Ripple = 0.5 dB n

R

C1

L2

2

1.9841 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.9827 0.9086 0.5635 0.3754 0.2282 0.1052 1.3067

1.9497 2.1030 3.1647 4.4111 6.6995 13.3221 0.9748

C3

L4

3

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

1.8636 1.9175 1.9965 2.1135 2.2889 2.5571 2.9854 3.7292 5.2543 9.8899 1.5720

1.2804 1.2086 1.1203 1.0149 0.8937 0.7592 0.6146 0.4633 0.3087 0.1534 1.5179

1.8636 2.0255 2.2368 2.5172 2.8984 3.4360 4.2416 5.5762 8.2251 16.1177 0.9318

4

1.9841 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.9202 0.8452 0.5162 0.3440 0.2100 0.0975 1.4361

2.5864 2.7198 3.7659 5.1196 7.7076 15.3520 1.8888

1.3036 1.2383 0.8693 0.6208 0.3996 0.1940 1.6211

1.8258 1.9849 3.1205 4.4790 6.9874 14.2616 0.9129

n

1/R

L1

C2

L3

C4

271

272

Table A.4 Chebyshev Response: Ripple = 0.5 dB (Continued) n

R

C1

L2

C3

L4

C5

5

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

1.8068 1.8540 1.9257 2.0347 2.2006 2.4571 2.8692 3.5877 5.0639 9.5560 1.6299

1.3025 1.2220 1.1261 1.0150 0.8901 0.7537 0.6091 0.4590 0.3060 0.1525 1.7400

2.6914 2.8478 3.0599 3.3525 3.7651 4.3672 5.2960 6.8714 10.0537 19.6465 1.9217

1.3025 1.2379 1.1569 1.0582 0.9420 0.8098 0.6640 0.5075 0.3430 0.1731 1.5138

1.8068 1.9701 2.1845 2.4704 2.8609 3.4137 4.2447 5.6245 8.3674 16.5474 0.9034

L6

C7

6

1.9841 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.9053 0.8303 0.5056 0.3370 0.2059 0.0958 1.4618

2.5774 2.7042 3.7219 5.0554 7.6145 15.1862 1.9799

1.3675 1.2912 0.8900 0.6323 0.4063 0.1974 1.7803

2.7133 2.8721 4.1092 5.6994 8.7319 17.6806 1.9253

1.2991 1.2372 0.8808 0.6348 0.4121 0.2017 1.5077

1.7961 1.9557 3.1025 4.4810 7.0310 14.4328 0.8981

7

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

1.7896 1.8348 1.9045 2.0112 2.1744 2.4275 2.8348 3.5456 5.0070 9.4555 1.6464

1.2961 1.2146 1.1182 1.0070 0.8824 0.7470 0.6035 0.4548 0.3034 0.1513 1.7772

2.7177 2.8691 3.0761 3.3638 3.7717 4.3695 5.2947 6.8674 10.0491 19.6486 2.0306

1.3848 1.3080 1.2149 1.1050 0.9786 0.8377 0.6846 0.5221 0.3524 0.1778 1.7892

2.7177 2.8829 3.1071 3.4163 3.8524 4.4886 5.4698 7.1341 10.4959 20.6314 1.9239

1.2961 1.2335 1.1546 1.0582 0.9441 0.8137 0.6690 0.5129 0.3478 0.1761 1.5034

1.7896 1.9531 2.1681 2.4554 2.8481 3.4050 4.2428 5.6350 8.4041 16.6654 0.8948

n

1/R

L1

C2

L3

C4

L5

C6

L7

NORMALIZED ELEMENT VALUES FOR LOWPASS FILTERS

Table A.5 Bessel Response: Maximally Flat Delay n

R

C1

L2

2

1.0000 1.1111 1.2500 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.5755 0.5084 0.4433 0.3801 0.3191 0.2601 0.2032 0.1486 0.0965 0.0469 1.3617

2.1478 2.3097 2.5096 2.7638 3.0993 3.5649 4.2577 5.4050 7.6876 14.5097 0.4539

C3

L4

3

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

0.3374 0.3708 0.4124 0.4657 0.5365 0.6353 0.7829 1.0283 1.5176 2.9825 1.4631

0.9705 0.8650 0.7609 0.6584 0.5576 0.4587 0.3618 0.2673 0.1752 0.0860 0.8427

2.2034 2.3745 2.5867 2.8575 3.2159 3.7144 4.4573 5.6888 8.1403 15.4697 0.2926

4

1.0000 1.1111 1.2500 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.2334 0.2085 0.1839 0.1596 0.1356 0.1120 0.0887 0.0658 0.0434 0.0214 1.5012

0.6725 0.7423 0.8292 0.9406 1.0886 1.2952 1.6040 2.1174 3.1416 6.2086 0.9781

1.0815 0.9670 0.8534 0.7410 0.6299 0.5202 0.4120 0.3056 0.2013 0.0993 0.6127

2.2404 2.4143 2.6304 2.9066 3.2727 3.7824 4.5430 5.8048 8.3185 15.8372 0.2114

n

1/R

L1

C2

L3

C4

273

274

Table A.5 Bessel Response: Maximally Flat Delay (Continued) n

R

C1

L2

C3

L4

C5

5

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

0.1743 0.1926 0.2154 0.2447 0.2836 0.3380 0.4194 0.5548 0.8251 1.6349 1.5125

0.5072 0.4542 0.4016 0.3494 0.2977 0.2465 0.1958 0.1457 0.0964 0.0478 1.0232

0.8040 0.8894 0.9959 1.1323 1.3138 1.5672 1.9464 2.5768 3.8352 7.6043 0.7531

1.1110 0.9945 0.8789 0.7642 0.6506 0.5382 0.4270 0.3174 0.2095 0.1036 0.4729

2.2582 2.4328 2.6497 2.9272 3.2952 3.8077 4.5731 5.8433 8.3747 15.9487 0.1618

L6

C7

6

1.0000 1.1111 1.2500 1.4286 1.6667 2.0000 2.5000 3.3333 5.0000 10.0000 ∞

0.1365 0.1223 0.1082 0.0943 0.0804 0.0666 0.0530 0.0395 0.0261 0.0130 1.5124

0.4002 0.4429 0.4961 0.5644 0.6553 0.7824 0.9725 1.2890 1.9209 3.8146 1.0329

0.6392 0.5732 0.5076 0.4424 0.3775 0.3131 0.2492 0.1859 0.1232 0.0612 0.8125

0.8538 0.9456 1.0600 1.2069 1.4022 1.6752 2.0837 2.7633 4.1204 8.1860 0.6072

1.1126 0.9964 0.8810 0.7665 0.6530 0.5405 0.4292 0.3193 0.2110 0.1045 0.3785

2.2645 2.4388 2.6554 2.9325 3.3001 3.8122 4.5770 5.8467 8.3775 15.9506 0.1287

7

1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 ∞

0.1106 0.1224 0.1372 0.1562 0.1815 0.2168 0.2698 0.3579 0.5338 1.0612 1.5087

0.3259 0.2923 0.2589 0.2257 0.1927 0.1599 0.1274 0.0951 0.0630 0.0313 1.0293

0.5249 0.5815 0.6521 0.7428 0.8634 1.0321 1.2847 1.7051 2.5448 5.0616 0.8345

0.7020 0.6302 0.5586 0.4873 0.4163 0.3457 0.2755 0.2058 0.1365 0.0679 0.6752

0.8690 0.9630 1.0803 1.2308 1.4312 1.7111 2.1304 2.8280 4.2214 8.3967 0.5031

1.1052 0.9899 0.8754 0.7618 0.6491 0.5374 0.4269 0.3177 0.2100 0.1040 0.3113

2.2659 2.4396 2.6556 2.9319 3.2984 3.8090 4.5718 5.8380 8.3623 15.9166 0.1054

n

1/R

L1

C2

L3

C4

L5

C6

L7

NORMALIZED ELEMENT VALUES FOR LOWPASS FILTERS

275

BIBLIOGRAPHY G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures. New York: McGraw-Hill Book Company, 1964. A. I. Zverev, Handbook of Filter Synthesis. New York: Wiley-Interscience, 1967.

276

Appendix B MICROSTRIP CHARACTERISTIC IMPEDANCE AND EFFECTIVE r VERSUS w/h A microstrip transmission line consists of a metal strip on a dielectric substrate with a metal ground plane. A cross-section of such a line is shown in Figure B.1. The metal strip has width w and thickness t that is assumed to be negligible. The dielectric substrate has relative dielectric constant r , height h over the ground plane, and conductivity σd which is usually assumed to be zero. As you will recall from Section 2.5, the characteristic impedance Z0 and effective dielectric constant  of transmission lines are important design parameters. Calculating these values for microstrip is especially difficult because it is not symmetric; the conductor sits on top of the dielectric rather than being surrounded by it. Since part of the field around the line flows through the dielectric and part fringes in air, a closed-form solution is quite complicated. When r for the dielectric substrate and the dimension w/h of the microstrip line are known, the equations in Table 2.5 can be used to calculate values for  as

ε0

w t=0

εrε0 , σd Figure B.1

Cross-section of a microstrip line.

277

h

278

= and for Z0 as

r + 1 r − 1 1 p + 2 2 1 + 12h/w

Z0 = ηF (g) =

r

µ0 1 ln  2π



8h w + w 4h

(B.1)



(B.2)

for cases where w/h ≤ 1, or Z0 = ηF (g) =

r

µ0 1  w/h + 1.393 + 0.667 ln(w/h + 1.444)

(B.3)

for cases where w/h ≥ 1. When designing a microstrip line for a particular Z0 on a dielectric substrate with known r , a useful set of equations is presented in Pozar that can be used to calculate the required dimension w/h. Because these equations are approximations, different equations must be used for different ranges of w/h. Equation B.4 calculates the ratio w/h for given Z0 for cases where w/h < 2. w 8eA = 2A h e −2

with Z0 A= 60

r

r + 1 r − 1 + 2 r + 1



(B.4) 0.11 0.23 + r



(B.5)

Equation B.6 calculates the ratio w/h for given Z0 for cases where w/h > 2.    w 2 r − 1 0.61 = B − 1 − ln(2B − 1) + ln(B − 1) + 0.39 − h π 2r r with B=

377π √ 2Z0 r

(B.6)

(B.7)

Equations B.1, B.4, and B.6 were used to calculate  and w/h for varying board thicknesses for a range of values of Z0 using the relative permittivity of three common dielectrics in microwave circuit boards: polytetrafluoroethylene with r = 2.2 (also known as Teflon, a registered trademark of E. I. du Pont de Nemours and Company), polyethylene with r = 2.32, and epoxy-glass with r = 4.4. The

MICROSTRIP CHARACTERISTIC IMPEDANCE

279

first column gives the characteristic impedance of the microstrip line in ohms. The second column gives the effective dielectric constant. The last four columns give w in inches for board thicknesses h = 1 inch, h = 1/8 inch, h = 1/16 inch, and h = 1/32 inch. BIBLIOGRAPHY I. J. Bahl and D. K. Trivedi, “A Designer’s Guide to Microstrip Line.” Microwaves, vol. 16, no. 5, pp. 174–182, May. 1977. D. M. Pozar, Microwave Engineering, 2nd ed. Hoboken, NJ: John Wiley and Sons, 2011, p. 162. H. A. Wheeler, “Transmission-Line Properties of a Strip on a Dielectric Sheet on a Plane.” IEEE Transactions on Microwave Theory and Techniques, vol. 25, no. 8, pp. 631–647, Aug. 1977.

280

Table B.1 Microstrip Data for Dielectric Constant r = 2.20 Z0

ef f

w/h

w/8h

w/16h

w/32h

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

2.1044 2.0946 2.0851 2.0760 2.0673 2.0589 2.0508 2.0429 2.0354 2.0280 2.0210 2.0141 2.0075 2.0010 1.9948 1.9887 1.9828 1.9771 1.9715 1.9661 1.9608 1.9557 1.9507 1.9458 1.9410 1.9364 1.9319 1.9274 1.9231 1.9189 1.9147 1.9107 1.9067 1.9028 1.8990 1.8953 1.8917 1.8881 1.8846 1.8811 1.8778 1.8745 1.8712

28.9059 25.4306 22.6556 20.3894 18.5045 16.9126 15.5507 14.3727 13.3438 12.4378 11.6340 10.9162 10.2714 9.6892 9.1609 8.6796 8.2392 7.8348 7.4623 7.1180 6.7990 6.5026 6.2265 5.9688 5.7276 5.5015 5.2891 5.0893 4.9010 4.7232 4.5552 4.3960 4.2452 4.1020 3.9659 3.8364 3.7131 3.5955 3.4832 3.3759 3.2734 3.1752 3.0812

3.6132 3.1788 2.8319 2.5487 2.3131 2.1141 1.9438 1.7966 1.6680 1.5547 1.4542 1.3645 1.2839 1.2111 1.1451 1.0849 1.0299 0.9794 0.9328 0.8898 0.8499 0.8128 0.7783 0.7461 0.7159 0.6877 0.6611 0.6362 0.6126 0.5904 0.5694 0.5495 0.5306 0.5127 0.4957 0.4796 0.4641 0.4494 0.4354 0.4220 0.4092 0.3969 0.3851

1.8066 1.5894 1.4160 1.2743 1.1565 1.0570 0.9719 0.8983 0.8340 0.7774 0.7271 0.6823 0.6420 0.6056 0.5726 0.5425 0.5149 0.4897 0.4664 0.4449 0.4249 0.4064 0.3892 0.3730 0.3580 0.3438 0.3306 0.3181 0.3063 0.2952 0.2847 0.2748 0.2653 0.2564 0.2479 0.2398 0.2321 0.2247 0.2177 0.2110 0.2046 0.1985 0.1926

0.9033 0.7947 0.7080 0.6372 0.5783 0.5285 0.4860 0.4491 0.4170 0.3887 0.3636 0.3411 0.3210 0.3028 0.2863 0.2712 0.2575 0.2448 0.2332 0.2224 0.2125 0.2032 0.1946 0.1865 0.1790 0.1719 0.1653 0.1590 0.1532 0.1476 0.1423 0.1374 0.1327 0.1282 0.1239 0.1199 0.1160 0.1124 0.1088 0.1055 0.1023 0.0992 0.0963

MICROSTRIP CHARACTERISTIC IMPEDANCE

Table B.1 Microstrip Data for Dielectric Constant r = 2.20 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

1.8680 1.8649 1.8618 1.8588 1.8558 1.8528 1.8500 1.8471 1.8443 1.8416 1.8389 1.8362 1.8336 1.8310 1.8285 1.8264 1.8239 1.8214 1.8190 1.8166 1.8143 1.8120 1.8097 1.8075 1.8053 1.8031 1.8010 1.7988 1.7967 1.7947 1.7926 1.7906 1.7886 1.7867 1.7847 1.7828 1.7809 1.7791 1.7772 1.7754

2.9910 2.9045 2.8215 2.7417 2.6649 2.5911 2.5200 2.4515 2.3855 2.3219 2.2604 2.2011 2.1438 2.0884 2.0348 1.9912 1.9407 1.8919 1.8447 1.7990 1.7548 1.7119 1.6704 1.6301 1.5910 1.5531 1.5163 1.4806 1.4458 1.4121 1.3793 1.3474 1.3164 1.2862 1.2568 1.2282 1.2003 1.1732 1.1467 1.1210

0.3739 0.3631 0.3527 0.3427 0.3331 0.3239 0.3150 0.3064 0.2982 0.2902 0.2826 0.2751 0.2680 0.2610 0.2544 0.2489 0.2426 0.2365 0.2306 0.2249 0.2194 0.2140 0.2088 0.2038 0.1989 0.1941 0.1895 0.1851 0.1807 0.1765 0.1724 0.1684 0.1646 0.1608 0.1571 0.1535 0.1500 0.1467 0.1433 0.1401

0.1869 0.1815 0.1763 0.1714 0.1666 0.1619 0.1575 0.1532 0.1491 0.1451 0.1413 0.1376 0.1340 0.1305 0.1272 0.1245 0.1213 0.1182 0.1153 0.1124 0.1097 0.1070 0.1044 0.1019 0.0994 0.0971 0.0948 0.0925 0.0904 0.0883 0.0862 0.0842 0.0823 0.0804 0.0786 0.0768 0.0750 0.0733 0.0717 0.0701

0.0935 0.0908 0.0882 0.0857 0.0833 0.0810 0.0788 0.0766 0.0745 0.0726 0.0706 0.0688 0.0670 0.0653 0.0636 0.0622 0.0606 0.0591 0.0576 0.0562 0.0548 0.0535 0.0522 0.0509 0.0497 0.0485 0.0474 0.0463 0.0452 0.0441 0.0431 0.0421 0.0411 0.0402 0.0393 0.0384 0.0375 0.0367 0.0358 0.0350

281

282

Table B.1 Microstrip Data for Dielectric Constant r = 2.20 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

1.7736 1.7718 1.7700 1.7683 1.7665 1.7648 1.7631 1.7615 1.7598 1.7582 1.7566 1.7549 1.7534 1.7518 1.7502 1.7487 1.7472 1.7457 1.7442 1.7427 1.7413 1.7398 1.7384 1.7370 1.7356 1.7342 1.7328 1.7315 1.7301 1.7288 1.7275 1.7262 1.7249 1.7236 1.7224 1.7211 1.7199 1.7187 1.7174 1.7162

1.0959 1.0714 1.0476 1.0243 1.0016 0.9795 0.9579 0.9369 0.9163 0.8962 0.8767 0.8575 0.8389 0.8206 0.8028 0.7855 0.7685 0.7519 0.7357 0.7198 0.7043 0.6892 0.6744 0.6600 0.6459 0.6320 0.6185 0.6053 0.5924 0.5798 0.5675 0.5554 0.5436 0.5321 0.5208 0.5097 0.4989 0.4884 0.4780 0.4679

0.1370 0.1339 0.1310 0.1280 0.1252 0.1224 0.1197 0.1171 0.1145 0.1120 0.1096 0.1072 0.1049 0.1026 0.1004 0.0982 0.0961 0.0940 0.0920 0.0900 0.0880 0.0862 0.0843 0.0825 0.0807 0.0790 0.0773 0.0757 0.0741 0.0725 0.0709 0.0694 0.0680 0.0665 0.0651 0.0637 0.0624 0.0611 0.0598 0.0585

0.0685 0.0670 0.0655 0.0640 0.0626 0.0612 0.0599 0.0586 0.0573 0.0560 0.0548 0.0536 0.0524 0.0513 0.0502 0.0491 0.0480 0.0470 0.0460 0.0450 0.0440 0.0431 0.0422 0.0413 0.0404 0.0395 0.0387 0.0378 0.0370 0.0362 0.0355 0.0347 0.0340 0.0333 0.0326 0.0319 0.0312 0.0305 0.0299 0.0292

0.0342 0.0335 0.0327 0.0320 0.0313 0.0306 0.0299 0.0293 0.0286 0.0280 0.0274 0.0268 0.0262 0.0256 0.0251 0.0245 0.0240 0.0235 0.0230 0.0225 0.0220 0.0215 0.0211 0.0206 0.0202 0.0198 0.0193 0.0189 0.0185 0.0181 0.0177 0.0174 0.0170 0.0166 0.0163 0.0159 0.0156 0.0153 0.0149 0.0146

MICROSTRIP CHARACTERISTIC IMPEDANCE

Table B.1 Microstrip Data for Dielectric Constant r = 2.20 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

1.7150 1.7139 1.7127 1.7115 1.7104 1.7093 1.7082 1.7070 1.7059 1.7049 1.7038 1.7027 1.7017 1.7006 1.6996 1.6986 1.6976 1.6966 1.6956 1.6946

0.4580 0.4484 0.4389 0.4296 0.4206 0.4117 0.4030 0.3945 0.3862 0.3781 0.3702 0.3624 0.3547 0.3473 0.3400 0.3328 0.3259 0.3190 0.3123 0.3058

0.0573 0.0561 0.0549 0.0537 0.0526 0.0515 0.0504 0.0493 0.0483 0.0473 0.0463 0.0453 0.0443 0.0434 0.0425 0.0416 0.0407 0.0399 0.0390 0.0382

0.0286 0.0280 0.0274 0.0269 0.0263 0.0257 0.0252 0.0247 0.0241 0.0236 0.0231 0.0227 0.0222 0.0217 0.0213 0.0208 0.0204 0.0199 0.0195 0.0191

0.0143 0.0140 0.0137 0.0134 0.0131 0.0129 0.0126 0.0123 0.0121 0.0118 0.0116 0.0113 0.0111 0.0109 0.0106 0.0104 0.0102 0.0100 0.0098 0.0096

283

284

Table B.2 Microstrip Data for Dielectric Constant r = 2.32 Z0

ef f

w/h

w/8h

w/16h

w/32h

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

2.1945 2.1846 2.1750 2.1658 2.1568 2.1482 2.1399 2.1319 2.1241 2.1166 2.1093 2.1022 2.0954 2.0887 2.0823 2.0760 2.0699 2.0640 2.0583 2.0527 2.0472 2.0419 2.0367 2.0316 2.0267 2.0219 2.0172 2.0126 2.0080 2.0036 1.9993 1.9951 1.9909 1.9869 1.9829 1.9790 1.9752 1.9714 1.9677 1.9641 1.9605

22.0649 19.8388 17.9873 16.4238 15.0866 13.9303 12.9210 12.0326 11.2449 10.5420 9.9111 9.3418 8.8258 8.3560 7.9266 7.5327 7.1702 6.8356 6.5258 6.2382 5.9706 5.7210 5.4878 5.2693 5.0643 4.8716 4.6901 4.5190 4.3573 4.2044 4.0595 3.9221 3.7915 3.6674 3.5493 3.4367 3.3293 3.2267 3.1287 3.0349 2.9450

2.7581 2.4799 2.2484 2.0530 1.8858 1.7413 1.6151 1.5041 1.4056 1.3177 1.2389 1.1677 1.1032 1.0445 0.9908 0.9416 0.8963 0.8544 0.8157 0.7798 0.7463 0.7151 0.6860 0.6587 0.6330 0.6089 0.5863 0.5649 0.5447 0.5255 0.5074 0.4903 0.4739 0.4584 0.4437 0.4296 0.4162 0.4033 0.3911 0.3794 0.3681

1.3791 1.2399 1.1242 1.0265 0.9429 0.8706 0.8076 0.7520 0.7028 0.6589 0.6194 0.5839 0.5516 0.5223 0.4954 0.4708 0.4481 0.4272 0.4079 0.3899 0.3732 0.3576 0.3430 0.3293 0.3165 0.3045 0.2931 0.2824 0.2723 0.2628 0.2537 0.2451 0.2370 0.2292 0.2218 0.2148 0.2081 0.2017 0.1955 0.1897 0.1841

0.6895 0.6200 0.5621 0.5132 0.4715 0.4353 0.4038 0.3760 0.3514 0.3294 0.3097 0.2919 0.2758 0.2611 0.2477 0.2354 0.2241 0.2136 0.2039 0.1949 0.1866 0.1788 0.1715 0.1647 0.1583 0.1522 0.1466 0.1412 0.1362 0.1314 0.1269 0.1226 0.1185 0.1146 0.1109 0.1074 0.1040 0.1008 0.0978 0.0948 0.0920

MICROSTRIP CHARACTERISTIC IMPEDANCE

Table B.2 Microstrip Data for Dielectric Constant r = 2.32 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

1.9570 1.9536 1.9502 1.9468 1.9436 1.9404 1.9372 1.9341 1.9310 1.9280 1.9250 1.9221 1.9192 1.9164 1.9136 1.9109 1.9082 1.9055 1.9029 1.9003 1.8978 1.8953 1.8928 1.8904 1.8880 1.8857 1.8833 1.8811 1.8788 1.8766 1.8744 1.8723 1.8702 1.8681 1.8660 1.8640 1.8620 1.8601 1.8581 1.8562

2.8590 2.7764 2.6971 2.6210 2.5478 2.4774 2.4096 2.3443 2.2813 2.2206 2.1621 2.1055 2.0509 1.9981 1.9470 1.8976 1.8498 1.8035 1.7587 1.7152 1.6731 1.6322 1.5926 1.5541 1.5167 1.4804 1.4452 1.4109 1.3776 1.3453 1.3138 1.2832 1.2534 1.2244 1.1961 1.1686 1.1418 1.1158 1.0903 1.0656

0.3574 0.3470 0.3371 0.3276 0.3185 0.3097 0.3012 0.2930 0.2852 0.2776 0.2703 0.2632 0.2564 0.2498 0.2434 0.2372 0.2312 0.2254 0.2198 0.2144 0.2091 0.2040 0.1991 0.1943 0.1896 0.1851 0.1806 0.1764 0.1722 0.1682 0.1642 0.1604 0.1567 0.1530 0.1495 0.1461 0.1427 0.1395 0.1363 0.1332

0.1787 0.1735 0.1686 0.1638 0.1592 0.1548 0.1506 0.1465 0.1426 0.1388 0.1351 0.1316 0.1282 0.1249 0.1217 0.1186 0.1156 0.1127 0.1099 0.1072 0.1046 0.1020 0.0995 0.0971 0.0948 0.0925 0.0903 0.0882 0.0861 0.0841 0.0821 0.0802 0.0783 0.0765 0.0748 0.0730 0.0714 0.0697 0.0681 0.0666

0.0893 0.0868 0.0843 0.0819 0.0796 0.0774 0.0753 0.0733 0.0713 0.0694 0.0676 0.0658 0.0641 0.0624 0.0608 0.0593 0.0578 0.0564 0.0550 0.0536 0.0523 0.0510 0.0498 0.0486 0.0474 0.0463 0.0452 0.0441 0.0431 0.0420 0.0411 0.0401 0.0392 0.0383 0.0374 0.0365 0.0357 0.0349 0.0341 0.0333

285

286

Table B.2 Microstrip Data for Dielectric Constant r = 2.32 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

1.8544 1.8525 1.8507 1.8489 1.8472 1.8454 1.8437 1.8421 1.8404 1.8388 1.8372 1.8356 1.8340 1.8325 1.8310 1.8295 1.8280 1.8266 1.8251 1.8237 1.8224 1.8210 1.8196 1.8183 1.8170 1.8157 1.8145 1.8132 1.8120 1.8108 1.8096 1.8084 1.8072 1.8061 1.8049 1.8038 1.8027 1.8016 1.8006 1.7995

1.0414 1.0179 0.9950 0.9726 0.9508 0.9295 0.9088 0.8885 0.8687 0.8494 0.8306 0.8122 0.7943 0.7768 0.7597 0.7430 0.7267 0.7107 0.6952 0.6800 0.6651 0.6506 0.6364 0.6226 0.6090 0.5958 0.5828 0.5702 0.5578 0.5458 0.5339 0.5224 0.5111 0.5001 0.4893 0.4787 0.4684 0.4583 0.4485 0.4388

0.1302 0.1272 0.1244 0.1216 0.1188 0.1162 0.1136 0.1111 0.1086 0.1062 0.1038 0.1015 0.0993 0.0971 0.0950 0.0929 0.0908 0.0888 0.0869 0.0850 0.0831 0.0813 0.0796 0.0778 0.0761 0.0745 0.0729 0.0713 0.0697 0.0682 0.0667 0.0653 0.0639 0.0625 0.0612 0.0598 0.0586 0.0573 0.0561 0.0549

0.0651 0.0636 0.0622 0.0608 0.0594 0.0581 0.0568 0.0555 0.0543 0.0531 0.0519 0.0508 0.0496 0.0485 0.0475 0.0464 0.0454 0.0444 0.0434 0.0425 0.0416 0.0407 0.0398 0.0389 0.0381 0.0372 0.0364 0.0356 0.0349 0.0341 0.0334 0.0326 0.0319 0.0313 0.0306 0.0299 0.0293 0.0286 0.0280 0.0274

0.0325 0.0318 0.0311 0.0304 0.0297 0.0290 0.0284 0.0278 0.0271 0.0265 0.0260 0.0254 0.0248 0.0243 0.0237 0.0232 0.0227 0.0222 0.0217 0.0212 0.0208 0.0203 0.0199 0.0195 0.0190 0.0186 0.0182 0.0178 0.0174 0.0171 0.0167 0.0163 0.0160 0.0156 0.0153 0.0150 0.0146 0.0143 0.0140 0.0137

MICROSTRIP CHARACTERISTIC IMPEDANCE

Table B.2 Microstrip Data for Dielectric Constant r = 2.32 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170

1.7985 1.7974 1.7964 1.7954 1.7944 1.7935 1.7925 1.7916 1.7906 1.7897 1.7888 1.7879 1.7870 1.7861 1.7853 1.7844 1.7836 1.7828 1.7819 1.7811 1.7803 1.7795 1.7788 1.7780 1.7772 1.7765 1.7757 1.7750 1.7743 1.7736 1.7729 1.7722 1.7715 1.7708 1.7701 1.7695 1.7688 1.7681 1.7675 1.7669

0.4294 0.4201 0.4111 0.4023 0.3936 0.3852 0.3769 0.3689 0.3609 0.3532 0.3456 0.3382 0.3310 0.3239 0.3170 0.3102 0.3036 0.2971 0.2907 0.2845 0.2784 0.2725 0.2667 0.2610 0.2554 0.2500 0.2446 0.2394 0.2343 0.2293 0.2244 0.2196 0.2149 0.2104 0.2059 0.2015 0.1972 0.1930 0.1889 0.1849

0.0537 0.0525 0.0514 0.0503 0.0492 0.0481 0.0471 0.0461 0.0451 0.0442 0.0432 0.0423 0.0414 0.0405 0.0396 0.0388 0.0379 0.0371 0.0363 0.0356 0.0348 0.0341 0.0333 0.0326 0.0319 0.0312 0.0306 0.0299 0.0293 0.0287 0.0281 0.0275 0.0269 0.0263 0.0257 0.0252 0.0247 0.0241 0.0236 0.0231

0.0268 0.0263 0.0257 0.0251 0.0246 0.0241 0.0236 0.0231 0.0226 0.0221 0.0216 0.0211 0.0207 0.0202 0.0198 0.0194 0.0190 0.0186 0.0182 0.0178 0.0174 0.0170 0.0167 0.0163 0.0160 0.0156 0.0153 0.0150 0.0146 0.0143 0.0140 0.0137 0.0134 0.0131 0.0129 0.0126 0.0123 0.0121 0.0118 0.0116

0.0134 0.0131 0.0128 0.0126 0.0123 0.0120 0.0118 0.0115 0.0113 0.0110 0.0108 0.0106 0.0103 0.0101 0.0099 0.0097 0.0095 0.0093 0.0091 0.0089 0.0087 0.0085 0.0083 0.0082 0.0080 0.0078 0.0076 0.0075 0.0073 0.0072 0.0070 0.0069 0.0067 0.0066 0.0064 0.0063 0.0062 0.0060 0.0059 0.0058

287

288

Table B.2 Microstrip Data for Dielectric Constant r = 2.32 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

1.7662 1.7656 1.7650 1.7644 1.7638 1.7632 1.7626 1.7620 1.7615 1.7609 1.7603 1.7598 1.7592 1.7587 1.7581 1.7576 1.7571 1.7566 1.7560 1.7555 1.7550 1.7545 1.7540 1.7535 1.7531 1.7526 1.7521 1.7516 1.7512 1.7507

0.1809 0.1771 0.1733 0.1696 0.1660 0.1625 0.1590 0.1556 0.1523 0.1491 0.1459 0.1428 0.1397 0.1368 0.1339 0.1310 0.1282 0.1255 0.1228 0.1202 0.1177 0.1152 0.1127 0.1103 0.1080 0.1057 0.1034 0.1012 0.0991 0.0970

0.0226 0.0221 0.0217 0.0212 0.0207 0.0203 0.0199 0.0195 0.0190 0.0186 0.0182 0.0178 0.0175 0.0171 0.0167 0.0164 0.0160 0.0157 0.0154 0.0150 0.0147 0.0144 0.0141 0.0138 0.0135 0.0132 0.0129 0.0127 0.0124 0.0121

0.0113 0.0111 0.0108 0.0106 0.0104 0.0102 0.0099 0.0097 0.0095 0.0093 0.0091 0.0089 0.0087 0.0085 0.0084 0.0082 0.0080 0.0078 0.0077 0.0075 0.0074 0.0072 0.0070 0.0069 0.0067 0.0066 0.0065 0.0063 0.0062 0.0061

0.0057 0.0055 0.0054 0.0053 0.0052 0.0051 0.0050 0.0049 0.0048 0.0047 0.0046 0.0045 0.0044 0.0043 0.0042 0.0041 0.0040 0.0039 0.0038 0.0038 0.0037 0.0036 0.0035 0.0034 0.0034 0.0033 0.0032 0.0032 0.0031 0.0030

MICROSTRIP CHARACTERISTIC IMPEDANCE

Table B.3 Microstrip Data for Dielectric Constant r = 4.40 Z0

ef f

w/h

w/8h

w/16h

w/32h

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

3.9712 3.9399 3.9101 3.8817 3.8545 3.8285 3.8037 3.7799 3.7570 3.7351 3.7141 3.6938 3.6743 3.6554 3.6373 3.6197 3.6028 3.5864 3.5705 3.5551 3.5401 3.5256 3.5115 3.4978 3.4845 3.4715 3.4589 3.4466 3.4346 3.4229 3.4115 3.4003 3.3894 3.3788 3.3684 3.3583 3.3483 3.3386 3.3292 3.3199 3.3108

15.6544 14.0456 12.7086 11.5804 10.6162 9.7832 9.0565 8.4174 7.8512 7.3463 6.8933 6.4850 6.1150 5.7783 5.4707 5.1887 4.9293 4.6899 4.4683 4.2626 4.0713 3.8929 3.7261 3.5700 3.4234 3.2857 3.1559 3.0336 2.9179 2.8085 2.7049 2.6066 2.5132 2.4244 2.3398 2.2592 2.1823 2.1088 2.0386 1.9714 1.9071

1.9568 1.7557 1.5886 1.4475 1.3270 1.2229 1.1321 1.0522 0.9814 0.9183 0.8617 0.8106 0.7644 0.7223 0.6838 0.6486 0.6162 0.5862 0.5585 0.5328 0.5089 0.4866 0.4658 0.4462 0.4279 0.4107 0.3945 0.3792 0.3647 0.3511 0.3381 0.3258 0.3141 0.3030 0.2925 0.2824 0.2728 0.2636 0.2548 0.2464 0.2384

0.9784 0.8779 0.7943 0.7238 0.6635 0.6114 0.5660 0.5261 0.4907 0.4591 0.4308 0.4053 0.3822 0.3611 0.3419 0.3243 0.3081 0.2931 0.2793 0.2664 0.2545 0.2433 0.2329 0.2231 0.2140 0.2054 0.1972 0.1896 0.1824 0.1755 0.1691 0.1629 0.1571 0.1515 0.1462 0.1412 0.1364 0.1318 0.1274 0.1232 0.1192

0.4892 0.4389 0.3971 0.3619 0.3318 0.3057 0.2830 0.2630 0.2453 0.2296 0.2154 0.2027 0.1911 0.1806 0.1710 0.1621 0.1540 0.1466 0.1396 0.1332 0.1272 0.1217 0.1164 0.1116 0.1070 0.1027 0.0986 0.0948 0.0912 0.0878 0.0845 0.0815 0.0785 0.0758 0.0731 0.0706 0.0682 0.0659 0.0637 0.0616 0.0596

289

290

Table B.3 Microstrip Data for Dielectric Constant r = 4.40 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

3.3019 3.2933 3.2848 3.2765 3.2683 3.2603 3.2526 3.2449 3.2374 3.2301 3.2230 3.2160 3.2091 3.2024 3.1958 3.1893 3.1830 3.1768 3.1708 3.1648 3.1590 3.1533 3.1477 3.1422 3.1369 3.1316 3.1265 3.1214 3.1165 3.1117 3.1069 3.1023 3.0977 3.0932 3.0888 3.0845 3.0803 3.0762 3.0721 3.0681

1.8455 1.7863 1.7296 1.6751 1.6227 1.5724 1.5239 1.4772 1.4323 1.3890 1.3472 1.3069 1.2681 1.2305 1.1943 1.1592 1.1254 1.0926 1.0610 1.0303 1.0007 0.9720 0.9442 0.9173 0.8912 0.8659 0.8414 0.8177 0.7947 0.7723 0.7507 0.7297 0.7093 0.6895 0.6703 0.6517 0.6336 0.6160 0.5990 0.5824

0.2307 0.2233 0.2162 0.2094 0.2028 0.1965 0.1905 0.1847 0.1790 0.1736 0.1684 0.1634 0.1585 0.1538 0.1493 0.1449 0.1407 0.1366 0.1326 0.1288 0.1251 0.1215 0.1180 0.1147 0.1114 0.1082 0.1052 0.1022 0.0993 0.0965 0.0938 0.0912 0.0887 0.0862 0.0838 0.0815 0.0792 0.0770 0.0749 0.0728

0.1153 0.1116 0.1081 0.1047 0.1014 0.0983 0.0952 0.0923 0.0895 0.0868 0.0842 0.0817 0.0793 0.0769 0.0746 0.0725 0.0703 0.0683 0.0663 0.0644 0.0625 0.0607 0.0590 0.0573 0.0557 0.0541 0.0526 0.0511 0.0497 0.0483 0.0469 0.0456 0.0443 0.0431 0.0419 0.0407 0.0396 0.0385 0.0374 0.0364

0.0577 0.0558 0.0540 0.0523 0.0507 0.0491 0.0476 0.0462 0.0448 0.0434 0.0421 0.0408 0.0396 0.0385 0.0373 0.0362 0.0352 0.0341 0.0332 0.0322 0.0313 0.0304 0.0295 0.0287 0.0278 0.0271 0.0263 0.0256 0.0248 0.0241 0.0235 0.0228 0.0222 0.0215 0.0209 0.0204 0.0198 0.0193 0.0187 0.0182

MICROSTRIP CHARACTERISTIC IMPEDANCE

Table B.3 Microstrip Data for Dielectric Constant r = 4.40 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

3.0642 3.0604 3.0566 3.0529 3.0493 3.0457 3.0423 3.0388 3.0355 3.0322 3.0289 3.0257 3.0226 3.0195 3.0165 3.0135 3.0106 3.0078 3.0049 3.0022 2.9995 2.9968 2.9941 2.9916 2.9890 2.9865 2.9841 2.9816 2.9793 2.9769 2.9746 2.9723 2.9701 2.9679 2.9657 2.9636 2.9615 2.9595 2.9574 2.9554

0.5663 0.5507 0.5355 0.5208 0.5065 0.4926 0.4790 0.4659 0.4531 0.4407 0.4287 0.4169 0.4055 0.3945 0.3837 0.3732 0.3630 0.3531 0.3435 0.3342 0.3251 0.3162 0.3076 0.2993 0.2911 0.2832 0.2755 0.2680 0.2608 0.2537 0.2468 0.2401 0.2336 0.2272 0.2211 0.2151 0.2092 0.2036 0.1981 0.1927

0.0708 0.0688 0.0669 0.0651 0.0633 0.0616 0.0599 0.0582 0.0566 0.0551 0.0536 0.0521 0.0507 0.0493 0.0480 0.0467 0.0454 0.0441 0.0429 0.0418 0.0406 0.0395 0.0385 0.0374 0.0364 0.0354 0.0344 0.0335 0.0326 0.0317 0.0308 0.0300 0.0292 0.0284 0.0276 0.0269 0.0262 0.0254 0.0248 0.0241

0.0354 0.0344 0.0335 0.0325 0.0317 0.0308 0.0299 0.0291 0.0283 0.0275 0.0268 0.0261 0.0253 0.0247 0.0240 0.0233 0.0227 0.0221 0.0215 0.0209 0.0203 0.0198 0.0192 0.0187 0.0182 0.0177 0.0172 0.0168 0.0163 0.0159 0.0154 0.0150 0.0146 0.0142 0.0138 0.0134 0.0131 0.0127 0.0124 0.0120

0.0177 0.0172 0.0167 0.0163 0.0158 0.0154 0.0150 0.0146 0.0142 0.0138 0.0134 0.0130 0.0127 0.0123 0.0120 0.0117 0.0113 0.0110 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0086 0.0084 0.0081 0.0079 0.0077 0.0075 0.0073 0.0071 0.0069 0.0067 0.0065 0.0064 0.0062 0.0060

291

292

Table B.3 Microstrip Data for Dielectric Constant r = 4.40 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170

2.9534 2.9515 2.9496 2.9477 2.9458 2.9440 2.9422 2.9404 2.9387 2.9369 2.9352 2.9336 2.9319 2.9303 2.9287 2.9271 2.9255 2.9240 2.9224 2.9209 2.9194 2.9180 2.9165 2.9151 2.9137 2.9123 2.9109 2.9096 2.9082 2.9069 2.9056 2.9043 2.9030 2.9018 2.9005 2.8993 2.8981 2.8969 2.8957 2.8945

0.1875 0.1824 0.1775 0.1726 0.1680 0.1634 0.1590 0.1547 0.1505 0.1464 0.1425 0.1386 0.1349 0.1312 0.1277 0.1242 0.1208 0.1176 0.1144 0.1113 0.1083 0.1054 0.1025 0.0997 0.0970 0.0944 0.0919 0.0894 0.0870 0.0846 0.0823 0.0801 0.0779 0.0758 0.0738 0.0718 0.0698 0.0679 0.0661 0.0643

0.0234 0.0228 0.0222 0.0216 0.0210 0.0204 0.0199 0.0193 0.0188 0.0183 0.0178 0.0173 0.0169 0.0164 0.0160 0.0155 0.0151 0.0147 0.0143 0.0139 0.0135 0.0132 0.0128 0.0125 0.0121 0.0118 0.0115 0.0112 0.0109 0.0106 0.0103 0.0100 0.0097 0.0095 0.0092 0.0090 0.0087 0.0085 0.0083 0.0080

0.0117 0.0114 0.0111 0.0108 0.0105 0.0102 0.0099 0.0097 0.0094 0.0092 0.0089 0.0087 0.0084 0.0082 0.0080 0.0078 0.0076 0.0073 0.0071 0.0070 0.0068 0.0066 0.0064 0.0062 0.0061 0.0059 0.0057 0.0056 0.0054 0.0053 0.0051 0.0050 0.0049 0.0047 0.0046 0.0045 0.0044 0.0042 0.0041 0.0040

0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0050 0.0048 0.0047 0.0046 0.0045 0.0043 0.0042 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0030 0.0029 0.0028 0.0027 0.0026 0.0026 0.0025 0.0024 0.0024 0.0023 0.0022 0.0022 0.0021 0.0021 0.0020

MICROSTRIP CHARACTERISTIC IMPEDANCE

Table B.3 Microstrip Data for Dielectric Constant r = 4.40 (Continued) Z0

ef f

w/h

w/8h

w/16h

w/32h

171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

2.8934 2.8922 2.8911 2.8900 2.8889 2.8878 2.8867 2.8857 2.8846 2.8836 2.8825 2.8815 2.8805 2.8795 2.8785 2.8776 2.8766 2.8756 2.8747 2.8738 2.8728 2.8719 2.8710 2.8701 2.8692 2.8684 2.8675 2.8666 2.8658 2.8650

0.0626 0.0609 0.0592 0.0576 0.0561 0.0546 0.0531 0.0517 0.0503 0.0489 0.0476 0.0463 0.0450 0.0438 0.0426 0.0415 0.0404 0.0393 0.0382 0.0372 0.0362 0.0352 0.0342 0.0333 0.0324 0.0315 0.0307 0.0299 0.0291 0.0283

0.0078 0.0076 0.0074 0.0072 0.0070 0.0068 0.0066 0.0065 0.0063 0.0061 0.0059 0.0058 0.0056 0.0055 0.0053 0.0052 0.0050 0.0049 0.0048 0.0046 0.0045 0.0044 0.0043 0.0042 0.0041 0.0039 0.0038 0.0037 0.0036 0.0035

0.0039 0.0038 0.0037 0.0036 0.0035 0.0034 0.0033 0.0032 0.0031 0.0031 0.0030 0.0029 0.0028 0.0027 0.0027 0.0026 0.0025 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0020 0.0019 0.0019 0.0018 0.0018

0.0020 0.0019 0.0019 0.0018 0.0018 0.0017 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 0.0014 0.0013 0.0013 0.0013 0.0012 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 0.0010 0.0010 0.0009 0.0009 0.0009

293

294

ABOUT THE AUTHORS Dr. William T. Joines is a professor in the sensing and signals group, Department of Electrical and Computer Engineering, Duke University, where his research and teaching focuses on electromagnetic field and wave interactions with materials and structures. Previously he was a member of the technical staff with Bell Telephone Laboratories, where he was engaged in research and development of microwave components and systems for military applications. He also served as a radar technician in the U.S. Air Force. He has authored over 200 technical papers on electromagnetic wave theory and applications, and holds 20 U.S. patents. Dr. Joines received a B.S.E.E. degree (with high honors) from North Carolina State University, and M.S. and Ph.D. degrees in electrical engineering from Duke University. He is a Life Fellow of the IEEE, a recent recipient of the Outstanding Engineering Educator Award from the IEEE, and the Scientific and Technical Achievement Award presented by the Environmental Protection Agency. Dr. W. Devereux Palmer is the program manager for electromagnetics, microwaves, and power at the U.S. Army Research Office, responsible for a portfolio of basic research projects focused on creating the innovations that will drive the next generation of DoD systems for radio communications, sensing, and electronic warfare. He has developed and sustained relationships between academic researchers, industry technologists, and Army scientists and engineers that have led to significant technology transitions, including advanced development programs and fielded applications. Dr. Palmer earned a B.A. in physics and M.S. and Ph.D. degrees in electrical engineering from Duke University. A Fellow of the IEEE, he received the U.S. Army Research Laboratory Award for Program Management in 2010 and the U.S. Army Superior Civilian Service Medal in 2011.

295

296

Dr. Jennifer T. Bernhard is a professor in the Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Illinois at UrbanaChampaign, where she is active as a researcher and educator in electromagnetics and antennas. She has authored over 200 journal articles, conference papers and presentations, books, and book chapters, and is inventor on ten patents and patent applications on antennas and ancillary components. Dr. Bernhard received a B.S.E.E. degree from Cornell University and M.S. and Ph.D. degrees in electrical engineering from Duke University, with support from a National Science Foundation Graduate Fellowship. She is an IEEE Fellow and served as president of the IEEE Antennas and Propagation Society in 2008.

Carrier wave, 12, 218 Cascade connection, 83, 101, 177 Channel attenuation, 10 Channel capacity, 19 Characteristic impedance, 28 Chebychev filter response, 134 polynomials, 135 Circulator, 6 Coaxial line, 46 Communications satellite, 11 system, 10, 212 Conjugate impedance match, 54 Conjugate matching, 115 Coplanar stripline, 52 Coplanar waveguide, 51 Corporate combiner, 199 Coupled line even mode, 161 filter, 160 odd mode, 161 Coupler 90◦ hybrid, 204 branch-line, 204 ring or “rat-race”, 211

INDEX ABCD matrix, 29, 100, 138, 181 ABCD parameters, 100, 164, 225 conversion to S parameters, 112, 234 Admittance short-circuit, 98 Amplifier ABCD matrix of, 102 bilateral, 230 low-noise, 15 potentially unstable, 231 transistor, 225 unconditionally stable, 230 unilateral, 226 Amplitude modulation (AM), 12 Attenuation, 45 Attenuation constant, 27 Bandpass filter, 141, 150, 164 Bandpass filters, 16 Bandstop filter, 144, 150 Bandwidth, 11 Barkhausen criterion, 243 Baseband, 12, 218 Bessel polynomials, 135 use in Thompson filters, 136 Binary digit (bit), 16, 257 Branch-line coupler, 204 Butterworth filter response, 133 polynomials, 134, 135

DC block, 230 Demodulation, 13 Demodulator, 218 Diode current-voltage curve, 250 PIN, 250, 253 rectifier, 251 Schottky, 250 square-law detector, 252

297

298

INDEX

varactor, 160, 259 Discrete element, 69 Distortion, 10 Doppler effect, 6 radar, 7 velocity detection, 7 Down-conversion, 253 Downlink, 12, 212 Dynamic range, 235 Eighth-wavelength transformer, 182 Electromagnetic spectrum, 2 Equal power division, 197, 204, 211 Feedback network, 244 oscillator, 243 Fiber optic link, 11 Filter bandpass, 10, 16, 141, 150, 164 bandstop, 144, 150 center frequency, 141 coupled line, 160 cutoff frequency, 140, 141 equi-ripple, 134 highpass, 144, 212 ideal lowpass, 132 lowpass, 16, 174, 212 maximally flat delay, 136 maximally flat magnitude, 133 transformations, 142 Frequency bands, 1 Frequency diplexer, 212 Frequency modulation (FM), 12 Frequency-modulated continuous-wave(FMCW) radar, 8 Function of geometry, 40 Gain compression point (1 dB), 238 Half-wavelength transformer, 175 Heaviside, Oliver, 27 Highpass filter, 144, 212 Human audio range, 7

Ideal lowpass filter, 132 Impedance input, 54, 131, 173 load, 32, 173, 191, 233 open-circuit, 97 source, 53, 173, 191, 233 wave, 32 Impedance inverter, 54 Impedance matching, 45 conjugate, 54, 115 distributed element, 174, 175 lumped element, 174, 182 multi-section, 177 network, 173 Input impedance, 54, 131, 173 Insertion gain, 103 Insertion loss, 103 Intermodulation distortion, 240 Intermodulation products, 238 Isolation between ports, 197, 199, 204 Load impedance, 32, 131, 173, 191, 233 Loaded-line phase shifter, 259 Lossy transmission lines, 36 Lowpass filter, 16, 74, 132, 174, 212 Lumped element, 69, 70 transmission line equivalent, 72, 80, 84 Mason’s gain rule, 121 Maximally flat delay response, 136 Maximally flat magnitude response, 133 Maximum stable gain (MSG), 234 Microstrip, 50, 74, 160 Microwaves, 1 Millimeter waves, 1 Minimum detectable signal, 236, 237 Mixer, 7, 252 Modulation AM, 12 FM, 12 index, 13, 252 PCM, 16 Modulator, 7, 218 90◦ hybrid coupler, 204 Negative resistance oscillator, 246

299

INDEX

Neper, 38 Noise, 10, 15 Noise figure, 227, 236 180◦ hybrid coupler, 208 Open circuit stub admittance, 70 Open-circuit impedance, 97 Oscillator feedback, 243 negative resistance, 246 Parallel-plate line, 48 Parallel-to-series transformation, 83 Phase constant, 27 Phase shifter loaded-line, 259 reflection, 258 switched-line, 257 PIN diode switch, 253 Power divider/combiner, 7 N -way, 199 corporate, 199 two-way, 197 unequal output, 202 Wilkinson, 199 Propagation constant, 27 Pulse code modulation (PCM), 16 Pulse radar, 6 range detection, 4 Q, 107, 143, 152, 154 Q tapering, 153, 174 QT , 141 Quantization error, 16 Quarter-wavelength transformer, 54, 175 Radar, 1 Doppler, 7 Frequency-modulated continuous-wave (FMCW), 8 pulse, 6 Radio, 11 Receiver, 10 Reflection coefficient, 31, 131, 258

lossless line, 32 Reflection phase shifter, 258 Repeater, 11 Return loss, 70 Ring or “rat-race” coupler, 211 S parameters, 110, 225 conversion to ABCD parameters, 112, 234 Scattering matrix, 110 Series-to-parallel transformation, 83 Shannon’s equation, 18 Short circuit stub impedance, 71 Short transformer, 57 Short-circuit admittance, 98 Siemens, 38 Signal flow graph, 117 Signal-to-noise ratio, 15 Single stub tuner, 60 Smith chart, 58, 77 Source impedance, 53, 131, 173, 191, 233 Stability factor (K factor), 232 Standing wave, 33 current, 35 voltage, 34 Stripline, 49, 160 Studio-to-transmitter links, 11 Sub-millimeter waves, 1 Switched-line phase shifter, 257 Tapped stub half-wavelength, 157 resonator, 155 Telegrapher equations, 27 Television analog, 15 broadcast, 11 cable, 11 digital, 15 TEM waves, 23 Terahertz waves, 1 Thompson filter, 136 use of Bessel polynomials, 136 Transducer gain, 104, 122, 225, 232 Transducer loss, 104

300

Transistor BJT, 234 FET, 229, 247 MESFET, 226 Transmission coefficient, 131, 259 Transmission line, 23 coaxial, 40, 46 coplanar stripline, 52 coplanar waveguide (CPW), 51 lumped element equivalent, 80, 84 microstrip, 50, 160 parallel plate, 48 planar, 23 stripline, 49, 160 two-wire or twin-lead, 23, 47 Transmission loss, 70 Transmitter, 10 Traveling wave, 33 Two-port networks, 97 Two-wire or twin-lead, 47 Unequal power division, 202, 204, 211 Up-conversion, 252 Uplink, 11, 212 Upper input power limit, 238 Varactor diode, 160 Varactor diodes (varactors), 259 Vector electric field intensity, 23, 40 Vector magnetic field intensity, 23, 40 Voltage reflection coefficient, 31 Voltage standing wave ratio, 34 Wave impedance, 32 Wave velocity, 30 Wavelength, 1 Waves incident, 29 reflected, 29 standing, 33 traveling, 33 Wilkinson power divider/combiner, 199 Wilkinson, Ernest, 199 Y matrix, 98 Z matrix, 98

INDEX

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