78897620 Electronic Applications of the Smith Chart SMITH P 1969

ELECTRONIC APPLICATIONSOF THE SMITH CHARTIn Waveguide, Circuit, and Component Analysis

PHII.LIP H. SMITHMember of the Technical StaffBell Telephone Laboratories, Inc.

Prepared under the sponsorship and direction ofKay Electric Company

McGRAW-HILL BOOK COMPANY New York St. Louis

San Francisco London Sydney Toronto Mexico Panama

ELECTRONIC APPLICATIONS OF THE SMITH CHART

Copyright © 1969 by Kay Electric Company, Pine Brook, NewJersey. All Rights Reserved. Printed in the United States ofAmerica. No part of this publication may be reproduced, stored ina retrieval system, or transmitted, in any form or by any means,electronic, mechanical, photocopying, recording, or otherwise,withou t the prior written permission of the publisher. Library ofCongress Catalog Card Number 69-12411

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Preface

The purpose of this book is to provide the student, the laboratorytechnician, and the engineer with a comprehensive and practical

source volume on SMITH CHARTS and their related overlays.In general, the book describes the mechanics of these charts in

relation to the guided-wave and circuit theory and, with examples,their practical uses in waveguide, circuit, and component applications.It also describes the construction of boundaries, loci, and forbiddenregions, which reveal overall capabilities and limitations of proposedcircuits and guided-wave systems.

The Introduction to this book relates some of the modifications ofthe basic SMITH CHART coordinates which have taken place since itsinception in the early 1930s.

Qualitative concepts of the way in which electromagnetic waves arepropagated along conductors are given in Chap. 1. This is followed inChaps. 2 and 3 by an explanation of how these concepts are related totheir quantitative representation on the "normalized" impedance coordinates of the SMITH CHART.

Chapters 4 and 5 describe the radial and peripheral scales of thischart, which show, respectively, the magnitudes and angles of variouslinear and complex parameters which are related to the impedancecoordinates of the chart. In Chap. 6 an explanation is given of equivalent circuit representations of impedance and admittance on the chartcoordinates.

Several uses of expanded portions of the chart coordinates aredescribed in Chap. 7, including the graphical determination therefromof bandwidth and Q of resonant and antiresonant line sections.

The complex transmission coefficients, their representations on theSMITH CHART, and their uses form the subject of Chap. 8. It isshown therein how voltage and current amplitude and phase (standingwave amplitude and wave position) are represented by these coefficients.

Impedance matching by means of single and double stubs, by singleand double slugs, and by lumped L-circuits is described in Chaps. 9 and

vii

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"'II,

viii PREFACE

10. Chapter 11 provides examples, illustrating how loci, boundaries,and forbidden areas are established and plotted.

The measurement of impedance by sampling voltage or current alongthe line at discrete positions, where a slotted line section would beexcessively long, is described in Chap. 11.

The effect of negative resistance loads on transmission lines, andthe construction and use of the negative SMITH CHART and its specialradial scales, are described in Chap. 12. Stability criteria as determinedfrom this chart are indicated for negative resistance devices such asreflection amplifiers.

Chapter 13 discusses, with examples, a number of typical applicationsof the chart.

Chapter 14 describes several instruments which incorporate SMITHCHARTS as a basic component, or which are used with SMITH CHARTSto assist in plotting data thereon or in interpreting data therefrom.

For the reader who may desire a more detailed discussion of anyparticular phase of the theory or application of the chart a bibliography is included to which references are made as appropriate throughout the text.

Fundamental mathematical relationships for the propagation ofe~ectromagnetic waves along transmission lines are given in AppendixA and details of the conformal transformation of the original rectangularto the circular SMITH CHART coordinates are included in Appendix B.A glossary of terms used in connection with SMITH CHARTS followsChap. 14.

Four alternate constructions of the basic SMITH CHART coordinates,printed in red ink on translucent plastic, are supplied in an evelope inthe back cover of the book. All of these are individually described inthe text. By superimposing these translucent charts on the generalpurpose complex waveguide and circuit parameter charts describedthroughout the book, with which they are dimensionally compatible,it is a simple matter to correlate them graphically therewith and totransfer data or other information from one such plot to the other.

The overlay plots of waveguide parameters used with these translucent SMITH CHARTS include the complex transmission and reflectioncoefficients for both positive and negative component coordinates,normalized voltage and current amplitude and phase relationships,normalized polar impedance coordinates, voltage and current phase andmagnitude relationships, loci of current and voltage probe ratios, L-typematching circuit components, etc. These are generally referred to as"overlays" for the SMITH CHART because they were originally published as transparent loose sheets in bulletin form and because theywere so used. However, as a practical matter it was found to bedifficult to transfer the parameters or data depicted thereon to theSMITH CHART, which operation is more generally required. Accord-

PREFACE ix

ingly, they are printed here on opaque bound pages and used as thebackground on which the translucent SMITH CHARTS in the backcover can be superimposed. The bound background charts are printedin black ink to facilitate visual separation of the families of curveswhich they portray from the red impedance and/or admittance curveson the loose translucent SMITH CHARTS. The latter charts have amatte finish which is erasable to allow pencil tracing of data or otherinformation directly thereon.

Phillip H. Smith

Acknowledgments

The writer is indebted to many of his colleagues at Bell TelephoneLaboratories for helpful discussions and comments, in particular, in

the initial period of the development of the chart to the late Mr. E. J.Sterba for his help with transmission line theory, and to Messrs. E. B.Ferrell and the late J. W. McRae for their assistance in the area ofconformal mapping. Credit is also due Mr. W. H. Doherty for suggestingthe parallel impedance chart, and to Mr. B. Klyce for his suggested useof highly enlarge portions of the chart in determining bandwidth ofresonant stubs. Mr. R. F. Tronbarulo's investigations were helpful inwriting sections dealing with the negative resistance chart.

The early enthusiastic acceptance of the chart by staff members atMIT Radiation Laboratory stimulated further improvements in designof the chart itself.

Credit for publication of the book at this time is principally due toencouragement provided by Messrs. H. R. Foster and E. E. Crump ofKay Electric Company [14].

Phillip H. Smith

xi

,

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f

Introduction

1.1 GRAPHICAL VS. MATHEMATICAL REPRESENTATIONS

The physical laws governing natural phenomena can generally berepresented either mathematically or graphically. Usually the more

complex the law the more useful is its graphical representation. Forexample, a simple physical relationship such as that expressed by Ohm'slaw does not require a graphical representation for its comprehension oruse, whereas laws of spherical geometry which must be applied insolving navigational problems may be sufficiently complicated to justifythe use of charts for their more rapid evaluation. The ancient astrolabe,a Renaissance version of which is shown in Fig. 1.1, provides an interesting example of a chart which was used by mariners and astronomers forover 20 centuries, even though the mathematics was well understood.

The laws governing the propagation of electromagnetic waves alongtransmission lines are basically simple; ho~ever, their mathematicalrepresentation and application involves hyperbolic and exponentialfunctions (see Appendix A) which are not readily evaluated withoutthe aid of charts or tables. Hence these physical phenomena lendthemselves quite naturally to graphical representation.

Tables of hyperbolic functions published by A. E. Kennelly [3]in 1914 simplified the mathematical evaluation of problems relatingto guided wave propagation in that period, but did not carry thesolutions completely into the graphical realm.

1.2 THE RECTANGULAR TRANSMISSIOI\J LINE CHART

The progenitor of the circular transmission line chart was rectangularin shape. The original rectangular chart devised by the writer in 1931 isshown in Fig. 1.2. This particular chart was intended only to assist inthe solution of the mathematics which applied to transmission lineproblems inherent in the design of directional shortwave antennas for

xiii

xiv INTRODUCTION

Fig. 1.1. A Renaissance version of the oldest scientific instrument in the world. (Danti des Renaldi, 1940.1

INTRODUCTION xv

Bell System applications of that period; its broader application washardly envisioned at that time.

The chart in Fig. 1.2 is a graphical plot of a modified form of J. A.Fleming's 1911 "telephone" equation [2], as given in Chap. 2 and inAppendix A, which expresses the impedance characteristies of highfrequency transmission lines in terms of measurable effects of electromagnetic waves propagating thereon, namely, the standing-waveamplitude ratio and wave position. Since this chart displays impedanceswhose complex components are "normalized," i.e., expressed as afraction of the characteristic impedance of the transmission line underconsideration, it is applicable to all types of waveguides, including openwire and coaxial transmission lines, independent of their characteristicimpedances. In fact, it is this impedance normalizing concept whichmakes such a general plot possible.

Although larger and more accurate rectangular charts have subsequently been drawn, their uses have been relatively limited because ofthe limited range of normalized impedance values and standing-waveamplitude ratios which can be represented thereon. This stimulatedseveral attempts by the writer to transform the curves into a moreuseful arrangement, among them the chart shown in Fig. 7.7 whichwas constructed in 1936.

1.3 THE CIRCULAR TRAI\lSMISSION LINE CHART

The initial clue to the fact that a conformal transformation of thecircular orthogonal curves of Fig. 1.2 might be possible was provided bythe realization that these two families of circles correspond exactly tothe lines of force and the equipotentials surrounding a pair of equal andopposite parallel line charges, as seen in Fig. 1.1. It was then a simplematter to show that a bilinear conformal transformation [55,109]would, in fact, produce the desired results (see Appendix B), and thecircular form of chart shown in Fig. 1.3, which retained the normalizingfeature of the rectangular chart of Fig. 1.2, was subsequently devisedand constructed. All possible impedance values are representablewithin the periphery of this later chart. An article describing theimpedance chart of Fig. 1.3 was published in January, 1939 [101 ] .

During World War II at the Radiation Laboratory of the MassachusettsInstitute of Technology, in the environment of a flourishing microwavedevelopment program, the chart first gained widespread acceptance andpublicity, and first became generally referred to as the SMITH CHART.

Descriptive names have in a few instances been applied to theSMITH CHART (see glossary) by other writers; these include "Reflection Chart," "Circle Diagram (ofImpedance)," "Immittance Chart," and"Z-plane Chart." However, none of these are in themselves sufficiently

xvi INTRODUCTION

IMPED. ALONG 'rRANS. LINE VS. STANDING WAVE RATIO (r) AND DISTANCE (D),IN WAVELENGTHS, TO ADJACENT CURRENT(OR VOLTAGE) MIN. OR MAX. POINT.

RDIST. TO FOLLOWING Imin OR Emax Zo DIST. TO FOLLOWING Imax OR Emin

ix 1.4. 1.2 1.0 .8

-Zo

.6 .4 .2 0 .2 .4 .6NORMALIZED REACTANCE

.8 1.0 1.2 1.4.+JX

P.H.S.4·22-31 Zo

Fig. 1.2. The original rectangular transmission line chart.

INTRODUCTION xvii

lOA.D- (I)~~~iTla;~~3l~Z<r

I I , I I I I I I I Iflf~f"" ~ ~ ~ " ? " " ~ " 0» -)( Z IIb: u; 0 ~ :;: 2 )( Z g

____-" c....... PlVOT AT CENTER OF CALCULATOR

O __-_II_I-_I_II_~_E_~E_R_~_TO_R_I --------'-,'=-'--1

I I i I I I ~ I

TRANSPARENTSLIDER FOR ARM

Fig. 1.3. Transmission line calculator. (Electronics, January, 1939.)

xviii INTRODUCTION

definitive to be used unambiguously when comparing the SMITHCHART with similar charts or with its overlay charts as discussed inthis text. For these reasons, without wishing to appear immodest, thewriter has decided to use the more generally accepted name in theinterest of both clarity and brevity.

Drafting refinements in the layout of the impedance coordinateswere subsequently made and additional scales were added showing therelation of the reflection coefficient to the impedance coordinates,which increased the utility of the chart. These changes are shown inFig. 1.4. A second article published in 1944 incorporated these improvements [102]. This later article also described the dual use of the chartcoordinates for impedances and/or admittances, and for convertingseries components of impedance to their equivalent parallel componentvalues.

In 1949 the labeling of the chart impedance coordinates was changedso that the chart would display directly either normalized impedanceor normalized admittance. This change is shown in the chart of Fig.2.3. On this later chart the specific values assigned to each of thecoordinate curves apply, optionally, to either the impedance or to theadmittance notations.

In 1966 additional radial and peripheral scales were added to portraythe fixed relationship of the complex transmission coefficients to thechart coordinates, as shown in Fig. 8.6.

1.4 ORIENTATION OF IMPEDANCE COORDINATES

The charts in Figs. 1.2 and 1.3 as originally plotted have theirresistance axes vertical. It became apparent shortly after publicationof Fig. 1.3, as thus oriented, that a horizontal representation of theresistance axis was preferable since this conformed to the acceptedconvention represented by the Argand diagram in which complexnumbers (x ± iy) are graphically represented with the real (x) component horizontal and the imaginary (y) component vertical.

Therefore, subsequently published SMITH CHARTS have generallybeen shown, and are shown throughout the remainder of this book,with the resistance (R) axis horizontal, and the reactance (± jX) axisvertical; inductive reactance (+ jX) is plotted above, ·and capacitivereactance (- jX) below the resistance axis.

1.5 OVERLAYS FOR THE SMITH CHART

Axially symmetric overlays for the SMITH CHART were inherentin the first chart, as represented by the peripheral and radial scales

INTRODUCTION xix

,:g ..

~ ~..

~, c IiL ---- --- -' f f 0 is i': Of ,.

I~, , , ,, I ,,

~ i I ~" N :. ;,. " 6 66 8 ~

iN $ ~ ~ ? l': ~g ~f ? , ! ? , <i'I~ I, , , , " , ,

~I

" N ;. a. 6 . ~ <3 .8 ~I~.. " " " 0

!\ffA,Rq, lOAO-- -- TpWARD GENERATOR ~g ,Ii ~1 ! I ,I , , ,~

, , , I'" !::

"- ;. b 6 0

N 8g~'~I>!N .. ~ " 0

!p

~ " ~ ? 6 " ° 8;5 Ii:! ~, , , , I , , , ,, , ',' l.m~ ~ <5

( ----- - --""1° il i5 ~ ~ ~ <) ~ :g '" § ~ z

Fig. 1.4. Improved transmission line calculator. (Electronics, January, 1944.1

xx INTRODUCTION

for the chart coordinates. These overlays include position and amplituderatio of the standing waves, and magnitude and phase angle of thereflection coefficients. Additionally, overlays showing attenuation andreflection functions were represented by radial scales alone (see Fig.1.4).

In the present text 26 additional general-purpose overlays (bothsymmetrical and asymmetrical) for which useful applications exist andwhich have been devised for the SMITH CHART are presented.

Contents

PREFACE

ACKNOWLEDGMENTS

INTRODUCTION

1.1 Graphical vs. Mathematical Representations1.2 The Rectangular Transmission Line Chart1.3 The Circular Transmission Line Chart1.4 Orientation of Impedance Coordinates1.5 Overlays for the SMITH CHART

CHAPTER 1 - GUIDED WAVE PROPAGATION

1.1 Graphical Representation1.2 Waveguide Structures1.3 Waveguide Waves1.4 Traveling Waves1.5 Surge Impedance1.6 Attenuation1.7 Reflection1.8 Standing WavesProblems

CHAPTER 2 - WAVEGUIDE ELECTRICAL PARAMETERS

2.1 Fundamental Constants2.2 Primary Circuit Elements2.3 Characteristic Impedance

2.3.1 Characteristic Admittance2.4 Propagation Constant2.5 Parameters Related to SMITH CHART Coordinates2.6 Waveguide Input Impedance2.7 Waveguide Input Admittance2.8 Normalization2.9 Conversion of Impedance to Admittance

vii

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xiii

xiiixiiixv

xviiixviii

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112333457

11

11111214141517181920

xxi

xxii CONTENTS

CHAPTER 3 - SMITH CHART COI\ISTRUCTION 21

3.1 Construction of Coordinates 213.2 Peripheral Scales 21

3.2.1 Electrical Length 243.2.2 Reflection Coefficient Phase Angle 25

3.3 Radial Reflection Scales 253.3.1 Voltage Reflection Coefficient Magnitude 263.3.2 Power Reflection Coefficient 263.3.3 Standing Wave Amplitude Ratio 263.3.4 Voltage Standing Wave Ratio, dB 30

CHAPTER 4 - LOSSES AND VOLTAGE-CURRENTREPRESENTATIONS 33

4.1 Radial Loss Scales 334.1.1 Transmission Loss 344.1.2 Standing Wave Loss Factor 364.1.3 Reflection Loss 374.1.4 Return Loss 38

4.2 Current and Voltage Overlays 38Problems 41

CHAPTER 5 - WAVEGUIDE PHASE REPRESENTATIONS 43

5.1 Phase Relationships 435.2 Phase Conventions 445.3 Angle of Reflection Coefficient 465.4 Transmission Coefficient 465.5 Relative Phase along a Standing Wave 505.6 Relative Amplitude along a Standing Wave 52Problems 52

CHAPTER 6 - EQUIVALENT CIRCUIT REPRESENTATlOI\ISOF IMPEDANCE AI\ID ADMITTAI\JCE 57

6.1 Impedance Concepts 576.2 Impedance-admittance Vectors 576.3 Series-circuit Representations of Impedance and

Equivalent Parallel-circuit Representations ofAdmittance on Conventional SMITH CHARTCoordinates 58

6.4 Parallel-circuit Representations of Impedance andEquivalent Series-circuit Representations of Admittanceon an Alternate Form of the SMITH CHART 60

6.5 SMITH CHART Overlay for Converting a Series-circuitImpedance to an Equivalent Parallel-circuit Impedance,and a Parallel-circuit Admittance to an EquivalentSeries-circuit Admittance 63

CONTENTS xxiii

6.6 Impedance or Admittance Magnitude and AngleOverlay for the SMITH CHART 63

6.7 Graphical Combination of Normalized PolarImpedance Vectors 66

CHAPTER 7 - EXPANDED SMITH CHARTS 71

7.1 Commonly Expanded Regions 717.2 Expansion of Central Regions 737.3 Expansion of Pole Regions 73704 Series-resonant and Parallel-resonant Stubs 767.5 Uses of Pole Region Charts 79

7.5.1 Q of a Uniform Waveguide Stub 817.5.2 Percent off Midband Frequency Scales on

Pole Region Charts 817.5.3 Bandwidth of a Uniform Waveguide Stub 82

7.6 Modified SMITH CHART for Linear SWR Radial Scale 827.7 Inverted Coordinates 82

CHAPTER 8 - WAVEGUIDE TRAI\JSIVIISSIOI\JCOEFFICIENT (,.) 87

8.1 Graphical Representation of Reflection andTransmission Coefficients 87

8.1.1 Polar vs. Rectangular Coordinate Representation 878.1.2 Rectangular Coordinate Representation of

Reflection Coefficient p 888.1.3 Rectangular Coordinate Representation of

Transmission Coefficient 'T" 888.104 Composite Rectangular Coordinate

Representation of p and 'T" 898.2 Relation of p and 'T" to SMITH CHART Coordinates 918.3 Application of Transmission Coefficient Scales on

SMITH CHART in Fig. 8.6 94804 Scales at Bottom of SMITH CHARTS in Figs. 8.6

and 8.7 95

CHAPTER 9 - WAVEGUIDE IMPEDANCE ANDADMITTANCE MATCHING 97

9.1 Stub and Slug Transformers 979.2 Admittance Matching with a Single Shunt Stub 97

9.2.1 Relationships between Impedance Mismatch,Matching Stub Length, and Location 98

9.2.2 Determination of Matching Stub Length andLocation with a SMITH CHART 100

9.2.3 Mathematical Determination of Required StubLength of Specific Characteristic Impedance 100

xxiv CONTENTS

9.2.4 Determination with a SMITH CHART of RequiredStub Length of Specified CharacteristicImpedance 101

9.3 Mapping of Stub Lengths and Positions on aSMITH CHART 102

9.4 Impedance Matching with Two Stubs 1029.5 Single-slug Transformer Operation and Design 1079.6 Analysis of Two-slug Transformer with a

SMITH CHART 1109.7 Determination of Matchable Impedance Boundary 112

CHAPTER 10 - NETWORK IMPEDANCETRANSFORMATIONS 115

10.1 L-type Matching Circuits 11510.1.1 Choice of Reactance Combinations 11610.1.2 SMITH CHART Representation of Circuit

Element Variations 11610.1.3 Determination of lrtype Circuit Constants

with a SMITH CHART 11810.2 T-type Matching Circuits 11910.3 Balanced lr or Balanced T-type Circuits 128

CHAPTER 11 - MEASUREMENTS OF STANDING WAVES 129

11.1 Impedance Evaluation from Fixed Probe Readings 12911.1.1 Example of Use of Overlays with Current Probes 130

11.2 Interpretation of Voltage Probe Data 13011.3 Construction of Probe Ratio Overlays 136

CHAPTER 12 - I\IEGATIVE SMITH CHART 137

12.1 Negative Resistance 13712.2 Graphical Representation of Negative Resistance 13812.3 Conformal Mapping of the Complete SMITH CHART 13912.4 Reflection Coefficient Overlay for Negative

SMITH CHART 14112.5 Voltage or Current Transmission Coefficient Overlay

for Negative SMITH CHART 14212.6 Radial Scales for Negative SMITH CHART 144

12.6.1 Reflection Coefficient Magnitude 14412.6.2 Power Reflection Coefficient 14512.6.3 Return Gain, dB 14512.6.4 Standing Wave Ratio 14912.6.5 Standing Wave Ratio, dB 14912.6.6 Transmission Loss, I-dB Steps 15012.6.7 Transmission Loss Coefficient 150

CONTENTS xxv

12.7 Negative SMITH CHART Coordinates, Exampleof Their Use 150

12.7.1 Reflection Amplifier Circuit 15012.7.2 Representation of Tunnel Diode Equivalent

Circuit on Negative SMITH CHART 15112.7.3 Representation of Operating Parameters of

Tunnel Diode 15212.7.4 Shunt-tuned Reflection Amplifier 155

12.8 Negative SMITH CHART 155

CHAPTER 13 - SPECIAL USES OF SMITH CHARTS 157

13.1 Use Categories 15713.1.1 Basic Uses 15713.1.2 Specific Uses 157

13.2 Network Applications 15813.3 Data Plotting 15913.4 Rieke Diagrams 16113.5 Scatter Plots 16113.6 Equalizer Circuit Design 162

13.6.1 Example for Shunt-tuned Equalizer 16313.7 Numerical Alignment Chart 16613.8 Solution of Vector Triangles 166

CHAPTER 14 - SMITH CHART INSTRUMENTS 169

14.1 Classification 16914.2 Radio Transmission Line Calculator 16914.3 Improved Transmission Line Calculator 17114.4 Calculator with Spiral Cursor 17314.5 Impedance Transfer Ring 17414.6 Plotting Board 17514.7 Mega-plotter 17614.8 Mega-rule 179

14.8.1 Examples of Use 18014.8.2 Use of Mega-rule with SMITH CHARTS 182

14.9 Computer-plotter 18214.10 Large SMITH CHARTS 182

14.10.1 Paper Charts 18214.10.2 Blackboard Charts 183

14.11 Mega-charts 18414.11.1 Paper SMITH CHARTS 18414.11.2 Plastic Laminated SMITH CHARTS 18414.11.3 Instructions for SMITH CHARTS 184

xxvi CONTENTS

GLOSSARY - SMITH CHART TERMS

Angle of Reflection Coefficient, DegreesAngle of Transmission Coefficient, DegreesAttenuation (I-dB Maj. Div.)Coordinate ComponentsImpedance or Admittance CoordinatesNegative Real PartsNormalized CurrentNormalized VoltagePercent off Midband FrequencyPeripheral ScalesRadially Scaled ParametersReflection Coefficients, E or IReflection Coefficient, PReflection Coefficient, X or Y ComponentReflection Loss, dBReturn Gain, dBReturn Loss, dBSMITH CHARTStanding Wave Loss Coefficient (Factor)Standing Wave Peak, Const. PStanding Wave Ratio (dBS)Standing Wave Ratio (SWR)Transmission Coefficient E or ITransmission Coefficient PTransmission Coefficient, X and Y ComponentsTransmission Loss CoefficientTransmission Loss, I-dB StepsWavelengths toward Generator (or toward Load)

APPENDIX A - TRAI\lSIVlISSION LINE FORMULAS

1. General Relationships for Any Finite-lengthTransmission Line(a) Open-circuited lines(b) Short-circuited lines

2. Relationships for any Finite-length LosslessTransmission Line(a) Lines terminated in an impedance(b) Open-circuited lines(c) Short-circuited lines

APPENDIX B - COORDINATE TRANSFORMATION

Bilinear Transformationa. The Lines v = a constant

185

185186186186187187187188188188188189189189189189189189190190190190190190191191191191

193

194194194

194194195195

197

197198

CONTENTS xxvii

b. The Lines u = a constant 198c. The Circles of Constant Electrical Line Angle 199d. The Circles of Constant Standing Wave Ratio 199

SYIVIBO LS 201

REFERENCES 207

BIBLIOGRAPHY FOR SMITH CHART PUBLICATIOI\JS 209

PART I - BOOKS 209Encyclopedias 209Handbooks 210Textbooks 210

PART II - PERIODICALS 213PART III - BULLETINS, REPORTS, etc. 216

INDEX 219

1.1 GRAPHICAL REPR.ESENTATION

The SMITH CHART is, fundamentally, agraphical representation of the interrela

tionships between electrical parameters of aunifonn waveguide. Accordingly, its designand many of its applications can best be described in accordance with principles of guidedwave propagation.

The qualitative descriptions of the electricalbehavior of a waveguide, as presented in thischapter, will provide a background for betterunderstanding the significance of various interrelated electrical parameters which are morequantitatively described in the following chapter. As will be seen, many of these parameters are represented directly by the coordinates and associated scales of a SMITHCHART, and their relationships are basic toits construction.

1.2 WAVEGUIDE STRUCTURES

The tenn waveguide, as used in this book,will be understood to include not only hollowcylindrical uniconductor waveguides, but allother physical structures used for guiding

CHAPTER 1Guided

WavePropagation

electromagnetic waves (except, of course, heatand light waves). Included are multi-wiretransmission line, strip-line, coaxial line, triplate, etc. Waveguide tenns as they firstappear in the text will be italicized to indicatethat their definitions are in accordance withdefinitions which have been standardized bythe Institute of Electrical and ElectronicsEngineers [ 11] , and are currently accepted bythe United States of America Standards Institute (USASI). (Also, terms and phrases aresometimes italicized in lieu of underlining toprovide emphasis or to indicate headings orsubtitles.)

Although it may ultimately be of considerable importance in the solution of any practical waveguide problem, the particular waveguide structure is of interest in connectionwith the design or use of the SMITH CHARTonly to the extent that its configuration, crosssectional dimensions in wavelengths, and modeof propagation establish two basic electricalconstants of the waveguide, viz., the propagation constant and the characteristic impedance.Both of these constants are further discussedin Chap. 2.

1

2 ELECTRONIC APPLICATIONS OF THE SMITH CHART

1.3 WAVEGUIDE WAVES

Waveguide waves can propagate in numerous modes, the exact number depending uponthe configuration and size of the conductor(or conductors) in wavelengths. Modes ofpropagation in a waveguide are generallydescribed in terms of the electric and magnetic field pattern in the vicinity of theconductors, with which each possible modeis uniquely associated.

As is the case for the waveguide structure,the mode of propagation does not playa directrole in the design or use of the SMITH CHART.

ELECTRIC FIELD -----MAGNETIC FIELD --

Its importance, however, lies in the fact thateach mode is characterized by a differentvalue for the propagation constant and characteristic impedance to which the variablesof the problem must ultimately be related.

A specific waveguide structure may providethe means for many different modes of propagation although only one will generally beselected for operation. Field patterns for themore common dominant mode in a two-wireand in a coaxial transmission line are shownin Figs. 1.1 and 1.2, respectively. In thismode both electric and magnetic field components of the wave lie entirely in planes

Fig. 1.1. Dominant mode field pattern on parallel wire transmission line.

ELECTRIC FIELD ----MAGNETIC FIELD --

Fig. 1.2. Dominant mode field pattern on coaxial transmission line.

transverse to the direction of propagation,and the wave is, therefore, called a transverseelectromagnetic (TEM) wave. There is nolongitudinal component of the field in thismode.

When the mode of propagation is known ina particular uniconductor waveguide, andoperation is at a particular frequency, thewaveguide wavelength can readily be computed. The SMITH CHART can then be usedin the same way in which it is used forproblems involving the simple TEM wave intwo-wire or coaxial transmission lines.

A further discussion of the subject ofpropagation modes and their associated fieldpatterns will not be undertaken herein sincethe reader who may be interested will findadequate discussions of this subject in theliterature [32,52].

1.4 TRAVELING WAVES

The propagation of electromagnetic waveenergy along a waveguide can perhaps best beexplained in terms of the component travelingwaves thereon.

If a continuously alternating sinusoidal voltage is applied to the input terminals of a waveguide, a forward-traveling voltage wave will beinstantly launched into the waveguide. Thiswave will propagate along the guide as acontinuous wave train in the only directionpossible, namely, toward the load, at thecharacteristic wave velocity of the waveguide.Simultaneously with the generation of aforward-traveling voltage wave, an accompanying forward-traveling current wave is engendered, which also propagates along the waveguide. The forward-traveling current wave is inphase with the forward-traveling voltage waveat all positions along a lossless waveguide.These two component waves make up theforward-traveling electromagnetic wave.

GUIDED WAVE PROPAGATION 3

1.5 SURGE IMPEDANCE

The input impedance that the forwardtraveling electromagnetic wave encounters asit propagates along a uniform waveguide iscalled the surge impedance or the initialsending-end impedance. In the case of auniform waveguide it is called, specifically,the characteristic impedance. The input impedance has, initially, a constant value independent of position along the waveguide. Itsmagnitude is independent of the magnitude orof the phase angle of the load reflectioncoefficient, it being assumed that in this briefinterval the forward-traveling electromagneticwave has not yet arrived at the load tenninals.At any given position along the waveguidethe forward-traveling electromagnetic wave hasa sinusoidal amplitude variation with time asthe wave train passes this position.

1.6 ATTENUATION

As the forward-traveling voltage and current waves propagate along the waveguidetoward the load, some power will be continuously dissipated along the waveguide. Thisis due to the distributed series resistance ofthe conductor (or conductors) and to thedistributed shunt leakage resistance encountered, respectively, by the longitudinal currentsand the transverse displacement currents inthe dielectric medium between conductors.This dissipation of power is called attenuation,or one-way transmission loss. Attenuationdoes not change the initial input impedanceof the waveguide as seen by the forwardtraveling wave energy as it progresses along thewaveguide, since it is uniformly distributed,but it nevertheless diminishes the wave powerwith advancing position along the waveguide.Attenuation should not be confused with thetotal dissipation of a waveguide measured


under steady-state conditions, as will be explained later.

Upon arrival at the load, the forwardtraveling voltage and current wave energy willencounter a load impedance which mayormay not be of suitable value to "match" thecharacteristic impedance of the waveguideand, thereby, to absorb all of the energywhich these waves carry with them.

1.7 REFLECTION

Assume for the moment that the load impedance matches* the characteristic impedance of the waveguide, and that the load,therefore, absorbs all of the wave energyimpinging thereon. There will then be noreflection of energy from the load. Inthis case the forward-traveling wave will bethe only wave along the waveguide. Avoltmeter or ammeter placed across, or atany position along the waveguide, respectively, would then indicate an rms value inaccordance with Ohm's law for alternatingcurrents in an impedance which is equivalentto the characteristic impedance of the waveguide.

If, on the other hand, the load impedancedoes not match the characteristic impedanceof the waveguide, and therefore does notabsorb all of the incident electromagneticwave energy available, there will be reflectedwave energy from the mismatched load impedance back into the waveguide. This resultsin a rearward-traveling current and voltagewave. The complex ratio of the rearwardtraveling voltage wave to the forward-travelingvoltage wave at the load is called the voltagereflection coefficient. Likewise, the complex

*1£ the waveguide is lossy its characteristic impedance willbe complex, that is, Ro - jXo. For a conjugate match, thecondition which provides maximum power transfer, the loadimpedance, should be R I + jX I, where R I = Ro andXI = Xo·

ratio of the rearward-traveling current waveto the forward-traveling current wave at theload is called the current reflection coefficient.If the waveguide is essentially lossless, themagnitude of the voltage reflection coefficientwill be constant at all points from the loadto the generator and will be equal to thevoltage reflection coefficient magnitude at theload. If there is attenuation along the waveguide, the magnitude of the voltage reflectioncoefficient will gradually diminish with distance toward the generator, due to the additional attenuation encountered by the reflected-wave energy, as compared to thatencountered by the incident-wave energy inits shorter path. Thus, there will appear to beless reflected energy at the input to such awaveguide than at the load.

The voltage reflection coefficient will alsovary in phase, with position along the waveguide. The phase angle of the voltage reflection coefficient at any point along awaveguide is determined by the phase shiftundergone by the reflected voltage wave incomparison to that of the incident voltagewave at the point under consideration, including any phase change at the load itself.If the reflection coefficient phase angle exceeds1800 (representing a path difference of morethan one-half wavelength between reflectedand incident-wave energy), the relative phaseangle of the voltage reflection coefficient onlyis usually of interest. This is obtainable bysubtracting the largest possible integer multipleof 1800 (corresponding to integer lengths ofone-quarter wavelengths of waveguide) fromthe absolute or total phase angle of the voltagereflection coefficient, to yield a relative phaseangle between plus and minus 1800

•

The traveling current waves on a waveguide,which accompany traveling voltage waves, willlikewise be reflected by a mismatched loadimpedance. The magnitude of the currentreflection coefficient at all points along awaveguide is identical to that of the voltage

r

reflection coefficient in all cases. The phaseangle of the current reflection coefficient atany given point along the waveguide may, however, differ from that of the voltage reflectioncoefficient by as much as ± 1800

, dependingupon the relative phase changes undergone bythe two traveling waves at the load.

An open-circuited load will, for example,cause complete reflection of both current andvoltage incident traveling waves. The voltageand current reflection coefficient magnitudeat the load will both be unity in this case. Thephase angle of the voltage reflection coefficientat the load will be zero degrees under thiscondition, since the same voltage is reflected(as a wave) back into the waveguide at thispoint. However, the current reflection coefficient phase angle at the open-circuitedload will be -1800 since the incident currentwave amplitude upon arriving at such a loadwill suddenly have to drop to zero (there beingno finite value of load impedance) and willbuild up again in opposite polarity to berelaunched as a reflected wave into the waveguide.

The collapse of the magnetic field attendingthe incident current wave, when it encountersan open-circuited load, results in an accompanying buildup of the electric field and in acorresponding buildup of voltage at the opencircuited load to exactly twice the value ofthe voltage attending the incident voltagewave. This is because electromagnetic waveenergy along a uniform waveguide is dividedequally between the magnetic and the electricfields. The reflected wave voltage builds upin phase with the incident wave voltage andthe voltage reflection coefficient phase angleis, accordingly, zero degrees at this point.

1.8 STANDING WAVES

At this point in the order of events, theload is in the process of engendering standing


waves along the waveguide, for there suddenlyappears a maximum voltage and a minimumcurrent at the load terminals, when just priorto this, the voltage and current at all pointsalong the line were constant (except for thediminishing effects of attenuation). Themodified load voltage now launches back intothe waveguide a rearward-traveling voltage andcurrent wave in a manner similar to the initiallaunching of these waves at the generator end.These rearward-traveling waves combine withthe respective forward-traveling waves and,because of their relative phase differences atvarious positions along the line (changing phaseangle of voltage and current reflection coefficients), cause alternate reinforcement andcancellation of the voltage and current distribution along the line. This phenomenonresults in what has previously been referredto as standing or stationary waves along thewaveguide. The shape of these standing waveswith position along a waveguide is shown inFig. 1.3. It will be seen that their shape issinusoidal only in the limiting case of complete reflection from the load, i.e., when thestanding wave ratio of maximum to minimumis infinity. A graphical representation of thecombination of the two traveling waves isshown in Fig. 104.

Upon arrival back at the generator terminals,the rearward-traveling voltage wave combinesin amplitude and phase with the voltage beinggenerated at the time, to produce a change inthe generated voltage amplitude and phase.At this instant, the generator is first presentedwith a change in the waveguide input impedance and readjusts its output accordingly.This is the beginning in time sequence of aseries of regenerated waves at each end ofthe waveguide, which after undergoing multiple reflections therefrom, eventually combineto produce steady-state conditions and become, in effect, a single forward-traveling anda single rearward-traveling current and voltagewave. Thus, in practice, it is not usually

r


SWR= 00

I V '\. /..... r'\

/ \ V 1\1/ 1\ / \

SWR =5.0

/ - ............ \ / - \V .".- .........

I / II" '\ \II J

SWR=2.0\ 1\ I / ' \

/ - \ \ I 1/ - \I 1/ / ......... \\ II / "V I'\. \ 1\'I J '\ \ / V \. ~\

!J V SWR=1.0 1'\\ I1// \ \\r/ \~\ U/ 1\'

) \. 11 '~VI \~ /' \'..- II l r-- J \ ........

~ 1\ J ,~ II

/ \'~I I\./ \I \ I \

II V

.2

oo .04 .08 .12 .16 .20 .24.28 .32 .36 .40 .44 .48 .52 .56 .60 .64 .68 .72 .76 .BO .B4 .BB .92 96 1.00

RELATIVE POSITION ALONG LINE - WAVELENGTHS

6

2.0

ILl<J)« 1.8I-..J0>I-

1.6zILl0

U~

I-1.4

z«I-UIz 1.20u

a::0 1.0l.L

ILl0::::l .8I-

..JQ.:::E« .6w>I-« .4..JWa::

Fig. 1.3. Relative amplitudes and shapes of voltage or current standing wajes along a 1055le55 waveguide (constant inputvoltage).

and when i---. T, S---. 00. Also, if the incidentvoltage or current wave amplitude is held

degrees. The accompanying standing voltagewave has a maximum-to-minimum wave amplitude ratio of infinity. The position of themaximum point of the voltage standing waveis at the open-circuited load terminals of thewaveguide, at which point the input impedanceis infinity.

Expressed algebraically, in terms of theamplitude of the incident i and reflected T

traveling waves, the standing wave ratio Sis seen to be their sum divided by their difference, i.e.,

necessary to consider transient effects ofmultiple reflections between generator andload beyond that of a single reflection at theload end of the waveguide.

The magnitude and phase angle of thevoltage and current reflection coefficient beara direct and inseparable relationship to theamplitude and position of the attendingstanding waves of voltage or current alongthe waveguide, as well as to the input impedance (or admittance) at all positions alongthe waveguide. Through suitable overlays thisrelationship can very simply be described onthe SMITH CHART.

As a specific example of this relationship,the magnitude of the voltage reflection coefficient at the open-circuited load previouslyconsidered is unity. Its phase angle is zero

s l + T

i - T(1- I)


The voltage (or current) reflection coefficient magnitude p is, by definition, simply

Fig. 1.4. Graphical representation of incident and reflected traveling waves of voltage or current forming standingwaves.

1-1. Assuming constant incident voltage amplitude, and using the graphical construction in Fig. 1.4, plot the spatial shape of

a voltage standing wave whose ratio S is3.0, along a uniform, loss-free waveguide.Plot points at one-sixteenth-wavelengthintervals toward load from a voltageminimum (current maximum) point.

Solution:As shown in Fig. 1.5:

1. Draw a horizontal axis (0-2.0) ofsome convenient length, and divide intoten equal parts; label these 0, 0.2, 0.4,etc., to 2.0.2. Draw a circle, in which the above horizontal axis is a diameter, representingthe boundary of a SMITH CHART within which all possible waveguide voltage orcurrent vectors can be represented.3. Draw three straight lines intersectingthe center point (1.0) of the horizontalaxis (real axis of the SMITH CHART) atangles of 45, 90, and 1350 from thehorizontal, corresponding to onesixteenth-wavelength intervals along awaveguide. (On the SMITH CHART 3600

corresponds to one-half wavelengths.)4. Draw a circle centered at 1.0 on thehorizontal axis, representing the locus ofthe voltage standing wave vector alongthe waveguide, intersecting two values(1.5 and 0.5) on the horizontal axis,the ratio S of which will be equal to3.0. (From Eq. (1-3), if S =3.0, r =0.5;and if i = 1.0, i + r = 1.5, and i - r = 0.5.)5. Draw voltage vectors from each of theintersections of this circle (4) with therespective angle lines (3) to the origin (0)on the horizontal axis, and number asshown in Fig. 1.5. Measure with horizontal axis scale, and tabulate voltagevector lengths 0 - CD ,0 - @, 0 -®,etc.to [email protected]. Starting at CD (the voltage minimumpoint of the standing wave) pl~t thevoltage standing wave on rectangular coordinates from the tabulated amplitudesat eight equally spaced positions (CDthrough@).

(1-4)

(1-3)

(1-2)

1.0.5

REFL.COEF MAGNITUDE

1.667(SWR MAX.)

II

//

,/......-o

ri

1 + r

1 - r

S - 1S + 1

p

PROBLEMS

or

s

constant at unity (the case of a well-"padded"oscillator), as the reflected wave amplitude rchanges (accompanying changes in load reflections)

~1:

G(<"0..,

.,,~

r-''2,

~_~------.:....-_l.333 INCI D. WAVE

(SWR MIN.)

\\

\.

" ...... --


(

C::wo(!)LL<fwI-0-.1~O1->

-.10Q.- I.~~cf-

wI->2_cf1-1-cfl/l-.lZwOC::U

00 1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16

WAVELENGTHS FROM Emin TOWARD LOAD

VOLTAGE AMPLI-VECTOR TUDE

o - <D 0.50

0 - ® 0.66

0 - @ I. 06

o - @ I. 40

0 - ® I. 50

o - ® I. 40

o - <.V I. 06

0 - ® 0.66

Fig. 1.5. Solution for Prob. 1-1, construction of standing wave shapes.

0.5~ In.o= 0.5&

1.0


traveling voltage waves, i.e.,

(b) What is the magnitude and angle ofthe current reflection coefficient at thesame point (maximum point (G)) of thecurrent standing wave)?

Solution:The vector ratio of the reflected to theincident traveling current waves, i.e.,

1-2. Plot the corresponding current standingwave shape.

Solution:Construct the dashed curve in Fig. 1.5,exactly duplicating that for the voltagestanding wave except with its null displaced one-quarter wavelength towardload, representing a 1800 phase shiftwith respect to the voltage standingwave.

1-3. (a) What is the magnitude and angle ofthe voltage reflection coefficient at theminimum point (CD) of the voltagestanding wave?

Solution:The ratio of the reflected to the incident

0.5~

1.00.5/180 0


In summary, as oriented in Figs. 11.1$rough 11.4 with respect to the overlayetharts A, B, or C, in the cover envelope, theintersection point of any pair of currentratio curves gives the impedance at P f , whilethe intersection point of any pair of voltageratio curves gives the admittance at P p. Whenrotated 1800 about their axes with respect tothe coordinates of the overlay charts A, B,or C, the intersection point of any pair ofcurrent ratio curves gives the admittance atPf, while the intersection point of any pairof voltage ratio curves gives the impedanceat P f .

11.3 CONSTRUCTION OF PROBE RATIOOVERLAYS

Information is given in Fig. 11.5 for plotting probe ratios which correspond to anydesired probe separation. All such plotsrequire only straight lines and circles fortheir construction. The outer boundary ofsuch a construction corresponds to the boundary of the SMITH CHART coordinates.

As shown in Fig. 11.5 the separation of anytwo sampling points S/A determines the anglea as measured from the horizontal R/Zo axis.This angle establishes the position of a straightline through the center of the constructionwhich represents the locus of impedances atthe probe position Pf when the currentstanding wave ratio is varied from unity toinfinity while the wave is maintained in sucha position along the transmission line withrespect to the position of the two samplingpoints that they always read alike, that is,that P /Pp = 1.0.

A construction line perpendicular to thelocus P/Pp = 1 and passing through theinfinite resistance point on the R/Zo axis willthen lie along the center of all of the P /P p

circles which it may be desired to plot.The ratio of each of these circular arcs (R 1

and R2 ) which corresponds to a particularcurrent ratio, and the distance of their centersfrom the chart rim (Dl and D2 ), is given by theformulas in Fig. 11.5 as a function of theratio of Pg to Pf and the SMITH CHARTradius R.

2.1 FUI\IDAMENTAL CONSTANTS

Two fundamental waveguide constants, thecharacteristic impedance and the propaga

tion constant, will next be discussed in tennsof traveling voltage and current waves, as wellas in terms of primary circuit elements. Therelationship of these two waveguide constantsto the normalized input impedance characteristics of a waveguide will be shown. This relationship is the basis for the coordinatearrangement of the SMITH CHART. Following this, the use of the SMITH CHARTcoordinates in converting from nonnalizedimpedances to nonnalized admittances willbe described.

2.2 PRIMARY CIRCUIT ELEMEI\ITS

It is well known that the impedance characteristics of any electrical circuit may becompletely described in tenns of four primary

CHAPTER 2Waveguide

ElectricalParameters

circuit elements: resistance R, inductance L,capacitance C, and conductance G.

Waveguides may be regarded as specificfonns of electrical circuits composed, basically,of these four primary circuit elements. If thewaveguide is of uniform configuration alongits length the primary circuit elements areunifonnly distributed and, what is equally ifnot more important, are always related one tothe other as constant ratios, such as the ratiol;IG, RIG, etc., per unit length of waveguide.One may develop directly from these primarycircuit elements the two previously mentionedfundamental waveguide constants by anyoneof several methods. The results only will begiven here.

To the same extent that four primary circuitelements are required and used for analysis ofcircuit impedance characteristics, the twofundamental waveguide constants, characteristic impedance and propagation constant, arerequired and used for analysis of waveguideimpedance characteristics. The importance of

11


these two waveguide constants cannot be overemphasized. They provide a means for completely expressing the impedance characteristics of any uniform waveguide in relation toits length and terminating impedance.

Each of the two fundamental waveguideconstants is, in general, a complex quantityhaving a real and an imaginary component.It is possible, and from a graphical point ofview useful, to attach a physical significanceto these constants, as well as to their real andimaginary parts, as will be seen.

At all radio frequencies the resistance Rper unit length of practical waveguides isgenerally negligible in comparison with theinductive reactance jwL per this same unitlength of waveguide. Likewise, the conductance G per unit length is negligiblein relation to the capacitive susceptance jwCper unit length. The radio frequency characteristic impedance of a uniform waveguideis, therefore, frequently expressed by thesimpler relationship

2.3 CHARACTERISTIC IMPEDANCE

"v (~)1/2Zo =C

(2-2)

where W is 217 times the frequency f in Hz.

The complex characteristic impedance is awaveguide constant which is equivalent to theinitial input (or surge) impedance encounteredby the forward-traveling electromagnetic wavealong a uniform waveguide. (This was discussed in Chap. 1.) Stated another way, it isthe ratio of the voltage in the forward-travelingvoltage wave to the current in the forwardtraveling current wave.

Although the characteristic impedance ofa waveguide has the dimensions and someof the properties of an impedance, it isimportant to note that these properties arenot physically equivalent to the impedanceproperties of a single-port circuit. For example, although it has a real and an imaginarypart, its real part, per se, does not dissipateenergy nor does its imaginary part storeenergy.

The characteristic impedance Zo of anyuniform waveguide, in terms of the aforementioned distributed primary circuit constants,may be expressed as

Zo = (R + ~WL)1I2G + JWC

(2-1)

This latter expression is seen to be independent of frequency. A small imaginary component, which would be contributed to thecharacteristic impedance by loss terms, maygenerally be neglected in computations involving the impedance characteristics of highfrequency waveguides. However, if moreaccuracy is required, Eq. (2-1) may be expanded to give the following expression forthe high-frequency characteristic impedanceof a waveguide in complex form [50]. Thisexpression neglects only those terms above thesecond powers in Rand G:

()1/2[ ( 2"v L R 3G2 RGZo - - 1+ -- - +- C &l2L2 &l2C2 4w2LC)

(2-3)

Although the characteristic impedance ofany waveguide is definable in terms of itselectrical parameters, its real part can, particularly in the case of transmission line-typewaveguides, also be expressed in terms of itsphysical configuration, and the dimensions ofthe conductors of which a given waveguide is

WAVEGUIDE ELECTRICAL PARAMETERS 13

240

230

220

210

200

190

IBO

170

160

OUTER CONDUCTORINSIDE DIAMETER IN INCHES

10150

5 MAXIMUM ANTI-_~_~_?.RESONANT - 130IMPEDANCE

120

2 I 10

100

90

-4 MINIMUM LDSS ._____.~.O

0 70-6 0.5z MINIMUM VOUAGE--60

W BREAKDOWN-B N 50

iiiw 0.2 40

'0 <ri MAXIMUM POWER,,"30

-'2 II>20cl5 0.1

-14 en 10

0

1.• 5

4

2

3

1.0~ 0.9% 0.8

! 0.1

!!: 0.6

~ 0.5w:. 0.4«a~ 0.3

e0.25:>~ 0.2e~ 0.15oz8~ 0.10Z 0.C9~ 0.08

0.07

, .06

o .C5 16

0.04 16

composed. This is seen to be the case sincesuch physical properties will establish theprimary circuit element values.

Nomographs from which the characteristicimpedance of coaxial and balanced-to-groundtwo-wire transmission lines may be obtainedfrom the physical size and spacing of conductors are shown in Figs. 2.1 and 2.2, respectively. Similar monographs could, of course, bedrawn for other conductor configurations.Formulas for less common configurations areavailable in handbooks [35].

The characteristic impedance of uniconductor waveguides is not as clearly defined asthat of transmission line-types. Three approved methods exist for defining the characteristic impedance of lossless rectangularuniconductor waveguides in which the dominant CfE ID ) wave is propagated. Thesemethods yield slightly different results, all of

0.03 20

Fig. 2.2. Characteristic impedance of coaxial lines.

(2-5)

Method 2 utilizes the total power Wand thetotal current I on a wide face of a rectangularwaveguide in the longitudinal direction:

which are acceptable if properly used in relation to the electrical parameters from whichthey are derived.

The evaluation of a specific characteristicimpedance for a uniconductor waveguide isfurther complicated by the fact that it varieswith frequency.

Method I for evaluating the characteristicimpedance Zw E utilizes the total power Wand the maxim~m rIDS voltage E:

(2-4)ZW,E

0.021,000

0.03900 20

IB 0.04

BOO 16 0.05

ci14

0.06z0.07

II> 700 ~ 12 O.OB2 iiiJ: ... 0.09 II>0 <r 10 0.10 w~ i %

600 <J

W B !<J II>Z ell ~« ell 60 ...... 500 <rQ.

0.20 i~ 4

~lL

20.. 400

II> Cl:0.30 ...

0: 0 ...... ..... 2'-' 300 o 40 «« a0:.. o 50J:'-'

200 0.60

0.70

0.80

, 00 0.90I 00

I 50

2 00

20

30

40

50

0.5

10

~

'"Z<J 5..Q.

II> 4...J..X..

Fig. 2.1. Characteristic impedance of open-wire lines.


2.3.1 Characteristic Admittance

Method 3 utilizes the maximum rms voltage E and the total current I on a wide face ofa rectangular waveguide in the longitudinaldirection:

The characteristic admittance of any typeof waveguide is the reciprocal of its characteristic impedance. It may therefore be used fornormalizing the conductance and susceptancecomponents of the input admittance of thewaveguide in the same way that the character- .istic impedance is used for normalizing theresistance and reactance components.

Method 3 yields a value of characteristic impedance which is the geometric mean betweenthe values obtained by methods 1 and 2.

Although in most engineering applicationsit is important to have some knowledge of theapproximate value of the characteristic impedance of the waveguide, in practice the need todetermine accurately the characteristic impedance of a waveguide, particularly that of auniconductor waveguide, is frequently avoidedthrough a process called normalization. Thisis a very useful dodge, which makes it possibleto essentially eliminate further considerationof this constant and to represent the impedance characteristics of all types of waveguideson a single set of normalized coordinates onthe SMITH CHART. This will be described inmore detail in the latter part of this chapter.

(2-7)[(R + jwL> (G + jwC) 11/2

a + j(3

P

a = (~)1I2 {[(H2 + w2L2)(G2 + w2C2)1112

1/2+ (HG - w 2 LC)} (2-8)

put current in the forward-traveling wave,where the input and output terminals areseparated by a unit lehgth of the waveguide.

As its name implies, the propagation constant describes the propagation characteristicsof electromagnetic waves which may be propagated along a waveguide. These propagationcharacteristics include the attenuation, andthe current and voltage phase relationships.

In general, the propagation constant is acomplex number. The real part, expressed innepers per unit length, is called the attenuation constant. This constant determines theenergy dissipated in the waveguide per unitlength. The imaginary part is called the phaseconstant, or the wavelength constant, and isexpressed in radians per unit length.

In terms of the primary circuit elements,the complex propagation constant Pis

The real and imaginary components of thepropagation constant a and j(3 are obtainedby expanding Eq. (2-7) and separating the realand imaginary parts.

The attenuation constant, in terms of theprimary circuit elements per unit length [35],is expressed as:

(2-6)EZE I = -

, I

2.4 PROPAGATION CONSTANT

The attenuation constant at high frequencies (aHF) can be adequately represented bythe simpler relationship

The propagation constant of a uniformwaveguide is most simply stated as the naturallogarithm of the ratio of the input to the out-

aH -+ !i (Q)1/2 + Q (~)1I2F 2 L 2 C

(2-9)


When the leakage conductance G can beneglected, as in the case of a uniconductorwaveguide operating in its dominant mode atmicrowave frequencies, the attenuation constant reduces to

(2-10)

The attenuation constant of a high-frequency type of waveguide, such as a two-wireor a coaxial transmission line composed ofcopper or other conductors of known resistivity, may also be expressed as a functionof the conductor dimensions and spacing [ 10] .

The phase constant f3 determines the wavelength in the waveguide and the velocity ofpropagation. Expressed in terms of the primary circuit elements per unit length, the .phase constant is

The propagation constant of a waveguidecontinues to vary as the frequency is increased,and loss terms, which generally increase withfrequency, cannot always be neglected (as wasgenerally permissible for the loss terms in thecharacteristic impedance) if the effect of attenuation on input impedance is to be evaluated.

Iflosses cannot be neglected, a good approximation for the phase constant f3 may beobtained by expanding Eq. (2-7) and neglectingall imaginary terms above the second power,to give [50]

f3 ~ (U (LC) 1/2 ~ + ! (~ - ~)2J (2-14)L 2 2.wL 2.wC

2.5 PARAMETERS RELATED TOSMITH CHART COORDINATES

As the frequency is increased to the pointwhere losses are small in relation to Land C,

the phase constant expressed in Eq. (2-1l)can be represented by the simpler relationship

A wavelength is defined as the length ofline l such that f3l .= 277. The length of linecorresponding to one wavelength A is, therefore, equal to 277/f3, and f3 may be expressedas

1/2f3HF -~ (U (LC)

The coordinates of the SMITH CHART(Fig. 2.3) are basic to its construction. Theycomprise a set of normalized input resistanceand reactance, and/or normalized conductanceand susceptance curves of constant valuesarranged in a unique graphical relationshipwithin a circular boundary. Upon this uniquecoordinate system all possible values of complex impedance and/or complex admittancemay be represented. As will be seen, the relationship of the normalized impedances andadmittances will satisfy the conditions encountered along any uniform waveguide.

Other important and useful waveguide parameters will also be seen to be graphically related to these basic coordinates as eitherperipheral or radial scales, or as asymmetricaloverlays. These other parameters and theirgraphical representations on the SMITHCHART coordinates will be further discussedin subsequent chapters.

(2-13)

(2-12)

(2-11 )

277

A

( )

1/2~ {[(R2 + (U

2L2)(G2

1/2+ «(U

2 LC - RG>}

f3

f3


Fig. 2.3. SMITH CHART coordinates displaying rectangular components of equivalent series-circuit impedance (or ofparallel-circuit admittance) (overlay for Charts Band C in cover envelope).

/

2.6 WAVEGUIDE INPUT IMPEDANCE

One may simplify this relationship byeliminating the effect of losses on the input

The input impedance Zs and load impedanceZr may be normalized to a common characteristic impedance Zo by dividing Eq. (2-15) byZo to give

(2-17)

Z s (Z/Zo) + j tan f31

Zo 1 + j(Z/Zo) tanf31

(Z/Zo) + j tan(2771/,\)

1 + j(Z/Zo) tan(2771/'\)

This latter relationship involves only threevariables, viz., the normalized input impedanceZ/Zo' the normalized load impedance Z/Zo'

and the length 1/'\, expressed as a fractionalpart of the wavelength. Two of these variables(Z/Zo and Z/Zo) are complex and each ofthese is composed of two other variables, viz.,the normalized resistance R/Zo and the normalized reactance ±jX/Zo' For any fixed loadimpedance R/Zo ± jX/Zo' one may plot onany two-variable coordinate system the locusof magnitudes for the variables R/Zo and±jX/Zo as 1/,\ is varied.

It will be observed that when 1/,\ is 0, orany integral number of one-half wavelengths,the input impedance equals the load impedance. It makes an excursion, controlled bythe tangent function, and returns exactly toits initial value for every incremental halfwavelength added to 1/'\, regardless of itsinitial value. An infinite number of twovariable coordinate systems exists whereuponone could trace the locus of R/Zo and ±j X/Zo,

but at this point it is sufficient to say thatonly one such coordinate system existswhereupon the loci of these variables, as thelength changes, are concentric circles whichclose on themselves in one-half wavelength.These are the coordinates of the SMITHCHART (Fig. 2.3). The derivation will befurther discussed in later chapters.


impedan~computations (except to theextent that these losses involve Zo) by simplysetting a equal to O. The hyperbolic term(tanhjf3l) may then be expressed as its trigonometric equivalent (j tanf3D, and Eq. (2-16)reduces to

(2-15)

(2-16)(Z/Zo) + tanh (al + jf3l)

1 + (Z/Zo) tanh(al + jf3l)

Z + Zo tanhPlZ _r _o

Zo + Zr tanh Pi

The input impedance of a waveguide isusually defined as the impedance betweenthe input terminals with the generator disconnected. This is sometimes called the finalsending-end impedance when it is desired tostress the fact that steady-state conditionshave been established. The input terminalsmay be selected at any chosen position alongthe waveguide. Thus, a waveguide may beconsidered to have an infinite number ofinput impedances existing simultaneously,since there is an infinite num ber of positionsalong any finite length of waveguide.

Waveguide input impedance coordinates aredepicted on the central circular area of thechart in normalized form, as described in Sec.2.8 entitled "Normalization." These coordinates are arranged to portray the series components of the normalized input impedancewith respect to a waveguide, as a function ofthe position of observation.

The inpu t- or sending-end impedance Zs

of any waveguide may be completely expressedin terms of the two fundamental waveguideconstants Zo and P, the complex load impedance Z , and the length 1. The relationshipr

[21 is as follows:


It will be evident from a consideration ofthe physics of wave propagation (as discussedin Chap. I) that the input impedance of awaveguide cannot in any degree be changed oraffected by the internal impedance of anygenerator which may be connected across theinput terminals of the waveguide. This is so,even though there may be a serious mismatchof impedances between the waveguide characteristic impedance and the generator internalimpedance. The generator impedance, therefore, does not have to be considered in asteady-state analysis of waveguide propagationcharacteristics, except as it affects the level ofpower available from the generator to thewaveguide. This is a rather important pointto bear in mind as it is quite commonly notfully appreciated.

The input impedance terminals of a uniconductor waveguide, like the terminals of theprimary circuit elements of which the waveguide is composed, are somewhat undefinable.However, the input impedance concept is stilla useful one. Where energy is propagated inthe dominant mode (Fig. 2.4), the accuracy issufficient for many engineering evaluations, ifone regards the waveguide terminals as beinglocated at the centers of the wide interior facesof the waveguide, between which the voltageis a maximum in any given transverse plane.

2.7 WAVEGUIDE II\IPUT ADMITTANCE

The final input admittance, or simply inputadmittance of a waveguide, is the reciprocal of

ELECTRIC FIELD --

MAGNETIC FIELD--

TRPoNS

"ERSE

Fig. 2.4. Dominant mode field pattern in a rectangular uniconductor waveguide.

the input impedance. The coordinate component curves of the SMITH CHART (shownon Fig. 2.3) display both normalized input impedance and normalized input admittance, inorder to broaden its application. As will beseen, this is possible because of the use ofnormalized values to designate the individualcomponent curves, and because of the wayin which the coordinate component curvefamilies are labeled. Thus, the real coordinatecomponent curve family is labeled "ResistanceComponent R/Zo, or Conductance Component

G/Y0," and the imaginary coordinate component family is labeled "Inductive ReactanceComponent +jX/Zo' or Capacitive SusceptanceComponent +jB/Yo," and "Capacitive Reactance Component - jX/Zo' or Inductive Susceptance Component - jB/Y0'"

The SMITH CHART is most convenientlyused as an admittance chart when the effects ofshunt elements on the waveguide are to beconsidered, and as an impedance chart whenthe effects of series elements are to be considered. One may also transfer the problemfrom the admittance coordinates to the impedance coordinates, and vice versa as occasiondemands, when both shunt and series elementsare involved. This is further discussed inChap. 6.


To further clarify the normalization process,consider the specific example of a coaxialtransmission line having a characteristic impedance of 50 + j a ohms, terminated in a loadimpedance of 75 +j 100 ohms. This particularload impedance, when normalized with respectto this particular transmission line characteristic impedance, would be expressed as 1.5 +j 2.0 ohms, which would appear on theSMITH CHART coordinates (Fig. 2.3) at theintersection of the two families of impedancecomponent curves, viz., where R/Zo = 1.5 and

+jX/Zo = 2.0. On the other hand, this sameload impedance (75 + j 100 ohms), whennormalized with respect to the characteristicimpedance of a two-wire transmission line of,say, 500 ohms, would be expressed as 0.15 +j 0.20 ohms. This would appear on the chartimpedance coordinates at a different position,viz., at the intersection of the impedancecomponent curves where R/Zo = 0.15 and+jX/Zo = 0.20. The same chart is thus seen tobe applicable to transmission lines of differentcharacteristic impedances.

The characteristic impedance of 50 ohms isequivalent to a characteristic admittance of1/50 or 0.020 mho, and a load impedance of75 + j 100 ohms is equivalent to a load admittance of

2.8 NORMALIZATION 1

75 + jl00or 0.0048 - jO.OO64 mho

Normalized load admittance for this assumedtransmission line is then, by defmition,

which, as previously stated, is the reciprocal ofthe normalized complex load impedance, viz.,1.5 + j 2.0 ohms.

Normalized impedance (with respect to awaveguide) is defined as the actual impedancedivided by the characteristic impedance of thewaveguide. The input impedance coordinateson the SMITH CHART (Fig. 2.3) are, as previously stated, designated in terms of "normalized" values. Normalizing is done to make thechart applicable to waveguides of any and allpossible values of characteristic impedance. Inaddition, as has been pointed out, this makesthe coordinate component curves applicable toeither impedances or admittances.

0.0048 - jO.0064 h------=---- m 0

0.020or 0.24 - jO.32 mho


2.9 CONVERSION OF IMPEDANCETO ADMITTANCE

The conversion from impedance to admittance, or vice versa, is readily accomplished onthe SMITH CHART by simply moving from aninitial impedance point on the normalizedimpedance coordinates of the chart to a pointdiametrically opposite and at equal distancefrom the center of the chart. Equivalent admittance values are then read on the normal-

ized admittance coordinates. Any two suchdiametrically opposite points at the same chart

radius give reciprocal normalized values of thecoordinate components and, thereby, convertnormalized impedances directly to normalizedadmittances and vice versa. Thus, in theexample given for the normalization of impedance and admittance, the respective coordinate points, viz., 1.5 + j 2.0 and 0.24 j 0.32, will be seen to lie diametricallyopposite each other at equal distance fromthe center of the chart in Fig. 2.3.

3.1 CONSTRUCTION OF COORDINATES

This chapter describes the construction ofthe basic coordinates of the SMITH CHART

(Fig. 2.3) and then discusses some of the moreimportant waveguide electrical parameters related thereto. All of these related parametersmay be graphically portrayed as either radialor peripheral scales.

The construction of the basic series impedance or parallel admittance coordinates ofthe SMITH CHART is shown in Figs. 3.1 and3.2. Figure 3.1 applies to the normalized resistance circles R/Zo' while Fig. 3.2 applies tothe superimposed normalized reactance circles±jX/Zo' The same construction is applicableto the basic admittance coordinates, by substituting G/Yo for R/Zo in Fig. 3.1, and bysubstituting +jB/Yo for +jX/Zo (and -jB/Yofor-jX/Zo) in Fig. 3.2.

CHAPTER 3SmithChart

Construction

3.2 PERIPHERAL SCALES

The impedance or admittance coordinatesof the SMITH CHART would be of little usewere it not for the accompanying related peripheral and radial scales which have generalapplication to waveguide propagation problems, and which serve as the entry and exit tothe chart coordinates. Peripheral scales generally have to do with chart coordinate quantitieswhich vary with position along the waveguide,while radial scales have to do with chart coordinate quantities which vary with reflection andattenuation characteristics of the waveguide.

Two important linear peripheral scales areshown on the SMITH CHART in Fig. 3.3. Theoutermost of these is the electrical lengthscale, the other is the phase angle of the voltage reflection coefficient scale. A nonlinearperipheral scale showing the angle of the

21


ALL CIRCLESPASS THROUGH

THIS POINT.

II+(RIZO)

l>'j..\S

~\1.r::;~

~~

~~

/j

~gG... ...

~ 0 CHARTGi- R"-I'-'Z"-"-=A-"X.!..:IS'----_ ~t-----..:C.:.EN;.;T;.cE:.:R-'------~o----------....:!l ..

X ~ ~'= v;~ UJ I----~ '"

~~~v

"b-(~."

~"'0

.}~O

Fig. 3.1. Construction of normalized resistance circles for SMITH CHART ofunit radius.

t-~ CHART5r- --"R-'-I'"'Z......,A'-'X"-IS"---- ...:C:..:E::.N..T..:E.:.:R -=:::;;_~

xt-

:E~

Fig. 3.2. Construction of normalized reactance circles for SMITH CHART ofunit radius.

SMITH CHART CONSTRUCTION 23

IMPEDANCE OR ADMITTANCE COORDINATES

..RADIALLY SCALED PARAMETERS§:; ~ e £! ~ ~

0 :;l ..~ ~ N 9p 1I ~ ~ ~ S i;I II i! ~ "11~~~8 9 oj ;; ..

°12~ ~

! .. r lTOWARD GENERATOR_ '4---- TOWARD LOAD 0 2 iii ~ il " ;. 3 l!l & l!: ill i;I II II ~ ~I:::~.-

~ ~ ~ I ~I~~

i~~8gg g !! 2 ~ ~ ~ ~ c~N N

:0 " c ~.. " .. l; .. l': N n,.

" " 0 0 " " b ..i~~ 0 .. N ~o !I~

0

9 0 e £! :il ~ ::: .. :! e g b N " .. .. is e~[ilz. .,

" 0 0 0z > '" CENTER~ I

Fig. 3.3. SMI TH CHART (1949) with coordinates shown in Fig. 2.3.


3.2.1 Electrical Length

voltage transmission coefficient is describedin Chap. 8.

In the case of waveguides along which energyis propagated in the TEM mode (which excludes uniconductor waveguides), the wavelength A, in meters, will be

where f is the frequency in Hz, and v is thevelocity of phase propagation in the waveguidein m/sec. For such waveguides, which includecoaxial and open-wire transmission lines withair insulation, v is generally very close to300,000,000 m/sec.

Linear length scales around the periphery ofthe SMITH CHART (Fig. 3.3) indicate valuesof electrical length of waveguide or transmission line, in wavelengths. Two scales are usedto indicate distances in either direction fromany selected point of entry of the chart, suchas a point corresponding to the nodal point

of a voltage standing wave, or the input terminals, or load terminals of the waveguide. Thesescales relate the position along the waveguideto the input impedances (or admittances)encountered at these points as read on thenormalized input impedance (or admittance)coordinates. These two scales are labeled,respectively, "Wavelengths Toward Generator"and "Wavelengths Toward Load." For convenience, the zero points are oriented (Fig.3.3) so that they are always aligned with avoltage standing wave minimum point. If ameasurement of distance is to be made fromsome reference point other than a voltageminimum position, the scale value as readradially in line therewith must be interpolatedand subtracted from subsequent scale valuereadings, since these length scales are notphysically rotatable with respect to the coordinates on a single sheet of paper.

The length scales on the periphery of theSMITH CHART encompass only one-halfwavelength (180 electrical degrees) of waveguide and this corresponds to a physical rotation of 3600 in either direction around thechart periphery. However, a graphical representation of the conditions along one-halfwavelength of waveguide is all that it is necessary to consider, since the input impedancevalues along any uniform waveguide or transmission line repeat cyclically at precisely thisinterval of distance, if one ignores the effect ofattenuation, which effect may be taken intoaccount in a manner to be described later.Any waveguide electrical length in excess ofone-half wavelength may always be reduced toan equivalent length less than one-half wavelength, to bring it within the scale range of thechart, by subtracting the largest possibleintegral number of half wavelengths.

The electrical length of a uniform loss-lesssection of high-frequency open wire, or coaxial transmission line, having predominantlya gas dielectric, is only slightly longer than itsphysical length when the latter is expressed in

(3-1)

(3-2)

217

v

f

It was stated in Chap. 2 that the phaseconstant {3 of a waveguide determines thewavelength in the waveguide and the velocityof propagation. The velocity of propagation,as therein stated, more accurately refers tothe velocity of phase propagation, or simplythe wave velocity. The electrical length isderived from considerations of phase velocityand wavelength.

As is also stated in Chap. 2, a wavelengthis defined as the length of waveguide l suchthat {3l = 217. The length of waveguide corresponding to one wavelength A is, therefore,defined as

terms of the wavelength in free space. However, the electrical length of practical low-losscoaxial transmission line with a solid dielectricinsulating medium, such as polyethylene, ismaterially increased from its electrical length,in the absence of solid insulation, due to theslower velocity of propagation of electromagnetic waves in dielectric media, by a factorwhich is equivalent to the refractive index ofthe dielectric medium between conductors, orby the factor /i, where f is the dielectricconstant of the medium.

For uniconductor waveguides, the electricallength is expressed with reference to waveguidewavelength, which, in turn, depends upon themode of propagation of the energy (fieldpattern) within the waveguide as well as thespecific dimensions and configuration of thewaveguide. For all hollow uniconductorwaveguides, including those of rectangular orcircular cross section, the waveguide wavelength is longer than the free space wavelength.Thus, a given section of rectangular or circularuniconductor waveguide is electrically shorterthan its physical length.

The waveguide wavelength is directly measurable in uniconductor waveguides or transmission lines when standing waves are present.It is equal to twice the distance between adjacent voltage or current nodal points. Inuniconductor waveguides the waveguide wavelength is also calculable from the physicaldimensions of the waveguide at the frequencyof interest [10].

3.2.2 Reflection Coefficient Phase Angle

The phase angle of the voltage reflectioncoefficient was described briefly under theheading "Reflection" in Chap. 1. This changeslinearly with distance along a waveguide asdoes the absolute phase of the traveling wave.However, the former changes twice as rapidlywith position as does the latter because of the


fact that the reflected wave, which is thereference for the reflection coefficient (ratherthan a fixed point along the waveguide) hastraveled twice as far as the incident wave inany given length of waveguide.

The phase angle of the voltage reflectioncoefficient is portrayed on the SMITH CHART(Fig. 3.3) as a linear peripheral scale ranging invalue from 0, along the resistance axis in thedirection of maximum resistances, to 1800

along this same axis in the opposite direction.Progressing clockwise from the zero-degreepoint on the voltage reflection coefficientphase angle scale, which is the direction inwhich one moves on the chart when movingtoward the generator, the voltage reflectioncoefficient phase angle increases negativelyfrom 0 to -1800

, indicating an increasing lagin the reflected voltage wave as compared tothe incident wave. The reverse is, of course,true in the opposite direction.

3.3 RADIAL REFLECTION SCALES

The radial scales to be described in thischapter apply to all angular positions on thebasic coordinates of the SMITH CHART.These are, therefore, equivalent to families ofcoaxial overlays, since the radial scales couldbe graphically represented as concentric circlesabout the center of the SMITH CHART. Eachcircle of each family of concentric circlescould then be assigned a specific value corresponding to the value of the radially scaledparameter which it represented. However, ifthis were done the resulting large number ofconcentric circles would mask the basic chartcoordinates and result in confusion. Radialscales avoid this.

Figure 3.4 is a plot of four radial reflectionscales which will be discussed in this chapter.These include (1) the voltage reflection coefficient magnitude, (2) the power reflectioncoefficient, (3) the voltage standing wave


ratio, and (4) the voltage standing wave ratioin dB. They are applicable to the coordinatesof Fig. 2.3.

3.3.1 Voltage Reflection CoefficientMagnitude

Although one may correctly refer to thereflection coefficient in any transmission medium as the ratio of a chosen quantity associatedwith the reflected wave to the correspondingquantity in the incident wave, the usualquantity referred to, for an electromagneticwave along a waveguide, is voltage. The voltage (and current) reflection coefficient wasdiscussed in Chap. I in the section on "Waveguide Operation." The voltage reflectioncoefficient is defined as the complex ratioof the voltage of the reflected wave to that ofthe incident wave.

On the SMITH CHART, the magnitude ofthe voltage reflection coefficient is representedas a radial scale, starting with zero at the centerand progressing linearly to unity at the outerboundary circle (see Fig. 3.4). Since thisscale is linear it is a convenient mathematicalreference for equating all radial scale parameters.

If the waveguide is uniform and lossless,the voltage reflection coefficient is constantin magnitude throughout its length. If, onthe other hand, the waveguide has attenuation,the voltage reflection coefficient will be maximum at the load end and will diminish withdistance towards the generator in accordancewith the attenuation characteristics of thewaveguide.

Figure 3.5 shows a family of constant magnitude circles and a superimposed family ofconstan t phase lines (radial) graphically depicting the voltage reflection coefficient magnitude and phase angle, respectively, along aunifonn waveguide.

The polar coordinates formed by these twofamilies of circles may be superimposed onthe basic SMITH CHART coordinates (Fig.2.3, or Chart A, B, or C in the cover envelope)to obtain the relationship of the complex reflection coefficient at any point along a waveguide to the complex impedance or admittance.

3.3.2 Power Reflection Coefficient

The power reflection coefficient is definedsimply as the ratio of the reflected to theincident power in the waveguide. Numericallyit is equivalent to the square of the voltagereflection coefficient. However, unlike thevoltage reflection coefficient, the power reflection coefficient has magnitude only, since"phase" as applied to power is meaningless.Like the voltage reflection coefficient, thepower reflection coefficient is constantthroughout the length of a uniform losslesswaveguide, but in a waveguide with attenuation it diminishes with distance toward thegenerator from a maximum value at the loadend.

The power reflection coefficient, expressedin dB, is called return loss. This will be discussed further in the next chapter.

Fig. 3.4. Radial reflection scales for SMITH CHART coordinates in Fig.2.3 (see radial loss scales in Fig. 4.1).

o

ICENTER

I

o

-'" :.N

".. .." 0 '"o

'"o '"o'"o

3.3.3 Standing Wave Amplitude Ratio

The standing wave amplitude ratio of maximum to minimum voltage along the waveguide(symbolized as VSWR) is a measure, indirectly,of the degree of mismatch of the waveguidecharacteristic impedance and load impedance. Limiting values are unity for a matched


Fig. 3.5. Complex voltage or current reflection coefficient along a waveguide (overlay for Charts A, B, or C in cover envelopel.


but

and

(3-6)

(3-5)

(3-4)

receiving-end voltagereceiving-end currentcharacteristic impedance

EI cos{31 + j~ sin{31r 2

o

cos{31 + jIr sin{31

sin ~l ± ~) = += cos {31

Therefore

cos ~l ± ~) = += sin{31

Is = Ir cos{31 + j sin{31

Considering the current I: at a point onequarter wavelength removed from the voltageEs ' that is, 77/2 radians,

and

For comparative purposes E r and 20 can beconsidered to be a constant equal to unity, inwhich case

Er =I r =20 =

where

waveguide and infinity for a lossless open(or short-circuited) waveguide.

As discussed in the section on "WaveguideOperation" in Chap. I, a standing (or stationary) wave may be thought of as theresultant of two traveling waves moving inopposite directions along the waveguide, i.e.,the incident (or direct) and the reflectedwaves. A single reflection from the load

only need be considered under steady-stateconditions, since multiple load reflections dueto several round trips of the wave betweenthe load and the generator can, under steadystate conditions, be considered to add up to asingle effective load reflection. Multiplegenerator reflections likewise can be considered to add up to a single effective generator output.

Standing waves are always accompanied bya change in input impedance as a function ofthe position of observation along the waveguide which would otherwise be constant andequal to the characteristic impedance. If thewaveguide is lossless the locus of input impedances, for a given standing wave ratio, isrepresented on the SMITH CHART as a circleconcentric to the center of the chart. Theradius of the circle is a function of the standing wave ratio and consequently the standingwave ratio may be represented as a radialscale on the chart. This scale is shown inFig. 3.4. Individual points on an impedancecircle give the input impedance of the waveguide at corresponding individual positions.

It will be shown by reference to Eqs. (A-II)and (A-I 2) in Appendix A that the voltage andcurrent wave shapes are identical along a lossless waveguide, for the sending-end voltage Esand current Is as a function of the length 1,

namely,

(3-3)= += j (cos {31 ± jI r sin (3[) (3-8)

and

A comparison ofEq. (3-8) with Eq. (3-5) showsthat Es and I: have the same shape and

amplitude for unit E r and that they are displaced 90°, or one-quarter wavelength. Thecurrent minimum point always occurs coincident in position with the voltage maximumpoint along the waveguide and vice versa.

The position of a voltage standing waveminimum point along a waveguide alwayscoincides with the position of minimumnormalized input impedance (or maximumnormalized input admittance), whereas theposition of a minimum current coincides withthe position of maximum normalized inputimpedance (or minimum normalized inputadmittance). Except in the case of highlydissipative waveguides the input impedance(or admittance) is a pure resistance (or conductance) at both of these points. At allother points the input impedance (or admittance) is complex.


The shape of the voltage and currentstanding waves along a waveguide are plottedin Fig. 3.6 for standing waves of severalamplitude ratios. On this plot each wave istransmitting the same power to the load. Ifthe waveguide is lossless, the voltage at themaximum point of a standing wave whoseratio is infinity is also infinity. The waveguide would, of course, arc over long beforethis point could ever be reached.

A plot similar to that in Fig. 3.6 was shownin Fig. 1.3, wherein the relative amplitudesand shapes of several standing waves wereplotted for the condition where the same incident voltage is applied to the waveguide.Under these latter conditions, which prevailwhen the generator is decoupled from thewaveguide with sufficient attenuation, it wasshown that the voltage at the maximum point

3.6

3.4

3.2I-::l

3.0Q.

~

It:2.8

w~ 2.60a..

2.4I-z<t 2.2I-IIIZ 2.00(,)

It: 1.80lL

1.6w0::l 1.4I-

k ...J 1.2

~a..:::E<t 1.0'I;'~ -;w

:1> 0.8~

l <t...J 0.6wIt:

0.4

0.2

SWR: 10.0

1/r- i\.. ,1,.-1"\I \ V 1\

1/ \ / \

/ \ 1/ \II SWR: 5.0 \ / \-- - ....

/ / "" \ / / \. \I 7 '\ I \I \.

1/ 1/ r\ \ 7 I \ ,I J SWR:2.0 \ \ / 1/ 1\ 1\/ 1/ 1/ I''" r-.. II / V 7 ......

'" \ \/ I /

/ SWR:1.0 f::: \ 1\ II V " r\\i'...lit17 ~ \ I V ~~

f-t1 ~- r11 ~

ir1 ~ 'I' \:t-1V ,

oo .04 .08 .12 .16 .20 .24 .28 .32 .36 .40 .44 .48 .52 .56 .60 .64 .68 .72 .76 .80 .84 .88 .92 .96 1.0

RELATI VE POSITION ALONG LINE - WAVELENGTHS

Fig. 3.6. Relative amplitudes and shapes of voltage or current standing waves along a lossless waveguide (constant inputpowerl.


3.3.4 Voltage Standing Wave Ratio, dB

By definition the decibel is fundamentallya power ratio:

(3-9)

(3-10)

(3-11)

~1 - p

The position of the voltage standing wavemaximum point on a waveguide always corresponds to the position where the phase angleof the voltage reflection coefficient is zero.The adjacent voltage standing wave minimumposition on the waveguide one-quarter wavelength (90 electrical degrees) removed fromthe maximum point on either side always corresponds to the position where the voltagereflection coefficient phase angle is 1800

•

amplitude values of the circles of constantnormalized resistance (or normalized conductance) at the points where these cross theR/Zo (or G/Yo) axis. This may be seen bycomparing the VSWR scale in Fig. 3.3 withthe R/Zo coordinate labeling.

The voltage reflection coefficient magnitude p and phase at a point along a waveguideare uniquely related to the voltage (or current)standing wave amplitude ratio S and standingwave position. A simple amplitude relationship exists between these parameters, namely,

VIdB = 20 loglO -

Vz

It was correctly stated that if two voltages areimpressed across the same resistance the squareof their ratio will be equivalent to the ratio ofpowers in the resistance.

PIdB = 10 loglO -

PzAn extension of this use came into being soonafter introduction [4] of the unit in 1929,namely,

S

of a standing wave, whose ratio is infinity, isexactly twice the incident voltage.

It will be observed from the curves inFigs. 1.3 and 3.6 that the shape of a standingwave is different for each standing wave ratioand that although traveling waves are sinusoidal in shape when observed as a function oftime, standing waves are not sinusoidal inshape when their amplitude is plotted as afunction of position along the waveguide,except in the limiting case when the standingwave amplitude ratio is infinite. When thestanding wave ratio approaches unity the waveshape approaches that of a sine wave of twicethe frequency, becoming finally a straightline when the standing wave ratio reaches theunity value. When the standing wave ratioapproaches infinity the wave shape approachesthat of a succession of half sine waves.

The standing wave shape in a short length,such as one-quarter wavelength, of highfrequency waveguide is unaffected to anyappreciable extent by attenuation (one-waytransmission loss), although this may be sufficient to affect the standing Wave amplituderatio.

The standing wave amplitude and waveposition are readily measurable waveguideparameters using slotted sections of waveguide. This data is useful in providing anentry to the SMITH CHART. Like the inputimpedance, the standing wave ratio or positionin a waveguide cannot be affected by theinternal impedance of the generator. However, caution should be exercised to ensurethat the conductance and susceptance of anymeasuring probe used to sample VSWR willnot, as it moves along the waveguide, changethe impedance presented to the generator,thereby changing the input power or frequency and causing what appears to be adistortion of the standing wave shape or ratio.

It may be of interest to note that amplitudevalues for the voltage standing wave ratio(VSWR), as scaled radially on the SMITHCHART coordinates, are exactly the same as

This is obviously a misuse of the tenn dB asoriginally intended, since the maximum and

Later, it became popular to measure standing waves of voltage along a waveguide with avoltmeter calibrated in dB in accordance withEq. (3-11) and to define the VSWR in "dB" as

VmaxdBS = 20 loglO -

V .min

(3-12)


minimum voltages in the standing wave donot exist across the same input resistance. Tofurther perpetuate this misuse of the tenn,manufacturers quite generally calibrate standing wave indicating devices in dB.

Since standing wave ratios are, today, sofrequently measured in dB it seems advisableto recognize this special use of the tenn. Accordingly, the lowest scale in Fig. 3.4 showsthe VSWR scale in dB.

4.1 RADIAL LOSS SCALES

In Chap. 3 it was shown that entry and exitto the coordinates of the SMITH CHART are

conveniently accomplished through the use ofappropriately graduated peripheral and radialscales. The peripheral scales (which were described in Chap. 3) relate all angular positionson the chart coordinates, as measured fromits center, to corresponding physical positionsalong a waveguide. These scales include twolinear length scales, one progressing clockwiseand the other counterclockwise, from zero toone-half wavelength around the chart circumference. A third peripheral scale measures thephase angle of the voltage reflection coefficient in relation to chart coordinates. Eachpoint along each of the three peripheral scaleswas shown to apply to all chart positions radially in line therewith.

Radial scales on the SMITH CHART (described in Chap. 3) were shown to be related

CHAPTER 4Losses, and

Voltage-CurrentRepresentations

to the magnitude of reflections from the load,or from discrete reflection points along thewaveguide. These scales include voltage (andpower) reflection coefficient magnitude andvoltage (or current) standing wave ratio. Asimple relationship between the magnitude ofthe voltage reflection coefficient and the voltage standing wave ratio was given.

It will now be shown that the effect of bothdissipative and nondissipative losses encountered in a waveguide may also be representedon the SMITH CHART by appropriatelygraduated radial scales. Dissipative losseswhich will be considered include transmissionloss (two-way attenuation) and standing waveloss factor (transmission loss coefficient).Nondissipative losses include reflection lossand re turn loss.

A universal voltage-current overlay for theSMITH CHART impedance (and/or admittance) coordinates is described in the latterpart of this chapter.

33


4.1.1 Transmission Loss

Transmission loss is defined as the powerloss in transmission between two points alonga waveguide. It is measured as the differencebetween the net power passing the first pointand the net power passing the second. Thus,by definition, transmission loss includes diss~pative losses incurred in the reflected, as wellas in the forward-traveling wave, and it alsoincludes losses due to radiation of electromagnetic energy from the waveguide. Transmission loss excludes nondissipative losses dueto impedance mismatches, which will be discussed later. Alternatively, transmission lossmay be defined as the ratio (in dB) of the netpower passing the first point to the net powerpassing the second.

Transmission loss in waveguides is of threebasic types, namely, conductor losses, dielectric losses, and radiation losses. The firsttwo types result in power dissipation or heatloss in the waveguide. Except in open-wiretransmission lines, radiation losses are generallysmall enough to be neglected [5]. Conductorlosses in waveguides result from the flow ofcurrents in the conductor resistance, whereasdielectric losses are due to the flow of conduction current in the dielectric conductance.

At high frequencies conductor resistancelosses in waveguides wherein TEM waves arepropagated (which includes coaxial and openwire transmission lines) increase as the squareroot of the frequency due to skin effect [ 10] .Dielectric conductance losses, on the otherhand, are directly proportional to the numberof alternations of the electric field in the

.,~ 0 0 00 8 2 0

~ " 0z a: '" .. .., N 0 0

~~ 0 g 0 0 0 ~0 0 0 0 9 ~ !' £ ~ 0

..J a: N ,.; .. <Ii .- .. <ri ..~

CENTER'" I'" 0 03 8 £ 2 0 ::J ~ ~ ~

N

310 on .. ,.; -u

., '" TOWARD GENERATOR~.. ~ TOWARD LOAD0 "'- I-

'"Fig. 4.1. Radial loss scales for SMITH CHART coordinates in Fig. 2.3(see radial reflection scales in Fig. 3.4).

dielectric medium adjacent to the conductorsin a given interval of time since for a givenvoltage the dielectric absorbs a fixed amountof energy each cycle. Consequently, theselosses increase linearly with frequency. Thesetwo types of loss are essentially independentand additive, so that in this type of waveguide,the total heat loss increases with frequency ata rate which is between the square root andthe first power, depending upon the materialsof which the waveguide is constructed andthe frequency of operation.

In cylindrical uniconductor waveguides, inwhich the dielectric adjacent to the conductor is entirely air or gas and in whichwaves are propagated in the dominant mode(for example, TE lO waves), losses are relatedto frequency in a much more complicatedmanner. Below the cutoff frequency (lowesttransmittable frequency), losses approach infinity. Immediately above the cutoff frequency, transmission losses decrease with increased frequency at a much higher rate thanthe concurrent increase in conductor resistance resulting from skin effect.

The radial scale labeled "transmission loss"in Fig. 4.1 actually refers to one-way transmission loss (attenuation). One-way transmission loss is the loss in a conjugate-matchterminated waveguide, and is the minimumpossible dissipative loss. It is this one-waytransmission loss, only, which must be considered in the determination of the effect ofall dissipative losses upon input impedance.

The one-way transmission loss scale isplotted, for use on the SMITH CHART coordinates, in dB units. Specific values arepurposely omitted so that the zero-dB pointon this scale may be taken to be at anyposition along the scale corresponding to anyselected point of entry on the chart coordinates. One-way transmission loss between thisinitial point of entry and any other point alongthe waveguide may then be measured on this

LOSSES. AND VOLTAGE-CURRENT REPRESENTATIONS 35

Since only dB units are plotted on the one-waytransmission loss scale, it will always be necessary to interpolate between scale divisions forpoints of entry and exit on the SMITH CHARTcoordinates which do not fall exactly at thescale division points.

If the terminating impedance is other than onewhich produces complete reflection in a waveguide, such as an open- or short-circuit, or apure reactance (or susceptance), the one-waytransmission loss in the waveguide betweenany two sampling points where the voltagereflection coefficient is, respectively, PI and

Pz' is

reflection coefficient overlay (or SMITHCHART) corresponding to 100 percent reflection. As explained in Chap. 1, theforward-traveling wave energy encounters onlythe characteristic impedance of the waveguideafter leaving the initial point of entry. Uponarrival at the above termination the waveenergy is completely reflected. The reflectedwave energy continues to encounter thecharacteristic impedance of the waveguide.Since the propagation path is common toboth the forward and the reflected waveenergy, the latter, in its backward path to theinitial point of entry, will be attenuated inthe same ratio as was the incident waveenergy. At the initial point of entry the powerreflection coefficient is, thus, a measure ofthe two-way transmission loss, expressed as apower ratio. One-half of this, therefore,represents the one-way transmission loss, viz.,

scale, with its consequent effect upon the input impedance (or admittance) coordinates.The proper direction to proceed along theone-way transmission loss scale depends uponwhether one moves along the waveguide"toward generator" or "toward load."

For example, if one enters the SMITHCHART at a point corresponding to a knownload impedance point and then moves "towardgenerator" for an observation of input impedance, the one-way transmission loss will decrease the radial distance from the center ofthe chart at which the input impedance isindicated on the basis of length moved. Thescale of one-way transmission loss ''Transmission Loss-l dB steps" is labeled to showthe direction in which it is effective in relationto the direction which the observer movesfrom the point of entry on the chart.

The one-way transmission loss scale for theSMITH CHART is most easily derived from aconsideration of the magnitude of the powerreflection coefficient.. As discussed in Chap.3, the power reflection coefficient is the ratioof reflected to incident wave power at anyselected reference point along a waveguide.It is numerically equal to the square of thevoltage reflection coefficient P at that point.The power reflection coefficient may be represented on SMITH CHART coordinates byan overlay of concentric circles (similar tothose shown on the voltage reflection coefficient overlay in Fig. 3.5) whose radii varyfrom zero at the center to unity at the rimof the overlay. The radius of individual circlesin this family is given in Fig. 3.4.

At any selected point of entry on this powerreflection coefficient overlay, and consequently on the SMITH CHART coordinates,forward-traveling wave energy may be assumedto exist. This will be attenuated by theone-way transmission loss as the powerflows toward the termination. The termination may conveniently be taken to be atany point on the outer rim of the power

1 zdB = - (-10 loglO P )2

or

dB = -10 loglOP

dB = -10 loglO (pz - PI)

(4-1)

(4-2)

(4-3)


This coefficient provides the means fordetermining the smoothed distribution of the

Also, since dielectric losses are proportionalto the square of the voltage and since thedielectric conductance is uniformly distributed, there are always increased dielectricconductance losses (if the waveguide has anydielectric conductance losses to start with)in the region of voltage maxima points. Thismore than compensates for the reduction indielectric loss in the region of adjacent voltageminima points.

The dielectric conductance loss at any givenpoint along a waveguide is, as previously stated,proportional to the square of the voltage atthe point, while conductor loss is proportionalto the square of the current. Since thestanding voltage and current wave shapes areidentical, it is evident that the percentageincreased losses from these two causes are thesame. For this reason a single transmissionloss coefficient scale may be used to represent added percentage loss as a function onlyof the standing wave ratio. This will holdwhether the losses are initially composed ofconductor loss, dielectric loss, or any proportion of each. The standing wave lossfactor is represented as a single radial scaleon Fig. 4.1, under the caption "TransmissionLoss-Loss Coef."

The added loss, which may be evaluatedfrom the standing wave loss factor, does notenter into the calculation of the bput impedance \'haracteristic of the waveguide, and consequen;ly when the unattenuated input impedance value on the SMITH CHART is beingcorrected to take losses into account, thisfactor should not be considered.

Expressed in terms of the standing waveratio S, the standing wave loss factor (transmission loss coefficient) is

4.1.2 Standing Wave Loss Factor

As previously discussed, additional dissipative losses occur in waveguides when there isreflection from the load, over the one-waytransmission loss incurred when there are noreflections, even though the generator impedance may be conjugate-matched to the inputimpedance of the waveguide. This increaseddissipative loss (heat loss) is incurred by thereflected power from the load. Reflectedpower results in reflected voltage and currentwaves, resulting in standing waves of voltageand current.

If a waveguide is one or more wavelengthslong, the average increase in dissipative lossdue to standing waves in a region extendingplus or minus one-half wavelength from thepoint of observation may be expressed as acoefficient or factor of the one-way transmission loss per unit length.

The standing wave loss factor can be derived from a consideration of the amplitudeand shape of the standing waves along a waveguide. The existence of a standing-wave lossfactor and the fact that its amplitude willvary with the degree of mismatch of the load,and/or the standing wave ratio, is apparentfrom Fig. 3.6. This figure, as previously indicated, shows the relative amplitudes andshapes of several standing waves of current orvoltage, all of which will transmit the samenet power to the load along a lossless waveguide. For any given standing wave patternthe conductor losses vary along the waveguidein proportion to the square of the current atsuccessive points. Since a uniform waveguidehas uniform resistance there are always increased conductor losses in the region of current maxima points which more than counteract the reduction in conductor loss in theregion of adjacent current minima points.(Note that the area between the currentmaxima loops and unity is greater than thearea between the current minima loops andunity.)

mismatchedLoss ratio,

matched 2S(4-4)

LOSSES, AND VOLTAGE·CURRENT REPRESENTATIONS 37

I

transmission loss (total dissipation losses) in awaveguide when standing waves are present.Smoothing is accomplished by integrating thelosses over plus or minus one-half wavelengths from the point of observation, thenaveraging the results over the same length and,finally, expressing the increase over the minimum attenuation on the basis of an equivalentincrease in an elemental length at the point.Thus, spatially repetitive variations in transmission loss within each successive halfstanding wavelength are equated to an equivalent smoothed loss by the standing waveloss coefficient.

The standing wave loss coefficient scale canalso be used to obtain, to a close approximation, the ratio of transmission loss toattenuation for an entire waveguide length inwhich the standing wave ratio changes alongits length due to attenuation. For this thescale value is simply observed at a point midway between the values which represent conditions at the end points of the waveguidesection. For example, in a waveguide in whichthe standing wave ratio is 3.0 at the load endand 1.5 at the generator end, the standingwave loss coefficient is, respectively (fromFigs. 3.4 and 4.1, or from Fig. 14.9), 1.667and 1.083. A point on this scale positionedmidway between these two scale positions (notscale values) indicates an average loss increasefor the entire length of waveguide to be1.28 times the attenuation or 1.07 dB (l010glO 1.28 = 1.07 dB). This compares,within the limits of ability to read the scales,with the exact value indicated by the differences in the reflection losses at the same twopositions, viz., 1.25 dB - 0.18 dB = 1.07 dB.The maximum variation of loss within astanding wavelength, from the attenuationper unit length, falls between the limits of Sand 1/S where S is the standing wave ratio.It reaches these peak values only when theloss is due entirely to either conductor ordielectric loss and when radiation losses areabsent.

The dissipative loss per unit length ofwaveguide due to conductor resistance isproportional to the square of the current atthe point, whereas the dielectric loss per unitlength of waveguide is proportional to thesquare of the voltage. Since both conductorresistance and dielectric conductance areevenly distributed in a uniform waveguide,when standing waves are present these twosources of loss reach peak values one-quarterwavelength apart. Their combined effect atany position along the standing waves canbe evaluated by simply adding them at thatposition as illustrated in Prob. 4-1.

4.1.3 Reflection Loss

In a lossless waveguide transmission system,if a matched impedance condition is assumedbetween a generator and waveguide characteristic impedance, and between the waveguidecharacteristic impedance and the load, therewill be no reflected power in the waveguideand consequently no reflection loss. If theload impedance only is then changed, so thata mismatch between the waveguide characteristic impedance and load occurs, there will bea reduction of power delivered to the load.The reduction in load power as the loadimpedance is changed is a measure of thereflection loss. This may be expressed as aratio of the reflected to the absorbed power.

In terms of the voltage reflection coefficient, the reflection loss in dB is:

Refl. Loss, dB = -10 loglO(1 - p2) (4-5)

Under the mismatched load condition thegenerator will deliver less power to the waveguide. The reduction of power introduced tothe waveguide at the generator end is the sameas the reflection loss at the load. In otherwords, the reflection loss at the load can bereferred back along the waveguide to the


4.1.4 Return Loss

In other words, return loss is the powerreflection coefficient expressed in dB. Thus,

Return loss is a nondissipative loss termfrequently used to describe the degree ofmismatch of a load which is closely matchedto the characteristic impedance (or characteristic admittance) of a waveguide and ·which,therefore, produces a small reflection coefficient or standing wave ratio.

Return loss is a measure of the ratio, in dB,of the power in the incident and reflectedwaves, i.e.,

generator terminals and it will have a constantvalue at all positions which is independent ofthe waveguide length.

Reflection loss is a nondissipative type ofloss representing only the unavailability ofpower to the load due to the mismatch ofimpedances originating, in this case, betweenthe waveguide and load. If a lossless. transformer is inserted at any point in the transmission system between the load and generatorto provide a conjugate match of the impedances, as seen in either direction at the pointof its insertion, then the reflection loss maybe canceled by a negative reflection loss(sometimes called reflection gain) introducedby the transformer.

If the waveguide is not lossless, under theabove mismatched conditions at the load, thereflection loss will decrease from its initialvalue at the load to something less than thisvalue at the generator end of the waveguide.The difference between these two values ofreflection loss, as may be read on the radialreflection loss scale for the SMITH CHARTon Fig. 4.1, will correspond exactly to theincreased dissipation in the waveguide due tothe reflection of power from the load.

4.2 CURRENT AI\ID VOLTAGEOVERLAYS

For a given transmitted power, the inputimpedance locus on the SMITH CHART resulting in constant current magnitude coincides with the locus of constant series resistance. Similarly, the complex admittancelocus resulting in a given voltage magnitudecoincides with the locus of constant conductance. The constant current and constant voltage loci are thus represented by twoseparate families of circles, all of which arecentered on the resistance (and/or conductance) axis of an overlay chart for the SMITHCHART in Fig. 3.3. As shown in Fig. 4.2,the constant voltage circles are an exactimage of the constant current circles as reflected in a vertical plane passing through thecenter of the overlay chart.

Corresponding current and voltage curvesin Fig. 4.2 in each of the two families havethe same magnitude since they are "normalized" to correspond to the current and voltage in a waveguide having a characteristicimpedance of one ohm (or a characteristic admittance of one mho), when transmittingone watt of power. The actual voltage is

a waveguide which is nearly match-terminatedwill have a relatively large return loss, whereasone which is badly mismatched will have asmall return loss.

Return loss is a term which is frequentlyused when working with waveguide directionalcouplers whose output ports contain sampleportions of the incident and reflected powerin the main waveguide.

Return loss is plotted as a radiai scale forthe SMITH CHART coordinates, ranging between zero dB at the outer rim and infinitedB at the center of the chart. This scale isshown in Fig. 4.1.

(4-6)Return loss, dB = 10 10glO p2

LOSSES, AND VOLTAGE-CURRENT R EPR ESENTATIONS 39

Fig. 4.2. Normalized voltage or current at any point along a waveguide (overlay for Charts A. B, or C in cover envelope).


obtained by multiplying the normalized voltage by the square root of the product of thepower and the characteristic impedance (orby the square root of the ratio of the powerand the characteristic admittance). The actualcurrent is obtained by multiplying the normalized current by the square root of theratio of the power and the characteristicimpedance (or by the square root of theproduct of the power and the characteristicadmittance). In waveguide transmission in theTEM mode, such as propagated in coaxial andopen-wire transmission lines, the normalizedcomplex impedance into which power flowsis the ratio of the normalized steady-statevoltage to the normalized steady-statelongitudinal current times the cosine ofthe phase angle between the voltage andcurrent.

The phase angle between the voltage acrossa waveguide and the longitudinal currentobserved at the same point will depend uponthe waveguide input impedance, or admittance. This can vary between the limits ofplus and minus 90°. This phase angle may berepresented as a family of curves shown onthe voltage-current overlay in Fig. 4.2, whichapply to either impedance or admittancecoordinates of the SMITH CHART. Thisphase angle is independen t of the power, thecharacteristic impedance, or the characteristicadmittance of the waveguide.

When the voltage-current overlay is appliedto the impedance coordinates the zero voltagepoint on the overlay corresponds to zero inputimpedance. When applied to the admittancecoordinates the zero voltage point correspondsto infinite admittance. In either case, in theinductive region of the chart the voltagealways leads the current, and the phase anglebetween voltage and current is, by convention,positive. In this region the voltage across theinductive reactance (or inductive susceptance)component of the input impedance (or admittance) always leads the voltage across the

resistance (or conductance) component by90°.

Similarly in the capacitive region of theSMITH CHART the voltage always lags thecurrent, and the phase angle between voltageand current is negative. In this region thevoltage across the capacitive reactance (orcapacitive susceptance) component of theinput impedance (or admittance) always lagsthe voltage across the resistance (or conductance) component by 90°.

From Fig. 4.2 the shape of the current andvoltage standing waves of any amplitude maybe plotted (as shown on Fig. 3.6). This isaccomplished by drawing the desired standingwave circle on the SMITH CHART coordi

nates and by then observing its intersectingpoints with the various normalized currentor voltage loci on the overlay correspondingto positions along the waveguide (as measuredalong the peripheral length scale). It may beseen from this overlay that only when a waveguide is match-terminated is uniform currentand voltage obtained throughout its length.

The radial transmission loss scales describedin this chapter will, of course, apply to thecurrent and voltage overlay curves on Fig.4.2. However, since the transmission loss inone-half wavelength of waveguide is generallysmall at high frequencies, such losses will notgenerally have a significant effect upon thestanding wave shape in a given region along awaveguide, except in the minimum region ofa standing wave having a large amplituderatio.

When using the universal voltage-currentoverlay (Fig. 4.2) on the SMITH CHARTcoordinates, one should remember that theposition of all voltage minima points along awaveguide always coincides with the positionof minimum input impedance along the resistance axis. Thus, the peripheral length scaleon the SMITH CHART in Fig. 3.3, progressing clockwise and labeled "wavelengthstoward generator," indicates wavelengths

LOSSES, AND VOLTAGE·CURRENT REPRESENTATIONS 41

toward generator from a VSWR minimumposition when its zero position is alignedwith zero voltage at the left-hand end ofthe overlay axis.

If the voltage-current overlay is applied tothe admittance coordinates of the SMITHCHART the position of zero voltage on theoverlay coincides with the position of maximum admittance. In this case, the peripherallength scale indicates wavelength toward generator from a VSWR maximum position.

PROBLEMS

4-1. A lossy waveguide is conducting powerto a mismatched load with produces astanding wave ratio S near the load endwhose value is 3.0. One-third of thetotal attenuation is due to dielectric lossand the remaining two-thirds is due toconductor loss. Determine graphicallythe distribution of the combined lossesalong one-half wavelength of the standingwave near the load end of the guide.

Solution:

1. On SMITH CHART A in the coverenvelope, draw a 3.0 standing wave ratiocircle. The radius of this circle is ob-

Table 4.1. Data for Prob. 4-1.

tained from the SWR scale across thebottom of the chart.

2. Superimpose Chart A on the normalized voltage and current overlay (Fig.4.2), and at the intersection of the 3.0standing wave ratio circle with the normalized voltage and current curves on theoverlay, observe and tabulate their amplitudes, as in Table 4.1, at eight pointsspaced one-sixteenth wavelength apart.

3. From the above data plot, as in Fig.4.3, the normalized voltage and currentstanding wave shapes (curves C and A,respectively) .4. Square the amplitude values at thedata points on both voltage and currentcurves and tabulate the results, as inTable 4-1. Plot loss distribution curvesthrough the squared amplitude values(curves D and B, respectively) as inFig. 4.3. These latter curves representthe loss distribution due to dielectricand conductor losses, respectively, eachof which would represent the overalldistribution only in the complete absenceof the other.5. Multiply the loss distributions bythe originally specified 1/3 and 2/3factors, respectively, and tabulate theindividual and the combined losses, as

DI ST. IN WAVELENGTHS0/16 1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16

FROM E MI N TOWARD LOAD

NORMALIZED 1.732 1.622 1.302 0.852 0.577 0.852 1.302 1.622 1.732CURRENT (I)

CONDUCTOR LOSS 3.000 2.630 1.696 0.726 0.333 0.726 1.696 2.630 3.000(PROPORTIONAL TO r 2 )

NORMALIZED 0577 0.852 1.302 1.622 1.732 1.622 1.302 0.852 0577VOLTAGE IE)

DIELECTRIC LOSS0333 0726 1.696 2.630 3.000 2.630 1.696 0.726 0.333

(PROPORTIONAL TO E2 )

2/3 CON DUCTOR LOSS 2.000 I. 754 1.131 0.484 0.222 0.484 1.131 1.754 2.000

1/3 DIELECTRIC LOSS 0.111 0.242 0.565 0.876 1.000 0.876 0565 0.242 0.111

COM81NED CONDUCTOR 2.111 1.996 1.696 1.360 1.222 1.360 1.696 1.996 2.11 ,AND DIELECTRIC LOSS


in Table 4.1. Plot the combined lossdistribution as tabulated in Table 4.1as the dot-dash curve (E) in Fig. 4.3.Note that this curve varies cyclically

about a mean value whose amplitudeclosely approximates that of the "standing wave loss coefficient" (horizontalline F).

-.....

1/ ~ /'"", /, /,1/,

\ J ~ /\

/ \,

\ /\ /

\ /\ /

--""""'- ...\ / ", ..-\ / ..

\

1/ \ / ../' ... \.......... \ C ~../--- .........~ -- - ---

.... / ''', /...... F ~/../ 1\ '"..... , /'

PX ......... ' ...E ../ i,/"'.

~, \ ......._- )00 ......... / //',\,\. I~//

bf ~, /?~0 \ , ~ /

~, ./ I

" ~/" /\ ..... ...., -- -- /,/

V , ~/

~..... ....- _/

3.

a::ILl~oa.Iz~(/)z8

2.l:lo...J

oZ<l

ILlo:::JI-::::ia.~<l

~ 1.I<l...JILla::

oo 1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16

(

WAVELENGTHS FROM Emin TOWARD LOAD

A NORMALIZED CURRENT (I) WHEN S= 3.0

B CONDUCTOR LOSS (PROPORTIONAL TO r 2 )

C NORMALIZED VOLTAGE (E) WHEN S = 3.0

D DIELECTRIC LOSS (PROPORTIONAL TO E2)

E COMBINED CONDUCTOR AND DIELECTRIC LOSS

F STANDING WAVE LOSS COEFFICIENT, (I+S 2 )/(2S)

Fig. 4.3. Solution for Prob. 4-1 .. waveguide loss distribution when S =3.0.

5.1 PHASE RELATIONSHIPS

The phase relationship between the reflectedand incident traveling voltage waves, and

between the reflected and incident travelingcurrent waves, at various positions along awaveguide has been discussed briefly in Chaps.I and 3. In Chap. 4 an additional phaserelationship, namely, that between the steadystate (standing wave) voltage and the steadystate (standing wave) current at variouscoincident points along a waveguide wasdiscussed. The two former relationships are,respectively, the phase angle of the voltagereflection coefficient and the phase angle ofthe current reflection coefficient. The latteris the phase angle of the power factor-thepower factor itself being thus defined as thecosine of this angle [35]. All three of theforegoing phase relationships have been shownto be graphically representable as overlays onthe impedance coordinates of the SMITHCHART.

CHAPTER 5Waveguide

PhaseRepresentations

It has also been shown that the voltagecurrent phase relationship (phase angle ofpower factor) overlay (Fig. 4.2) can be applied to a SMITH CHART whose coordinatesare labeled to represent either impedances oradmittances (such as the SMITH CHART ofFig. 3.3). Directions were indicated thereinfor properly orienting the overlay on thespecific coordinates selected, and for interpreting the sign of the indicated phase angle.

In this chapter some fundamental waveguide phase conventions will first be reviewed.Following this, more generalized uses of theperipheral scale labeled "angle of reflectioncoefficient" (Fig. 3.3) will be presented.Next, a discussion of the voltage, current, andpower transmission coefficient, with generalized SMITH CHART overlays therefor, will begiven. Finally, some additional waveguidevoltage and waveguide current phase relationships will be discussed and presented in theform of general-purpose overlays for theSMITH CHART. These latter phase relation-

43


ships are of fundamental importance in thedesign of waveguide components and antennasemploying phased radiating elements excitedthrough waveguide feed systems.

5.2 PHASE CONVENTIONS

In waveguide terminology, phase describesthe particular stage of progress of any periodically alternating quantity. It may also describethe relative progress of two such quantities.The two quantities may be at the same or atseparate positions along a waveguide and theymayor may not be of the same kind. Waveguide quantities which are customarily relatedby phase include currents, voltages, or currentvs. voltage.

Absolute phase is expressed as the totalnumber of cycles (including any fractionalnumber) separating the two quantities, wherein one complete cycle is 217 radians or 360°.

Relative phase is frequently of more interest.This is the absolute phase less the largest integral number of 217 radians (or 360°) whichseparates the quantities. The unit of phase is,therefore, the radian, or the electrical degree.

Relative phase is represented graphically asthe angle between two periodically rotatingvector quantities. If the periodicity of thetwo quantities is the same, as in all waveguideapplications to be considered in this book, therelative phase is independent of real timevariation. If the two quantities are not exactly in-phase (0°) or out-of-phase (± 180°),one of them is considered to have a relativephase lead or lag over the other.

Increasing phase lag (or decreasing phaselead) is represented by a clockwise rotation ofa voltage or a current vector. Conversely,decreasing phase lag (or increasing phase lead)is represented by a counterclockwise rotationof a vector.

Table 5.1. Voltage or Current Phase Relations Along a Lossless Waveguide When SWR = 3.0(Refer to Fig. 5.1).

(SWR=3.0)

VOLTAGE (OR CURRENT)

STANDING WAVE #2 MAX.

GEN.....- MIN. -+LOAD

POINTS OF OBSERVATION#1 MAX. P3 P2 PI MIN. p' p' p' 1#2MIN

(along standing wave I .0625). apart) I 2 3

PHASE OF INCIDENT WAVE+90 +67.5 +45 +22.5 0 -22.5 -45 -67.5 -90

(reI. to incid. wave @ min.)

PHASE OF REFLECTED WAVE-90 -67.5 -45 -22.5 0 +22.5 + 45 +67.5 +90

(reI. to refl. wave @ min.)

PHASE OF REFL. COEFF.0 +45 +90 +135 ±180 -135 -90 -45 0

(ratio refl. to incid. wove)

PHASE Of TRANSM. COEFF.0 +14 +25.5 +29.5 0 -29.5 -25.5 -14 0

(ratio transm. to incid. wove)

PHASE OF STANDING WAVE AT A PT.90 82 71.5 52 0 -52 -71.5 -82 -90

(re I. to standing wave @ min.)

WAVEGUIDE PHASE REPRESENTATIONS 45

LEGEND

i-INCIDENTr- REFLECT£DR- RESULTANT

0.12 0.13

0.38 0.37

8£"0 LtL"O

Fig. 5.1. Vector representation of phase relations for voltages on impedance coordinates, or currents on admittance coordinates,of SMITH CHART when SWR = 3.0 (refer to Table 5.1).

l

A relative phase lag of one vector overanother is indicated by a negative sign (-)on the lagging vector, whereas a relative phaselead of one vector over another is indicated bya positive sign (+).

In accordance with the above convention,Fig. 5.1 shows a SMITH CHART upon whicheight specific vector representations of thevoltage or current on the impedance or admittance coordinates, respectively, are plotted.


These representations are for eight uniformlyspaced positions within one-half wavelengthof waveguide, accompanying a standing wavewhose amplitude ratio is 3.0. They illustratethe amplitude and phase relationships of theincident, reflected, and resultant wave components.

The various relationships involving phase atany position along a waveguide which 'havebeen, or will be, discussed herein include:

I. Phase of the incident wave relative tothe incident wave at the nearest standing waveminimum position.

2. Phase of the reflected wave relative tothe reflected wave at the nearest standing waveminimum position.

3. Phase of the reflection coefficients(phase of reflected voltage or current wave atany position relative to the incident wave atthe same position).

4. Phase of the transmission coefficien ts(phase of transmitted voltage or current waveat any position relative to the incident waveat the same position).

5. Phase of standing wave (phase of resultant of incident and reflected wave at apoint along a standing wave relative to resultant wave at the standing wave minimumposition).

All of the above relationships are representable, and may be evaluated by graphicalmeans for any specified position along a waveguide, and for any specified standing waveratio (or load impedance). The numericalresults for the eight examples are given inTable 5.1.

5.3 ANGLE OF REFLECTIONCOEFFICIENT

It was previously indicated that the angleof the reflection coefficient scale (Fig. 3.3)applies to the voltage reflection coefficientphase angle on impedance coordinates. Moregeneralized uses for this scale, as drawn, will

be shown which are consistent with thepreviously discussed conventions.

Zero relative phase angle for the voltagereflection coefficient occurs at all voltagemaxima positions along a waveguide, at whichpoints the impedance is maximum (and theadmittance is minimum). At these points thereflected and incident voltage waves are inphase. Likewise, zero relative phase angle forthe current reflection coefficient occurs at thecurrent maxima positions along a waveguide ,I

where the impedance is minimum (and the I

admittance is maximum). At these points thereflected and incident current waves are inphase.

Thus, as shown on Fig. 3.3, the angle of thereflection coefficient scale applies not only tothe voltage reflection coefficient on the impedance coordinates, but also to the currentreflection coefficient of the admittance coordinates.

A simple rule which applies to any combination of voltage or current reflection coefficient and impedance or admittance coordinates is that zero on the reflection coefficientphase angle scale should always be alignedwith a maximum of the corresponding standing wave.

5.4 TRANSMISSION COEFFICIENT

A term which is used less frequently thanreflection coefficient but which is neverthelessuseful in many waveguide applications istransmission coefficient. This term, like reflection coefficient, may be applied to any twoassociated quantities at any given positionalong a waveguide. The chosen quantitiesmust, of course, be specified. In waveguides,the term transmission coefficient is mostfrequently applied to voltage and current,although it may also be applied to power.The value of the transmission coefficient, likethat of the reflection coefficient, will dependupon the associated quantities selected, thefrequency, and the mode of transmission.


Similarly, the phase angle a in terms of rand

fJ is

At any given position along the waveguide thephase angle of rand p will always have thesame sign.

Both the voltage and the current transmission coefficient may be represented by a singleoverlay on the coordinates of the SMITHCHART shown in Fig. 3.3. Such a generalpurpose transmission coefficient overlay is

Since p is always between zero and unity, itwill be observed from Fig. 5.3, or from Eqs.(5-1) and (5-2), that the magnitude of T mustalways be between zero and two. Scalesrepresenting the magnitude of T as a voltage(or current) ratio, and in dB, are shown in Fig.5.2. In the region along a waveguide wherethe voltage transmission coefficient is greaterthan unity, resulting in a gain, the currenttransmission coefficient will be less thanunity, resulting in a compensating loss.

Also from the geometry of Fig. 5.3, thephase angle fJ in tenns of p and a is seen to be

2 1/2Ip I = [r + 1 - 2r cos fJ]

(5-3)

(5-4)

(5-2)

1800 _ tan-1 r sinfJ1 - r cosfJ

fJ = tan-1 p sin(lSO° - a)

1 - p cos(l80° - a)

a

I r I = [p2 + 1 - 2p cos(lSO° - a)]l!2 (5-1)

Similarly, the magnitude of p in tenns of r

and fJ is

The voltage transmission coefficient is defined as the complex ratio of the resultant ofthe incident and reflected voltage to theincident voltage. Similarly, the current transmission co.efficient is defined as the complexratio of the resultant of the incident andreflected current to the incident current.

Since power has no "phase," the powertransmission coefficient is simply the scalarratio of the transmitted to incident power.This is constant at all positions along a losslesswaveguide. The power transmission coefficientis numerically equal to one minus the powerreflection coefficient, which was described inChap. 3. Figure 5.2 shows a radial scale forthe SMITH CHART coordinates in Fig. 2.3representing this parameter. This scale appliesequally to the impedance and admittancecoordinates.

A graphical representation of the relationship between the complex voltage (or current)reflection coefficient pia and the complexvoltage (or current) transmission coefficientr/fJ is shown in Fig: 5.3.

From the geometry of Fig. 5.3 the magnitude of r in tenns of p and a is seen to be

POWER TRANSMISSION COEFFICIENT - RATIOi"'i" iii '''"'II'i'i'''''''' ",iiii

PIVOT AT CENTEROF CHART

+I

o

(GAIN)

I

<;

VOLTAGE (OR CURRENT) TRANSMISSION COEFFICIENT-DB(LOSS)

~ I " " ': I 'i !:' I i I:'! I 'I i i \ I'.!. i I Iii ',' I I I i I Iii j' I t' I ~' I '\ " "

f~p iN6~~~~g6 b b (; 0 0 ~ 6 6 0 b

iPIVOT AT 0 VOLT(OR 0 CURRENT) POINTAT RIM OF CHART

:: :.':' :': i : ' : r i,' : Iii i : I I f i : I ,

8Pf5~~~~ S S ;; ~

I POWER TRANSMISSION COEFFICIENT - DB

L--- CHART DIAMETER----------------------O'l

I VOLTAGE (OR CURRENT) TRANSMISSION COEFFICIENT - RATION u. ~ u. i:Jl :..... i:ll l.o:- ;-;-. N

~i! I" ~, , 'I " ! I ,,? ii" 0 \! I' 0 0 0 0 0 - f\) UII .,. U. (r, ..., iD (0 0

Fig. 5.2. Power. voltage. and current transmission coefficient magnitude scales for SMITH CHART coordinates in Fig. 2.3.


2.01.0 11.5 I. 667TRANSM. COEF. MAGNITUDE 1-1 (SWR MAX)

::tJ I

I~ If.~ /

/,8 II.'!i/ /J~

,..."7~- /,:;If

I/

.333 .5(SWR MIN)

\\\

""" ................. _-

oI

.5 .667I I II

REFL. COEF. MAGNITUDE

1.0I

Fig. 5.3. Representation of voltage or current reflection coefficient magnitude and phase, andvoltage or current transmission coefficient magnitude and phase on a SMITH CHART (diagram showsconditions at a point along a standing wave whose ratio is 5.0).

shown in Fig. 5.4. As oriented in Fig.5.4, in relation to Fig. 3.3, it represents thevoltage transmission coefficient on the impedance coordinates, or the current transmission coefficient on the admittance coordinates.When rotated 1800 from this orientation, theoverlay of Fig. 5.4 represents the voltagetransmission coefficient on the admittancecoordinates, or the current transmission coefficient on the impedance coordinates of thissame chart. The convergence point of alltransmission coefficient phase angles on thisoverlay should always be aligned radially withthe corresponding standing wave minimapoints.

From the transmission coefficient overlaythe shape vs. amplitude of all standing wavesof voltage or current along a waveguide maybe plotted for a constant incident waveamplitude.

Standing wave shapes for three specificstanding wave amplitude ratios have beenplotted in Fig. 1.3. Note that the maximumvoltage (or current) never exceeds twice theincident voltage (or current) at any pointalong a waveguide.

The requirement for a constant incidentwave on a waveguide is commonly satisfied byinserting a large, for example, 20-dB or more,dissipative attenuator, whose characteristic


Fig. 5.4. Complex voltage or current transmission coefficient along a waveguide loverlay for Charts A. B. or C in cover envelope).


impedance matches the waveguide, betweenthe generator and the waveguide. In this way reflected waves from the load do not return tothe generator with sufficient power to significantly affect its output power. Consequently,the incident wave on the waveguide is heldconstant and independent of the load impedance or admittance characteristics.

The phase angle of the voltage (or current)transmission coefficient is a measure of theextent that the resultant wave departs in itsphase relationship with the incident wave.The maximum departure from an in-phaserelationship depends upon the standing waveratio, being smallest for small standing wavesand reaching a maximum of 900 for an infinitestanding wave ratio.

5.5 RELATIVE PHASE ALONG ASTANDING WAVE

While any reference position is possible, themost generally useful position is the minimumposition of the standing wave nearest to thepoint of measurement, either toward the loador toward the generator. The selection of aminimum reference position along a standingwave on a waveguide makes it possible to plot,and to unambiguously identify, the phaseangle on a family of relative-phase curves forall standing voltage or current waves along awaveguide. Such a plot, as shown in Fig. 5.5,may conveniently be used as an overlay forthe SMITH CHART impedance or admittancecoordinates. From the reference phase shiftat each of the two points the relative phase ofthe standing wave voltage or current at thetwo points is readily obtainable.

Mathematically, the relationship for thephase of the standing wave voltage or currentat a point relative to that at the nearestminimum point is

where II>.. is + in the direction of the generatorfrom the minimum point. Thus, if II>.. (thedistance from the standing wave minimumpoint to the point in question) is in thedirection of the generator (positive direction)the phase angle at the point in question ispositive, i.e., it leads the phase at the minimum point. In the opposite direction, i.e.,toward the load from the minimum, l!>" isnegative and the phase at all points in theregion between the minimum and the following maximum lags that at the minimum.

The overlay of Fig. 5.5 was machineplotted. Equation (5-5) was first rewritten toexpress ¢ in terms of the voltage re(lectioncoefficient magnitude p and its phase angle a.

(See Eq. (3-8).) The resulting equation is

Another useful waveguide phase relationship is the relative phase angle between thestanding wave voltages (or currents) at anytwo positions along a waveguide. This phaserelationship is particularly important in thedesign of phased-array antennas. It shouldnot be confused with the relative phase at twopoints along a traveling wave, nor with therelative phase angle of the voltage (or current)reflection (or transmission) coefficients at thetwo points.

The standing voltage (or current) wave ateach of two separated positions along a waveguide is the vector sum of the incident and thereflected voltage (or current) waves at therespective positions (Fig. 5.1). The relativephase between these two resultant voltages(or currents) is not linearly related to theirphysical separation. It is a function of thedegree of mismatch of the waveguide andload (standing wave ratio) and the positionsalong the waveguide relative to the positionof some reference point on the standing wave.

psin[¢ - (aI2)]

sin [¢ + (aI2)]

(5-5)

(5-6)


Fig. 5.5. Phase of voltage or current at any point along a waveguide relative to voltage or current. respectively. at nearestminimum point loverlay for Charts A. B. or C. in cover envelope).

The x and y components of the phase angle ¢

5.6 RELATIVE AMPLITUDE ALONG ASTANDING WAVE

The relative amplitude of the voltage (orcurrent) at any two points along a standing

One radiating element of an array antennais connected via a coaxial cable to ajunction point of a "corporate" feedsystem. The known conditions are: (1)electrical length of the cable is 6.279wavelengths, (2) attenuation is 1.0 dB,(3) normalized load impedance of cableis 1.6 - j 1.35 ohms. (See insert in Fig.5.7.)

Using a SMITH CHART, determine(1) the standing wave ratio in the cableat its load end, (2) the normalized sending end impedance 2/20, (3) the standingwave ratio in the cable at the sendingend, (4a) the voltage transmission coefficient TE at each end of the cable, (4b)the voltage insertion phase <PE' (Sa) thecurrent transmission coefficient TI at eachend of the cable, and (5b) the currentinsertion phase <PI •

Solution:1. As shown in Fig. 5.7, locate onSMITH CHART A in the cover envelopethe normalized load impedance 2/20 =

1.60 - j 1.35, and construct a standingwave circle centered on the chart coordinates, passing through this impedance point. Observe that the radius ofthis circle, as laid out on the SWR scaleacross the bottom, shows that at theload end of the cable the standing waveratio S is 3.0.

PROBLEMS

5-1.

(5-8)

(5-7)


wave are uniquely related to the relative phaseat the two points. Hence it is possible to plotthe family of amplitude ratio curves in Fig.5.6 which are uniquely related to the familyof phase curves in Fig. 5.5.

All of the rules which have been given forapplication of the phase curves of Fig. 5.5 tovoltage or current on impedance or admittancecoordinates also apply to the amplitude curvesof Fig. 5.6.

p sina

are

x = p caSa

52

y

The relative phase of the current at anytwo positions along a standing current waveof a given amplitude ratio on a waveguide isidentical to the relative phase of the voltageat any two corresponding positions along astanding voltage wave of the same amplituderatio. Accordingly, the phase overlay of Fig.5.5 may be used to obtain either relative

. voltage or relative current phase relations.This overlay may also be used on either theimpedance or the admittance coordinates.

A simple rule to follow in orienting theoverlay of Fig. 5.5, for either voltage orcurrent phase relationships on the SMITHCHART impedance or admittance coordinates,is to observe that "the overlay is always orientedso that the phase change vs. length of waveguide is greatest in the region of the respectivestanding wave minima.

The eight vector phase relationships plottedin Fig. 5.1 may conveniently be used as checkpoints on the proper orientation of any ofthe phase overlays which have so far beendiscussed. As shown, they represent voltageson impedance coordinates or currents on admittance coordinates. When rotated, as agroup, through 1800 with respect to theirpresent positions on the coordinates, theyrepresent voltages on the admittance coordinates or currents on the impedance coordinates.

WAVEGUIDE PHASE R EPR ESENTATIONS 53

Fig. 5.6. Amplitude of voltage or current at any point along a waveguide relative to voltage or current, respectively, at nearestminimum point (overlay for Charts A, B, or C in cover envelope).



o 0~ Nu u

CONDITIONS:ELEC. LENGTH

=6.279A

IZ IZ: I

=====A=T=T=E=N=.=== 1.:- i 1.~5= I.dB

1.412

0.38 0.37

0.12 0. 3

...7...2

ORIGIN

FROM CONSTRUCTION:AT LOAD;

S: 3.0TE" 1.43 112.70

T1: .681 27.50

AT GEN:S: 2.3TE= .87~

Tl: I. 245rrs:o:

4>E =64.30

4> I : 143.90

Fig. 5.7. Construction for Prob. 5-1, transmission coefficients and insertion phase.

2. Move clockwise around this standingwave circle (toward generator) a distanceequal to 0.279 wavelength, as measuredon the outermost peripheral scale, to thesending end position, where in the absence of attenuation the sending endimpedance would be 0.42 + j 0.50.(Note: the largest integral number ofhalf wavelengths is subtracted from thetotal electrical length since relative-notabsolute-phase is required.) Correct for1.D-dB attenuation by moving radiallytoward the center of the chart coordinates a distance! corresponding to 1.0-dBmajor divisions as obtained from theattenuation scale across the bottom, tothe position where the normalizedsending-end impedance Z/Zo is 0.54 + j

0.49.3. The standing wave ratio at Z/Zo =0.54 + j 0.49 is seen to be 2.3 whenmeasured as the radial distance on thechart coordinates, employing the SWRscale across the bottom.4a. Construct straight lines from theorigin of the chart coordinates throughthe load and sending-end impedancepoints, respectively, intersecting the innermost peripheral voltage transmissioncoefficient angle scale. Determine themagnitudes of the voltage transmissioncoefficient at each end of the cable usingthe magnitude scale across the bottom,and thus observe that at the load end ofthe cable TE = 1.43/ 12.7° , and at thesending end T E =0.87/23.4°.


4b. The voltage insertion phase 'PE is thephase change undergone by the voltagetraveling wave (total angle of a matchterminated cable reduced to an equivalentelectrical length less than one-half wavelengths) plus the net difference in theangles of the voltage transmission coefficients at the load and sending ends,respectively. Thus, in this example,

= 64.3°

Sa. Use the complex transmission coefficient overlay (Fig. 5.4) to represent thecurrent transmission coefficients by inverting it (rotating it 180°) with respectto the impedance coordinates of Fig. 5.7.Then observe from this overlay that atthe normalized load impedance point,where Z/Zo = 1.6 - j 1.35, the current

transmission coefficient TI is 0.68/27.5°,and at the sending-end impedance point,where Z/Zo = 0.54 + j 0.49, the currenttransmission coefficient is 1.245/16.0°.Sb. The current insertion phase 'P I isobtained analogously to that of the volt-

age insertion phase as described in Prob.4b, and is found to be

'PI

= {0.279 x 360° + [(+27.5) - (-16.00 n}= 143.9°

6.1 IMPEDANCE CONCEPTS

In this chapter the concept of waveguideinput impedance (and admittance) is pre

sented first. This is followed by a consideration of the input impedance (or admittance)relationships of a waveguide to those ofsimple series or parallel circuits which presentequivalent impedance (or admittance) at agiven frequency.

Two overlays for conventional SMITHCHART coordinates which provide alternativecoordinate forms, useful in specific waveguideapplications, will be described. One of thesecoordinate overlays displays normalized inputimpedance components presented by theequivalent parallel-circuit (rather than the conventional series-circuit) elements, and/or normalized input admittance components -presented by the equivalent series-circuit (ratherthan the conventional parallel circuit) elements. The other overlay displays normalized

CHAPTER 6Equivalent

CircuitRepresentations of

Impedance andAdmittance

polar coordinate components of impedance,Le., magnitude and phase angle.

A graphical method for combining twonormalized polar impedance vectors in parallel,which utilizes special polar coordinates, isincluded.

6.2 IMPEDANCE-ADMITTANCEVECTORS

The input impedance at any specified position along a waveguide is, fundamentally, thecomplex ratio of voltage to current at thatposition; the input admittance is the reciprocal of this ratio. Any sinusoidally varyingvoltage or current at any point in a waveguidemay be represented by the projection of auniformly rotating vector on a fixed axis. Theratio between voltage and current is a stationary vector whose magnitude is the ratio ofvoltage magnitude to current magnitude, and

57


Similarly, the normalized input admittancepresented by parallel-circuit combinations of

6.3 SERIES-CIRCUIT REPRESENTATIONSOF IMPEDANCE AND EQUIVALENTPARALLEL-CIRCUIT REPRESENTATIONS OF ADMITTANCE ON CONVENTIONAL SMITH CHARTCOORDINATES

Waveguide input impedance and input admittance vectors may be expressed in eithercomplex or polar notation. On conventionalSMITH CHART coordinates, such as the coordinates in Fig. 6.1, the complex notation isused. On this chart the normalized impedancepresented by series-circuit combinations ofresistive and inductive (or capacitive) circuitelements is expressed in complex notation by

(6-2)

these respective circuit elements is expressedby

The vector diagrams superimposed on theconventional SMITH CHART coordinates ofFig. 6.1 show normalized input impedancevectors Z/Zo and (diametrically opposite)normalized input admittance vectors Y/Yoateight positions equally spaced along one-halfwavelength of waveguide. In this example, awaveguide termination is arbitarily chosenwhich produces a standing wave ratio of 3.0.The normalized impedance vector ZI/ZO' for,example, represents the impedance of a series circuit which is equivalent, at a givenfrequency, to the parallel circuit representedby the diametrically opposite admittance

vector Y1/Yo- Similarly, the normalizedimpedance vector Z2/Z0 represents the impedance of a series circuit which is equivalent,at a given frequency, to the parallel circuitrepresented by the diametrically. oppositeadmittance vector Y2/YO'

In the upper half of the SMITH CHARTof Fig. 6.1 normalized impedances are represented at a single frequency by series circuitcombinations of resistive and inductive elements, that is, R/Zo + jX/Zo' The voltageacross any inductive circuit leads the currentflowing into the circuit by a phase anglebetween 0° and 90°. As discussed in Chap. 5,a lead is indicated graphically by a counterclockwise rotation of a vector, and the sign ofthe angle is positive. Impedance vectors inthe upper half of the SMITH CHART ofFig. 6.1 are therefore assigned positive phaseangles. Conversely, impedances in the lowerhalf of Fig. 6.1 are assigned negative phaseangles.

The above convention for the representation of impedance vectors also applies to the

(6-1 )R "Xs J s-±-Zo Zo

whose angle is the difference between thephase angles of the voltage and currentvectors. Thus, waveguide input impedanceand its inverse (waveguide input admittance)may be regarded as stationary vectors. At agiven position along a waveguide these twovectors have reciprocal magnitudes and equalphase angles of opposite sign.

The terms waveguide input impedance andwaveguide input admittance were defined anddiscussed briefly in Chap. 2. The "normalization" of these terms to the waveguide characteristic impedance or characteristic admittance, respectively, was also discussed therein.Also in Chap. 2, the conversion of a normalizedwaveguide input impedance to an equivalentnormalized waveguide input admittance, andvice versa, on conventional SMITH CHARTcoordinates was described.

EQUIVALENT CIRCUIT REPRESENTATIONS OF IMPEDANCE AND ADMITTANCE 59

representation of admittance vectors. Thus, admittances in the upper half of Fig. 6.1 are represented by parallel-circuit combinations ofconductance and capacitance, that is, GIY0 +jBIYo' This follows from the fact that in allcapacitive circuits the current flowing intothe circuit leads the voltage across the circuit

0.12

0.38

90

by an angle between 0° and + 90°. Sinceadmittance is the reciprocal of impedance,i.e., the ratio of current to voltage, capacitiveadmittance vectors have a positive angle andlie in the upper half of the SMITH CHART.

The complex impedance and complex admittance represented, respectively, by series

0.13

0.37

o 1. ~ ~ ~ ~ LI ! I ! I I I

NORMALIZED VECTOR MAGNITUDE SCALE

Fig. 6.1. Normalized impedance and admittance vector relationships along a waveguide when SWR = 3.0.


combinations of resistive and reactive circuitelements, and parallel combinations of conductive and susceptive circuit elements, areshown in Fig. 6.2(a) and 6.2(b). A specificexample is illustrated in Fig. 6.3. The twoequivalent circuit positions (A and B) shownin this latter figure are diametrically oppositeone another at equal chart radius.

11

011

Z +'JS=Rs-IX s

- ±jxs

circuit elements [Fig. 6.2(c)]. Similarly, it issometimes useful to represent waveguide inputadmittance by series combination of conductive and susceptive primary circuit elements [Fig. 6.2(d)].

Figure 6.2(c) shows a parallel combinationof a resistance Rp and an inductive or capacitive reactance ± jXp' resulting in an inputimpedance Z. The input impedance of this,as of any parallel circuit, is equal to the reciprocal of the sum of the reciprocals of theresistive and reactive components, viz.,

Fig. 6.2. Equivalent series and parallel circuits.

•

~~. ~JsZ = I I -JX Rp Y = I I _.- + -.- -+ -.- +JBsRp ±IXp Gs +ISs

(6-3)1Z =

Equation (6-3) can be rationalized to obtainthe component parts of Z which representresistance and reactance of the equivalent seriescircuit shown in Fig. 6.2(a) (R s ± jX), viz.,

lid IIIle"

Similarly, as shown in Fig. 6.2(d), a particular combination of a conductance Gs inseries with an inductive or capacitive susceptance =+= jBs results in an input admittance Y,which is equal to the reciprocal of the sum ofthe reciprocals of the conductive and susceptive components of the series circuit, viz.,

As may be seen from Fig. 6.2(a) and (b),the usual representation of waveguide inputimpedance (in terms of its series circuit component values) facilitates the calculation ofthe resultant input impedance when additionalseries elements are to be added. Similarly, theusual representation of waveguide input admittance in terms of its parallel-circuit componentvalues facilitates the calculation of the resultant input admittance when additionalparallel elements are to be added.

R 2X± j-_P__P-

R 2 + X 2P P

(6-4)

Y 1

l/G s + l/=+=jBs(6-5)

6.4 PARALLEL-CIRCUIT REPRESENTATIONS OF IMPEDANCE AND SERIESCIRCUIT REPRESENTATIONS OFADMITTANCE ON AN ALTER-NATE FORM OF THE SMITHCHART

Equation (6-5) may similarly be rationalizedto obtain the component parts of Y whichrepresent the conductance and susceptance ofthe equivalent parallel circuit shown in Fig.6.2(b) (GP and Bp )' viz.,

In some applications it is useful to represent

waveguide input impedance by a parallelcombination of resistive and reactive primary

G =+= Bp 1 p

Gs2 Bs

=+= j---G 2 + B 2

s s

(6-6)


The conventional SMITH CHART coordinates shown in Fig. 6.1 are not suitable forportraying, directly, the parallel-circuit component values of impedance or the seriescircuit component values of admittance. However, these latter component values may beplotted directly on an alternate form ofcoordinates shown in Fig. 6.4. The modifica-

tion of the conventional SMITH CHART toobtain this alternate form involves the redesignation of all normalized coordinate valueswith their reciprocal values, and the rotationof coordinates through 1800 with respect tothe peripheral scales. Rotation of the coordinates through an angle of 1800 is necessary sothat fractional wavelength designations on the

Fig. 6.3. Series-circuit representation for an impedance (AI and equivalent parallel-circuit representation for an equivalentadmittance (BI at the same position along a waveguide, on SMITH CHART in Fig. 3.3.


Fig. 6.4. Alternate form of SMITH CHART coordinates displaying rectangular components of equivalent parallel-circuiimpedance (or of series-circuit admittancel (overlays for Charts A and C in cover envelope).


peripheral scales of this alternate form SMITHCHART refer to voltage nodal points on itsimpedance coordinates (and to current nodalpoints on its admittance coordinates), as onthe conventional SMITH CHART. Also, aswill be shown later, this permits the use ofthis alternate form of SMITH CHART coordinates to be used as an overlay for theconventional coordinates for converting seriesto equivalent parallel-circuit components.

The two positions A and B indicated on thealternate form of SMITH CHART coordinates(Fig. 6.6) give, respectively, the parallel-circuitimpedance components, and the series-circuitadmittance components of equivalent circuits.Equivalent circuit positions on the coordinatesof either Fig. 6.3 or Fig. 6.6 must always belocated diametrically opposite each other atequal chart radius.

6.5 SMITH CHART OVERLAY FOR CONVERTING A SERIES-CIRCUIT IMPEDANCE TO AN EQUIVALENTPARALLEL-CIRCUIT IMPED-ANCE, AND A PARALLEL-CIRCUIT ADMITTANCETO AN EQUIVALENTSERIES-CIRCUIT AD-MITTANCE

On the conventional SMITH CHART ofFig. 6.3, point A represents the impedance ofa circuit whose normalized series componentvalues may be read directly from the chartcoordinates. On the alternate form of SMITHCHART of Fig. 6.6, point A, at the samelocation, represents the impedance of a circuit

whose equivalent normalized parallel component values may also be read directly from thechart coordinates. Thus, any two coincidentpoints on these respective charts correspondto equivalent series and parallel-circuit combinations whose normalized resistive and reactive components are readable from therespective charts at a single frequency.

Thus, the alternate form of SMITH CHARTcoordinates shown in Fig. 6.6 (drawn without circuit diagrams or peripheral scales inFig. 6.4) provides an overlay for conventionalSMITH CHART coordinates (Fig. 3.3) whichpermits graphically converting component values ofseries-circuit normalized impedance toequivalent component values of parallel-circuitnormalized impedance. Similarly, this overlayprovides a convenient means for graphicallyconverting component values of parallel-circuitnormalized admittance to equivalent component values of series-circuit normalized admittance.

When the overlay of Fig. 6.4 is providedwith the peripheral scales on Fig. 3.3 the res'jlting chart (Fig. 6.5) becomes an alternateand basic form of SMITH CHART to whichall of the overlays applicable to the seriescomponent chart (Fig. 3.3 or Fig. 8.6) areequally applicable.

The general-purpose, parallel-impedance (orseries-admittance) chart shown in Fig. 6.5 isreproduced as one of four translucent plasticcharts whose function is described in thePreface and which is contained with the otherthree in an envelope in the back cover' of thisbook (Chart B).

6.6 IMPEDANCE OR ADMITTANCE MAGNITUDE AND ANGLE OVERLAYFOR THE SMITH CHART

Complex impedance or admittance vectorsR ± jX or G + jB, respectively, may also be expressed in polar form, that is, z/±e or Y/+8.The magnitude of the normalized impedancevector Z/Za in terms of its rectangular components is

(6-7)

The phase angle eis


11 .. '" RADIALLY SCALED PARAMETERS3 .. ~ g ~

q 0 :;l .. N 3f 3 i!; t g ~ II I! ~ ~ I~ ..~ 38 8 2 . '" ~ ! <; a

~

~ .. Ie l TOWARD GENERATOR - ~ TOWARD LOAD f 2 2 " lil .. l!l is t is ~ a ~ II ~ ~ I:: ~

;;; ~0 ... .HI~

~

i- In

~q 0 . i:: :g .. ,.

~ l; ; 1::N "~~9Sg g !! Q ~ . N 0<> Pi :- !'i <; " 0 " b ~

8 ~I:0

I ~~ 8 ,,; g 2 ~0

~ :;l ..~

. ~ ~o 3 0 ;; N 1:: • g j5 z. ., .. " 0>",

CENTER~ I

Fig. 6.5. Alternate form of SMITH CHART with coordinates of Fig. 6.4 (see Chart B in cover envelope).


() ±arctan(~) (6-8)(6-9)

Similarly, the magnitude of the normalizedadmittance vector YIY0 in terms of its rectangular components is

The phase angle () is

() = +arctan (~) (6-10)

Fig. 6.6. Parallel·circuit representation for an impedance (A) and equivalent series-circuit representation for an admittance (8) atthe same position along a waveguide, on SMITH CHART in Fig. 6.5.


The normalized resistance component R/Zo'and reactance component X/Zo' respectively,of an impedance expressed in polar notation,that is, Z/±O, is

~ = I.!:... I cos (±O)Zo Zo

and

I~ \ sin(±O)

(6-11)

(6-12)

This general-purpose polar impedance (oradmittance) chart shown in Fig. 6.8 is reproduced as the third of four translucent plasticcharts whose function is described in thePreface and which is contained with the otherthree in an envelope in the back cover of thisbook (Chart C).

6.7 GRAPHICAL COMBINATION OFNORMALIZED POLAR IMPEDANCE VECTORS

The phase angle of the normalized impedance vector (angle between the rotating voltage and current vectors) is also the angle ofthe power factor. This is represented graphically as an overlay for the SMITH CHARTcoordinates by the family of dotted curvesin Fig. 4.2. This overlay also displays loci ofthe normalized voltage and normalized currentvector extremities.

Both the magnitude and the phase angle~

of the normalized impedance and admittancevectors are plotted in Fig. 6.7, which is usefulas an overlay for SMITH CHART A or B inthe cover envelope. As oriented in Fig. 6.7,with relation to the orientation of the SMITHCHART coordinates this overlay permits theconversion of the normalized rectangular components of impedance to equivalent normalized magnitude and phase angle. When rotatedthrough 1800 from the orientation shown inFig. 6.7, with respect to these same SMITHCHART coordinates, it permits the conversionof the normalized rectangular components ofadmittance to equivalent normalized magnitude and phase angle. Thus, Fig. 6.7 providesa useful means for graphically convertingcomponents of any complex impedance oradmittance from the rectangular to the equivalent polar form, and vice versa. With theaddition of the peripheral scales, Fig. 6.7becomes a complete alternate form of transmission line chart commonly known as theCarter chart [9] .

The resultant of two normalized impedancevectors, expressed in polar notation, e.g.,

whose representative circuit elements are connected in series, may readily be obtainedgraphically by the familiar parallelogram construction. In this case, all vector magnitudesare directly proportional to their lengths, andthe plot is most conveniently made on ordinary polar coordinate paper with a linearradial scale. The left half of a complete polarplot is used only for impedances havingnegative real parts (see Chap. 12).

Figure 6.9 shows a polar coordinate systemon which it is possible to plot directly the

resultant of two normalized impedance vectorswhose representative circuit elements are connected in parallel. The resultant is determinedby the same parallelogram construction usedfor series elements. On such a plot all vectormagnitudes are inversely related to theirlengths by a constant. The value of this constant is selected on the plot of Fig. 6.9 toprovide a convenient general-purpose scalerange. This may be varied as required toextend the range of the plot.

The significance of polar impedance vectorswhose angles lie between + 90 and + 1800

, orbetween - 90 and - 1800

, is that the resistive


Fig. 6.7. Carter Chart coordinates displaying polar components of equivalent series-circuit impedance (or of parallel-circuitadmittance) (overlay for Smith Charts A and B in cover envelope).


~g)~ RADIALLY SCALED PARAMETERS

~ 9~ 8 g 2 :;: C! :;l ~ ..~ :!; ..

~Co (; 3 jl; ~ t 8 a 8 II b ~ In :It

.. U.

11:0: ~~ .. Ie 1TOWARD GENERATOR_ ~ TOWARD LOAD 0 2 2 i i Ii • 3 is $ II 8 a 8 II (; l;: r-... ~~1il~ ~ia~9gg 2 !i 2 ~ ~ ~ ::J c~ ~ !:l :- " p ~

..~

.. :; ~ t: ..0;'" 0 '" 0I .0 ci g 2 :;: ~ :;l ~ ~ ~

. ~ So e ~ 0; ;: t: ~ g j5 8 ~ Ii Zg5 8 . 0

j! CENTER.. I

Fig. 6.8. Carter Chart with coordinates of Fig. 6.7 (see Chart C in cover envelope).


INDUCTIVE

CAPACITIVE

__ EXAMPLE_

6.9. Coordinates for graphically combining two normalized polar impedance vectors representing two circuits in parallel.

----~: =1.17/85.5·


component of the impedance represented

thereby is negative. Thus, in the exampleshown in Fig. 6.9, the normalized polarimpedance vector Z/Zo = 1.2 /120° represents an impedance having a negative resistance component -R/Zo = 0.6, and an

inductive reactance component + jX/Zo= 1.04,that is, 1.2/120° = - 0.6 + j 1.04, as obtainedfrom Eqs. (6-11) and (6-12), respectively.

Figure 6.9 is useful in problems involvingpolar impedance vectors, such as those represented in Fig. 6.7.

7.1 COMMONLY EXPANDED REGIONS

The accuracy which can be obtained inplotting and reading out data on the coordi

nates of the SMITH CHART, as with any otherchart, can be improved by simply increasingits size to provide space for a finer and moreexpanded coordinate grid. Where it is impractical to increase the overall chart size to thedesired extent, small regions of special interestmay be enlarged as much as is desired.

In this chapter, enlargements of the morefrequently used regions (see Fig. 7.1) of theSMITH CHART will be presented and discussed, and special applications for these willbe given. The graphical representation of theproperties of stub sections of waveguide whichare operated near their resonant or antiresonant frequency, as may be readily portrayedon two of these expanded charts, will bediscussed in some detail.

The specific region on a SMITH CHARTwhich is most commonly expanded is perhapsa circular region at its center concentric withits perimeter. This region, typically as shownin Fig. 7.2 or 7.3, is of special interest becauseit encompasses all possible input impedances,and equivalent admittances, along a waveguide

CHAPTER 7Expanded

Smith Charts

when the load reflection coefficient or standing wave ratio is less than the value at theperimeter of the expanded central area. Theexpanded central portion of a SMITH CHARTutilizes the same peripheral scales as the complete chart; all radial scales have the samevalue at the center and are linearly expandedto correspond to the linear radial expansionof the coordinates.

Other regions of the SMITH CHART whichare sometimes expanded to provide greaterplotting accuracy are the small (approximatelyrectangular) regions shown at the top of Fig.7.1 which encompass the two diametricallyopposite poles of the chart and which arebisected by the real axis. These regions (Figs.7.4 and 7.5) are particularly useful for therepresentation of the electrical properties ofwaveguide stubs operated near their resonantor antiresonant frequency. These electricalproperties include input impedance (or inputadmittance), frequency, bandwidth, attenuation, and Q, as will be more fully describedherein.

In addition to the above conventionalSMITH CHART expansions, two nonconformal transformations of the SMITH CHARTcoordinates will be described which, in effect,

71


//

/,/'

~--CAPACITIVE REACTANCE OR

INDUCTIVE SUSCEPTANCE REGION

\\

"-"

INVERTED (FIG. 7.8)

/---B(--..."/ EXPANDED (FIG. 7.2) '\

I ~ \I HIGHLY ~~PANDED (FIG.7.3) \

.a. OR 2Zo Yo

CONVENTIONAL SM~'\ Of I.,.,.,

~Qll'O~~ INDUCTIVE REACTANCE OR

CAPACITIVE SUSCEPTANCE REGION

FOR RESONANT STU8S: FOR ANTIRESONANT STUBS:

woZ4oWQ.~

VOLTAGE

1-"£....--

" --; --- --;)C:~--

.............. _-_#" ....... _--- --- ,,- -- ......

~~ "X~ ':........... _---.,," ......_-~/

--..... .".---'«)«

woZ4oWQ.

~

o I 234

QUARTER WAVELENGTHSo I 2 3 4

QUARTER WAVELENGTHS

Fig. 7.1. Commonly expanded regions of the SMITH CHART.

result in expansion of certain regions of thechart coordinates at the expense of a compression of adjoining regions. One of these(Fig. 7.7) results from a transformation ofthe usually linear radial reflection coefficientmagnitude scale into a linear radial standingwave ratio scale. On these coordinates thestanding wave ratio is expressed as a numberranging from a to 1.0 rather than 1.0 to 00 asis more customary in this country,* and thecircular coordinates are distorted to permitthis standing wave ratio scale, rather than thereflection coefficient scale, to vary linearlyfrom unity at the chart center to zero at itsrim. This, in effect, radially expands theregion near the center of the conventionalchart and radially compresses the region nearits rim.

The other transformation of the usualSMITH CHART coordinates (Fig. 7.8) resultsfrom their being radially inverted about thelinear reflection coefficient magnitude circlewhose value is 0.5. This transforms thecircular central region to a band adjacent tothe perimeter, and vice versa; no radialexpansion is provided, however.

7.2 EXPANSION OF CENTRALREGIONS

While any degree of linear expansion of thecentral area of the coordinates of the SMITHCHART is possible, two expansions of thisregion have been selected which satisfy thelarge majority of applications. The first ofthese is shown in Fig. 7.2. This incorporates a4.42/1 radial expansion of the central area ofthe SMITH CHART of Fig. 3.3, and displaysreflection coefficient magnitudes between aand 1/4.42, that is, 0.226. This chart is suitable for displaying waveguide input impedanceor admittance characteristics accompanying

*In the United Kingdom, for example, the standing waveratio is usually expressed as a number less than unity.

EXPANDED SMITH CHARTS 73

mismatches which produce standing waveratios less than 1.59 (4 dB).

The chart of Fig. 7.3 is a more highlyexpanded version of Fig. 7.2. It incorporatesa 17.4/1 radial expansion of the central areaof the SMITH CHART of Fig. 3.3 and displays reflection coefficient magnitudes between a and 1/17.4, that is, 0.0573. It issuitable for displaying waveguide input impedance and admittance characteristics accompanying very small mismatches (standing waveratios of less than 1.12, or I dB).

Like the complete SMITH CHART ofFig. 3.3, both of the above expanded chartsdisplay series components of impedance and/or parallel components of admittance. Theperipheral scales are unchanged from those onthe complete chart, and indicate distances fromvoltage nodal points when used to displayimpedances, and distances from current nodalpoints when used to display admittances. Theradial scales have the same center values asthose for the complete chart but are linearlyexpanded by the radial expansion factorsindicated above.

7.3 EXPANSION OF POLE REGIONS

Figure 7.4 is a linear expansion of theSMITH CHART coordinates in the pole regionwhere the normalized impedance, or normalized admittance, has a real component lessthan 0.0333 and/or a reactive component lessthan ±j 0.02355. Similarly, Fig. 7.5 is alinear expansion in the opposite pole regionwhere the normalized impedance, or normalized admittance, has a real component greaterthan 30 and/or a reactive component greaterthan ±j 42.46.

At the scale size plotted in Figs. 7.4 and 7.5the radius of the coordinates of a completeSMITH CHART would be 84.6 inches. Figures7.4 and 7.5 can, therefore, display as muchdetail per square inch within its area as can



..o

..o

~2 2CENTER

RADIALLY SCALED PARAMETERS

i ~ ~ 2 ~~ TOWARD LOAD

N.,..o

.9

..9

.. 0 ..~ ..,.; ,.; .... . .. N

-' -'

To."'''D GfNERATOR_.. Ow......I ~:1-0

Fig. 7.2. Expanded central region of SMITH CHART.



I

II I

II I

,

I

I 1 I

I I I

I I I

I I I I

II

O.~8 0.37

0.12 O.l!

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I I I I I!il I ~ I ~ I ~ 19 I I~ I~ 18

~o.

'. 0o

RESiSTANCE COMPONENT (toloR CONQUCTANCE CQt.APON NT(fo,.---+-I----III-----r--+_---,---'--+-+-r----+_-+__1Y-I----.-+-+-,------f__t_nI I I I I

o s:

l 1 I I I I H-- I fI\§n~~ I I' I 1\~~oog I I

f-~ I I I ,

I

C[NTtR

z~ ~ " ~ ~0 0

~ :: .. .- '. ::G~" " " "w

~ ~ ~ ~

~ 0 0 0w«


~ 8 § § 08 ~ ~0

~,; ~'g 0

0 e e ~~

" 0 TOWARD LOAD------..0

;~~

II a (; ii o· ~

~ ~ ~0- l~ :: ".. z

Fig. 7.3. Highly expanded central region of SMITH CHART.


be displayed on the corresponding area of acomplete SMITH CHART whose coordinatesare over 150 square feet in area.

The regions of the SMITH CHART shownin Figs. 7.4 and 7.5 are particularly useful fordetermining the input impedance (or admittance) characteristics vs. frequency of waveguide stubs operating near resonance or antiresonance.

In Fig. 7.5 short portions only of a largespiral, which could be drawn in its entiretyonly on a complete SMITH CHART (nearlyvertical dashed curves), have been drawn.These curves trace the input impedance (orinput admittance) locus on the enlarged chartcoordinates as the frequency is varied within± 1.5 percent of the resonant frequency. Dueto the high degree of enlargement of coordinates the dotted spiral curves closely approximate arcs of circles centered on the 84.6-inchradius SMITH CHART. Their departure isso slight that it may, for all practical purposes,be ignored and each curve may thus be used torepresent a particular value of waveguide attenuation (one-way transmission loss) determined by its intersection with this scale atthe bottom, which value holds essentiallyconstant over the length of each arc. OnFig. 7.4 the curves of constant series resistance approximate very closely the inputimpedance (or input admittance) locus so thatthe dotted portions of the large spirals representing the locus of constant attenuation arenot required to be drawn as they are on Fig. 7.5.

The use of Figs. 7.4 and 7.5 is described inmore detail following a brief discussion ofthe relationship of waveguide attenuation tothe series and parallel resonant impedanceand/or admittance of short- and open-circuitedwaveguides.

7.4 SERIES-RESONANT ANDPARALLEL-RESONANT STUBS

Resonance (specifically, electrical amplituderesonance) may be defined as a condition that

exists in any passive electrical circuit containing inductance and capacitance when its combined reactance is zero. This reactance maybe lumped as in conventional circuit elements or it may be distributed as along awaveguide.

Any uniform stub section of waveguidewhose characteristic impedance (or characteristic admittance) is (1) essentially real, (2)short- or open-circuited at its far end, and (3)an integer number of quarter wavelengthslong electrically has electrical properties whichclosely parallel those of simple series or parallel resonant circuits. At the resonant (midband) frequency the input impedance, oradmittance, to such a stub is purely resistive,or purely conductive, respectively. If thisinput resistance is appreciably lower than thecharacteristic impedance of the waveguide itis called a resonant stub; if it is appreciablyhigher than the characteristic impedance ofthe waveguide it is called an antiresonantstub. Similarly, if the input conductance isappreciably higher than the characteristicadmittance of the waveguide the stub iscalled a resonant stub, or if it is appreciablylower the stub is called an antiresonant stub.

The combination of the electrical length ofa waveguide stub and its termination (open- orshort-circuited) determines whether it willbe resonant or antiresonant. In either case,the electrical length of the stub must be aninteger number of quarter wavelengths. (SeeFig. 7.1.)

Near its resonant frequency the equivalentcircuit for a uniform waveguide stub is a seriesresonant circuit; near its antiresonant frequency the equivalent circuit is a parallelresonant circuit.

The attenuation (one-way transmission loss)of any uniform waveguide uniquely determinesthe normalized resonant impedance or admittance (Zmin/ZO or Ymax/YO) and/or the normalized antiresonant impedance or admittancetZmax/ZO or Ymi/YO) of stubs constructedthereof. Thus


2 LOSS), dBATTE~UATION (ONE-WAY TRANSMISSION

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~

.I.I v 0

~r,

I , , I , If') v

I I II I I

, ,I I N

~ I I I , i I I II ,

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I£J I'- ~~.£t -

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~ ............... I£JUZ

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9 I£J

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0 q q q qRESISTANCE COMPONENT (fo) OR CONDUCTANCE COMPONENT (~)

f---- -71>-

f---- -~Z

f-- I£JZ

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f---- Z

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.,N

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~ZLtJ

- Z00-2

~ 0(,)

~ LtJUZ

...; ~<l....

$u<lLtJ0:

I--- LtJ>

~~

I--- (3 I<l I I I I I I I I I I I I~

, I , , I i ..,....- 0-

i i i i i I<l I i i(,) i i I I i It), i 0 ~i , I i i i It)

~- i i

N

ro

o

Q

~ ~ 8 g g 8 0 0 n (n. NO. OF QUARTER WAVE LENGTHS IN STUB)

~ 0

CID

+~

t>uZ

~NaI£J0:I&.

oZ~II)

6 0

iL&L&o~ztj~0:I£J

1

Fig. 7.4. Expanded zero-pole region of SMITH CHART.


+

-~ I

t

g

~ 0

_ 0

ATTENUATION (ONE-W~Y TRANSMISSION LOS~. dB

Q

n

>oZL&J

N=>·0

L&J0:u..oz«III

1--_......... ----4--.....L..-...L.........L..-l....-...L...-...J.......L-~:---I-----L.-.1....-......L-----L.----L.......:...--=--~-:----...--+--+-+--+--+--+--++--+--+I t- 0 6i

Fig. 7.5. Expanded infinity-pole region of SMITH CHART.


Z. Ymin log-l (dB/10> - 1mIn

Zo YO log-l (dB/10> t 1

and

Zmax Ymax log-l (dB/IO> + 1

Zo YO log-l (dB/I0) - 1

(7-1)

(7-2)

its far end). For the same size and type ofwaveguide the half-wavelength stub will therefore have a resonant impedance which is twiceas high as that of the quarter-wavelengthstub.

7.5 USES OF POLE REGION CHARTS

Similarly, if Zmax » Zo and Ymax » Yo, agood approximation for Eq. (7-2) is

If Zmin « Zo and Ymin « YO' a good approximation for Eq. (7-1) is

Equation (7-3) is accurate to within 0.3percent if the normalized resonant impedance (Zmin/ZO) or antiresonant admittance(Ymin/YO) is less than 1/30, and Eq. (7-4) isaccurate to within 0.3 percent if the normalized antiresonant impedance Z /ZO or reso-maxnant admittance YmaJYOis greater than 30.

The shortest possible resonant or antiresonant waveguide stub is one-quarter wavelength long. For a given size and type ofwaveguide this length stub has the lowestattenuation. As seen from Eq. (7-3) it willtherefore have the lowest resonant impedance or admittance; or, as seen from Eq.(7-4), it will have the highest antiresonantimpedance or admittance.

For example, a resonant waveguide stubwhich is one-half wavelength long (shortcircuited at its far end) has twice the attenuation of a resonant waveguide stub which isa quarter wavelength long (open-circuited at

As previously stated, the charts of Figs.7.4 and 7.5 are enlargements of the poleregions of the SMITH CHART.

The chart of Fig. 7.4 is of particular usefor tracing the normalized input impedancelocus of waveguide stubs as a function offrequency deviations from the resonant (midband) frequency. Alternatively, it is usefulfor tracing the normalized input admittancelocus of stubs as a function of frequencydeviations from the antiresonant (midband)frequency.

Similarly, the chart of Fig. 7.5 is of particular use for tracing the normalized inputimpedance locus of waveguide stubs as afunction of frequency deviations from theantiresonant (midband) frequency. His alsouseful for tracing the normalized input admittance locus of stubs as a function of frequencydeviations from the resonant (midband) frequency.

The nearly horizontal dashed lines of Fig.7.5 (omitted in Fig. 7.4 because they parallelthe lines of constant resistance and are therefore not necessary) are drawn as a conveniencein translating chart values of normalized impedance or admittance to the nearly verticalscale on the chart coordinates which indicatesdeviation from the midband frequency.

On both Figs. 7.4 and 7.5 the stub attenuation is indicated directly on scales across thebottom. The scale values refer to the totalattenuation of a stub regardless of the numberof quarter wavelengths therein. To obtain avalue of Q from the Q/n scales across the topof either of these charts it is only necessary

(7-4)

(7-3)dB

8.686

Ymax 8.686'V

Yo dB

Y .mIn__ 'V

Z.mIn


to multiply the Q/n scale values by thenumber n of quarter wavelengths in the stub.Similarly, to obtain a specific value for M fromthe "percent off midband frequency" scale(M x n) at the side of these charts it is onlynecessary to divide the scale values by thenumber n of quarter wavelengths in the stub.

The use of these charts is further illustratedby an example: Assume that we have a waveguide stub which is one-quarter wavelengthlong and short-circuited at its far end. Assumealso that the antiresonant input impedance ofthe stub is known to be 200 times Zo' thecharacteristic impedance of the waveguide, andwe wish to know its impedance at a frequency0.5 percent off resonance. Since the stub isonly one-quarter wavelength long (n = I),enter the chart of Fig. 7.5 directly at 0.5percent on the nM scale. (Had the stubbeen three-quarter wavelengths long (n = 3),for example, the chart would be entered at

3 x 0.5 percent or 1.5 percent on this scale.)Erect a perpendicular to the chart perimeterthrough the 0.5 percent point on the nl1f

scale and extend it to intersect the nearlyvertical dashed curve which passes throughR/Zo= 200. At this latter point on the chartcoordinates the impedance in complex notation is found to be (58 - j 90) Zo' and theabsolute magnitude is the square root of thesum of the squares or 107 Zo. This procedureis repeated to obtain as many data points onthe impedance-frequency relationship as desired. The result, in the example given, is aplot as shown on Fig. 7.6.

If it is found that the range of the chartis too small it may be extended, within limits,by multiplying the scales properly. Forinstance, suppose that the antiresonant impedance of the waveguide in question is2,000 ZOo The "200" curve in Fig. 7.5 may beused if the resistance and reactance circle

Fig. 7.6. Normalized input impedance vs. frequency of quarter-wave antiresonant stub as obtained from Fig. 7.5.

I I I I I_ I

-1->-/4

R/Z o ± J X/Z o-: 1Zo

I---ATTEN = .04 db

ZAR = 200 ZoQ = 157. I---

/ \ B.W. = .636%

I I IICAPACITIVE REACTANCE

l lfT-.636%-f\I rINDUCTIVE REACTANCECOMPONENT (-JX/Z o ) COMPONENT (+JX/Z o )

1...... °1 ""' 1/ \" r-...r,/ II 1\ 1\ ........

./ \ I ""-./ / \ I \ ...........

./ / I \ ...........V V 1\ ............

/ \j

"'"V \RESISTANCE

""COMPONENT

V IR 1 i0) I

""20

30

40

100908070

60

50

300

200

l.LI(,)

Z<l:ol.LI[l.

~

en~

Zl.LIZo[l.

~o(,)

oN...... 400x.,+1oZ<l:o

N......0:

ol.LIN

..J<l:~

gs 10z -1.6 -1.4 -1.2 -1.0 -.8 -.6 -.4 -.2 0 +.2 +.4 +.6 +.8 +1.0 +1.2 +1.4 +1.6

t::. f IN PERCENT

500


7.5.1 Q of a Uniform Waveguide Stub

(7-7)

7.5.2 Percent Off Midband FrequencyScales on Pole Region Charts

Peripheral scales in Figs. 7.4 and 7.5 indicating "percent off midband frequency"show specific values of the function100 nM/fo' where M is the frequency deviation from the resonant or antiresonant (midband) frequency.

It is possible to plot such scales on thesecharts since this function is linearly related tothe electrical length of the waveguide stubwhich is a linear radial parameter on theSMITH CHART. If, for example, the frequency of operation of a stub one-quarterwavelength long is changed I percent, fromfo to fo ± M, the electrical length of the

Along any lossless uniform waveguideRmin/ZO or Gmi/YO are the reciprocal valuesfor Rmax/ZO or Gmax/YO' respectively. Furthermore, all pairs of reciprocal relationships whenplotted on a SMITH CHART are at equalradius along the real axis. Thus a single scalefor Q/n is applicable to both Figs. 7.4 and7.5.

In the example given, the antiresonantimpedance of a stub one-quarter wavelengthlong (n = 1) was assumed to be 200 ZOo ItsQ is determined by projecting this antiresonant impedance value vertically along thedashed line to the Q/n scale where it is foundto be 157.

Similarly, if the real part of the normalizedantiresonant input impedance or normalizedresonant admittance (Rmax/ZO or Gmax/YO)of any uniform waveguide stub is more thanapproximately 30.0, for which condition Fig.7.5 is applicable,

(7-5)

(7-6)nTT

4Q

fobandwidth

Rmin

values (and the Q/n scale values) are multipliedby 10 and the "percent off midband frequency" (n().f) scale is divided, i.e., expanded,by 10. This can be checked by noting that theimpedance point (50 - j 50) Zo occurs at 0.64percent off resonance while the point (500 j 500) Zo occurs at 0.064 percent off resonance. The range of the "percent off midbandfrequency" scale may be increased a limitedamount by division of the resistance andreactance circle values. Caution must beused, however, as the errors may be large ifthe extension is carried too far. Figure 7.5is similarly used for obtaining the inputadmittance characteristics of stub waveguidesat frequencies near resonance.

If the real part of the normalized resonantinput impedance or normalized antiresonantinput admittance (Rmin/ZO or Gmin/YO) ofany uniform waveguide stub an integer numbern of quarter wavelengths long (including anylumped resistive loading at either end) is lessthan approximately 0.03, for which conditionFig. 7.4 is applicable,

The Q of a uniform waveguide stub may bedefined as the ratio of its resonant or antiresonant (midband) frequency fo to its bandwidth.The bandwidth is defined as the total widthof the frequency band within which the realpart of the input impedance equals or exceedsthe imaginary part. This corresponds to thehalf-power (3 dB) criterion for bandwidth ofconventional tuned circuits; thus

Q


stub is changed 1 percent of one-.:}uarterwavelength, or ± 0.0025 wavelength. If thestub were three-quarter wavelengths long,this 1 percent frequency change would resultin a stub length change of 3 x ± 0.0025 wavelength, and so on. The percent off midbandfrequency scales in Figs. 7.4 and 7.5 readdirectly for one-quarter wavelength stubs. Foruse at other stub lengths the scale values mustfirst be divided by n.

7.5.3 Bandwidth of a UniformWaveguide Stub

From Eq. (7-5) it will be seen that the Q ofa uniform waveguide stub is simply its midband frequency divided by its bandwidth.Thus its bandwidth is its midband frequencydivided by Q. In the foregoing example, theQ of the one-quarter wavelength stub was157; its bandwidth is, therefore, 1/157 =0.636 percent.

The band edges are equally- removed on apercentage basis from the midband frequency;thus they occur in the above example at± 0.318 percent. At the band edge frequencies, the stub input resistance and reactance is equal by definition. This may bechecked by projecting these peripheral scalevalues (± 0.318 percent) nearly horizontal(parallel to the dashed lines) to the pointswhere they intersect the nearly vertical dashedline passing through the antiresonant impedance value of 200 20 , at which points theimpedance values (100 ± j 100) 2 0 will befound.

7.6 MODIFIED SMITH CHART FORLINEAR SWR RADIAL SCALE

The chart shown in Fig. 7.7 is one of themany possible nonconformal transformationsof SMITH CHART coordinates. Historically

this chart, developed in 1936, was the firstSMITH CHART. Although difficult to draw,the chart offers at least one significant advantage, in plotting data involving small standingwave ratios, over the conventional chart,viz.,the region near its center is expanded in relation to the overall size of the chart. Forthe same overall chart diameter standingwave ratios up to 1.5 occupy an area 2.86times as large as that occupied for this samestanding wave ratio on the more easily drawnconventional SMITH CHART coordinates.

7.7 INVERTED COORDINATES

Another unorthodox method for achievingan expansion of the central area of the SMITHCHART is to invert the coordinates about thedotted line in Fig. 7.1, where the reflectioncoefficient magnitude equals 0.5. An invertedcoordinate SMITH CHART is shown in skeleton form in Fig. 7.8.

All radial scales of the original chart (suchas Fig. 3.3) remain unchanged but read in theopposite direction. The peripheral scales,their designations, and their sense of ~irection

also remain unchanged. The large outer areaof the inverted SMITH CHART coordinatesrepresents an area of small reflection coefficients (standing wave ratios near unity). Pointson the perimeter of the inverted coordinatesrepresent a matched impedance termination.

Figure 7.9 shows a complete invertedSMITH CHART [14] with a finer coordinategrid structure than that shown in Fig. 7.8.The radial scales at the top of this figure areidentical to those shown in Figs. 3.4 and 4.1except that they are inverted and their lengthsare reduced to correspond to the invertedchart radius.

Although inverted SMITH CHART coordinates may have advantages in special applications, they have several disadvantages. Oneof the disadvantages is that the families of


SMt TH CHART (19361 with linear standing wave ratio scale.

o'i ',' I

1.0~

CENTER

.9


o .1 .2 .3 .4 .5 .6 .7 .8VOLTAGE STANDING WAVE RATIO

VOLTAGE REFLECTION COEFFICIENT1.0.9 .8 .7 .6 .5 .4 .3 .2 .1I!.' I,' ',! I' I 'I II' 'I' Itt \' II' 't' 'I' 'i 1 t' 'i 1

Fig. 7.7.


Fig. 7.8. Series impedance and parallel admittance components on inverted SMITH CHART (coordinates inverted about 0.5reflection coefficient).

,o

"


:1 ~ 0 ..... - 0 0

1

Fig. 7.9. Inverted SMITH CHART with peripheral and radial scales. (Reproduced by permission of W C. Blanchard of the Bendix

Corporation and the Editors of Microwaves.)


impedance or admittance component curvesof the inverted SMITH CHART are not circlesas on the conventional chart of Fig. 3.3, andthey are not orthogonal families of curves;consequently, like the chart of Fig. 7.7, it isnot easy to draw. Also, the magnitude of thereflection coefficient (or of the standing waveratio), which is measured from the perimeterof the chart toward the center, is not expanded,whereas the phase angle of the reflectioncoefficient is highly expanded. For these

reasons inverted coordinates for the SMITHCHART are not generally useful.

The later coordinate transformations illustrate only two of many possible nonconformaltransformations which may have special applications. In each of these, the circular orthogonal SMITH CHART coordinates are transformed into noncircular nonorthogonalcoordinates which are difficult to constructand therefore not likely to achieve widespreaduse.

8.1 GRAPHICAL REPRESENTATION OFREFLECTION AND TRANSMISSIONCOEFFICIENTS

Numerous attempts have been made to finda simpler grid [125] than that of the

SMITH CHART on which waveguide transmission and reflection functions could bedisplayed. This chapter presents one solutionto the problem which does not alter the basicSMITH CHART coordinates and which because of its inherent simplicity has certainpractical advantages.

8.1.1 Polar ys. Rectangular CoordinateRepresentation

Polar coordinates are generally chosen tographically portray the voltage and currentreflection and transmission coefficients alonga waveguide as shown in Figs. 8.1 and 8.2, andas described more fully in Chaps. 3 and 5,respectively. Their component values (magnitude and phase) are directly related to thevoltage and current at any position along a

CHAPTER 8Waveguide

TransmissionCoefficient T

standing wave which can readily be observedand measured. Accordingly, they have awell-defined physical significance. It is notthe purpose of this chapter to depreciate thepolar coordinate representation of these coefficients since this is extremely useful, butrather to show how an alternative rectangularcoordinate representation can offer simplifications and advantages in some applications.This chapter will also show how the transmission coefficient components (both polarand rectangular) can be represented on theSMITH CHART in a manner which will notobscure the basic impedance or admittancecoordinates with additional superimposedcurves.

Any vector can, of course, be graphicallyrepresented on either polar or rectangular coordinates. For the specific representation ofthe reflection and transmission coefficientsthe use of rectangular coordinates makes itpossible to employ a single rather than a dualset of coordinates. This single set serves,alternatively, to represent either of these coefficients as they coexist at all positions along awaveguide. Furthermore, the rectangular

87


Fig. 8.1. Vol. refl. coeff.-polar coord.

component values of each of these coefficients are mathematically more simply related to each other, and to the impedanceor admittance coordinate components of the

SMITH CHART, than are the polar componentvalues; also, as with polar coordinates, theyare fully compatible with the SMITH CHARTcoordinates and therefore can be used as anoverlay thereon. Evaluation of these coefficients in terms of their rectangular components provides the data necessary to generatea spot or a trace on conventional cathode raytubes, and on mechanical (x, y) curve plotters.Finally, such a waveguide chart can be constructed with ordinary cross-section paperavailable to any engineer.

8.1.2 Rectangular Coord inate Representation of Reflection Coefficient p

If the voltage reflection coefficient whosepolar magnitude ranges between 0 and 1.0,and whose phase angle ranges between 0 and± 1800 (see Fig. 8.1), is expressed in equivalent complex notation, it can then be plottedon rectangular coordinates as shown in Fig.8.3. In the latter plot the X componentranges uniformly from -1.0 to + 1.0, and the Y

Fig. 8.2. Vol. transm. coeff.-polar coord.

component ranges uniformly from + J 1.0 to- j 1.0. As in the polar representation, thezero value for this coefficient (no-reflectionpoint) lies at the center of the plot.

The same radial scale as used on polarcoordinates, representing the magnitude ofthe voltage reflection coefficient, and rangingfrom 0 at the center of the chart to 1.0 at itsperiphery, is applicable to the equivalentrectangular coordinate representation. Also,the same peripheral scales, representing thephase angle of the voltage reflection coefficient and the relative positions along the waveguide, are applicable thereto. Thus, the rectangular coordinate representation of Fig. 8.3is directly applicable as an overlay for themore usual polar coordinate representation ofFig. 8.1.

8.1.3 Rectangular Coordinate Representation of Transmission Coefficient 7

In a manner somewhat analogous to thatdescribed above for the voltage reflectioncoefficient the voltage transmission coefficient, whose polar magnitude ranges betweeno and 2.0 and whose phase angle ranges from+ 90 to - 900 (see Fig. 8.2), may be expressed

WAVEGUIDE TRANSMISSION COEFFICIENT T 89

j 0I.

-j I.

-j .5

.5

Fig. 8.4. Vol. transm. coeff.-rect. coord.

+j I.

+' .5

8.1.4 Composite Rectangular CoordinateRepresentation of p and T

It will be noted from Figs. 8.1 and 8.2that, in polar form, the coordinate grids forthe voltage reflection and transmission coefficients are not coincident with each other(one being displaced with respect to the otherby the chart radius), while in the respectiverectangular forms (Figs. 8.3 and 8,4) thecoordinates, per se, are one and the samethe difference in this latter case being thatthe labeling thereon is different for thereflection and for the transmission coefficient real parts. Thus, a composite representation of those two coefficients on a commonrectangular grid is possible in this case. Figure8.5 shows a mutually compatible coordinategrid for both reflection and transmissioncoefficients p and r, respectively. Radial andperipheral scales of the SMITH CHARTrelate points on this common grid to thestanding wave ratio and wave position, andto other uniquely related guided-wave parameters, each of which has been discussedindividually in previous chapters.

Through a comparison of the geometry ofany common point on the polar coordinates

oj 0

-j I.

-j .5

+' .5

-.5

Fig. 8.3. Vol. refl. coeff.-rect. coord.

+j I.

in equivalent complex notation and thenplotted on rectangular coordinates, as shownin Fig. 8,4. In this latter plot, however, theX component ranges uniformly from 0 to + 2.0,and the Y component ranges uniformly from+ il.O to - i 1.0. As in the polar representationof the transmission coefficient, and in bothpolar and rectangular replesentations of thereflection coefficient, the no reflection pointlies at the center of the respective circularplots.

The magnitude scale for the voltage transmission coefficient, whose zero value lies onthe rim of the chart at the origin of the polarcoordinates (see Fig. 8.2), is similarly applicable to the equivalent rectangular representation, and its zero value also lies at the originof these latter coordinates. As in the case forthe reflection coefficient, the peripheral anglescale representing the phase angle of the voltage transmission coefficient (referenced to theorigin of the polar coordinates at the peripheryof the chart) is applicable to this rectangularrepresentation. Thus, the rectangular coordinate representation for the transmission coefficient in Fig. 8,4 is directly applicable as anoverlay for the polar coordinate representationof Fig. 8.2.


rl'r:J

\0

TRANS~ISSION COEFF CIENT.2 .1+

1 .6.6

:".4

rl0'0

I

1 -2 1.4..2

EFLECTION COEFFICIENT

1.6

1.8.8

2.1.

Fig. 8.5. Rectangular coordinate representation of transmission and reflection coefficients (overlay for Charts A. B. or C in coverenvelope).


well-known vector relationship which plotsthe SMITH CHART

of Figs. 8.1 and 8.2 it will be seen that inpolar form the magnitudes and phase anglesof p relative to T are given by the followingquadratic trigonometric equations, as wasfurther described in Chap. 5, viz.,

z 1 + P

1 - p(8-7)

where T and p are the magnitudes, and f3 anda are the respective phase angles, of the voltage transmission and voltage reflection coefficients, respectively, and in which 0 'S. T ;:> 2,and 0 'S. p ;:> l.

On the rectangular coordinates of Figs.8.3 and 8.4, and on the composite rectangularcoordinates of Fig. 8.5, the same relationshipsare given by two simple linear equations, viz.,

where the subscripts Xand Y indicate real andimaginary components, respectively, of therespective vector coefficients T and p, and inwhich -1 'S (Tyor py) ? + 1.

From Eqs. (8-5) and (8-6) one observes theinteresting fact that the real componentsof T and p invariably differ by unity, and theimaginary components are invariably equalto each other at any given position along awaveguide.

The conversion of the component values ofp to equivalent normalized waveguide complexinput impedances is possible by virtue of the

Ty = Py

where Z is the normalized complex inputimpedance and p is the complex voltagereflection coefficient.

The relationships expressed by Eqs. (8-1)through (8-6) are consistent with accepteddefinitions [I 1] for the voltage transmissioncoefficient and the voltage reflection coefficient (in a transmission medium), as appliedto a transmission line or waveguide.

8.2 RELATION OF p AND TTO SMITHCHART COORDINATES

The SMITH CHART of Fig. 8.6 includesthe basic impedance or admittance coordinates, shown in Fig. 2.3, with the addition offour peripheral scales. The two outermostof these scales indicate the relative positionalong the waveguide in either direction froma voltage null position, and the two innermostscales indicate the angle of the reflection coefficient and the angle of the transmissioncoefficient, as plotted in polar form. Asdiscussed in Chap. 5, when the SMITHCHART is used to represent impedances theperipheral reflection coefficient angle scalethereon refers to the voltage reflection coefficient and when this chart is used to representadmittances this scale refers to the currentreflection coefficient. The same rule appliesto the peripheral transmission coefficientangle scale in Fig. 8.6. A rotation of eitherof these scales about the center of the chartreverses its application, Le., the scale will thenapply to current as related to the impedancecoordinates or voltage as related to admittancecoordinates.

(8-1)

(8-4)

(8-3)

(8-6)

(8-2)

(8-5)

t -1 P sinaanp COSa + 1

VT 2 - 2T cosf3 t- 1

tan -1 T sin f3

T cosf3 - 1

p

T = Vp 2 + 2p COSa t- 1

TX = Px + 1

and

a




1.8 I.' TOWARD LOAD ----

.. 4 2. I.'12. 14. 20.

.0' .01 1.7.2 .1 ••

CENTER.2 .3 .4 .. .. .7 .8 I. 12 1.3 1.4

ORIGIN

Fig. 8.6. SMITH CHART (1966) with transmission coefficient scales (see Chart A in cover envelope).



E-< .9

'"~'"0 .8P;:s0u

.7:>-<

.6

1..6 .7 .8 .9X - COMPONENT

~-j1.

.5

X - COMPONENT

1.5 1.6 I.} 1.8 1.9 2 •

1.4I.'1.2

0.38 0.37

.0

0.12 0-13


..

.5

.4

. 4

-,

.5 .3.7 .6

SMI TH CHART (1966) with polar and rectangular coordinate transmission and reflection coefficient scales.

TRANSMISSION

o .1 .2- _.3

-1 -.9 -.8REFLECTION

ORIGIN

j1.

.9E-<

'".8 ~

'"0P;:s

.7 0u

.6 :>-<

Fig. 8.7.


and the polar angle from

.5

fZwzoo a.~ou>-

w(/)

o <IJ:a.

.4

30·

-.4

-30·

(8-8)

(8-9)

.2 .3WAVELENGTHS

.1

~V UOll ) V .......

\II ~v \ \"-~

0--- -;;0~'i>\ II <?~ 1\

\ V / \./

./ "'-V '" /'

~~

/ "b ll P\ ""-I J

W l& 1\ \0""

~q Jj<S \~ ~

~vO

/v...: '"./ "- --....... ,/ "-

o

.6

1.4

.5

1.4

wo::::lf-1.0...Ja.~<I

fZwzoa. 1.0~

oux

Across the bottom of Fig. 8.6 all of theradial scales described in previous chapters aredisplayed. Also displayed is a polar coordinatetransmission coefficient magnitude scale measuring in all cases from the origin of thecoordinates.

The rectangular coordinates for p and T

described in this chapter can be added toFig. 8.6 to produce the composite chart ofFig. 8.7. This addition involves only rectangular coordinate component scales for p andT, to which points on the chart coordinatescan readily be projected. The superpositionof Figs. 8.5 and 8.6 to produce Fig. 8.7 isthus in effect accomplished without confusingthe basic impedance or admittance coordinates.

As was the case with the polar coordinatescales, the rectangular component scales forp and T apply to voltage reflection and transmission coefficients in relation to impedancecoordinates and to current reflection aridtransmission coefficients in relation to admittance coordinates; and a rotation of either ofthese latter scales through 1800 about thecenter of the chart reverses its application.

8.3 APPLICATION OF TRANSMISSIONCOEFFICIENT SCALES ON SMITHCHART IN FIG. 8.6

Fig. 8.8. Polar and equivalent rectangular components ofwaveguide transmission coefficient for SWR =3.0.

An example of the application of the polarcoordinate transmission coefficient scales inFig. 8.6 is shown in Fig. 8.8(a), which is a plotof the amplitude and phase of the voltage orcurrent transmission coefficient attending astanding wave whose ratio is 3. The plot ofthis coefficient is identical to that of thestanding wave. Figure 8.8(b) shows anequivalent plot for the rectangular components X and Y. If the rectangular component values X and Y of the voltage orcurrent transmission coefficient T are known,the polar magnitude is readily obtained from

The X- and Y-component curves (Fig.8.8(b)) are seen to have identical shapes whichare sinusoidal regardless of their maximum andminimum amplitude value, whereas the shapesof the amplitude and phase component curves(Fig. 8.8(a)) are a function of the magnitudeof the SWR to which they apply, and approachsinusoidal shapes only when the SWR approaches infinity. Furthermore, X- and Ycomponent curves are always displaced, onefrom the other, by 45 0 and consequently maybe drawn by inspection, whereas the displacement of the amplitude and phase curves ofFig. 8.8(a) varies with the SWR between the

TRANSMISSION - REFLECTION FORMULAS

Table 8.1. Waveguide Transmission-Reflection Formulas

SCALE FUNCTION TRAVELING WAVES REFLECTIONSTANDING WAVESCOEFFICIENT

(1) VOLTAGE REFL.COEF. r P S -IT S+T

(2 ) POWER REFL. COEF. (+)2 p 2 ( ..!=..!..)21 S+ I

(3) RETURN LOSS, dB 10, 10910 (+)2 -1010910p2S-I 2

- 1010910 (S+T)

'2-10 10910( l_p2) -10109 10[1-( ~~: )2]( 4) REFLECTION LOSS, dB 1010910-1-

12_ r 2

(5 ) STJ>G. WAVE LOSS COEF.1+ [( 1+r)/(i-r)] 2 l_p+p2_p3 I+S2

2[(iH)/(I-r)]l_p_p2+p3 2"S

(6 ) STDG. WAVE RATlO,dB 20 i+r i+P10910 "'"'i=r 20 10910-- 20 10910 SI-P

(7) MAX. OF STDG. WAVE ( i+r ,'12 (,+p )1/2 .ISI-r I-p

(J..=f...)1/2 I(8 ) MIN. OF STDG WAVE ( I-r )1/2 .fll+r I+P

(9) STANDING WAVE RATIO I+r b.f... ST:r I-Ii

-10 10910+S-I

(10) ATTENUATION, dB -10 10910 P -10 10910-S+I

VOLTAGEOR

CURRENT

which stems from the origin of the coordinatesof the SMITH CHART, was seen to be relatedto both the magnitude and the angle of thevoltage reflection coefficient by Eq. (8-1).

All of the remaining radially scaled parameters are related by simple algebraic formulasto the magnitude only of the voltage or current (l) reflection coefficient, (2) standingwave, and (3) traveling waves. While theserelationships have been discussed individuallyin previous chapters for convenience, theformulas are grouped in Table 8.1.


I = INCIDENT WAVE AMPLITUDE Jr= REFLECTED WAVE AMPLITUDEP= REFLECTION COEFFICIENTS= STANDING WAVE RATIO

8.4 SCALES AT BOTTOM OF SMITHCHARTS IN FIGS. 8.6 AND 8.7

The linear scale representing the voltage orcurrent transmission coefficient magnitude,

limits of 0 and 90°. Thus when the transmission coefficient is expressed in rectangularcomponents a simple sine function plot ofthe standing wave ratio serves to display theshape of the X-and Y-component curves fora current or voltage standing wave of anyratio of maximum to minimum (SWR).

9.1 STUB AI\ID SLUG TRANSFORMERS

As discussed in previous chapters, a mis-match between the load impedance and the

characteristic impedance (or between the loadadmittance and the characteristic admittance)of a waveguide or transmission line causes areflection loss at the load and increased dissipative lo~ses along the entire length of theline. Also, a mismatch termination causes thetransmission phase to vary nonlinearly withchanges in frequency or line length, and increases the tendency of the waveguide or lineto overheat and arc over at the current andvoltage maximum points, respectively, whenoperating under high power.

Several commonly used devices for obtaining a match at the load end of a transmissionline will be described in this chapter. Theseinclude the single and the dual matching stub[65], and the single and the dual matchingslug transformers. Matching stubs or buildingout sections, as they are sometimes called, ares~ctions of transmission line, frequently of thesame characteristic impedance (or characteristic admittance) as that of the main line,

CHAPTER 9Waveguide

Impedance andAdmittance Matching

and either open- or short-circuited at theirfar end, connected in shunt with the mainline at anyone of several permissible positions in the general location where it isdesired to provide the match. Slug transformers, on the other hand, are sections ofline of appropriate characteristic impedance(or characteristic admittance) and length connected in series with, and forming a continuation of, the main line.

These devices are described in some detailherein since it is through its terminationsalone that the transmission characteristics ofwaves along any given waveguide can be controlled. Furthermore, the SMITH CHARTprovides the ideal medium for visualization ofthe principle of operation of such transformersand for quantitatively determining their specific design constants.

9.2 ADMITTANCE MATCHING WITH ASINGLE SHUNT STUB

The single open- or short-circuited shuntmatching stub whose length is continuously

97


in which A is the wavelength.Similarly, for an open-circuited stub line,

the relationship is

main line of characteristic impedance ZO' therelationship between its required length L', tlJedistance D toward the generator from a voltagemaximum position along the main line (atwhich position the stub must be attached),·and the standing wave ratio 5 are given by [8]

The two possible positions within each halfwavelength of main line for matching stubs,which are always located at equal distances Dbut in opposite directions from a voltagemaximum (and therefore minimum) position,are a function only of the standing wave ratio5, as can be seen from Fig. 9.1. The "forbidden" stub locations in Fig. 9.1 withinone-eighth wavelength of either side of avoltage maximum point correspond to thepositions along the line where no shunt susceptance can provide a matched condition.

The mathematical relationship between Dand S is simply

adjustable over a range of one-quarter wavelength, and whose position along the mainline is adjustable over a range of one-halfwavelength, is capable of correcting any mismatched condition whatsoever along the mainline [8].

For its principle of operation refer to theSMITH CHART of Fig. 9.1. Therein it willbe observed that the circle of unit conductance (C, A, P, B) passes uninterruptedlythrough both the center and the periphery ofthe admittance coordinates, at points C andP, respectively, and is centered on the zeroconductance axis. Thus two admittances,such as at points A and B, of equal magnitudes and of opposite sign, whose normalizedconductance component is unity, always coexist within a quarter wavelength of eachother along any transmission line regardlessof the extent of the mismatch. At either ofthose positions a susceptance, whose magnitude is equal to that of the input susceptancebut of opposite sign, can be shunted acrossthe main line to cancel the main line susceptance, and thereby provide a match (unitconductance) for that section of the mainline between the point of attachment of thestub and the generator end of the line.

Although the susceptance provided by areactive circuit element will of course servethe purpose, a short-circuited or an opencircuited stub transmission line of suitablelength in relation to its chosen characteristicadmittance provides a practical, convenient,and controllable shunt susceptance. Thismatching principle is made use of in the socalled slide-screw tuner often used in microwave plumbing.

j (Zos /ZO) tan (21TL '/A)

(Zo/Zo) + j tan (21TL'/A)

,.. j(Zos/Zo) COt(21TL'/A)

(Zo/Zo) - j COt(21TL'/A)

tan 21TD = ±< S)1/2

A

5 + j tan (21TD /A)

1 +- jS tan (21TD /A)

(9-1)

5 + j tan (21TD /A)

1 + jS tan (21TD /A)

(9-2)

(9-3)

9.2.1 Relationships between ImpedanceMismatch, Matching Stub Length,and Location

For a short-circuited matching stub ofcharacteristic impedance ZOs in shunt with a

The required length for a matching stub fora given value of S will depend upon whether ashort-circuited or an open-circuited stub is tobe used. The lengths of these two possiblestub types will also depend upon the selectedratio of ZOs to ZOo Stub lengths less than

Fig. 9.1.

WAVEGUIDE IMPEDANCE AND ADMITTANCE MATCHING 99

ADMITTANCE COORDINATES

"FOR SHORTED STUBS AT SAMEPOSITIONS ADD OR SUBTRACT0.2S WAVELENGTHS

Determination of required length and position of a single shunt matching stub.


in which L is the required length of stub ofcharacteristic impedance Zo' Similarly, Eq.(9-2), for the open stub, reduces to

one-quarter wavelength are generally preferredto equivalent stub lengths an integral numberof half wavelengths longer since the shorterstub provides the wider operating bandwidth.Also the characteristic impedance of the stub isfrequently made the same as that of the mainline, so that the relationship between L'and Sas derived from Eqs. (9-1) and (9-2) is independent of the characteristic impedances ofeither the main line or the stub, in whichcase Eq. (9-1), for the shorted stub, reduced to

Because of frequency dispersive effects asingle stub is capable of providing a perfectimpedance match only at a single frequency.However, for the applications in which radiofrequency lines are generally employed, theoperable bandwidth of a single stub matchingtransformer is generally adequate. Broaderoperating bands can be obtained through theuse of two or more stages of impedance transformation which can be provided by the useof multiple stubs. In this latter case eachstub may be adjusted to provide an incremental share of the overall match. A sufficient number of stubs may thus be used toachieve the desired bandwidth.

j tan (27TL/;")

1 + j tan (27TL/;")

- j COt(27TL/;")

1 - j COt(27TL/;")

S +- j(S)1I2

1 + jS(S)V2

S + j(S)1I2

1 + jS(S)1/2

(9-4)

(9-5)

specifically the method for determinationof stub length and position directly from aSMITH CHART.

1. Draw a circle, centered on the chartcoordinates, whose radius corresponds to thestanding wave ratio S on the main line.(Example: S =4.0.)

2. Draw two radial lines each of whichintersects the circle G/Y0 = 1.0, and notethe values at the intersection of these lineswith the outer wavelength scale. (Example:0.176 wavelength at A, and 0.324 wavelengthat B.) These are the two locations "towardthe generator" from a voltage maximumposition where a matching stub can be located.

3. The intersections of the radial lines withthe susceptance circles where they intersectthe unit conductance circle is then followedout to the rim where the inner wavelengthsscale values are noted. (Example: 0.344wavelength at A and 0.156 wavelength at B.)

These are the lengths of open-circuited stubswhich may be placed at the positions A andB, respectively, to provide an admittancematch.

If short-circuited stubs are desired, theirequivalent length is obtainable by addingor subtracting one-quarter wavelength fromthe length required for the open-circuitedstubs. Example: short-circuited stubs shouldbe 0.344 ± 0.25 = 0.094 or 0.594 wavelengthlong at A or 0.156 ± 0.25 =- 0.094 or 0.406wavelength long at B. In this latter case thenegative stub length is of course impossibleand must be discarded as a possible choice.

9.2.2 Determination of Matching StubLength and Location with aSMITH CHART

The SMITH CHART in Fig. 9.1 will serveboth to describe generally and to illustrate

9.2.3 Mathematical Determination ofRequired Stub Length of SpecificCharacteristic Impedance

The required length of a matching stub asdetermined in (3) above is for a stub whosecharacteristic impedance is the same as thatof the main line. If the stub has a different


stubs which have the same input reactancecan be obtained from

L' ~ tan-1 tan(217L/..\.)

..\. 2" K

characteristic impedance its required lengthwill be different from the values indicatedabove but of such length as will provide thesame absolute input reactance. The mathematical determination of stub lengths ofdifferent characteristic impedance from thatof the main line which will provide equivalent absolute input reactance can be shownas follows:

The absolute input reactance X of anylossless short-circuited stub line is

tan (217L/..\.)

tan (2"L '/..\.)

from which

K (9-10)

(9-11 )

X 'Z t 217L') an-Os ..\.

(9-6)

and similarly the ratio of the lengths of twoopen-circuited stubs which have the sameinput reactance can be obtained from

and that of any lossless open-circuited line is cot (2"L/..\.)

cot (217L '/..\.)K (9-12)

X 'Z t 217L'-) 0 co -s ..\.

(9-7)from which

As an example, the open-circuited stub required at B in Fig. 9.1 has the same characteristic impedance as that of the main line andwas determined from this chart to be 0.156wavelength long. If the characteristic impedance of another stub which it may be desiredto use is 1.25 times that of the main line,that is, K = 1.25, we obtain the required newlength L' by substitution of the above valuesin Eq. (9-13); thus

Designating:

L = length of stub of characteristic impedance

ZoL' = length of stub of characteristic impedance

ZosX = absolute input reactance of stub of char-

acteristic impedance ZoX'= absolute input reactance of stub of char

acteristic impedance Zos

we may write, for the case of short-circuitedstubs,

L'

..\.

1 t-1 cot(2"L/..\.)- co217 K

(9-13)

'Z t 2"L) 0 an-..\.

'Z t 2"L') 0 an--s ..\.

(9-8)L' = 0.172..\. (9-14)

and for the case of open-circuited stubs,

'Z t 2"L-) 0 co -..\.

'Z 2"L'-) 0 cot--s A

(9-9)

9.2.4 Determination with a SMITH CHARTof Required Stub Length of SpecifiedCharacteristic Impedance

We also note that in any particular problem. the ratio Zo/Zos is a constant K. Thus, the

ratio of the lengths of two short-circuited

A method of finding stub lengths of different characteristic impedance having equivalentinput reactance by means of the SMITHCHART can be shown by use of the above

102 ELECTRONIC APPLICATIONS OF TH E SMITH CHART

example. The required length of an opencircuited stub of characteristic impedanceequal to that of the main line was found tobe 0.156 wavelength, and it was desired tofind the length of a stub whose characteristicimpedance was 1.25 times that of the mainline which would have the same absolute inputreactance. Proceed as follows:

1. Enter the SMITH CHART (Fig. 8.6) atthe open-circuit position on the impedancecoordinates, where R/Zo = 00.

2. Move "toward the generator" 0.156wavelength (to the outer wavelengths scaleposition, 0.156 + 0.25 = 0.406 wavelength)and observe at the perimeter of the impedancecoordinates that the input reactance of thestub is-jO.67 ZOo

3. Divide -j 0.67 Zo by 1.25 to obtain-j0.536 ZOs' which is the input reactance of astub of the desired characteristic impedancehaving this same length.

4. Move to .new position at 0.536 at theperiphery of the coordinates and observe anew required stub length of characteristicimpedance ZOs to be 0.172"-<0.422"- - 0.25"-),

which checks the value obtained in Eq. (9-14).

A summary of the several relationshipsdescribed above between stub position D, stublength L, standing wave ratio S, and type ofstub (short-circuited or open-circuited) isprovided in Fig. 9.2. Figure 9.2 can beplotted point by point from data obtainedfrom the SMITH CHART of Fig. 9.1 bythe method described above. For conveniencein plotting, the standing wave ratio scale inFig. 9.2 corresponds to that on the SMITHCHART in Fig. 9.1.

9.3 MAPPING OF STUB LENGTHS ANDPOSITIONS ON A SMITH CHART

Figures 9.3 and 9.4 are overlays for theSMITH CHART impedance or admittancecoordinates. The curves thereon show loci

of constant required stub lengths and distancesof attachment of the stub, toward the generator, from the load terminals. Figure 9.3applies to short-circuited and Fig. 9.4 toopen-circuited matching stubs, respectively.For the lengths indicated,the stubs must havethe same characteristic impedance or characteristic admittance as that of the main line. Ifthis is different, the stub length must be suchas will provide the same absolute input reactance as that of the specified stub, inaccordance with methods previously outlined.

If the normalized complex load impedanceor load admittance is known, these curvesmake it possible to directly determine therequired point of attachment of the stub,without reference to the voltage minimumposition along the main line.

Figures 9.3 and 9.4 can be overlaid directlyon the impedance coordinates of Charts A,B, or C in the cover envelope. When overlayedon the admittance coordinates of these lattercharts they must first be rotated through anangle of 1800

•

9.4 IMPEDANCE MATCHING WITHTWO STUBS

Two stubs, each of which is adjustable inlength and fixed in position along a transmission line, may be used in lieu of a singlestub which is adjustable in length and position,to provide a general-purpose impedance matching device. If each of these two stubs is permitted to be either open-circuited or shortcircuited at its far end and separately adjustablein length over a range of one-half wavelength,the combination will provide the means formatching the characteristic impedance of themain line to the impedance of the load withina specified range of load impedance valueson the SMITH CHART.

The matching capability of such a pairof stubs is a function of the stub spacing


Fig. 9.2. Allowable locations and lengths of a single short-circuited, or open-circuited, matching stub.

only and can be represented on the SMITHCHART of Fig. 9.5 as a family of "forbidden" areas each of which correspondsto a specific spacing. In Fig. 9.5 two suchf~milies are shown, one representing forbiddenadmittances and the other equivalent forbidden impedances.

Should the load normalized impedanceZt/Zo or admittance Yt /Y0 of a two-stubmatching circuit fall within their respective forbidden areas no adjustment of the two stubs ispossible which will provide a match. However,if the two stubs are moved as a pair along thesame transmission line in either direction a


GEN _ ZI. OR--~-+""---r-----l Zo

'------1

DANDL INWAVELENGTHS

Fig. 9.3. Required length and position for a short-circuited matching stub (overlay for Chart A. B. or C in cover envelope).


GEN __ ZL OR .!.!.---+-+.......---y-----! Z 0 Yo

D AND L INWAVELENGTHS

Fig. 9.4. Required length and position for an open-circuited matching stub (overlay for Chart A. B. or C in cover envelope).


IMPEDANCE OR ADM ITTANCE COORDINATES

Fig. 9.5. Normalized waveguide impedances or admittances, except those in "forbidden areas," are transformable to 2 0 or to Yo,with two adjustable-length stubs at fixed spacing S.


distance of one-quarter wavelength, the loadimpedance will then fall outside of the forbidden area and an adjustment of the stublengths will be possible which will provide thematch.

An example of a typical problem is indicatedin Fig. 9.6 for a stub spacing of one quarterwavelength.

The point of entry to the SMITH CHART inFig. 9.6 is the normalized input admittance ofthe transmission line at Yf /Y0 =0.4 - j 0.18, andit can be seen that this occurs 0.216 wavelength "toward the generator" from a voltageminimum point on the line. The transformingeffect of shunting Yr/Yo (see insert in Fig.9.6) with an inductive susceptance B1 whosenormalized value is - jB1/YO = 0.31 is tomove Yt /Y0 to the position Y1/Y0 . Theadditional transforming effect of the mainline section between the stubs S/II. = 0.25is to move Y1 /Y0 to the position Y2 /Y0

on the unit conductance circle. Finally, thetransforming effect of shunting Y2 = 1.0 + j 1. 22with an inductive susceptance B2 , whosenormalized value - jB2 /YO = 1.22, is to moveY2 /YO to Yo = 1.0 ± jO, at which point thedesired match is accomplsihed.

In all cases, Y2 /Y0 must be on the unitconductance circle, and the distance betweenY1/Y0 and Y2 /Y0' as measured on the peripheral scales to which these points are projectedradially, must correspond to the spacing of thestubs.

9.5 SINGLE-SLUG TRANSFORMEROPERATION AND DESIGN

An elementary type of transmission lineimpedance transformer consists of a onequarter wavelength section of transmissionline of appropriate characteristic impedance.In operation such a transformer is alwaysinserted in the main line between a voltage

r maximum and adjacent minimum as thesepositions exist prior to its insertion. Such

positions along the line will provide inputand output impedances which are representable on the SMITH CHART along the realaxis.

The characteristic impedance Z t of thequarter-wave /Islug" transformer must in allcases be made equal to the geometric meanbetween the characteristic impedance Zo ofthe main line and the real impedance R f

at the load terminals of the slug. Thus

(9-15)

From Eq. (9-15) it is seen that for a fixedvalue of Z t only a single load resistance Rf canbe transformed to ZOo

If the ratio Z/Zo is between zero and unitythe load terminals of the transformer must bepositioned at the voltage (and consequentlyimpedance) minimum point along the line, inwhich case the transformer will step up thelow impedance of its load to match thecharacteristic impedance of the line. Conversely, if it is between unity and infinityits load terminals must be positioned at avoltage (and consequently impedance) maximum point, and it will then step down theload impedance to match the line characteristic impedance.

For a general-purpose matching device theposition of the quarter-wave slug transformermust be continuously adjustable over a rangeof one-half wavelength in order to ensure thatit can always be properly positioned withrespect to the standing wave which it is required to eliminate.

The design and operation of a single quarterwave transformer can be shown on the SMITHCHART in Fig. 9.7 as follows:

1. Assume that a mismatched transmissionline has a standing wave whose ratio S is3.333. Draw the circle for S = 3.333 centeredon the chart coordinates, and note that this


ADMITTANCE COORDINATES

Fig. 9.6. Representation of an admittance transformation from Ye!Y0 to YO with two shunt matching stubs spaced one-quarterwavelength.


s= 1.0r----- A14 ----ojf----- -----1 s =3.33

I i

0.12 0." f , I '\Zt /Z o=·547

R,t/Zo= 0.3

Fig. 9.7. Impedance matching with a single quarter-wave slug transformer.


circle intersects the real axis at points A andB where Rf/ZO =0.3 and 3.333, respectively.Assuming that it is desired to use a transformer which steps up the load resistance, selectthe lower of these two possible values as theload impedance for the transformer, that is,R /Zo = 0.3 at point A.

2. Draw a construction line connectingpoint C on the perimeter of the chart withpoint A, and at its intersection with the S =3.333 circle (point D) erect a perpendicularto intersect the real axis of the chart atpoint E. The required normalized characteristic impedance Z/Zo of the slug transformeris found, in this example, to be 0.547. (Hada step-down transformer been selected thenormalized value of its characteristic impedance would have been 1.83, as indicated atthe extremity of the dotted construction lineat point F. The results thus obtained will beseen by inspection to satisfy Eq. (9-15).

9.6 ANALYSIS OF TWO-SLUG TRANSFORMER WITH A SMITH CHART

A more versatile matching circuit whosedesign and performance characteristics are described herein is the two-slug transformer. Theanalysis of this transformer is carried out in aspecific example on the SMITH CHART ofFig. 9.8.

The two slugs shown schematically in Fig.9.8 are generally constructed in the form oflongitudinally adjustable enlargements of theinner conductor diameter of a coaxial line,as shown, or sleeve-type reductions of its outerconductor diameter. Either case results in areduction of the characteristic impedance ofthe line section occupied by the slug. Alternatively, the slugs may be constructed of adielectric material.

Given sufficient space along the Hne foradjustment of slug position, this type oftransformer is capable of eliminating standing

waves of any amplitude ratio which does notexceed its maximum design value, which valuedepends only upon the fixed value of char- 0

acteristic impedance for the slugs.Individual conducting slugs are frequently

made one-quarter wavelength long, and dielectric slugs, which completely fill the spacebetween inner and outer conductor, are usually made mechanically shorter by the factor1/Ii to take into account the reduction invelocity of a wave in a dielectric materialwhose dielectric constant f is greater thanunity.

The two slugs are usually fixed in lengthand in characteristic impedance for a particular application in accordance with a predetermined maximum standing wave matchingrequirement. The slugs are adjustable alongthe inner conducter (I) with respect to eachother, and (2) with respect to a fixed refer- i

ence position along the line.If minimum overall length of a two-slug

transformer is important in a given application,one-quarter wavelength is not necessarily themost desirable length for the individual slugs.For this reason the analysis which followswill consider slugs whose length ranges betweenzero and one-quarter wavelength, and whosecharacteristic impedance ranges between zeroand one times that of the main line, to determine how the overall matching capability isrelated to these design variables. From thisdata the range of transformable load impedances for a given slug geometry as a functionof frequency may readily be found.

A specific combination of two identicalone-eighth wavelength-long slugs with a normalized characteristic impedance Z /Zo =0.4 is represented by the construction inFig. 9.8. By a successive consideration ofother slug lengths and normalized characteristic impedances by the same method, thedata in Fig. 9.9 is obtained. The use ofidentical slugs in a two-slug transformer isjustified on the basis that it simplifies the


ig.9.8. Determination of impedance matching capability of two one-eighth-wavelength-Iong slugs when 2 t /20 = 0.4.


analysis while still not restricting the methodof analysis for two slugs of arbitrary lengthsor characteristic impedances.

Let it be required to find the boundariesof impedance areas on the SMITH CHARTwhich define the impedance transformingcapabilities of two-slug radio frequency impedance transformers as a function of (1) the sluglength in wavelengths, (2) the ratio of characteristic impedance of the slugs to thecharacteristic impedance of the main line,that is, Z/Zo' (3) the spacing D betweenslugs in wavelengths, and (4) the maximumrequired overall length of slugs plus positionadjustments.

For the purpose of analysis the conditionsof the problem may be reversed and it maythen be assumed that the load is a constantimpedance which perfectly matches the normalized characteristic impedance of the line.The boundary of all possible input impedancevectors for all possible slug positions is thenfound. The principle of impedance conjugates may then be applied to the matchingproblem, that is, for example, if a normalizedinput impedance of 2.0 +j 4.0 falls within thisboundary of impedances which are availableat the input terminals, then this transformeradjustment will provide a match for a normalized load impedance of 2.0 - j4.0. The inputSWR for the conditions calculated will be thesame as the output SWR in the actual case. Itwill be found that in all cases the matchableimpedance area is bounded by a circle centeredat ZOo

9.7 DETERMINATION OF MATCHABLEIMPEDANCE BOUNDARY

1. On the circular transmission line impedance chart (Fig. 9.8) draw a circle centered atZo cutting the R axis through the valueassigned to Z/Zo' assumed in this example

to be 0.4. This is the locus of impedancesalong that slug which is nearest to the load,normalized with respect to Zt' Label the load(maximum) impedance point Zl/Zt.

2. Layoff the point Zz/Zt on this circle ata distance of 0.125 wavelength, correspondingto the length of a single transformer slug inwavelengths towards generator, i.e., clockwisefrom point Z/Zt.

3. Normalize Zz with respect to ZO' bymultiplying each component of the compleximpedance Zz/Zt' that is, 0.62 - j 0.72, bythe numerical ratio, 004, which was assignedto Z/Zo' Plot the resulting point, that is,0.276 - j 0.288, at Zz/Zo.

4. Draw a circle centered at Zo and passingthrough Zz/Zo. This is the locus of impedancesZs/Zoas the separation of the slugs is changed.

5. Normalize the above locus circle Zs/Zowith respect to Zt' that is, find the locus ofZs/Zt' by dividing each component of eachcomplex impedance value ZS/ZO on the periphery of this locus circle by the numericalratio assigned to Z/Zo'

6. Rotate the locus circle Zs/Zt clockwiseabout point Zo by an amount correspondingto the length of a single transformer slug, thatis, 0.125 wavelength, to obtain the locus of

Z4/Z t·7. Normalize and replot the above locus

circle Z4/Zt with respect to ZO' that is, findthe locus of Z4/Z0' by multiplying eachcomponent of each complex impedance valueZ4/Zt on the periphery of this locus circleby the numerical ratio assigned to Z/Zo' thatis, 004.

8. Draw a circle representing the Z4/Z0boundary. This circle is centered at Zo andis tangent to the outermost point of locuscircle Z4/Z0' It will be observed from Fig.9.8 that the locus of Z4/Z0 IS a circle passingthrough unity (the center point of the SMITHCHART).

This, in effect, states that regardless of theslug length or slug characteristic impedance,


it is always possible to find a slug spacing such that the reflection introduced byone slug just cancels that from the other.The result is an overall 1(1 impedance transformation and the slug spacing required issuch that 22/20 and 2 3/20 are conjugate.Any departure it, slug spacing in eitherdirection from the spacing which resultsin a 1/1 impedance transformation introducesa standing wave of some amplitude ratio otherthan unity. The maximum amplitude of thisstanding wave is reached when the slug spacingdeparts from the above setting by a quarterwavelength. The position of the standing waveintroduced by the slugs is adjusted with

respect to any arbitrary fixed reference pointalong the line by sliding the slugs as a pair(maintaining constant spacing). For a generalpurpose two-slug transformer it is necessarythat the position of the standing wave beadjustable throughout a full one-half wavelength. Thus the minimum required overalllength of a general purpose two-slug transformer is seen to be composed of the sum ofthe following lengths:

1. One-half wavelength required for phaseadjustment.

2. The spacing required between slugs.3. The combined lengths of the slugs.

CD 40.20. 10. 5. 1.5

.12

LS

.13

.24

.25

LS: INDIVIDUAL SLUG LENGTHS IN WAVELENGTHS

LT = OVERALL TRANSFORMER LENGTH IN WAVELENGTHS

ZT = RATIO OF SLUG TO MAIN LINE CHARACTERISTIC IMPEDANCE

S : MAXIMUM SWR WHICH CAN BE REDUCED TO UNITY

ig.9.9. Impedance matching capability of two-slug impedance transformer.


Figure 9.9 is a summary of data obtainedas described from a SMITH CHART, showingthe overall impedance matching capability ofa two-slug transfonner. In this figure, Ls isthe slug length and LT is the overall transformer length in wavelengths, and ZT is thecharacteristic impedance of the slugs normalized with respect to the characteristic impedance of the main line. The chart shows themaximum value of the standing wave ratioS which can be reduced to unity.

The analysis on the SMITH CHART of thetwo-slug transformer, as described, reveals thepossibility of using slugs much shorter thana quarter wavelength in applications where

it is desirable to minimize the overall transfonner length, or to limit the maximumvalue of the SWR which the two-slug combination can reduce to unity, thereby improvingthe operating bandwidth of the transfonner.For example, it can be seen from Figs. 9.8and 9.9 that a pair of one-eighth-wavelengthlong slugs whose characteristic impedance is0.4 times that of the main line will match anypossible load impedance which could result instanding wave ratios between unity and 14/1.Furthermore, such a combination of slugs willprovide a transformer whose overall lengthnever exceeds 1.096 wavelengths (0.5 + 2 x

0.125 + 0.096 + 0.25).

10.1 L-TVPE MATCHING CIRCUITS

The advantages of matching impedances at ajunction between two circuits or media

have been pointed out in Chap. 9 and elsewherein this book and need no further explanationhere.

One of the most commonly used and generally satisfactory impedance transformingnetworks for radio frequency applications isthe half-section L-type circuit employing twoessentially pure reactance elements [ 19, 149] .At a single frequency and, for most practicalpurposes, embracing at least the sidebandfrequencies of a radio telephone transmitter,the simple L-type circuit may be used effectively to transform any load impedance to anydesired pure input resistance value. Conversely, the L-type circuit may be employedto accomplish the reverse transformation, thatis, to transform any load resistance to anydesired complex input impedance value. It isnecessary to consider only the former type of

Ctransformation, however, since the circuit canalways be reversed to make the transformation

CHAPTER 10Network

ImpedanceTransformations

in the opposite direction. This simplifies thepresentation of design information.

It will be seen by referring to Fig. 10.1 thatthere is a total of eight possible combinationsof reactance types, i.e., inductive and capacitive, in an L-type circuit. Each of these eightcircuits is capable of transforming a restrictedrange of complex load impedance values to agiven pure resistance value. The transformableimpedance values associated with each circuitcan conveniently be represented by the impedances within a bounded area on a SMITHCHART. A set of eight such representationswill therefore completely outline the capabilityand limitations of the eight possible reactancecombinations, and will furnish a comprehensive outline of the impedance transformingcapability of each reactance combination.

For radio-frequency applications, the lossesin an L-type circuit are usually small incomparison to the power which is beingconducted through the circuit. Thus, thecircuit losses generally will not limit to anyserious extent the range of load impedancevalues which can be transformed to a desired

115

On each of the eight diagrams shown inFig. ID.2 an example of the function of eachelement of the circuit is illustrated using anassumed load impedance vector Zf' Theinfluence of each of the circuit elements uponZ( may be regarded as forcing the latter tomove along an "impedance path" on a SMITHCHART from its initial position to a positionalong the R axis, with its extremity at positionZo- This impedance path followed by a singlevector is illustrated on each of the eightdiagrams of Fig. 10.2 by a heavy line and anaccompanying arrow.

For example, refer to Fig. 10.2(a). Here,any load impedance (such as Z ~ whose

10.1.2 SMITH CHART Representation ofCircuit Element Variations

10.1.1 Choice of Reactance Combinations

The eight SMITH CHART overlays inFig. ID.2 summarize the matching capabilitiesof the eight possible L-type circuit combinations of Fig. 10.1, and serve as a guide to theselection of a suitable L-type matching circuitfor any particular impedance transformation.

Fig. 10.1. Eight possible L.type circuits fortransforming a complex load impedance Z! to a pureresistance Zo0

~0------11 [ ~Zo

~ 0

(0) (b)

~ ~[If------o

Z.t

~ --0(el (d)

~r----r: ;: [ ~~(el (f)

~~ ~ [ :1-~(9) (h)


A shaded area is shown on each of the eightdiagrams in Fig. 10.2. This is to indicate thatany load impedance vector whose extremity ~

falls anywhere within this "forbidden" areacannot be transformed to Zo (the desired input resistance value) with the specific circuitto which the diagram applies, and that in thiscase one of the seven other L-type circuitsmust be selected. If the extremity of the loadimpedance vector falls anywhere inside of theunshaded area, the circuit is capable of performing the desired impedance transformation.

In cases where the impedance transformingcapabilities of two or more L-type circuitsoverlap, the particular circuit which calls forthe more practical circuit constants should,of course, be chosen. It is of interest tonote that the circuits shown in Fig. ID.2(e)and (g) are each capable of transforming thesame range of load impedances, although eachaccomplishes a given transformation withdifferent reactance values. The circuits inFig. 1O.2(f) and (h) are also both capable ofmaking the same impedance transformations.

resistance, nor will they ordinarily have amajor effect upon the reactance values for thecircuit elements which are theoreticallyrequired on the assumption that they are lossless. The design charts to be described are,therefore, plotted for the idealized case of lossless circuits. Having selected a suitable losslesscircuit and having obtained the reactancevalues required in such a circuit from thecharts, the probable resistance of the circuitelements which must be used and the resultinglosses will be more readily determinable.

NETWORK IMPEDANCE TRANSFORMATIONS 117

(e)SEE FIG. 107

(g)SEE FIG. 10.9

Fig. 10.2. Impedances in unshaded areas of SMITH CHARTS, represented by eight circular boundaries, aretransformable to a pure resistance 2 0 with specific L-type circuits indicated (transforming effect of each circuitelement is indicated by a heavy line with arrows).


in the cover envelope, with which it must beaccurately aligned.

The load impedance, indicated at the eXtremity of the load impedance vector, shouldbe spotted on the SMITH CHART coordinateswhich are superimposed on the appropriatedesign chart. The required circuit reactancevalues are then obtained from the design chartby interpolating between the indicated valueson the nearest reactance curves plotted thereon.

Problem 10-1, which follows, further illustrates this use of the design charts.

10-1 (a) Select an L-type circuit which willtransform a load impedance of 140 +j 60 ohms to a pure resistance of 50ohms.

Solution:From the foregoing we may write:

extremity falls in the unshaded area may beselected to be transformed with an L-typecircuit of the type indicated on this diagramto a chosen value of pure resistance ZOo Inthis case it will be noted that the effect ofthe shunt capacitive reactance on the impedance vector Z f is to rotate its extremity clockwise around a circular path leading to the pointZl' This path is always along a circle tangentto the X axis at X =0, and centered on the R

axis of the SMITH CHART. Zl representsthe extremity of a second impedance vector,the resistance component of which is equalto ZOo (To simplify the diagrams, only theextremities of the vectors are indicated.) Thecapacitive reactance component of the impedance vector Z1 is then canceled by thereactance of the series inductance element ofthe L-type circuit, which moves the vectoralong the path to position Zo, thus completingthe transformation.

The required inductive and capacitive reactances of the L-type circuit elements arenot shown in Fig. 10.2, which, as explained,is useful primarily to compare the matchingcapabilities of the various circuits.

Zo = 50

Z f = 140 + j60 2.8Zo + j1.2Zo

10.1.3 Determination of L-type CircuitConstants with a SMITH CHART

To obtain the proper value of the inductiveand capacitive reactances required in a givenL-type circuit to transform a given complexload impedance Zt to a given pure resistanceZo' select the design curves which havebeen plotted for the particular circuit chosen.These are plotted in Figs. 10.3 through 10.10for each reactance element of each of theeight possible circuits. The applicable circuitcan be identified by referring to the smallschematic diagram associated with each.Each of these families of design curves isused as an overlay for SMITH CHART A,

Refer to SMITH CHART A and Fig.10.2 and observe that the above loadimpedance vector Zf falls within theunshaded (transformable) area of diagrams a and b, and within the shadedor "forbidden" area of diagrams c to hinclusive. A choice of two circuits istherefore available for this transformation. Select one-for example, that ofdiagram b.(b) Determine the reactance value ofeach element of the circuit selected.

Solution:On SMITH CHART A locate the aboveimpedance value; then superimposethis chart on the design curves of Fig.10.4 (as directed on diagram b of Fig.10.2). Next determine the correct


reactance values for XL and Xc bynoting that the extremity of this loadimpedance vector Zt, as plotted onthe SMITH CHART, falls at the intersection of the design curves labeledXL/Zo = 3.0 and Xc/Zo = 1.5. SinceZo = 50 ohms, XL = 3.0 x 50 = 150ohms, and Xc =1.5 x 50 =75 ohms.

If a complex load impedance is not knownexactly but can be estimated within certainlimits, these limits may be mapped directlyon the SMITH CHART and the range ofcircuit reactances required can thus be completely bracketed.

This feature will be most appreciated whenan L-type circuit must be designed to accommodate anyone of a range of possible loadimpedance values. The design of a circuit tomatch the input impedance of a radio antenna,which is usually not definitely known in advance of its construction, to the characteristic impedance of a transmission line isreadily accomplished with this type of diagram. In such cases, the limitations of agiven circuit establish limiting requirementsfor the circuit elements. Problem 10-2, whichfollows, illustrates this case.

10-2 (a) Select an L-type circuit which canbe adjusted to match any load impedance falling within the range 25 to 75ohms resistance and 0 to 50 ohmspositive reactance to a pure resistanceof 100 ohms.

Solution:From the foregoing we may write

Zo 0

Zt (20 to SO> + j <0 to 60) ohms

<0.20 Zo to O.SO Zo)

+ j<O to 0.60Zo) ohms

From Figs. 10.3 to 10.10 and SMITHCHART A select a diagram upon

which the above "block" of impedance values all fall within a transform-

able (unshaded) area. The L-typecircuit of Fig. 10.6 is found to be theonly suitable circuit for this case.(b) Determine the limiting reactancevalues of each of the two circuitelements.

Solution:On SMITH CHART A outline theabove range of impedance values,then superimpose this chart on thedesign curves of Fig. 10.6. Determine the limiting values for XL andXc from the curves which just touchthe edges of the outlined area. Thefollowing limiting values will beobserved:

XL = 0.5 Zo to 2.0 Zo = 50 to 200 ohms

Xc = 0.5 Zo to 1.1 Zo = 50 to 110 ohms

10.2 T·TVPE MATCHING CIRCUITS

By the addition of a third reactance element in series with the chosen input resistanceobtained with an L-type impedance matchingcircuit, thus forming a T-type circuit, anycomplex load impedance value can be transformed to any desired complex input impedance value. The overlay charts of Figs. 10.3to 10.10 inclusive are applicable in this casealso. The reactance required in the thirdelement depends upon the value of inputreactance desired. If a circuit is chosen whichalready includes a series reactance elementin the input side, such as one of the circuitsshown in diagrams a, b, e, and f of Fig. 10. 1,the "third" reactance required would becombined algebraically with the former, resulting in a single net reactance value in thisposition.


Fig. 10.3.envelope).

Normalized reactances of L-circuit elements required to transform 2! to 2 0 (overlay for Charts A, B. or C in cover


Fig. :10.4.envelope).

Normalized reactances af L·circuit elements required to transform Ze to 2 0 (overlay for Charts A. B. or C in cover


Fig. 10.5.envelope!.

Normalized reactances of L-circuit elements required to transform Z 1 to Zo (overlay for Charts A. B. or C in cover

NETWOR K IMPEDANCE TRANSFORMATIONS 123

Fig. 10.6.envelope).

Normalized reactances of L-circuit elements required to transform Z 1 to 2 0 (over.lay for Charts A, B. or C in cover

Fig. 10.7.envelope),

ELECTRONIC APPLiCATIONS OF THE SMITH CHART

FORBIDDENAREA

"

Normalized reactances of L-circuit elements required to transform 2 t to 2 0 (overlay for Charts A, B, or C in cove

Fig. 10.8.

envelope).


FORBIDDENAREA

Normalized reactances of L -circuit elements required to transform Z, to 2 0 (overlay for Charts A, B, or C in cover

126 ELECTRONIC APPLICATIONS OF TH E SMITH CHART

C2

d. _l Ii !!:-re,

o----~ 0

Fig. 10.9. Normalized reactances of L-circuit elements required to transform Z, to Zo (overlay for Charts A, B, or C in coverenvelopel.

Fig. 10.10."nvelopel.


FORBIDDENAREA

Normalized reactances of L-circuit elements required to transform 2! to 2 0 (overlay for Charts A, S, or C in cover


10.3 BALANCED L- OR BALANCEDT-TVPE CIRCUITS

If the input impedance of a matching circuit must be balanced with respect to ground,the L-type circuit design curves can be used todesign a suitable impedance matching circuit

by treating the problem as an unbalanced one.The required series reactance obtained fromthe diagrams is simply divided into two parts, )each having one-half of the value called for onthe charts. These two halves of the necessarytotal series reactance are then connected inseries with each side of the circuit to preservethe balanced-to-ground arrangement.

11.1 IMPEDANCE EVALUATION FROMFIXED PROBE READII\IGS

At radio frequencies where slotted wave-guide or transmission line sections would be

excessively long, probe measurements of therelative current (or relative voltage) amplitudesat discrete sampling points along the waveguideprovide a convenient and practical techniquefor measuring the complex impedance and related parameters. The SMITH CHART is usefulfor interpreting and evaluating data obtainedfrom such measurements [ 112, 208] as will bedescribed herein. The principle is made use ofin the SMITH CHART plotting board shownin Fig. 14.5.

In a waveguide or transmission line propagating electromagnetic wave energy in a singlemode, relative amplitude measurements ofeither current or voltage at three fixed probepositions uniquely determine the standingwave ratio and the wave position, providedthat no two of the probes are separated anexact multiple of one-half wavelength. The

CHAPTER 11Measurements of

StandingWaves

standing wave pattern, in turn, is related tothe impedance and other waveguide parameters, as described in previous chapters. Theelectrical separations of the probes neednot be uniform but must be known. This isgenerally calculable [10] from the physicalconstruction of the waveguide or transmissionline, as described in Chap. 3.

As was shown in Chap. 3, the shape ofstanding current and voltage waves of a givenratio are identical, and both are different andunique for each different standing wave ratio,varying from a succession of half sine wavesas the amplitude ratio approaches infinity toa sinusoidal shape as the amplitude ratioapproaches unity. It is possible for this reasonto plot unique families of impedance locuscurves, each family corresponding to a fixedprobe spacing, and each curve of each familycorresponding to a fixed ratio of current (orvoltage) probe readings. Figures 11.1 through11.4 show four families of such probe ratiocurves, corresponding to fixed probe spacingsof one-sixteenth, one-eighth, three-sixteenths,and one-quarter wavelength.

129


For flexibility in the application of thesecurves to voltage-probe as well as currentprobe readings, the SMITH CHART impedance coordinates with which they are associated have been omitted. However, whenrequired these impedance coordinates canconveniently be supplied as an overlay thereonby the use of the translucent SMITH CHART(Chart A) in the back cover of this book.

Any two families of probe ratio curves incombination with the translucent SMITHCHART coordinates are capable of determining values of all possible load impedances alonga waveguide. In any given situation, however,a single value is indicated at Py on theSMITH CHART coordinates, which pointoccurs at the intersection of a single pair ofprobe ratio curves corresponding to the actualreadings.

At 1800 (one-half wavelength) separationtwo probe readings along a uniform losslesswaveguide are identical and the impedanceloci curves for the two families of probe ratiocurves become coincident with one another.One-half wavelength spacing for any of thethree sampling points must, therefore, beavoided. Sampling point spacings of less thanabout one-sixteenth wavelength or betweenseven- and nine-sixteenths wavelength, fifteenand seventeen-sixteenths wavelength, etc., areinadvisable, since in these cases the familiesof ratio curves so nearly parallel each otherthat small errors in readings can representlarge errors in the determination of theimpedance.

11.1.1 Example of Use of Overlays withCurrent Probes

Three current probes are spaced one-eighthwavelength apart along a transmission line; theoverall separation of probes is, therefore, onequarter wavelength. Two probes, each designated Pg' are located on the generator side, and

a third probe Py on the load side. The ratio ofthe current in the center probe Pg to the current in the probe on the load side of the groupPy is assumed, for example, to be 0.5. (Notethat this pair of probes is spaced one-eighthwavelengths.) Simultaneously, the ratio of currents in the probe on the generator side of thegroup P g to the probe on the load side of thegroup P y is assumed to be 0.7. (Note thatthis pair of probes is spaced one-quarter wavelength and that Py is common to both proberatios.)

Find the impedance on the SMITH CHARTat Py:

1. For the pair of probes which is spacedone-eighth wavelength, select Fig. 11.2, whichapplies specifically to probe ratios at thisspacing, and trace the locus curve P/Py =0.5 onto the superposed impedance coordinates of the translucent SMITH CHART(Chart A) in the back cover of this book.

2. For the pair of probes which is spacedone-quarter wavelength, select Fig. 11.4 andsimilarly trace the locus curve P/Py = 0.7,thereon, onto the same impedance coordinates.

3. Observe the intersection of these twotraces on the SMITH CHART impedancecoordinates to be at the common referencepoint P y where the normalized impedanceof the transmission line is (0.5 +j 0.5) ZOo

11.2 II\1TERPRETATION OF VOLTAGEPROBE DATA

It is equally feasible and practical to employFigs. 11.1 through 11.4 in combination withthe SMITH CHART coordinate overlay, todetermine the impedance corresponding toany three voltage probe readings. As wasshown in Chap. 4, the normalized voltage atany position along a uniform loss1ess waveguide is exactly equivalent to the normalized current one-quarter wavelength removed

MEASUREMENTS OF STANDING WAVES 131

Pg___________- LOAD

f- }./ 16 -l

Fig. 11.1. Loci of constant ratios of current or voltage on a SMITH CHART at two probe points P g and p! spaced A/16 along atransmission line (overlay for Charts A. B. or C in cover envelope).


____________- LOAD

Fig. 11.2. Loci of constant ratios of current or voltage on a SMI TH CHART at two probe points P g and P f spaced A/8 along atransmission line (overlay for Charts A. B. or C in cover envelope).


____________- LOAD

f-- 3A/16-1

Fig. 11.3. Loci of constant ratios of current or voltage on a SMITH CHART at two probe points P g and P f spaced 3A/16 along atransmission line (overlay for Charts A, B, or C in cover envelope).


----:- - LOA D

~ >'/4-j

0 C\I 'it; ~ CIO 0 It) 0...; ...; -= C\I C\I ..,II

'"'Q.

......

'"Q.

Fig. 11.4. Loci of constant ratios of current or voltage on a SMI TH CHART at two probe points P g and P t spaced X/4 along atransmission line (overlay for Charts A, B, or C in cover envelope).

therefrom. Also, anyone-quarter wavelengthtransfer of reference point along a waveguide isrepresented by a rotation around the centerof the SMITH CHART coordinates an angulardistance of 1800

• Therefore, if voltage probesare used instead of current probes, for whichthe ratio curves in Figs. 11.1 through 11.4


apply directly, it is only necessary to rotatethe ratio curves through an angle of 1800

with respect to the SMITH CHART impedance coordinates to make them applicable.The desired impedance is then indicated atPe, as before, from any intersecting pair ofvoltage probe ratio curves.

1------ RADIUS OF SMITH CHART I R) -----~~ ~t-:o-----------.----------+---------/----------~.. :.,.~

0,

WHEN20~ Pg/P.t< 1.0: I

0, ~-2lPg/P.L) SIn a/[IPg/P.t )-IJ

R,~IPg/P.L) 0,

WHEN 1.0 < Po/Pl < Q);

0, ~ 2.in a/[IPg/P.t )-1]

R, ~ lPg/P.L) 0,

Fig. 11.5. Construction of loci of constant ratios of current on impedance coordinates of a SMITH CHART at two points spaced

5//\ along a lossless transmission line.


In summary, as oriented in Figs. 11.1through 11.4 with respect to the overlayCharts A, B, or C, in the cover envelope, theintersection point of any pair of currentratio curves gives the impedance at Py, whilethe intersection point of any pair of voltageratio curves gives the admittance at P p. Whenrotated 1800 about their axes with respect tothe coordinates of the overlay charts A, B,or C, the intersection point of any pair ofcurrent ratio curves gives the admittance atPr, while the intersection point of any pairof voltage ratio curves gives the impedanceat Py•

11.3 CONSTRUCTIOI\l OF PROBE RATIOOVERLAYS

Information is given in Fig. 11.5 for plotting probe ratios which correspond to anydesired probe separation. All such plotsrequire only straight lines and circles fortheir construction. The outer boundary ofsuch a construction corresponds to the boundary of the SMITH CHART coordinates.

As shown in Fig. 11.5 the separation of anytwo sampling points S/>.. determines the anglea as measured from the horizontal R/Zo axis.This angle establishes the position of a straightline through the center of the constructionwhich represents the locus of impedances atthe probe position Py when the currentstanding wave ratio is varied from unity toinfinity while the wave is maintained in sucha position along the transmission line withrespect to the position of the two samplingpoints that they always read alike, that is,that P iPy = 1.0.

A construction line perpendicular to thelocus PiPy = 1 and passing through theinfinite resistance point on the R/Zo axis willthen lie along the center of all of the PiPYcircles which it may be desired to plot.

The ratio of each of these circular arcs (R 1and Rz ) which corresponds to a particularcurrent ratio, and the distance of their centersfrom the chart rim (01 and Dz), is given by theformulas in Fig. 11.5 as a function of theratio of P g to P y and the SMITH CHARTradius R.

12.1 NEGATIVE RESISTANCE

In all passive waveguide structures, the pri-mary circuit elements resistance and con

ductance act like absorbers of electromagneticwave energy. As such, their mathematicalsign is, by convention, positive. However,certain active electrical devices exist whoseequivalent circuit can most conveniently berepresented by negative resistance or negativeconductance elements. Negative resistance andnegative conductance circuit elements act morelike sources than absorbers of electromagneticwave energy.

Electrical devices such as parametric andtunnel diode reflection amplifiers, which canbe schematically represented by negative resistance or negative conductance in combination with conventional circuit elements, arefrequently used in connection with waveguides. In order to properly represent theircharacteristics on a SMITH CHART for graphical analysis, it is important to understand the

(effect of negative resistance or negative conductance upon the associated waveguide elec-

CHAPTER 12Negative

SmithChart

trical characteristics. This will be discussedmore fully herein. It will be found, forexample, that when a waveguide is terminatedin a circuit which has a resultant negativeresistance or negative conductance component, the complex reflection coefficients andthe complex transmission coefficients at allpoints along the waveguide are different fromwhat they would have been had the sign ofthe resistance or conductance been positive.

Negative resistances and negative conductances do not actually produce energy, butrather act to transform and release energyfrom an associated source (such as a battery)into electromagnetic energy whose frequencyis controlled by a primary source of smallerenergy level. Negative resistance is defined[7] as the property of a two-terminal devicewith an internal source of energy which iscontrolled either by current through or byvoltage across the terminals, but not by both.Negative conductance is defined as the reciprocal of negative resistance.

The mathematical distinction between positive and negative resistance (or positive and

137


Pm

Fig. 12.1. Electrical characteristics of a typical tunneldiode.

BIAS VOLTAGE

IZWa:a:::)u

II

w I>

~ ~ LRESllsNT~~wvR~GIONJ~+I---_--+------r-

I -----+1--~ (b)a: W Iu >

< 5 I~ I

II

with either positive or negative resistancecomponents and, in fact, how a single representation best serves both purposes. Sincethe sign of a negative resistance is, byconvention, identical to the sign of the equivalent conductance, it follows that no changeis required in the admittance coordinates ofthe conventional SMITH CHART to representadmittances with either positive or negativeconductance components. All peripheralscales of the conventional SMITH CHARTincluding the reflection coefficient phase anglescale are independent of the sign of the realcomponents of impedance or admittance.

All radial scales of the conventional SMITHCHART which have been described in previouschapters except the reflection coefficientmagnitude scale and the reflection loss scaleapply in all cases where the real components

12.2 GRAPHICAL REPRESENTATION OFNEGATIVE RESISTANCE

The graphical representation on the SMITHCHART of impedances or admittances withnegative resistance or negative conductancecomponents, respectively, presents no specialproblems, as may have been inferred from thevarious transformations of the conventionalchart coordinates which have been suggestedto accomplish this [12] .

It will be shown in this chapter how theconventional SMITH CHART impedance coordinates and associated peripheral scales(Fig. 3.3) are applicable without any transformation to the representation of impedances

negative conductance) is portrayed by theslope at a point on a plot of the voltagecurrent relationship, where current is represented as the ordinate and voltage as theabscissa on a cartesian coordinate grid. Figure12.1(a) illustrates the voltage-current characteristics of a typical negative-resistance device such as a tunnel diode. If the currentthrough a resistance or conductance increaseswith increased applied voltage (positive slope),the sign of the resistance or conductance isconsidered to be positive in this portion of itsoperating range. On the other hand, if thecurrent decreases with increased applied voltage (negative slope), the sign of the resistanceor conductance is considered to be negativein this portion of its operating range.

In Fig. 12.1(a) the points Pm and Pn' wherethe slope is zero, illustrate conditions wherethe resistance is plus and minus infinity,respectively. The inflection point P, wherethe slope changes from a decreasing negativevalue to an increasing negative value, representsconditions where the negative resistance isminimum (negative conductance is maximum).The effective value of negative resistance atany point on the voltage-current plot of Fig.12.1(a) is shown in Fig. 12.1(b).

NEGATIVE SMITH CHART 139

On the negative (left) half of the plot ofFig. 12.2(b) the magnitude of the reflectioncoefficient is the reciprocal of the magnitudeof the reflection coefficient at the corresponding position on the positive resistancehalf, as reflected through the origin. Thus,on the negative half it is

It will be observed from Fig. 12.2(b) thatboth the complex reflection coefficient andthe related complex impedance componentcurves extend to infinity in all directions fromthe origin. This seriously limits the usefulrange of such a plot of this relationship.

To overcome the above difficulty a bilineartransformation of coordinates may be made,the results of which are shown in Fig. 12.2(c).

In such a transformation the circular shapeof all individual curves and the angle betweenrespective curves of the two families at all

normalized values of positive and negativeresistance, and positive and negative reactance.(In order to simplify this discussion theoptional admittance component designationson these coordinates are omitted.) Upon thiscoordinate system the loci of the reflectioncoefficient amplitude and phase componentsmay be plotted (Fig. 12.2(b». These lociare observed to be orthogonal families ofcircles.

On the positive resistance (right) half ofthe plot in Fig. 12.2(b) the reflection coefficient magnitude is at all points less thanunity. In terms of the normalized impedancecomponents R/Zo and + jX/Zo the complexreflection coefficient on the positive half ofthis plot is given by

of impedance or admittance are designatedwith positive or negative values.

Interpretation of the use of these previouslydescribed radial scales in connection withnegative real-component coordinates may behelpful and, accordingly, will be discussedherein. A new radial scale for reflectioncoefficient magnitude, to be used exclusivelywith negative real-component coordinates,will be fully described. The radial "reflectionloss" scale described in Chap. 4 is meaninglessin connection with real-component coordinates.

The reflection coefficient and transmissioncoefficient magnitude and phase overlays(Figs. 3.5 and 5.4, respectively) which havebeen described for positive real-componentcoordinates of impedance and admittance arenot applicable to negative real-componentcoordinates. In their place, a new reflectioncoefficient magnitude and phase overlay (Fig.12.6) and a new transmission coefficientmagnitude and phase overlay (Fig. 12.7),specifically applicable to negative real-component coordinates, will be presented anddiscussed.

The SMITH CHART negative real-component coordinates are particularly applicable tothe design and performance analysis of reflection-type oscillators and amplifiers employing,for example, tunnel diodes which function inconnection with waveguides, well up into themicrowave frequency range. The gain andstability characteristics of such devices mayreadily be determined from an input impedance or input admittance plot of operatingparameters, such as frequency and bias voltage,on negative real-component SMITH CHARTcoordinates.

12.3 CONFORMAL MAPPING OF THECOMPLETE SMITH CHART

A set of cartesian coordinates is shown inFig. 12.2(a) whose axes are designated with

p

p

m/zo + jX/Zo) - 1

m/zo + jX/Zo) + 1

(-R/Zo + jX/Zo) - 1

(-R/Zo + jX/Zo) + 1

\ple-je (12-1)

\-1 I -j8P . e

(12-2)


COMPLEX IMPEDANCE COORDINATESNORMALIZED - GENERAL PURPOSE

(PLOTTED ON CARTESIAN COORDINATES)

.t.2

+ JX/Z o

+1

.....-~ -2 -I 0 +1 +2 +~-

-R!Zo + R/lo

-I

-JX/Z o

2

TlIa ll

IWAVEGUIDE IMPEDANCE COORDINATES

(PLOTTED ON REFLECTION COEFFICIENT COORDINATES AT RIGHT)

NEGATIVE SMITH CHART POSITIVE SMITH CHART

WAVEGUIDE REFLECTION COEFFICIENTMAGNITUDE AND PHASE

(PLOTTED ON NORMALIZED IMPEDANCE COORDINATES AT LEFT)

IIb

l!

IWAVEGUIDE REFLECTION COEFFICIENT

MAGNITUDE AND PHASE

(AFTER TRANSFORMATION OF REFLECTION COEFFICIENT COORDINATES ABOVE)

lid II "en

Fig. 12.2. Conformal mapping of complete SMITH CHART.

points of intersection are unchanged. It willbe noted that, in contrast to Fig. 12.2(b), thefinite area occupied by the transformed coordinates of Fig. 12.2(c) includes all possiblevalues of the complex' reflection coefficient.

Using Fig. 12.2(c) as a new coordinategrid, the loci of normalized waveguide impedance components can be plotted from therelationship given in Eqs. (12-1) and (12-2).This results in a complete (positive andnegative real component) SMITH CHARTshown on Fig. 12.2(d).

At this point it is important to recognizethat any graphical representation of the interrelationship of several dependent variableswhich can be plotted on a flat sheet ofpaper is equally valid when viewed fromthe front, or through the paper on whichthe plot is made, i.e., from the back. Accordingly, one is free to choose the moreconvenient of these two representations. Withthis in mind, it will be observed that the lefthalf of the impedance plot of Fig. 12.2(d)(shown as solid' line curves) viewed throughthe paper from the back is identical to theright half, viewed from the front, with thesingle exception that the sign of the normalized resistance component values is negative.It will also be observed that when comparedin this way the sign of the normalizedreactance component is unchanged, i.e., thepositive reactance region of the plot remainson the upper half and the negative reactanceregion of the plot remains on the lower half.Also, the magnitude of the positive reactanceincreases in a clockwise direction and themagnitude of the negative reactance increasesin a counterclockwise direction on bothrepresentations.

Thus, by electing to view the two halves ofFig. 12.2(d) as indicated above, the sameplot may be used to represent either positiveor negative resistance components. However,

, in the same way that one must choose whetherthe SMITH CHART coordinates are to repre-


sent normalized impedance or normalizedadmittance before entering the chart, so musthe also choose whether these same coordinatesare to represent impedances (or admittances)having positive or negative real parts.

The advantages to be derived from acommon SMITH CHART representation ofimpedances or admittances with either positiveor negative real part components are apparentwhen it is appreciated that this permits theapplication of a common technique for thegraphical evaluation of impedance employingstanding wave amplitude and position measurements along a waveguide.

12.4 REFLECTION COEFFICIENTOVERLAY FOR NEGATIVESMITH CHART

As explained above, the conventionalSMITH CHART of Fig. 3.3 or Fig. 8.6 canserve for the representation of impedances oradmittances with either positive or negativereal parts. However, certain important differences exist in the respective complex reflection coefficient and transmission coefficient overlays. As previously stated, thephase angle of the voltage or current reflection coefficient overlay, displayed on theperipheral scale of the above chart and alsoas the family of radial lines on the overlay(Fig. 3.5), is unaltered in magnitude or senseof direction when the coordinates are chosento represent negative real parts. The magnitude of the voltage or current reflectioncoefficient, as represented on this overlay bythe family of concentric circles, applies onlyto SMITH CHART coordinates with positivereal components. For negative real components the voltage or current reflection coefficient magnitudes are represented in theoverlay of Fig. 12.6.

When SMITH CHART coordinates (Fig.12.3) are used for displaying impedances


with negative resistance components, the magnitude of the voltage or current reflectioncoefficient at any point on the chart, represented by the values designated on the concentric circles on the overlay of Fig. 12.6, isthe reciprocal of that shown at the samechart position on the overlay (Fig. 3.5) forthe positive resistance chart. This may beseen from a comparison of Eqs. (12-1) and(12-2). On the positive resistance SMITHCHART coordinates the magnitude of thevoltage or current reflection coefficient (radialscale) progresses linearly from unity at therim of the chart to zero at its center (asshown on the right half of Fig. 12.2(c», whileon the negative resistance SMITH CHARTcoordinates this scale is not linear (as shownon the left half of Fig. l2.2(c». Physically,this simply means that the magnitude of thereflected voltage (or current) waves along awaveguide is greater than the magnitude ofthe corresponding incident waves from thegenerator.

The net power flow along a waveguideterminated in a negative resistance (or negative conductance) is, accordingly, towardsthe generator. Such a termination thus actslike a secondary source of electromagneticenergy whose amplitude is equal to theamplitude of the primary source times themagnitude of the reflection coefficient.

For a given termination mismatch ratiothe standing wave ratio (SWR) accompanyinga negative resistance (or negative conductance)load component is the same as the SWR accompanying a positive resistance (or positiveconductance component) of equivalent amplitude. The sign of the SWR is negative in theformer case, which has no physical significance,and can, therefore, be ignored. Since it is notpossible to distinguish from standing waveratio and wave position measurements whetherthe load resistance (or conductance) component is positive or negative, to resolve thisquestion it may be necessary to measure the

absolute magnitude of the standing wavevoltage and/or current, and to determinefrom these measurements the actual powerflow in the waveguide in relation to themaximum available power output from theprimary source operating into a known positive resistance load. If the power flow in thewaveguide is more than the available powerfrom the primary source, the load impedanceor admittance has a negative resistance (orconductance) component. If the waveguidehas sufficient attenuation, another methodfor resolving this question is to observe,from standing-wave probe measurements,whether the input impedance vs. positionalong the waveguide spirals toward the chartcenter (positive resistance or conductancetermination) or toward the rim (negativeresistance or conductance termination) asthe fields are probed along the waveguidetoward the generator.

12.5 VOLTAGE OR CURRENT TRANSMISSION COEFFICIEI\lT OVERLAY FOR I\IEGATIVE SMITHCHART

The voltage, current, and power transmission coefficient as applied to waveguidesterminated in impedances or admittances withpositive real parts was discussed in Chap. 5.The definitions for these transmission coefficients as given therein apply regardless ofwhether the waveguide is terminated in impedances or admittances with positive ornegative real parts. The transmission coefficient is defined in all cases as the complexratio of the resultant of an incident andreflected quantity associated with the wave(such as voltage, current, or power) to thecorresponding incident quantity.

A voltage (or current) complex transmissioncoefficient overlay for the coordinates in Fig.12.3 is shown in Fig. 12.7. As oriented in


Fig. 12.3. SMITH CHART coordinates displaying rectangular component of equivalent series-circuit impedance (or ofparallel-circuit admittance) with negative real parts.


Fig. 12.7 in relation to Fig. 12.3, it representsthe voltage transmission coefficient on thenegative resistance coordinates or the currenttransmission coefficient on the negative conductance coordinates. When rotated 1800

from this orientation, the overlay of Fig.12.7 represents the voltage transmission coefficient on the negative conductance components or the current transmission coefficienton the negative resistance coordinates of thissame chart. The 1800 phase contour inFig. 12.7 should always be aligned with thecorresponding standing wave minima loci onthe negative impedance or admittance coordinates.

From the transmission coefficient overlayof Fig. 12.7 the shape and relative amplitudesof standing waves vs. SWR may be plottedfor a constant incident wave amplitude. Thisis accomplished by plotting the intercepts ofany desired standing wave circle with thosecircles in Fig. 12.7 which indicate the magnitude of the voltage or current transmissioncoefficient. The construction of standingwave circles on SMITH CHART coordinatesis described in Chap. 3. Standing waves ofthree different amplitude ratios are plottedin Fig. 12.8. Note that in this case, wherethe real part of the waveguide termination isnegative, the absolute value of voltage orcurrent along the waveguide can approachinfinity, as compared to a limiting value oftwice the incident wave amplitude for thecase where the real part of the termination ispositive (compare Fig. 12.7 with Fig. 1.3).

The SWR is unity in the limiting casewhere the transmission coefficient is infinity.

Since, as previously stated, power has no"phase," the power transmission coefficientis the scalar ratio of the transmitted (resultantof incident and reflected) to the incidentpower; this is constant for all positions alonga los~less waveguide regardless of whether thewaveguide termination has positive or negativereal components.

12.6 RADIAL SCALES FOR NEGATIVESMITH CHART

A set of radial scales for negative SMITHCHART coordinates* is shown in Fig. 12.4.The overall length of these scales correspondsto the radius of the negative resistance coordinates in Fig. 12.3 and to the reflectionand transmission coefficient overlays in Figs.12.6 and 12.7, respectively. All but one ofthese radial scales, viz., the voltage (or current)reflection coefficient magnitude scale, areidentical to those described in previous bulletins for use on positive SMITH CHARTcoordinates. (Since the reflection loss scalehas no physical significance it is omitted fromthe radial scales in Fig. 12.4.)

Although the radial scales are the same, aninterpretation of their special significance anduse in connection with negative SMITHCHART coordinates may be helpful. Thiswill be discussed in the following paragraphs.

12.6.1 Reflection Coefficient Magnitude

As previously shown, when used withnegative resistance (or conductance) loadsthe voltage or current reflection coefficientat any point on SMITH CHART coordinatesis the reciprocal of the corresponding valueof voltage or current reflection coefficientwhen the load resistance (or conductance) ispositive; thus, all points along this scale aregreater than unity. A voltage or currentreflection coefficient radial scale suitable foruse with negative SMITH CHART coordinatesis shown on the upper right-hand scale ofFig. 12.4. This scale is used with neg~tive

SMITH CHART coordinates in the same waythat the voltage-current reflection coefficient

*The tenn "negative SMITH CHART coordinates" willhenceforth be used to designate normalized impedance ornormalized admittance coordinates of a SMITH CHARTrepresenting negative real components.

magnitude scale of Fig. 3.4 is used withpositive coordinates.

A particular significance of the radial voltage-current reflection coefficient scale onnegative SMITH CHART coordinates is thatit provides a measure of the voltage gain ofany. negative resistance reflection amplifier.This is described in the last part of thischapter using a SMITH CHART impedance oradmittance representation of the amplifier'sinput characteristics.

12.6.2 Power Reflection Coefficient

As on a conventional positive SMITHCHART, the radial power reflection coefficientscale indicates the ratio of reflected to incident power. Its value at any point on thenegative SMITH CHART is, however, greaterthan unity. Values along this scale equal thesquare of the voltage-current reflection coefficient magnitude at the same point.

The power reflection coefficient scale isshown in Fig. 12.4 adjacent to the scale forvoltage-current reflection coefficient magnitude described above. Unlike the voltagecurrent reflection coefficient the powerreflection coefficient has amplitude only.


Both power and voltage-current reflectioncoefficients are constant throughout a uniform lossless waveguide, but in a waveguidewith attenuation they both increase withdistance toward the load from a minimumvalue at the generator end.

12.6.3 Return Gain, dB

A waveguide with a negative resistanceand/or negative conductance termination hasmore power reflected toward the generatorthan is incident on the load. Consequentlythe return loss is negative, i.e., the waveguidetermination results in a return gain. Thescalar values as obtained from the returnloss scale for positive resistance or conductance terminations (see Chap. 4) are unchanged. However, with negative resistanceor conductance terminations these valuesmust be interpreted to be the gain (in dB)afforded by the termination. It will benoted that this scale is also the powerreflection coefficient in dB.

The return gain scale is shown in Fig. 12.4directly below the scale for power reflectioncoefficient.

8~N

P :; ~ ... N_

i» ':; ij; :;p 0 0 0'0 '" '" N -8-gN 5 ..... N Ii ~ '" ..

~N i» ij; :. N -§ 8 p 00 p 0 0 0

~N PO

~ 1" P .. CD '" ~ :; b l; POb'" p 0 0 0 "

LCENTER I~ TOWARD GENERATOR

C! ~ C!.. N

0 "! ~ ! ~oi

Fig. 12.4. Radial scales for negative SMITH CHART coordinates of Fig. 12.3.


IMPEDANCE OR ADMITTANCE COORDINATESNEGATIVE REA~ PAR~S


_ TOWAltD GENEItATOft

~ ~ ~ ! ~ 9CENTER

Fig. 12.5. SMITH CHART with coordinates of Fig. 12.3 (see Chart 0 in cover envelope).


Fig. 12.6. Complex voltage or current reflection coefficient (overlay for negative SMITH CHART D in cover envelope).


Fig. 12.7. Complex voltage or current transmission coefficient (overlay for SMITH CHART D in cover envelope).

12.6.4 Standing Wave Ratio

The radial scales expressing voltage orcurrent standing wave ratio (both as a ratioand in dB), which are discussed in Chap. 3,are not altered by the fact that the loadre.sisjance or conductance is negative. Slottedline measuring techniques may be appliedin the usual way to waveguides tenninatedin negative resistance or negative conductanceloads. The shapes of voltage or currentstanding waves of several amplitude ratiosare shown in Fig. 12.8. Their shape (aswell as their position along the waveguide)is independent of the sign of the real component of the waveguide input impedance or


admittance. The standing wave ratio scale isthe lowest of the four scales on the left sideof Fig. 12.4.

12.6.5 Standing Wave Ratio, dB

This scale, like the standing wave ratioscale, is unaltered by the sign of the real partof the termination. However, as discussed inChap. 3, the use of the decibel (dB) toexpress standing wave ratio is not in accordancewith the true meaning of this term. The standing wave ratio scale in dB is shown on theleft side of Fig. 12.4 adjacent to the standingwave ratio scale.

'" ..........

"/v "'I "SWR; I.~/ \

\.

-'"V '"r--..... ./

/

/'..... .........

~

2/ '"j:V

" I/ \ /

/ '" /../ V

...... --'" "-V./ ~

V "\// '"/ ~ V

j /

1/ '\1/

6.0

w~ 5.0I...Jo>IZWe 4.0uz

IZ4I-

~ 3.0ou

a:o"-

~ 2.02ot

w>Iotjj 1.0a:

o 0.1 0.2 0.3 0.4 0.5RELATIVE POSITION ALONG WAVEGUIDE - WAVELENGTH

0.6

(Fig. 12.8. Relative amplitudes and shapes of voltage or current standing waves along aIossiess waveguide with negative resistance (or negative conductance) termination (constantinput voltage).


12.6.6 Transmission Loss, 1-dB Steps

The effects of dissipative losses along awaveguide on input impedance or input admittance, when the load resistance or conductance is negative, is opposite to the usualeffect encountered with loads having positivereal parts. With negative resistance or negativeconductance terminations, attenuation in awaveguide results in an outward spiral pathfrom the SMITH CHART center when progressing along a waveguide toward the generator, and an inward spiral path toward thechart center when progressing along the waveguide toward the load. The one-way transmission loss scale divisions are unaltered, butthe designated directions of movement alongthis radial scale (shown on the left side ofFig. 12.4 adjacent to the transmission losscoefficient scale) are reversed from thoseshown in Fig. 4.1.

12.6.7 Transmission Loss Coefficient

This scale shows the percentage increasein dissipative losses in a waveguide due to thepresence of standing waves as compared tothe losses in the same waveguide when transmitting the same power to a load withoutstanding waves. It is the ratio of mismatchedlosses to matched losses. The transmissionloss coefficient scale (shown on the left sideof Fig. 12.4 adjacent to the transmissionloss scale) is unaltered by the sign of the realpart of the termination.

12.7 I\IEGATIVE SMITH CHART COORDINATES, EXAMPLE OF THEIR USE

The representation of the electrical characteristics of a device with a negative resistance (or conductance) component on anegative SMITH CHART can perhaps best be

described with a practical example [13J. Atunnel diode reflection amplifier, which makesuse of the increase in reflected power in anassociated waveguide, will be selected forthis purpose.

The dc voltage-current characteristics of atypical tunnel diode, as shown in Fig. l2-l(a),were discussed earlier in this chapter. Itsnegative resistance region is shown in Fig.12.1 (b). The diode is nominally biased tooperate in the linear portion of its negativeresistance region, near point P. Since atunnel diode reflection amplifier dependsupon negative resistance to provide amplification, only the voltage interval for whichthe resistance is negative is of significance.

12.7.1 Reflection Amplifier Circuit

A basic waveguide circuit for a tunneldiode reflection amplifier is shown in Fig.12.9. An ac signal source, whose internalimpedance is matched to the characteristicimpedance of the waveguide to which it isconnected, is indicated thereon. For thisdiscussion it will be assumed that the internalimpedance of this source is constant over thefrequency band in which the amplifier isoperable. In practice this is not entirelytrue and the extent to which the above

ACSOuRCE

WAVEGUIDE

WAvEGUIDECIRCULATOR

LOAD

Fig. 12.9. Basic waveguide circuit for a tunnel diodereflection amplifier.

assumptions depart from true conditions willaffect the analysis.

The input power from the source to thewaveguide flows along the waveguide, throughthe circulator in the path indicated, throughthe tuner, and into the tunnel diode, whosebias is adjusted to provide a negative resistance.From the diode it is reflected back throughthe tuner into the waveguide toward thesource, around another section of the circulator, and into the load resistance. The loadis effectively isolated from the source by thecirculator if, as will also be assumed for thisdiscussion, its impedance matches that of thewaveguide characteristic impedance over theoperable band of the amplifier, and if thecirculator functions without reflection overthis same band.

The reflected voltage traveling wave whichreturns along the waveguide from the diode tothe load resistance is greater than the incidentwave on the diode from the source, since the

.diode is biased to operate in its negativeresistance region. Thus, the voltage reflectioncoefficient along the waveguide between thecirculator and the diode is greater than unity.The magnitude of the voltage reflection coefficient along this waveguide is a measure ofthe voltage gain of the amplifier. The usualstanding wave measuring technique, whereinthe standing wave ratio and wave position


are plotted on a SMITH CHART, is applicable for determining the negative inputimpedances vs. frequency locus and the related reflection coefficient (amplifier gain).In addition to the gain, the same data yieldsinformation concerning the stability of theamplifier, and the design of a tuner which isnecessary to peak the gain of the amplifier ata selected frequency, as will be seen.

12.7.2 Representation of Tunnel DiodeEquivalent Circuit on NegativeSMITH CHART

The first step in the analysis of thereflection amplifier characteristics is to represent the tunnel diode by its high-frequencysmall-signal equivalent circuit whose elementvalues are normalized to the positive characteristic impedance (or characteristic admittancewhere more convenient) of the waveguidewith which it is used. This representation,with typical values for the equivalent circuitelements, is shown in Fig. 12.10. As withany high-frequency negative-resistance device,there is an unavoidable shunt capacitance Cand series inductance L. These reactanceswill, of course, be frequency dependent,which effect must be taken into account inthe analysis. The small inherent capacitance

~ TOWARD SOURCE

WAVEGUIDE DIODE

___ --------.J1--------- ,1/r---..L..1-----..,

TERMINAL PLANE -IL = .5 nH- - - ----------------...........

Yo = .02 MhoI

02: g(v) 2: - .016 Mho----

IIrs = I Ohm

C =3.0 pF

Fig. 12.10. Small-signal equivalent circuit of a tunnel diode.


C of the diode varies slightly with bias;however, this variation may generally beignored. The diode conductance g (V) is afunction of its bias voltage V. The characteristic admittance Yo of the waveguide (coaxialtransmission line in this case) is assumed to be0.02 mho. Let it be required to plot the abovediode admittance characteristics on a SMITHCHART, at a frequency of 2.0 GHz.

The normalized negative conductance ofthe diode is -0.16/0.20 =-0.8 mho. This isplotted at point A on Fig. 12.11.

The normalized shunt capacitive susceptance + jB /Yo at 2.0 GHz is (21TfC)/Yo = + j 1.9mho. The normalized admittance of thisparallel combination is -0.8 + j 1.9 mho,which value is located at point B. Theeffect of adding 1 ohm of positive seriesresistance rs is next represented; however,before this is done point B, which is onnegative conductance coordinates, is transferred to equivalent negative resistance coordinates at point C. Since the normalizedseries resistance r/Zo = 0.02 ohm is positive,the effect of adding this to the negativeresistance at point C is to move it to point D(a less negative point on the negative resistance coordinates). The addition of normalized series inductance of 0.5 nH, whoseinductive reactance + jXL/ZO at 2.0 GHz is(21TfL>/Zo = 1.26 ohm, to the impedance atpoint D is to move it to point E, which isthe desired normalized impedance of thediode at its input terminals. The equivalentnormalized input admittance Y/Yo =-1.25 +j 2.5 is located at point F.

12.7.3 Representation of OperatingParameters of Tunnel Diode

The effect on the diode's input admittanceof changing two operating parameters, namely,the frequency and the negative conductance

(bias adjustment), can be determined bystanding wave measuring techniques, and theresults can be represented on the negativeSMITH CHART by two intersecting familiesof curves, shown for a typical untuned diodereflection amplifier on Fig. 12.12. Thedashed curves represent loci of input admittances resulting from changes in diode negativeconductance at various operating frequencies.The solid curves represent loci of input admittances resulting from changes in operatingfrequency at various negative conductancevalues established by the bias voltage. Allpoints within the shaded area occupied bythese two families of curves represent aspecific combination of operating frequencyand negative conductance. Point F is transferred directly from Fig. 12.11 to Fig. 12.12.Point go is the diode's negative conductanceat zero operating frequency (point A on Fig.12.11). Point g r is the conductance at theself-resonant frequency, at the diode terminals.Point gco is the diode admittance at theresistance cutoff frequency. This is thefrequency above which any negative-resistancedevice cannot amplify (or oscillate) because itstotal positive resistance equals its negativeresistance, and consequently the real part ofits input impedance equals zero.

From Fig. 12.12 it is possible to observeseveral significant facts concerning the stabilityand gain of this reflection amplifier. Aspreviously discussed, the center point of thenegative SMITH CHART coordinates is thepoint where the reflection coefficient is infinity. An infinite reflection coefficientcorresponds to infinite gain, and infinite gainin a reflection amplifier is the criterion foroscillation. If it is not possible, by means ofadjustment of the operating frequency or theconductance, to include the center point ofthe negative SMITH CHART coordinates(infinite reflection coefficient point) withinthe shaded operating area of Fig. 12.12, theamplifier will be stable. If the amplifier is


Fig. 12.11. Representation of circuit of Fig. 12.10 on negative SMITH CHART at F = 2.0 GHz.



___ TOWARD GENERATOR

;: ~ :::

::: ~ ~ ! ~

g "' .. p ~ ~ b .. - :;" :; .. " Npp o~ '" -

§ug ~:s ~ is 1': '" . '" .. .. :;; .. .. ~0 b C b

o~..

~ ;; fS P ~ 1: " .. '" ~'M ..

b,. 0 0 0 b 0 C

g

Fig. 12.12. Representation of admittance of an untuned reflection amplifier on a ll8lIative SMITH CHART.

operated at a low frequency and the bias isadjusted so that the diode admittance isessentially equivalent to a normalized negativeconductance of 0.8 mho (which conditionprevails at point go on Fig. 12.12) the amplifier gain, as measured on the radial powerreflection coefficient scale, is a maximumand, in this example, is 19.7 dB. At all otheroperating points (specific combination offrequency and bias settings) within the shadedarea the gain is less than 19.7 dB. At point G,for example, where the frequency is 1 GHzthe gain is 6.4 dB.

12.7.4 Shunt-tuned Reflection Amplifier

If it is required to peak the above amplifiergain at a particular frequency such as 1.0 GHz,this may be accomplished by employing atransforming circuit which causes point Gon Fig. 12.12 to move closest to the centerof the negative SMITH CHART. By addinga shunt inductance across the diode's equivalent circuit of Fig. 12.10, for example, pointG may be shifted downward along a line ofconstant conductance, to the zero susceptanceaxis of the chart. The required value ofnormalized inductive susceptance is seen fromthe chart to be - j 1.06 mho. The inductancecorresponding to this susceptance at 1 GHzis readily calculable from the simple relationship L = 1/ jwBYo to be 7.5 nH. All points onthe plot of Fig. 12.12 are similarly shifted bythe effect of this shunt inductance to theirrespective positions indicated on Fig. 12.13.The admittance at point G on Fig. 12.12 isthus transformed to a normalized negativeconductance of -0.92 mho on Fig. 12.13.From the power reflection coefficient scale,the amplifier gain at 1.0 GHz is found to be27 dB. It is peaked at this frequency since

( no other point within the shaded area ofFig. 12.13 is closer to the center of the chart.


Since the reflection gain is less than infinity,the amplifier will not oscillate at 1.0 GHz withany other bias adjustment. However, in thefrequency range between 1.2 GHz and 5.0GHz, as seen from Fig. 12.12, the normalizednegative conductance component of the diodeadmittance is greater than minus one. If, forexample, the diode's admittance was tunedto zero susceptance with a shunt inductancein this frequency range the resultant conductance would place the center of the SMITHCHART coordinates inside an area where thediode is operable. Thus, over a limited rangeof bias settings in this frequency range thediode reflection amplifier would oscillate.

There are, of course, other more complicated circuits, including distributed circuitscomposed of waveguide sections, which willpermit stable operation in the higher frequencyrange over which the diode exhibits negativeresistance or negative conductance characteristics. In one such circuit a shunt capacitanceis located at a position along the waveguidetoward the source from the diode's terminals,where the conductance is of the proper valueto provide the desired amplifier gain (bycontrol of reflection coefficient magnitude).The value of capacitance is adjusted to tuneout the inductive input susceptance of thewaveguide at the chosen frequency, and tothereby peak the amplifier gain at this frequency. Thus, this circuit allows some independent control of gain and operating frequency.

12.8 NEGATIVE SMITH CHART

The general-purpose negative SMITHCHART shown in Fig. 12.5 is reproduced asthe fourth of four translucent plastic chartswhose function is described in the Prefaceand which is contained with the other threein an envelope in the back cover of this book(Chart D).


b N .. :; " :;; .. :;; "" '" ..0 "

,.~ ~

N :;; :;; .. .. -" 0 "

p,. . ~ .. b ~

. Ni)0 " 0 0 0 Co


N:: ..:~p p g b

0 " ~.. .~

,. ,. ! N

o

'"! ~ ~ TOWARD LOAD - --- TOWARD GENERATOR §~8~ ti ~ ~

i~~:~.~'--T~w'-t-'r'---j:a-j-t-'I-'-1....!!-+-+---t-rl'-r/Q..,'-+-,-~..,'c--'T--+""'-r-+--'--'r---r'--,--',--_of~,-,--,l'l--'~'--'--~-':;;~;;;'-'----'-'-WW-'-'--'-'-'-~"---'----'----L----L----'--'_.L-~---'--=-j

1~~lIg~ 2 ~S ~ g

eFig. 12.13. Representation of admittances of a shunt tuned reflection amplifier on a negative SMITH CHART.

13.1 USE CATEGORIES

The many uses of the SMITH CHART mayconveniently be classified as "basic," "speci

fic," and "special." The more common usesin the first two categories, in which proceduresfor solving problems will be evident from thematerial which has been presented in previouschapters, are listed below. These lists arefollowed by descriptions of some of the moreimportant "special" uses of this chart.

13.1.1 Basic Uses

1. For evaluating the rectangular compo-< nents, or the magnitude and angle of the inputimpedance or admittance, voltage, current,and related transmission-reflection functions,at all positions along a uniform waveguide.These related transmission-reflection functionsinclude:

a. Complex voltage and current reflection coefficients.b. Complex voltage and current transmission coefficients.

CHAPTER 13Special

Uses ofSmith Charts

c. Power reflection and transmissioncoefficients.d. Reflection loss.e. Return loss.f. Standing wave loss factor.g. Amplitude of maximum and minImum of voltage and current standingwave, and standing wave ratio.h. Shape, position, and phase distribution along voltage and current standingwaves.

2. For evaluating the effects of waveguideattenuation on each of the above parametersand on related transmission-reflection functions at all positions along a waveguide.

3. For evaluating input-output transfer functions.

13.1.2 Specific Uses

1. For evaluating the input susceptance orreactance of open- or short-circuited waveguide stubs.

2. For evaluating the effects of shunt susceptances or conductances, or series reactances

157


havior of a uniform transmission line insofaras the terminals are concerned [10]. TheSMITH CHART of Fig. 8.6 is well suited to )determining the input impedance characteristics of any four-terminal symmetrical passivenetwork, filter, attenuator, etc.

The three constants which completely determine the operation of any such network are(1) the image impedances Zl1 and Z12 at eachend and (2) the image transfer constant e.The image impedances of a symmetrical network may be thought of as corresponding tothe characteristic impedance of a uniformline. When the network is not symmetrical,it has a different image impedance as viewedfrom each end. In this case the equivalentline is unsymmetrical and has a transformingaction upon the load impedances. Theimage transfer constant, which is the same ineither direction through the network, is analogous to the hyperbolic angle of the equivalent transmission line, the real part corresponding to the attenuation constant and theimaginary part to the phase constant of thetransmission line.

All three of these parameters can be evaluated from the open- and short-circuited impedance of the network according to thefollowing relations:

or resistances, on the input admittance orimpedance, respectively, of waveguides.

3. For displaying or evaluating the inputimpedance or input admittance characteristicsof resonant or antiresonant waveguide stubs,including bandwidth and Q, or inversely, fordetermining bandwidth or Q of waveguideresonators.

4. For design of impedance matching circuits employing single or multiple open- orshort-circuited stubs.

5. For design of impedance matching circuitemploying single or multiple slugs.

6. For design of impedance matching circuits employing single or multiple quarterwave line sections.

7. For displaying loci of passive lumpedcircuit impedance or admittance variationsattending changes in the circuit constants.

8. For displaying loci of waveguide inputimpedance or admittance variations attendingchanges in operating parameters of activeterminating circuits.

9. For converting impedances to equivalentadmittances.10. For converting series to equivalent parallel-circuit representations of impedance oradmittance.11. For converting complex numbers to theirequivalent polar form.12. For converting the series circuit representation of impedance to its equivalentparallel circuit and the parallel circuit representation of admittance to its equivalentseries circuit.13. For obtaining the reciprocal of a complexnumber of the geometric mean between twocomplex numbers. and

(Z' Z' )1/2oe se

(13-1 )

(13-2)

13.2 NETWORK APPLICATIONS (13-3)

It is generally known that the image impedance operation of any four-terminal passivesymmetrical network can be related to the be-

where Zoe and Zse are the impedances at theinput terminals of the network with the

output terminals open- and short-circuited,respectively, and Z~c and Z~c are the impedances at the output terminals with the inputterminals open- and short-circuited, respectively.

To obtain the input impedance, for example,of a four-terminal symmetrical passive networkusing SMITH CHART A, proceed as follows:

1. Normalize the load impedance with respect to the image impedance at the loadterminals of the network Z12 and enter thechart at this point.

2. Move radially toward the center of thechart an amount corresponding to the attenuation constant of the network in dB. (Useradial "Atten. 1 dB Maj. Div." scale at upperright, transversing the proper number of dBsteps.)

3. Move clockwise an amount equal to thephase angle of the image transfer constantusing the outermost peripheral scale labeled"Wavelengths Toward Generator" (one degreeequals 1/360 wavelength) and obtain thenormalized input impedance of the networkby multiplying the normalized input impedance by the image impedance at the inputterminals Z11'

13.3 DATA PLOTTING

Measured or computed data on waveguidecomponents is frequently plotted directly onthe coordinates of the SMITH CHART forthe purpose of analysis and evaluation ofthe characteristics of the device. For example,the envelope of the plotted data may beimportant for evaluation of the overall capabilities of impedance matching devices suchas stubs, slugs, etc. A single asymmetricallypointed tuning plug which is screwed intothe broad wall of a uniconductor waveguidein an off-center position is one type of

( matching device which will serve to illustratethis use of the chart.

SPECIAL USES OF SMITH CHARTS 159

If an asymmetrically pointed plug is allowedto protrude into a match-terminated waveguide it will produce a reflection whosemagnitude and phase will change in somesystematic way as the plug is screwed progressively deeper into the waveguide. A plotof this changing impedance on a SMITHCHART will reveal a scanned pattern of theinput impedance locus at some referenceposition along the waveguide.

In order to determine the envelope ofimpedances at a given frequency within whichany load impedance on the waveguide may bematched to its characteristic impedance withthe asymmetrical tuning plug the problemmay be turned around, for purposes ofgathering data, to one of determining whatresultant impedances can be obtained at afixed reference position along the waveguideas the plug is continuously screwed in. The"reference position" may be at the center ofthe plug or at half-wavelength intervals towardthe generator therefrom. The trace on theSMITH CHART taken by the resultant impedance vector is plotted in Fig. 13.1 for tenturns of the asymmetrical tuning plug configuration illustrated therein.

It can then be assumed that input impedances which are capable of being matchedto the characteristic impedance of the waveguide by this plug will have conjugate valuesto those plotted in Fig. 13.1. Thus, when theplug protrudes into the waveguide its capacitivesusceptance cancels the inductive input susceptance of a conjugate admittance. In itsmost outward position the plug is seen to beequivalent to an inductive susceptance whichaccounts for part of the area scanned lying inthis region of the chart.

The effective diameter of the plug in guidewavelengths will determine the angle of thesector of the impedance envelope on theSMITH CHART which is scanned. This isseen to be roughly 2000 in Fig. 13.1 whichcorresponds to an effective plug diameter of


Fig. 13.1. Locus of impedances resulting from insertion of asymmetrical tuning screw into match-terminated waveguide, ten turns(conjugate impedances may be matched to Z 0)'

200/360 times one-half wavelength, or approximately 0.28 wavelength. The largestcircle which can be inscribed within thescanned impedance envelope, and which iscentered on the SMITH CHART coordinates,in Fig. 13.1 corresponds to a standing waveratio of 1.58 (4 dB), indicating that thistuning plug is capable of eliminating reflections of any phase which would producestanding waves between unity and this maximum value.

The plot in Fig. 13.1 also shows the effectof the pitch of the thread. A finer threadwould cause the plug to advance more slowlyinto the waveguide as it is rotated, and willproduce a larger number of scan lines withinthe same scanned area.

Thus, one example is seen of how the plotting of measured data reveals information concerning the design of a particular waveguidedevice; specifically, in the case illustrated, thisincludes the plug diameter, depth of penetration, and pitch of the threads.

13.4 RIEKE DIAGRAMS

One of the early uses of the SMITH CHARTwas for plotting constant-power and constantfrequency contours of magnetron oscillatorsused in World War II radar equipments.Subsequently, the chart has been widely usedfor displaying these same load properties ofelectron tube oscillators. The reason forusing the SMITH CHART for this purpose isthat the load characteristics of electron tubeoscillators are in general a function of thecomplex load impedance, which is conveniently represented over its entire range ofpossible values by these coordinates.

A plot of the load characteristic of oscillators on a SMITH CHART is called a Riekediagram. A typical Rieke diagram [24,33]consists of two families of curves, one representing contours of constant power and the


other contours ofconstant frequency as shownin Fig. 13.2 for an X-band Klystron oscillatortube which operates in the frequency rangefrom about 8,500 to 9,650 GHz. From aRieke diagram one may select a load impedance which represents the best compromisebetween power output and frequency stability.For example, in Fig. 13.2 such a point isevidently near a normalized load impedanceof 2 - j 0.9 ohms.

In general, on a Rieke diagram the constantpower contours roughly parallel the contoursof constant conductance on a SMITH CHARTwhile the constant frequency contours roughlyparallel constant susceptance contours. Thedegree to which they depart from this generalization is representative of the degree to whichthe equivalent circuit of the oscillator departsfrom a parallel circuit combination of G andjB, G being the parallel combination of thenegative conductance of the oscillator, thepositive conductance of the load, and theelectronic conductance associated with energyconversion within the tube, the summation ofwhich under stable operating conditions iszero. This summation is also assumed to beconstant with frequency.

13.5 SCATTER PLOTS

Another application of the SMITH CHARTwhich is useful for determining differencesbetween intentionally alike circuit componentsis the scatter plot. Such a plot shows theeffects of these differences on any or all of theparameters which the SMITH CHART interrelates. One example is a plot of the measureddistribution of the input and output admittances of an L-band microwave transistor, asshown in Fig. 13.3. The area of the scatterplot can be made proportionately larger byshowing only that portion of the chart whichis of interest.


Fig. 13.2. Rieke diagram on a SMITH CHART for an UHF electron tube oscillator.

13.6 EQUALIZER CIRCUIT DESIGN

The transmission coefficient magnitude andangle scales for the SMITH CHART, as shownin Fig. 8.6, make it practical to design shuntimpedance or series-admittance equalizer circuits therefrom. A rotation of the transmission

coefficient scales with respect to the chart coordinates (which can readily be accomplishedby use of the transmission coefficient overlayof Fig. 5.4) extends the design possibilities ofthe SMITH CHART to include shunt-admittance or series-impedance equalizer circuitdesign. The method has been described in the


Fig. 13.3. Scatter plot on a SMITH CHART of input and output distribution of admittances for a microwave transistor(conductance coordinates, Yo =20 mU).

20 30 402

-,,' i--' -.....f''\.

,..,'~' C \....~

..1',

... -,' \B 1'_ -"\\

-10I 3 4 5 6 8 10

FREQUENCY IN kHz

Fig. 13.4. Typical response characteristics of an electrontube audio amplifier and equalizer circuit; before equalization(A), equalizer response (B), and after equalization (C).

4

2

0

VI.-l -2ILlIII(3 -4ILl<:>

-6

-8

13.6.1 Example for Shunt-tuned Equalizer

The response curve of an electron tubeaudio amplifier whose gain-vs.-frequency characteristic it is desired to flatten or "equalize"is shown in Fig. 13.4 (curve A). Thisparticular amplifier tube has an internal plate

literature [111]. It will suffice here toillustrate one such example of the use of theSMITH CHART.


(13-4)Ro = Rp + Rz

woCRo -1

0 (13-5)Wo (L/Ro)

and off resonance (5 kHz in the aboveexample)

and which in the above example is 4,000ohms.

2. Observe that at resonance (10kHz inthe above example),

along the zero susceptance axis a distancecorresponding to 3.5 dB (the loss needed at10kHz) on the voltage (or current) transmission coefficient (dB) scale in Fig. 5.2.

Note: A count of 3.5 dB from theextreme right-hand end of this scale (6-dBgain point) locates the proper position alongthis scale at its 2.5-dB point, correspondingto point B in Fig. 13.6.

Observe that point B falls on the 2.0 normalized conductance circle. Thus the normalizedconductance of the equalizer circuit must be2.0.

2. Obtain the required value of susceptancein the equalizer circuit which will provide aloss of 1 dB at a frequency of 5 kHz. To dothis, proceed from point B clockwise aroundthe 2.0 conductance circle to the point Cwhere this circle intersects the I-dB contourof the aforementioned transmission coefficient scale. Observe that point C falls on the4.0 normalized susceptance curve. Thus, thenormalized susceptance of the equalizer circuitmust be 4.0.

Finally, the following steps must be takento evaluate the required resistances R, inductance L, and capacitance C of the equalizercircuit elements.:

1. Determine the normalizing value Rowhich is the total loop resistance of the circuit,namely,

Shunt tuned equalizer circuit (in dotted

Independent control of these three equalizercircuit elements makes it possible to accomplish three independent results, viz., (l) totune the equalizer circuit to resonance at anydesired frequency, (2) to add any specifiedloss at the resonant frequency, and (3) to addsome other specified loss at a frequency whichis at a specified frequency off resonance.

In the example under consideration, itwould be desirable (1) to tune this circuitto resonance at 10kHz, that is, the frequencywhere the gain peaks, and to introduce 3.5 dBloss at this frequency, (2) to introduce aloss of 1.a dB at a frequency of 5 kHz.This will then result in an overall loss characteristic for the equalizer circuit as shownin Fig. 13.4 (curve B), which when combinedwith curve A in this figure will result in theflattened characteristic shown in curve C.

The procedure for establishing the correctvalues for each of the three shunt circuitelements from the SMITH CHART is (withreference to Fig. 13.6) as follows:

1. Obtain the required value of normalizedconductance in the equalizer circuit to prerduce 3.5-dB loss. To do this, enter the charton its admittance coordinates at the infiniteadmittance point A, and move to the left

resistance Rp of 3,000 ohms and works intoa load resistance R I of 1,000 ohms. Anequalizer circuit consisting of a parallel combination of resistance, inductance, and capacitance is to be inserted in series with Rand

p

RI as shown within the dotted rectangle inFig. 13.5.

Fig. 13.5.rectangle).

r---------------,I :I I

Rp ' 3000: I\ .-----+---+--t-~IIr_r__~I----.....,

II II Il J



Fig. 13.6. Evaluation on a SMITH CHART of conductance and susceptance of elements of shunt-tuned equalizer circuit.

CUI eRo - 14.0 (13-6)

4 (13-7)

from which we obtain 4.25 x 10-5 (13-8)


The angle a for the tangent and cotangentis projected to the chart coordinate boundaryline (at point L on Fig. 13.7) along a straight 'line from the peripheral transmission coefficient angle scale value to its origin at point0.0. The sine and cosine functions of a arenot directly obtainable from the SMITHCHART but should the need arise theirvalues can be computed from correspondingvalues of the tangent function from thefollowing relationships:

and

5.97 x 10-6

Thus, in the example given,

Ro = 4,000 ohms and GRo = 2.0

from which

1- or R = 2,000 ohmsG

Also, from Eq. (13-8),

C = 0.0106 IlF

and from Eq. (13-9),

( 13-9)

sina

COSa

tan a

1

(13-10)

(13-11)

L = 23.9 mH

13.7 NUMERICAL ALlGI\IMENT CHART

Although not specifically intended for general-purpose arithmetic and trigonometric calculations, the outer circular boundary line ofthe coordinates of the SMITH CHART, andthe resistance axis (with the normalized scaledesignations along it), can conveniently beused as an alignment chart to perform manyof the numerical operations of an ordinaryslide rule.

The more conventional of the mathematical operations are listed in Table .13.1along with corresponding geometrical constructions and numerical results of specificexamples as obtained from Fig. 13.7. Aswith the slide rule, the decimal point can bemoved to accommodate a wide range ofnumerical representations, for example, 4.0on the chart scales also can represent .04, .40,40., 400., etc., provided that the decimalpoint is correspondingly moved in the solution.

The symbol (I) at the intersection of aconstruction line with the resistance axisindicates that the line must always be drawnmutually at right angles to the resistance axis.

13.8 SOLUTION OF VECTOR TRIANGLES

If the magnitude and either the resistanceor reactance component of any impedance isknown, the remaining component can readilybe determined on the SMITH CHART. Thisis possible because of a unique property of aone-eighth wavelength section of losslesstransmission line. If such a line is terminatedin any pure resistance whatsoever the magnitude of its input impedance will equal itscharacteristic impedance.

To make use of this property of an eighthwavelength line in the solution of vectortriangles, a vertical line connecting points+ j 1.0 and - j 1.0 at the periphery of theimpedance coordinates is first drawn acrossthe coordinates of the SMITH CHART, suchas Chart A in the cover envelope. This linewill trace the locus of all input resistance and

reactance components of an eighth-wavelength line which is terminated in any loadresistance from zero to infinity.

For example, suppose that the magnitudeof an impedance vector is known to be 275ohms and its resistance component is 165ohms. The corresponding reactance component is desired. The procedure is to firstnormalize the known input impedance component with respect to the known impedancemagnitude; thus,

~ = 165 = 0.6IZo I 275

The next step is to find X/Zo by entering theSMITH CHART at the point along its resist-


ance axis where R/Zo = 0.6 and then bymoving upward along the 0.6 normalizedresistance circle to the point where it intersects the vertical line representing the oneeighth wavelength position. At this intersection, the normalized value of the desiredreactance component X/Zo is found to be

0.8

If the magnitude of the reactance component had been known and the resistancecomponent was desired, the procedure is quitesimilar. In this case the SMITH CHART isentered at its periphery where X/Zo = 0.8,

Table 13.1. Results of Numerical Operations from Fig. 13.7

OPERATION CONSTRUCTION EXAMPLE

PRODUCT A·a = C 4 .•. 5 = 2.

CIA = a 2./4. = .5QUOTIENT

cia = A 2.1.5 = 4.

110 = E I. 1.33 = 3.RECIPROCAL

liE = 0 1.1 3. = .33

SQUARE F 2 = G 2. 2 = 4.

SQUARE ROOT G1/2 = F 4. 1/2 = 2.

( 1J) 1/2 = H (.5' I. 5 )112 = .87GEOMETRIC MEAN

I/H = H/J 5/.87 = .87/1.5

TANGENT ton a = K ton 30· :: .58

. COTANGENT cot a = L cot 30· = 1.75

ARC TANGENT ton-I K = a ton-I .58 = 30·

ARC COTANGENT-I

L-I 1.75 = 30·cot =a cot


and this reactance circle is then followeddownward to the intersection with the verticalline representing the one-eighth wavelengthposition, where the value of the normalizedresistance component is observed to be

And again, sincecomponent

R 0.6 x 275

Zo 275,

165 ohms

the resistance

0.6

Vector triangles representing voltages, currents, and admittances are similarly solved onthe SMITH CHART.

Fig. 13.7 Constructions for numerical operations on a SMITH CHART as summarized in Table 13.1.

14.1 CLASSIFICATION

,One important class of instruments displaysdata electronically on a SMITH CHART.

Some of these instruments employ a translucent SMITH CHART attached to the face of acathode ray tube; others employ an opaqueSMITH CHART on an electrically operatedplotting table. In either of these the chartserves as the reference coordinate system.Such devices generally sample the amplitudesand phases of forward- and backward-travelingwaves, respectively, at the output ports of adirectional coupler. The samples are translated by sine and cosine function generatorsto control the position of the cathode rayspot, or of the pen on the plotting table. Byusing sweep-frequency oscillators such devicescan display a large amount of data in a briefinterval of time. Their detailed features andoperation are best described by the individualmanufacturer.

Other types of SMITH CHART "inst~

ments" which are described herein are purely

CHAPTER 14SmithChart

Instruments

mechanical in operation, and are thus moreanalogous to the ordinary logarithmic-scaledslide rule.

14.2 RADIO TRAI\ISMISSION LINECALCULATOR

The calculator shown in Fig. 14.1, constructed in 1939, employs a pair of cardboarddisks, and a radial arm pivoted from thecenter. This arrangement allows separatecontrol of the zero position on the outerperipheral scales which were printed on thelarger disk, and the position of the radial armwith its attenuation and SWR scale (expressedas a ratio less than unity). A slidable cursorpermits the establishment of reference pointson the chart coordinates. This calculator,currently out of print, was superceded in 1944by the improved version shown in Fig. 14.2.

The following instructions are printed onthe back of this early "Radio TransmissionLine Calculator":

169


RADIO TRANSMISSION LINE CALCULATORINSTRUCTIONS

The Calculator may be used for open-wire or coaxial transmission lines. It gives the impedance atany position along the line, standing-wave amplitude ratio, and attenuation. If the power is known,the voltage and current at any position along the line is readily calculable from the impedanceinformation obtained.

The curved lines on the central disk are simply resistance and reactance coordinates upon whichall impedances, both known and unknown, can be read. These coordinates indicate series components of the impedances and are labeled as a fraction of the characteristic impedance of the line used.

The separately rotatable scale around the rim of the calculator provides the means for measuringthe distance* along the transmission line between any two points in question. (Any distance inexcess of a half wavelength can be reduced to an equivalent shorter distance to bring it within thescale range of the calculator by subtracting the largest possible whole number of half wavelengths.)

Attenuation in decibel intervals and current or voltage ratio scales are plotted along a rotatableradial arm. This arm and its slider also provide a cross-hair-indicating mechanism.

EXAMPLE A. To find the impedance at any given point along a line when the impedance at anyother point is known:

1. Adjust the cross hair formed by the lines on the radial arm and slider to intersect over theknown impedance point.

2. Rotate only the outer "distance" scale until its 0 point lines up with the line on the radialarm.

3. Rotate only the radial arm (with its slider fixed thereon) the required distance as measuredalong the outer scale, either "towards load" or "towards generator" as the case may be,and read unknown impedance under the new cross-hair location.

*I'or \.:oaxiallincs ",ith insulation. distam:c is in effect increased by the factor /K where J\ = dielcc. canst.

Fig. 14.1. Radio transmission line calculator [101].

SMITH CHART INSTRUMENTS 171

4. To correct for attenuation, move the slider only along the radial arm in the directionindicated thereon the desired number of decibel intervals. before reading the unknownimpedance.

EXAMPLE B. To find the impedance at any given point along a line (with respect to the positionof a current or voltage minimum or maximum point) when the standing-wave amplitude ratio is known:

1. Point the radial arm along the resistance axis in either direction depending upon whethermeasurements are to be made from a minimum or maximum current or voltage point. (Acurrent minimum and voltage maximum occur simultaneously at a resistance maximumpoint, while a current maximum and voltage minimum occur simultaneously at a resistanceminimum point.)

2. Rotate only the outer "distance" scale until its 0 point lines up with the line on the radialarm.

3. Move the slider along the ratio scale on the radial arm to the known standing-wave amplituderatio.

4. Rotate only the radial arm (with its slider fixed thereon) the desired distance from theinitial pure resistance point either "towards load" or "towards generator" as the case maybe, and read the unknown impedance at the new cross hair location.

EXAMPLE C. To find the standing-wave amplitude ratio if the impedance at any point along theline is known:

1. Adjust the cross hair to intersect over the known impedance point and read directly thestanding-wave amplitude ratio on the ratio scale along the radial arm.

EXAMPLE D. To find the attenuation of a line when the input and load impedances are known:1. Adjust the cross hair, formed by the lines on the radial arm and slider, to intersect over the

known input impedance point, noting the position of the slider along the decibel scale onthe arm.

2. Readjust the cross hair to intersect over the load impedance point and again note theposition of the slider along the decibel scale.

3. The attenuation of the line is obtained by counting the number of decibel intervals along theradial arm across which it was necessary to move the slider.

Refer to: "Transmission Line Calculator"by P. H. Smith, Bell Telephone Laboratories,

January 1939, Electronics.

14.3 IMPROVED TRANSMISSION LINECALCULATOR

The all-plastic calculator shown in Fig.14.2 was fIrst constructed in 1944, and hasbeen commercially available [16] withoutalteration since that time. Its design is described in a magazine article [102] by theauthor in 1944 entitled "An Improved Trans-

mISSIon Line Calculator." It provided additional radial scales, shown more clearly inFig. 1.4, and its coordinate system was graduated in a more orderly way. A newer andlarger combined calculator and plotting devicecalled a "computer-plotter" is shown in Fig.14.11 [20].

The instructions printed on the back ofthe calculator in Fig. 14.2 are:

RADIO TRANSMISSION LINECALCULATOR INSTRUCTIONS

The calculator relates the series components of impedance at any position along an open-wire orcoaxial transmission line to (1) the impedance at any other point, (2) the standing-wave amplituderatio (SWR), and (3) the attenuation. It also relates the impedance to the reflection coefficient andto the admittance and, in addition, provides a means for determining the equivalent parallel components of impedance. If the power P is known the voltage and current at any position along the line arereadily calculable from impedance information obtained (see Example F).


All impedances are read on the resistance and reactance coordinates on the central disk. Thesecoordinates are designated as a fraction of the characteristic impedance Zo of the line in order toeliminate this parameter, which is a constant in any given problem.

The rotatable "wavelengths" scale around the rim provides the means for translating impedancesat a given point along the line to another point a given distance* away. (Any distance in excess of ahalf wavelength should be reduced to an equivalent shorter distance to bring it within the scale rangeof the calculator by subtracting the largest possible whole number of half wavelengths.)

Attenuation in 1.0 dB steps and current or voltage SWR scales are plotted on the rotatable radialarm (nearest index) together with scales which are a function of SWR. These include, to the left,scales for minimum and maximum values of current or voltage at node or antinode points (relative tocurrent or voltage on matched line), and the SWR expressed in dB; and to the right, loss coefficient dueto standing waves, reflection loss, and reflection coefficient magnitude. The cross hair index on thearm provides the indicating mechanism for relating all parameters at any given point along the line.EXAMPLE A. TO FIND SWR IF IMPEDANCE AT ANY POINT ALONG LINE IS KNOWN:

1. Adjust cross hairs to intersect over known impedance point and read SWR and relatedparameters on the scales on radial arm.

EXAMPLE B. TO FIND IMPEDANCE AT ANY GIVEN POINT ALONG LINE (WITH RESPECT TOPOSITION OF A CURRENT OR VOLTAGE MINIMUM OR MAXIMUM POINT)WHEN SWR IS KNOWN:

1. Align index on arm with resistance axis in the direction depending upon whether referencepoint is a minimum or maximum current or voltage point. (A current minimum and voltage maximum are always at the same position as the resistance maximum point, i.e., alongresistance axis in direction of R/Zo = 00, while a current maximum and voltage minimumare always at the same position as resistance minimum point, i.e., along resistance axis indirection of R/Zo = 0.)

2. Move slider index along arm to known SWR.3. Rotate only "WAVELENGTHS" scale until its 0 point is aligned with index line on arm.

"For coaxial tines with insulation. distance is effectively increased by Ii< where K = dielectric cons!.

Fig. 14.2. Improved transmission line calculator [102].


4. Rotate only arm (with slider fixed thereon) the desired distance in direction indicated on"WAVELENGTH" scale and read unknown impedance at new cross hair location.

5. Correct above impedance for attenuation by moving slider only along arm the desirednumber of decibel intervals in the proper direction indicated thereon.

EXAMPLE C. TO FIND IMPEDANCE AT ANY GIVEN POINT ALONG LINE WHEN IMPEDANCEAT ANY OTHER POINT IS KNOWN:

1. Adjust cross hairs to intersect over known impedance point.2. Same as B(3), B(4), and B(5).

EXAMPLE D. TO FIND ATTENUATION OF LINE WHEN INPUT AND LOAD IMPEDANCES AREKNOWN:

1. Adjust cross hairs to intersect over input impedance point and note position of slider indexon arm at scales designated "I dB STEPS" and "LOSS (dB)"

2. Readjust cross hairs to intersect over load impedance point and again note position ofslider index on above scales.

3. The number of decibel intervals between the two slider index positions above on scalesdesignated "I-dB STEPS" gives the nominal attenuation of the line due to resistance,etc. To this add the corresponding dB difference on the "LOSS (dB)" scale, which gives theadditional loss due to standing waves.

EXAMPLE E. TO CONVERT IMPEDANCES TO ADMITTANCES AND TO EQUIVALENT PARALLELCOMPONENTSOFIMPEDANCE

1. Equivalent admittances, as a fraction of the characteristic admittance, appear diametricallyopposite any impedance point. To read admittances, consider resistance components tobe conductance components and reactance components to be susceptance components ofadmittance. (Capacitance is considered to be a positive susceptance and inductance anegative susceptance.) No other change in scale designations, direction of rotation, etc., arerequired for using calculator to obtain admittance relations to other parameters.

2. Equivalent parallel components of impedance are the reciprocal of the equivalent admittance values. (Use reciprocal "LIMITS" scale on arm.)

EXAMPLE F. TO FIND VOLTAGE OR CURRENT AT ANY POINT ALONG LINE:

E IR par x P E min =VPs~:o

I /PIR ser Imin =VSWR~ ZO

R par obtained from E(2).

E max = Ip x Zo x SWR

= yP x SWRI maxZo

Refer to "Transmission Line Calculator"by P. H. Smith, Bell Telephone Laboratories

Jan. 1939 and Jan. 1944 Electronics

THE EMELOID CO., INC.HILLSIDE 5, N. J.

14.4 CALCULATOR WITH SPIRALCURSOR

The cursor mechanism shown in Fig. 14.3was devised by R. S. Julian in 1944 at BellTelephone Laboratories. Approximately 100calculators of this type were manufactured butthey were never made commerically available.However, the ingenious design of the dual

logarithmic (equiangle) spiral cursor is worthyof note.

The d~al spiral cursor served several purposes, one of the more important of whichwas to enable the attenuation scale to bemoved from the radial arm to the perimeterof the chart. In this position the scale couldbe expanded in length by a factor of at leastpi. Since the spiral cursor is adjustable with


J

Fig. 14.3. Transmission line calculator [18].

respect to the coordinates, as is also thestraight radial cursor, their intersection can bemade to coincide with any point on the chartcoordinates. Having thus set the two adjustablecursors the peripheral attenuation scale canbe rotated until its zero position correspondsto the position of the straight line segment atthe tip of the spiral. Finally, by rotating thespiral cursor only to a desired setting on theperipheral attenuation scale, the intersectionof the spiral and radial cursors will move a distance corresponding to the change of theattenuation. In this way it is not necessaryto count dB steps of attenuation as isrequired for a fixed radial attenuation scalewith a "floating" zero position.

By using the dual spiral cursor, points onthe chart coordinates which are diametrically

opposite and at equal chart radius from anyselected point can be located without swinging the single radial arm and its slidable cursorthrough 1800

• Thus, with this device impedances can more readily be converted toadmittances.

A minor difficulty with the spiral cursor isthe ambiguity caused by more than oneintersection of the spiral with the straightradial lines.

14.5 IMPEDANCE TRANSFER RING

A manual plotting device called an impedance transfer ring is shown in Fig. 14.4. Thisdevice [122, 213] consists of a circular protractor graduated in wavelengths, and a narrow

diametrical scale (with a slidable cursor) graduated in standing wave ratio. The impedancetransfer ring is intended for use with paperSMITH CHARTS with a coordinate radius of9.1 cm. (Kay form 82-BSPR shown in Fig.8.6.)

The construction and operation of the impedance transfer ring, with specific examplesof its use, is contained in a 29-page U.S.Naval Electronics Laboratory Report [213].A device which can accomplish the sameobjectives as the impedance transfer ring,called the Mega-Plotter and described in the


following paragraphs, is commercially available [14].

14.6 PLOTTING BOARD

A plotting board [204] designed to plotimpedances in the frequency range around 65MHz is shown in Fig. 14.5. Such a device canas well be designed to operate around anyother chosen frequency. This device dependson the fact that the impedance at a pointalong a transmission line is uniquely related

fig. 14.4. The impedance transfer ring and SMITH CHART mounted onmatching board [1221.


to the magnitude and phase relationshipsbetween the voltages Cl' currents along theline at discrete sampling points of knownseparation. (See Chap. lIon probe measurements.) Probe spacings of approximately oneeighth wavelength are used for the plot:ingboard shown, and probe voltages are measuredwith vacuum tube voltmeters.

The impedance at a point in question isdetermined from the SMITH CHART by theintersection of three arcs whose radii areproportional to the three respective probevoltages.

An expanded center SMITH CHART (Kayform 82-SPR shown in Fig. 7.2) is used withthis particular plotter, which enables a determination of the impedance with an accuracyof about 10 percent. The three dials controlthe position of a single pointer to which theyare connected through a system of cords andsprings. The angle between the cords is

TAKE-UP SPRING

FREQUENCYSCALE

FREQUENCYCONTROL

GANGING--t-_...,.,;-LEVERS

Fig. 14.5. Plotting board for SMITH CHARTS [204].

proportional to the phase relationships between the probe voltages, and can be adjustedto compensate for the change in electricallength of the transmission line between fixedprobe positions as the frequency is varied overa limited range. This adjustment is accomrlished for all three cords simultaneously byvarying the position of the pointer shown inthe upper left corner of Fig. 14.5. Thepointer is connected to all three arms on whichthe dia:s are mounted through a pantographwhich controls the angular separation of thecords.

The plotting board shown in Fig. 14.5 isnot commercially available.

14.7 MEGA-PLOTTER

The plotting board shown in Fig. 14.6 iscommercially available [1 4 ] under the trade


Fig. 14.6. Mega-Plotter [14].

name "Mega-Plotter." The separately rotatable scales for this device are shown In

Fig. 14.7.The Mega-Plotter is designed primarily to

facilitate the plotting of electrical transmissionline data on SMITH CHARTS such as Kaychart forms 82-BSPR (SWR = 1.0 to 00) and82-SPR (SWR = 1.0 to 1.59). However, thisplotting board accommodates other commonlyused SMITH CHARTS, and polar reflectioncharts of equal or smaller radius, if they areprinted on standard size (8Y2 x 11 in.) sheets.It may also be used for plotting or reading outdata in polar form from antenna radiation diagrams, and as a general-purpose polar plottingboard.

SMITH CHARTS are held securely in placeon the board with a handy snap-on nickel-silverfastener. A small hollowed point pin piercesthe paper chart center and provides a pivot fora compass (not supplied) and for independently rotatable peripheral and radial scales.

A protractor provides adjustable peripheralscales. On one side these are graduated in

wavelengths for measuring angular distanceson the coordinates of any SMITH CHARTfrom any initial point. On the other side,the scales are graduated in degrees for generalpurpose polar plots.

Two diametrical straightedges are provided,each with two commonly used radial scales ofcorrect length for the above SMITH CHARTS.One side of each of these straightedges isblank, with a matte surface, to permit pencilmarking or pasting thereon any of the morespecialized scales which are printed across thebottom of the aforementioned SMITHCHARTS, or for marking thereon radialscales for charts of other radii.

The straightedge on each of the diametricalscales, like the inside straightedge on theprotractor, is intentionally offset from centerby about the amount a line would be offset ifdrawn with a sharp pencil. When setting theseedges to the desired peripheral scale value, orto coordinate points on the chart, allowanceshould be made for this offset by using thepencil point as a gauge.

...'-I

0.12 0.13 ~ CO,,,,,~0.13 0.12 0./1 0.15

______ 0. 10 n

m,..mn-I:DoZ

n:to.".",..n:to-I

oZVI

o."

-I::E:mVIs:-I::E:

n::E:

0' '~I ~-I

TRANSM. LOSS I

RETURN LOSS DB 0

T~~DGENERATOR- 1.0 DB STEPSI , I I .' I I 1 I I I 1'1 I I I I

10 20 3 a 4.0 ,5.0 6.0 70 8.0 iiW 10.

--- TOIARD LQAD ~.oI I I I I I I I III

12 14. 20. 2.5 0

'2 ,.,30 '. 18 2.0

I 6~O30, ,

00,

'2

5.0, ,15

'0.n

2,0, ~O (~

30 00

VSWR

VSWR IN DB

TRANSM. LOSS

"RETURN LOSS DB 13

TQll.6l..RDGEl'£RATOR _I I I I I I I I

14 15 18 17. ,. 1.0 DB STEPSI I I I I

,. 2<l, , ,

50

11I I 11 I

1.0

Fig. 14.7. Protractor and radial scales for Mega-Plotter in Fig. 14.6.

14.8 MEGA-RULE

The "Mega-Rule" shown in Figs. 14.8,14.9, and 14.10 provides a convenient andrapid means for comparing and evaluating tencommonly used functions which specify transmission and reflection characteristics of electromagnetic waves on waveguides. The tenfunction values are plotted on a set of tenparallel interrelated scales including an attenuation scale printed in red. The voltage reflection coefficient scale thereon is linear, andtherefore is directly related to the radialdistance on a SMITH CHART. The relationship of individual values of these tenscales to each other, and to the attenuation


of the waveguide, is evaluated by means of aslidable cursor. The most commonly used ofthese functions is perhaps the standing waveratio S, to which all others are mathematicallyrelated by the formulas on the back of theMega-Rule.

The Mega-Rule is applicable to any type ofuniform waveguide, for example, uniconductorwaveguide, coaxial cable, strip-line, or parallelwire transmission line. The waveguide mayhave any characteristic impedance. The tenfunctions represented on the Mega-Rule scalesapply to any mode of propagation such asthe transverse electromagnetic (TEM) modein coaxial cables and parallel wire transmissionlines, or any of the many transverse electric

Fig. 14.8. Mega-Rule [20J .

0:: .1> I, I- .1, I. .1. .1. ,. (I) VOLTAGE REFL.COEF:Ow , .011 .0< .06 .O! I, ,.~, 2) POWER REFL. COEF:"-><Jl '~ll:> ,"-' ~. -I· 1. 0 (3) RETURN LOSS. DBo::<tz

I,J' ". " (4) REFLECTION LOSS DBO~O...- 1=1. ,., .. 2 O. ~ . (5) STDG. WAVE LOSS COEF:

j~o I. ,. ,. .. .. .." ". ". >0. I ~o. - (6) STDG. WAVE RATIO DB

WOZI. ,~I '~lJ '.t 'f' '.' '.' 2. ,'~ "",. (7) MAX. OF STDG. WAVE

0::- ::J.~, ..

1.61~ll, .. .. (8) MI N. OF STDG.WAVE((::llL'-

0'" ,., ",

- '.' '.' "2.' '"' .. ').10. 20. "- o. (9) STANDING WAVE RATIO

0--III 11111 111 11 jill IIII111 II ,-+ ATTEN. (MAJ. DIV.=IDB)

"I

Fig. 14.9. Mega-Rule Scale-Front Side.

<J) RATIO OF' (I) REFLECTED TO INCIDENT VOLTAGE - - - - - - - - - - (S-I)/(S+ I)o ll..~ (2) REFLECTED TO INCIDENT POWER - - [IS I)/(S+I)]'0:: o~ (3) I NC I DENT TO REFLECTED POWER IN DB - - - - - - - 20 log,o [(S-I)/(S+I)<l:",w

(4) INCIDENT TO TRANSMITTED POWER IN DB - - -IOIOQ,n(J-[(S-I)/(S+I)]')o z" - - - - - -Z Ow (5) UNMATCHED TO MATCHED GUIDE TRANSMISSION LOSS - (1+ S')/(2 S)~ ~~ (6) MAXIMUM TO MINIMUM OF STANDING WAVE IN DB - 20 lo~,oS<J) z~

(7) MAXIMUM TO MATCHED GUIDE VOLTAGE OR CURRENT (CONST. POWER) VS-"W"w (B) MINIMUM TO MATCHED GUIDE VOLTAGE OR CURRENT (CONST. POWER) I IllSw>WOof (9) MAXIMUM TO MINIMUM OF STANDING WAVE (VOLTAGE OR CURRENT) SW ;<

SCALE RELATING CHANGES IN GUIDE ATTENUATION TO CHANGES IN ABOVE RATIOS. 10 lo~,o[(S-I)/(S+I)] I-

Fig. 14.10. Mega-Rule Formulas-Back Side.


(TEmn) or transverse magnetic (TMmn) modesin uniconductor waveguides.

When using the Mega-Rule for problemsinvolving dissipative loss (attenuation) it isassumed that the waveguide is sufficientlylong so that the loss within a quarter wavelength is a negligible fraction of the total loss.

Individual graduations on the ten scales ofthe Mega-Rule are plotted from data computedto seven significant figures. The values inTable 14.1 may be interpolated should it bedesired to obtain more accuracy than ispossible by visual settings of the cursor onthe Mega-Rule scales.

The three columns in Table 8.1 expressthe transmission-reflection function values forthe ten respective scales on the Mega-Rule interms of:

1. Incident i and reflected r traveling wavesof voltage or current.

2. Voltage or current reflection coefficientmagnitude p.

3. Voltage or current standing wave ratio S.The ten functions with their definitions

printed on the back of the Mega-Rule andrepresented by the ten scales on the front areas follows:

1. Voltage reflection coefficient-the ratioof reflected to incident voltage.

2. Power reflection coefficient-the ratioof reflected to incident power.

3. Return loss in dB-the ratio of incidentto reflected power in dB.

4. Reflection loss in dB-the ratio of incident power to the difference betweenthe incident and reflected power in dB.

5. Standing wave loss factor-the ratio ofmismatched- to matched-guide transmission loss.

6. Standing wave ratio in dB-twenty timesthe logarithm to the base 10 of thestanding wave ratio (special use of dB).

7. Standing wave maximum-the ratio ofstanding-wave maximum to matchedguide voltage (or current).

8. Standing wave minimum-the ratio ofstanding-wave minimum to matchedguide voltage (or current).

9. Standing wave ratio-the ratio of maximum to minimum of the standing waveof voltage or current.

10. A ttenuation- this scale relates the effectof waveguide dissipative losses on allother scales.

All above definitions are printed on theback of the Mega-Rule.

14.8.1 Examples of Use

1. A given length of waveguide connectinga radio transmitter to an antenna has anattenuation (one-way dissipative loss) of 1.3dB at a particular operating frequency. Atthe antenna end of the waveguide, at thisfrequency, the standing wave ratio is observedto be 2.5 (8.0 dB).

It is desired to determine the total lossincurred by the load mismatch and thedissipative losses in the waveguide.

Enter the Mega-Rule by setting the cursorto 2.5 on the standing wave ratio scale (no. 9),or 8.0 dB on the standing wave ratio, dB,scale (no. 6). Observe from scale no. 4 thatthe reflection loss at the antenna (a nondissipative loss defined on the back of theMega-Rule) is 0.88 dB.

Next, slide the cursor 1.3-dB intervals inthe positive direction on the attenuationscale, at the bottom. Now observe fromscale no. 4 that the reflection loss at theinput end of the waveguide is 0.47 dB. Theincrease in dissipative loss, due to a reflectedwave in the waveguide, over the one-waydissipative loss (attenuation) is obtainablefrom the Mega-Rule by subtracting the reflection loss at the input end, that is, 0.47 dB,from the reflection loss at the load end, thatis, 0.88 dB, to yield 0.41 dB. The totaldissipative loss is, therefore, 1.3 dB plus0.41 dB, or 1.71 dB.


Table 14.1 Function Values at SWR Scale Divisions of Mega-Rule.

(1)

VOL.REFL.COEF.

(2 ) (3 ) (4) (5 ) (6 ) (7) (8) (9) (10)

:PWR. RETN REFL. SToo. SToo. STOO. SToo. STW. ATTEN-REFL. LOSS, LOSS, WAVE WAVE, WAVE WAVE WAVE UATIONCOEF. DB DB LOSS DB MAX. MIN. RATIO (I-WAY)

COEF.

0.000000 o.ocnoc~ INF. 0.000000 1.000000 o.ooooeo 1.00000~ 1.000000 1.00 INF.0.024390 O.OO~595 32.255676 O.OV2584 1.001190 0.423786 1.024695 0.97590) 1.05 16.1278380.047619 0.002268 26.444386 0.009859 T;l'-o454S- n o;-8278S4--- 1.04880-9 ---C.95346r -- -1.10 ---B-;~22193--

0.069767 O.0~4867 23.126944 0.021191 1.009783 1.213957 1.072381 0.932505 1.15 11.5634720.090909 0.'08264 2Q.821854 0.036041 1.016661 1.583625 1.095445 0.912911 1.20 10.413~21

0.111111 0.~12346 19.084850 0.053950 1.025000 1.938200 1.118034 0.894427 1.25 9.5424250.130435 0.017013 17.692132 0.074523 1.034615 2. 27886/--r.H71l'-5------O;-8nO;a---l~.o_--l,_~846066----0.148936 0.e22182 16.539996 0.097420 1.04537C 2.606675 1.161895 0.860663 1.35 8~269998

O. 166661 0.027778 15. 563('25 0.122345 1.0511 1,3 Z;92l5b1!-----. .T8nH - --o-:-tr4'5I,4--r;411---';-,sfSD--0.183673 0.033736 14.719071 0.149042 1.069821 3.227360 1.204159 (.830455 1.45 7.3595360.200000 0.040001 13.979400 0.177288 1.083333 3.521825 1.224745 0.816497 1.50 6.9897000.215686 0.046521 13.323550 0.206887 1.097581 3.8"6634 1.244990 0.8C3219 1.55 6.6617750.230769 0.053254 12.736442 0.237667 1. 1125G0 4.082399--1-;-2649n--O~t9-0569----L60 -/;;Y682210.245283 0.060164 12.206650 0.269478 1.128030 4.349679 1.284523 0.778499 1.65 6.1033250.259259 ('.067215 11.725314 O. 302186 1.1441184;Mf8971l---r;30~840---0:766965 1;7ll----s.s6-:16'T--0.272727 0.074380 11.285429 0.335673 1.160714 4.860760 1.322876 0.755929 1.75 5.6427140.285114 0.08163, 10.88[361 1).369836 1.171118 5.105450 1.34[641 0.145356 1.80 5.44j6800.298246 0.08895~ 10.508519 0.404580 1.195270 5.343434 1.360147 0.735215 1.85 5.2542590.310345 0.u96314 10.1631 Ie. 0.439824 1.2131;a---S-;>ts-crz-----r.3?1l4GS----O.'lS4r6-

n

--l-;-90---S-;OUS-55 -C.322034 0.103706 9.841968 0.475494 1.231410 5.800692 1.396424 0.716115 1.95 4.920984C.B3333 0.111111 9.542425 ~.5U525 1.250~6-:-020-599----1.4fi.-ii4---t-;1Ij1l07---z-;-60----~nf:nT-

0.354839 0.125911 8.999380 0.584441 1.288095 6.444386 1.449138 0.690066 2.10 4.4996900.375nco 0.140625 8.519374 ,.658173 1.327273 6.848453 1.483240 0.674200 2.20 4.2596870.393939 0.155188 8.091412 0.732401 1.367391 7.234557 1.516575 0.65938' 2.30 4.045706C.411765 0.169550 7.707018 O.B{\6866 1.408333 7.604225 y;5"9T93---6-.64549r---2-;i;O------3~85H69--

0.428571 ~.183673 7.359536 0.881361 1.450000 7.958800 1.581139 0.632456 2.50 3.67976B0.444444 C.197531 7.0436~C (1. 955117 1.49ZJOS-- 8.2994&6- - r:-612452-----0;62-oT74- - -Z~-611----- -3-;-;-218250.459459 0.211103 6.755056 1.029797 1.53;-185 8.627275 1.643168 0.608581 2.70 3.3775280.413684 0.224311 6.4~1222 1.103492 1.5185/1 8.943160 1.611320 0.591614 2.80 1.z451110.487179 0.237344 6.246220 1.176712 1.622414 9.247960 1.702939 0.587220 2.90 3.123110

-lr.50lrOO~---,O.--.--,2.-.5"'o"'07<Or0-----i6:-'-."0-,,-2,.-06L.{\"'0.-----'I-.....24"'9""3..-i8'"7r--·l--c.6"6"6"6"6-"-7- 9;S,,~-r:-'r3:105 r t-;57n5~-----3;-OO - -);III ncio0.512195 0.262344 5.811291 1.321460 1.711290 9.827234 1.760682 0.567962 3.10 2.905646

-O'-52T810 0.274376 5.616532 1.392886 1.756250 10.102999---r;-iiliflf54-- jj;-s5901r-----j-;zo---Z:e08266--0.534884 0.286101 5.434812 1.463630 1.801515 10.370278 1.816590 0.550482 3.30 2.7174060.545455 0.297521 5.264829 1.533664 1.847059 10.629578 1.843909 0.542326 3.40 2.6324140.555556 0.308642 5.105450 1.602970 1.892857 10.881360 1.870829 0.534522 3.50 2.552725(,.565217 0.319471 4.955690 1.671532 1.938889 11.126049 -----1-.-8-9''-j6i-6~52i6-~b-----3~--O----i~-"7f8~-5--0.574468 0.33e014 4.814682 1.739340 1.985135 11.364034 1.923538 0.519875 3.70 2.407341O. 583333 0.340278 4.681664 1.806389 2.031579 11.595671 -r;Q49359----0.512-989----3.8o---z.340e-n---0.591837 0.350271 4.555962 1.872676 2.078205 11.821292 1.974842 0.506370 3.90 2.277981~.600ijoO ~.360000 4.436975 1.938200 2.125000 12.041199 2.000000 0.500005 4.00 2.2[84870.615385 0.378698 ~.217067 2.066974 2.219048 12.464985 2.049390 0.487950 4.20 2.1085340.629630 0.396433 4.018297 2.192748 2.313636 12.869053 2.097618 0.476731 4:40--2-;O09~

0.642857 0.413265 3.837711 2.315582 2.408696 13.255156 2.144761 0.466252 4.60 1.9188550.655172 0.429251 3.672888 2.435547 2.504167 13.624824 2.190890 0.456435 4.80 ----f;83644,.--0.666667 0.444444 3.521825 2.552725 2.600000 13.979400 2.236068 0.447214 5.00 1.76)9130.692308 0.479290 3.194017 2.834040 2.840909 14.807253 2.345208 0.426401 5.50 1.5970080.714286 0.510204 2.922561 3.099848 3.083333 15.563025 2.449490 0.408248 6.00 1.4612800.733333 0.537778 2.693972 3.351492 3.326923 16.2i;82bf----2;i;495iO----OJ92hZ----~i;o--I--:-3469il-b-

0.750000 0.56250~ 2.498775 3.590219 3.571429 16.901960 2.645751 0.377964 7.00 1.249387----C;:-764706 0.584775 2.330111 3.817166 3.816667 17.501225 2.73!f6n~365I48---f:5-0--T;n5056

0.777778 0.604938 2.182889 4.033350 4.0625CO 18.061799 2.828427 0.353553 8.00 1.0914450.189474 0.623269 2.053247 4.239683 4.308824 18.588378 2.915476 C.342997 8.50 1.0266230.800000 0.640000 1.938200 4.436975 4.555556 19.084850 3.000000 0.333333 9.00 0.9691000.809524 0.655329 1.835408 4.625950 4.802632 19.554472 3.082207 0.324443 9.50 J~l'~0.818182 0.669421 1.743004 4.807254 5.050000 19. 999999---3; 1622f8---Q;)1~22if---rO.OO---;>;e'ris-020.846154 0.715976 1.451013 5.466454 6.041667 21.583624 3.464102 0.288675 12.00 0.7255070.866661 0.751111 1.242959 6.039945 7.035714 22.922559 3.74[657 0.267261 14.00 0.6214190.882353 0.778547 1.087153 6.547178 8.031250 2~.082399 4.000000 0.250~00 16.00 0.5435770.894737 0.800554 0.966094 7.001747 9.027778 25.105449 4.242641 --C~2357~2---n,-;~--O_;_483047-

0.904767 0.818594 0.869314 7.413486 10.025000 26.020599 4.472136 0.223607 20.00 0.4346570.935484 0.875130 0.579274 9.035"21 15.016667 29.542424 --S-:"7722b - O~f82574 30:00----0;2896370.951220 0.904819 0.434385 10.214476 20.012500 32.041199 6.324555 0.158114 40.00 0.2171920.960784 0.9231C6 0.347483 11.141102 25.010000 33.979399 7.071068 0.141421 50.00 0.1737410.967213 0.935501 0.289556 11.904483 30.008333 35.563024 7.745967 0.129099 60.00 0.1447780.971831 0.944455 0.248185 12.553585 35.007143 36.901959 8.36660-o-c~II952y--715:o-0 0.124093---0.975309 0.951227 0.217158 13.118200 40.006250 38.061798 8.944272 0.111803 80.00 ?1085790.978022 ~.95b527 0.193029 13.61780[ 45.005555 39.084849 9.481'-S3Y ·0.TCrS4Cf9--- 90-;CO-----ij:b-96~14

0.980198 0.960788 0.173724 14.065827 50.005000 39.999999 10.000000 0.100000 100.00 0.0868621.000000 1.000000 0.0000(,0 INF. INF. INF. INF. 0.000000 INF. 5.000000

The total10ss incurred by the nondissipativeload-mismatch at the input end of the waveguide (0.47 dB), plus the attenuation of thewaveguide (1.3 dB), plus the increase indissipative loss due to the reflected wave

(0.41 dB), is thus 2.19 dB. Of this total itis possible to recover only the reflection lossat the input end of the waveguide (0.47 dB),by conjugate matching the transmitter impedance to the input impedance of the waveguide


rather than to its characteristic impedance, asis sometimes done.

2. It is required to measure the attenuation(one-way dissipative loss) in a section ofcoaxial cable 50 feet long. For this measurement the cable may be either open-circuitedor short-circuited at its far end, at whichpoint the power reflection coefficient is unity.(This corresponds to a standing-wave ratio ofinfinity.) The standing wave ratio at the inputend of this cable is 2.8, or 9.0 dB.

Enter the Mega-Rule at the extreme rightend, where the power reflection coefficientis unity. The attenuation which the cablewould have if it were terminated in a matchedload is now obtained by sliding the cursor fromthe initial setting to the position where thestanding wave ratio is 2.8 on scale no. 9, or9.0 dB on scale no. 6, and then reading thenumber of dB on the attenuation scale atthe bottom traversed by the cursor. Theresult, in this case, is seen to be 3.25 dBattenuation for the 50-foot length, or 0.065dB per foot.

Alternatively, the attenuation may be obtained from the Mega-Rule by reading theround-trip loss on the return loss, dB, scaleno. 3, and taking one-half of the 6.5 dBvalue read thereon, namely, 3.25 dB.

14.8.2 Use of Mega-Rule withSMITH CHARTS

The voltage reflection coefficient scale(scale no. 1) on the Mega-Rule is linear fromzero at its left end to unity at its right end.Since individual values on all other scales arerelated by the cursor to individual values onscale no. 1, any or all of these scales may beused to determine radial distance on a SMITHCHART to the point where specific scalefunction values apply. The left end of allMega-Rule scales correspond to the center

point of a SMITH CHART; the right endcorresponds to any point around its periphery.

If the radius of the SMITH CHART is notthe same as the length of the Mega-Rule, allcorresponding intermediate positions on thechart and Mega-Rule can, of course, bedetermined by simple proportionality, orgraphically by the use of proportional dividers.

The Mega-Rule is commercially available[20] .

14.9 COMPUTER-PLOTTER

A combined transmission line calculator andplotting board is shown in Fig. 14.11. Therelatively large (lO-in.-diameter coordinates)plotting area is matte finished to permitplotting data directly thereon which can beeasily erased. Ten radial scales as shown onthe Mega-Rule are provided on a radial armwith a notched cursor for drawing circles witha pencil on the coordinates. This newest andlargest SMITH CHART instrument is commerciallyavailable [20].

14.10 LARGE SMITH CHARTS

Large SMITH CHARTS are available commercially in several forms, individually described below. Such charts are particularlyuseful for classroom instruction or for groupdiscussions around a table where viewingdistances are generally too great for theconventional size (8Y2 x 11 in.) charts. It isalso possible to achieve somewhat greateraccuracy with a larger chart.

14.10.1 Paper Charts

Large paper SMITH CHARTS are commercially available from one supplier [15]in 22Y2 x 35 in. pads of approximately 75

sheets each. The charts are printed on arelatively heavy paper in red ink, and havean actual coordinate diameter of 18Y2 in.These are direct enlargements of the formshown in Fig. 3.3, and are the most economiCc1 of the large charts which are available.Eight of the radially scaled parameters, whichhave been individually described herein, areprinted across the bottom of each chart.


Tnese pads have a clipboard backing andmay conveniently be supported on an easel,or laid flat on a table top.

14.10.2 Blackboard Charts

Large cloth SMITH CHARTS (30 x 45 in.}with a black matte finished black background

fig. 14.11. SMITH CHART Computer-Plotter [20].


for erasable chalk marking are also commerciallyavailable [20). (See Fig. 14.12.) Theseare printed with white characters and can berolled up and down on a curtain roller. Theblackboard chart is intended for basic instruction in a large classroom, and consequently designations are in a bold type anda coarse coordinate grid is employed. Thischart is not suitable for accurate solution ofspecific problems.

Fig. 14.12. Blackboard SMITH CHART for classroom use[20].

14.11 MEGA-CHARTS

14.11.1 Paper SMITH CHARTS

Regular size (8Y2 x 11 in.) SMITH CHARTSin the following several forms, each of which

is described herein, are commercially available[ 14]. These "Mega-Chart" forms are printedin red ink on 15 lb. (approximately 7 Ibs/l ,000sheets) translucent master paper, and packagedin clear plastic envelopes of 100 sheets each,either padded or loose:

1. Standard SMITH CHART-Form 82BSPR (9-66) (see Fig. 8.6).

2. Expanded Center SMITH CHARTS-Form 82-SPR (2-49) (see Fig. 7.2).

3. Highly Expanded Center SMITHCHARTS-Form 82-ASPR(see Fig. 7.3).

Also, SMITH CHARTS with coordinateshaving negative real parts (negative SMITHCHARTS) are available in the same paper andpackaging, printed in green ink, in the following form:

1. Negative SMITH CHART-Forrr.. 82CSPR (see Fig. 12.3).

14.11.2 Plastic Laminated SMITH CHARTS

All of the above chart forms (except thenegative SMITH CHART) are available [14]laminated to a thickness of 0.025 in., with amatte finish on the front for erasable pencilmarking. Abbreviated instructions for use ofSMITH CHARTS are printed on the back.

14.11.3 Instructions for SMITH CHARTS

Abbreviated sets of instructions for use ofthe SMITH CHART, containing an explanationof the chart coordinates and radial scales andprinted on single sheets, are available commercially [14] printed on 50 lb offset paper.These may be used for classroom instruction.

GlossarySmith Chart

Terms

The terms which appear on SMITH CHARTS as coordinate designations, radially scaledparameters, peripheral scale captions, etc., are individually defined and reviewed in this

glossary. A more complete discussion of these terms is found in applicable sections of the text.Although relevant to all SMITH CHARTS, these terms are specifically associated with the

basic chart forms printed in Chaps. 6, 8 and 12, and enlarged chart forms described in Chap. 7.Other SMITH CHARTS with which these terms are specifically associated include the normalizedcurrent and voltage overlay in Chap. 4, and the charts with dual (polar and rectangular) coordinatetransmission and reflection coefficients in Chap. 8.

In the definitions which follow, certain qualifying words and phrases are omitted when, in thecontext in which the terms are used, these words and phrases will be understood to apply. Forexample, the phrase "at a specified frequency" will apply to many of the definitions, and thephrases "normalized input impedance of a uniform waveguide" and "normalized input admittanceof a uniform waveguide" will generally be understood to be meant by the shorter terms, "waveguide impedance" and "waveguide admittance," respectively.

This glossary supplements definitions which have been formulated and published by theInstitute of Radio Engineers (IRE) [II] (presently the Institute of Electrical and ElectronicsEngineers, IEEE), and by the American Standards Association (ASA-C42.65-1957) (presentlythe United States of America Standards Institute, USASI), and upon which usage of such termsin this text is based.

Angle of Reflection Coefficient, Degrees

At a specified point in a waveguide, thephase angle of the reflected voltage or currentwave relative to that of the corresponding

r incident wave. The relative phase angle of the

reflection coefficient, i.e., the total anglereduced to a value less than ±1800

, is generally indicated by this term. This relativephase angle has a fixed relationship to aspecific combination of waveguide impedances or admittances and, accordingly, to a

185


specific locus on the impedance or admittancecoordinates of a SMITH CHART, this locusbeing a radial line.

Note: The angles of both the voltage andthe current reflection coefficients are represented on SMITH CHARTS by a single linearperipheral scale, with designated values ranging between 0 and ±1800

• The angle of thevoltage reflection coefficient is directly obtainable for any point on the impedance coordinates by projecting the point radially outwardto this peripheral scale, labeled "Angle ofReflection Coefficient, Degrees." Similarly,the angle of the current reflection coefficientis directly obtainable for any point on theadmittance coordinates. At any specifiedpoint along a waveguide the angle of thecurrent reflection coefficient always lags thatof the voltage reflection coefficient by 1800

•

Angle of Transmission Coefficient, Degrees

At a specified point along a waveguide, thephase angle of the transmitted wave relativeto that of the corresponding incident wave.The "transmitted" wave is the complex ratioof the resultant of the incident and reflectedwave to the incident wave. The angle of thetransmission coefficient has a fixed relationship to a specific combination of waveguideimpedances or admittances and, accordingly,to a specific locus on the impedance or admittance coordinates of a SMITH CHART, thislocus being a straight line stemming from theorigin of the coordinates.

Note: The angles of both the voltage andthe current transmission coefficients are represented on SMITH CHARTS by a single linearangle scale at the periphery, referenced to theorigin of the impedance or admittance coordinates, and ranging between 0 and ±90°.The angle of the voltage transmission coefficient is directly obtainable for any pointon the impedance coordinates by projectingthe point along a straight line stemming from

the origin of the impedance coordinates tothe intersection of the peripheral scale labeled"Angle of Transmission Coefficient, Degrees." )Similarly, the angle of the current transmissioncoefficient is directly obtainable for any pointon the admittance coordinates.

Attenuation (1 dB Maj. Div.)

The losses due to dissipation of powerwithin a waveguide and/or the radiation ofpower therefrom when the waveguide ismatch-terminated. On SMITH CHARTS attenuation is expressed as a ratio, in dB, of therelative powers in the forward-traveling wavesat two separated reference points along thewaveguide.

Note: On a SMITH CHART, "attenuation"is a radially scaled parameter. The attenuation scale is divided into dB (or fraction ofdB) divisions which are not designated withspecific values, with an arbitrarily assignable(floating) zero point. The number of attenuation scale units (dB) radially separating anytwo impedance or admittance points on theimpedance or admittance coordinates of aSMITH CHART is a measure of the attenuation in the length of waveguide which separates the two reference points.

Coordinate Components

The normalized rectangular components ofthe equivalent series or parallel input impedance or admittance of a waveguide or circuit,which are represented on SMITH CHARTS bytwo captioned families of mutually orthogonalcircular curves comprising the coordinates ofthe chart.

Note 1: Coordinate components on thethree SMITH CHARTS printed in red ontranslucent sheets in the back cover envelopeare:

Chart A1. the equivalent series circuit impedance

coordinates: resistance component R/Zo andinductive (or capacitive) reactance component ±j X/ZOo

2. the equivalent parallel circuit admittance coordinates: conductance componentG/Yo and inductive (or capacitive) susceptancecomponent +iB/Yo'

Chart B1. the equivalent parallel circuit impedance

coordinates: parallel resistance componentR/Zo and parallel inductive (or capacitive)reactance component ±jX/Zo'

2. the equivalent series circuit admittancecoordinates: series conductance componentsG/Y0 and series inductive (or capacitive)component +jB/Yo'

Chart C1. the equivalent series circuit impedance

or shunt circuit admittance coordinates withnegative real parts: negative resistance component -R/Zo and negative conductancecomponent -G/Yo'

Note 2: The inductive reactance andinductive susceptance coordinate componentsrepresent equivalent primary circuit elementswhich are capable of storing magnetic fieldenergy only. The resistance component andthe conductance component of the coordinates represent equivalent primary circuitelements which are capable of dissipatingelectromagnetic field energy. The negativeresistance component and the negative conductance component of the coordinates represent equivalent circuit elements which arecapable of releasing electromagnetic fieldenergy, as would be represented by theequivalent circuit of a generator.

Impedance or Admittance Coordinates

The families of orthogonal circular curves( representing the real and imaginary compo

nents of the waveguide or circuit impedance

GLOSSARY-SMITH CHART TERMS 187

and/or admittance, and comprising the mainbody of a SMITH CHART. The designatedvalues of the curves are normalized withrespect to the characteristic impedance and/orthe characteristic admittance of the waveguide,and the entire range of possible values lieswithin a circle. Enlarged portions of SMITHCHART coordinates are sometimes used torepresent or display a portion of the totalarea of the coordinate system, thereby providing improved accuracy or readability.

Note: Most commonly, SMITH CHARTimpedance or admittance coordinates expresscomponents of the equivalent series circuitimpedance or parallel circuit admittance. However, a modified form of SMITH CHARTexpresses components of the equivalent parallel circuit impedance or series circuit admittance. A coordinate characteristic which iscommon to all SMITH CHARTS is that acomplex impedance point on the impedancecoordinates and a complex admittance pointon the admittance coordinates which is diametrically opposite, and at equal chart radius,always represent equivalent circuits.

Negative Real Parts

On the SMITH CHART form in Fig. 12.5, adesignation of the sign of the normalizedresistance component of the impedance, orthe normalized conductance component ofthe admittance coordinates.

Note 1: See "Coordinate Components(Chart C)."

Note 2: A SMITH CHART whose impedance or admittance coordinates are designated with negative real parts is useful inportraying conditions along a waveguide onlywhen the returned power is greater than theincident power.

Normalized Current i/vP/Zo or i/VPYo

The rms current which would exist at aspecified point along a hypothetical waveguide


having a characteristic impedance of one ohm(or a characteristic admittance of one mho)and transmitting one watt of power to a load.This current is the vector sum of the incidentand reflected currents at the point.

Note 1: The actual current at any specifiedpower level in a waveguide is obtainable fromthe normalized current by multiplying it bythe square root of the ratio of the power andcharacteristic impedance (or by the squareroot of the product of the power and characteristic admittance).

Note 2: A plot of normalized currentand/or normalized voltage is provided as anoverlay for SMITH CHART impedance oradmittance coordinates in Fig. 4.2.

Normalized Voltage e/vPZo or e/vP/Yo

The rms voltage which would exist at aspecified point along a hypothetical waveguidehaving a characteristic impedance of one ohm(or a characteristic admittance of one mho)and transmitting one watt of power to a load.This voltage is the vector sum of the incidentand reflected voltages at the point.

Note 1: The actual voltage at any specifiedpower level in a waveguide is obtainable fromthe normalized voltage by multiplying it bythe square root of the product of the powerand characteristic impedance (or by the squareroot of the ratio of the power and characteristic admittance).

Note 2: See Note 2 in definition fornormalized current.

Percent Off Midband Frequency n· M

Captions for peripheral scales near the poleregions on expanded SMITH CHARTS, whichrelate specific values of the frequency deviation, from the resonant or antiresonant frequency, to the impedance or admittancecharacteristics of open- and short-circuited

stub transmission lines n quarter wavelengthslong. M is the deviation from the midbandfrequency in percent.

Peripheral Scales

The four scales encircling the impedance oradmittance coordinates of the SMITH CHART,individual graduations on each of which areapplicable to a straight line locus of points onthe impedance or admittance coordinates.

Note: Each graduation on each of thethree outermost of these scales is applicableto all points on the impedance or admittancecoordinates which are radially aligned therewith; each graduation on the innermost ofthese is applicable to all points on thecoordinates which are in line with the graduations and the point of origin of the impedance or admittance coordinates.

Radially Scaled Parameters

A set of guided wave parameters representedby a corresponding number of scales whoseoverall lengths equal the radius of a SMITHCHART, and which are used to measurethe radial distance between the center andthe perimeter of the impedance or admittance coordinates, at which point a specificvalue of the parameter exists.

Note 1: Radially scaled parameter valuesare mutually related to each other as wellas to a circular locus of normalized impedancesor admittances centered on these coordinates(see Chap. 14, Par. 14.8).

Note 2: The use of radial scales torepresent radially scaled parameter valuesavoids the need to superimpose families ofconcentric circles on the impedance or admittance coordinates which (if all parameterswere thus represented) would completelyobscure the coordinates.

Reflection Coefficients E or I

At a specified point in a waveguide, theratio of the' amplitudes of the reflected andincident voltage or current waves. If thewaveguide is lossless the magnitude of the"Reflection Coefficients E or I" is independentof the reference position. If it is lossy themagnitude will diminish as the referenceposition is moved toward the generator.

Note 1: At any specified reference positionalong any uniform waveguide the magnitudeof the voltage reflection coefficient is equalto that of the current reflection coefficient.

Note 2: On a SMITH CHART the "Reflection Coefficients E or I" is a radiallyscaled parameter.

Reflection Coefficient P

At a specified point in a waveguide theratio of reflected to incident power.

Note 1: In a uniform lossless waveguidethe "Reflection Coefficient P" is independentof the reference position.

Note 2: When expressed in dB the "PowerReflection Coefficient P" is equivalent to the"Return Loss, dB."

Note 3: On a SMITH CHART the "Reflection Coefficient, P" is a radially scaledparameter.

Reflection Coefficient, X or Y Component

In a waveguide, the in-phase or quadraturephase rectangular component, respectively, ofthe "Reflection Coefficients E or I" represented on a SMITH CHART as a rectangularcoordinate overlay. (See Chap. 8.)

Reflection Loss, dB

A nondissipative loss introduced at a discontinuity along a uniform waveguide, such


as at a mismatched termination. "ReflectionLoss, dB" can be expressed as a ratio, in dB,of the reflected to the absorbed power at thediscontinuity and/or at all other points alonga uniform waveguide toward the generatortherefrom.

Note 1: If the input impedance of a lossless waveguide is matched to the internalimpedance of the generator, a compensatinggain will occur at the generator end of thewaveguide. Any difference between the"Reflection Loss, dB" at each end of a waveguide corresponds to the increase in attenuation in a waveguide due to reflected powerfrom the load.

Note 2: On a SMITH CHART "ReflectionLoss, dB" is a radially scaled parameter.

Return Gain, dB

In a waveguide terminated in an impedanceor admittance with a negative real part, theratio in dB of the power in the reflected andincident waves.

Note 1: On a SMITH CHART whoseimpedance or admittance coordinates aredesignated with negative real parts this is aradially scaled parameter.

Return Loss, dB

In a waveguide, the ratio in dB of thepower in the incident and reflected waves.The term "Return Loss, dB" is synonymouswith "Power Reflection Coefficient" whenthe latter is expressed in dB.

Note: On a SMITH CHART this is aradially scaled parameter.

SMITH CHART

A circular reflection chart composed oftwo families of mutually orthogonal circularcoordinate curves representing rectangular


components of impedance or admittance,normalized with respect to the characteristicimpedance and/or characteristic admittance ofa waveguide. Peripheral scales completelysurrounding the coordinates include a setof linear waveguide position and phase anglereference scales. The SMITH CHART alsoincludes a set of radial scales representingmutually related radially scaled parameters.

Note: The SMITH CHART is commonlyused for the graphical representation andanalysis of the electrical properties of waveguides or circuits [25] .

Standing Wave loss Coefficient (Factor)

The ratio of combined dissipation andradiation losses in a waveguide when mismatch-terminated and when match-terminated.

Note 1: A specific value of this coefficientapplies to the transmission losses integratedover plus or minus one-half wavelengths fromthe point of observation, as compared to theattenuation in the same length of waveguide.Thus, spatially repetitive variations in transmission loss within each standing half wavelength are smoothed.

Note 2: On a SMITH CHART this is aradially scaled parameter.

Standing Wave Peak, Const. P

The ratio of the maximum amplitude of thestanding voltage or current wave along a mismatch-terminated waveguide to the amplitudeof the corresponding wave along a matchterminated waveguide when conducting thesame power to the load.


Standing Wave Ratio (dBS)

In a waveguide, twenty times the logarithmto the base 10 of the standing wave ratio (S).


Standing Wave Ratio (SWR)

The ratio of the maximum to the minimumamplitudes of the voltage (or current) along awaveguide.

Note 1: For a given termination, and in agiven region along a waveguide the SWR isidentical for voltage or current. If the waveguide is lossy the SWR will diminish as thepoint of observation is moved toward thegenerator.

Note 2: On a SMITH CHART the SWRis a radially scaled parameter.

Transmission Coefficient E or 1

At a specified point along a waveguide theratio of the amplitude of the transmittedvoltage (or current) wave to the amplitudeof the corresponding incident wave.

Note 1: The "transmitted voltage (orcurrent) wave" is the complex resultant ofthe incident and reflected voltage (or current)wave at the point.

Note 2: On the SMITH CHART the"Transmission Coefficient E or 1" is a linearscale equal in length to the diameter of theimpedance or admittance coordinates andpivoted from their origin. As so plotted, thisscale applies to voltage in relation to impedance coordinates or to current in relation toadmittance coordinates.

Transmission Coefficient P

In a waveguide, the ratio of the transmitted to the incident power. The "Transmission Coefficient P" is equal to unity minus the"Reflection Coefficient P."

Note: On a SMITH CHART the "transmission Coefficient P" is a radially scaledparameter.

Transmission Coefficient, X and YComponents

In a waveguide, the in-phase and quadrature-phase components of the "TransmissionCoefficient E or I" represented on a SMITHCHART as a rectangular coordinate overlay.(See Chap. 8.)

Transmission Loss Coefficient

In a waveguide, the "Standing Wave LossCoefficient (Factor)."

Note: On SMITH CHARTS this is aradially scaled parameter.

Transmission Loss, 1-dB Steps

A term used on SMITH CHARTS toindicate the total losses due to dissipation ofpower within a waveguide and/or the radiationof power therefrom when the waveguide ismatch-terminated. "Transmission Loss" isexpressed as a ratio in dB of the relativepowers in the forward-traveling waves at twoseparated reference points along the waveguide.


Note: As above defined and used onearlier SMITH CHARTS, the term is synonymous with the term "Attenuation (l dB Maj.Div.)" which is used on more recent SMITHCHARTS. The change in designation wasmade to avoid possible misunderstanding ofthe foregoing SMITH CHART usage of theterm which in other usage frequently meansthe total losses when the waveguide is mismatch-terminated. (See "Standing Wave LossCoefficient (Factor).")

Wavelengths Toward Generator (or TowardLoad)

In a waveguide, the relative distances anddirections between any two reference points,represented on SMITH CHARTS by the twooutermost peripheral scales expressing electricallengths in wavelengths from an arbitrarilyselected radial reference locus on the impedance or admittance coordinates.

Note: On SMITH CHARTS with fixedscales, the zero points of these scales arearbitrarily referenced to the position of avoltage standing wave minimum on the impedance coordinates and/or a current standingwave minimum or admittance coordinates. Ona SMITH CHART instrument, in which thesescales are rotatable with respect to the impedance or admittance coordinates, the zeropoint may be aligned with any other referenceposition on the coordinates.

The mathematical relationships of the vari-ous parameters involved in guided wave

propagation are basic to the construction ofthe SMITH CHART, or any other chart whichportrays these relationships. Accordingly,the applicable transmission line equations areincluded herein for those who may desire thisbackground information without the necessityof referring to other sources. Also, theserelationships serve to provide more exactsolutions to specific problems in cases wheregraphical solutions cannot provide the desiredaccuracy or when, for example, a computeris available.

In these formulas the sending-end impedance Zs is the input impedance of a transmission line looking toward the load. Exceptfor its mismatch effect on the transfer ofpower from generator to line (and/or to ioad),the value of Zs is completely independent ofthe impedance looking toward the generatorfrom this position, and is thus the impedancewhich would be seen if the line (or other

rconnections to the generator) were cut off atthis point. Zs is thus the impedance at anyspecified point along the line.

APPENDIX ATransmission

LineFormulas

There are an infinite number of inputor sending-end impedances along any finitelength of transmission line, and the positionof the specified sending-end impedance withrespect to the receiving-end or load impedanceZr must be known in order to evaluate it.The receiving-end impedance, like the sendingend impedance, can be referenced to anyposition along the line provided that it is onthe load side of the specified input impedance position. The length I is the distancebetween Zsand Zr' and must be expressedin the same units as the wavelength A, forexample, meters.

Es and I s are the voltage and current,respectively, at the position corresponding toZs' and represent root-mean-square values ofthe complex sinusoid at this point. Similarly,E r and I r are the voltage and current,respectively, at the position assigned to Zr'P is the propagation constant and Zo is thecharacteristic impedance, as defined in Chap.2.

A computer program for plotting impedanceor admittance data on a SMITH CHART hasbeen written [146,150] in FORTRAN IV to

193


which the reader who may be interested isreferred. An example is given in the referenced article.

The formulas given herein are extracted,with some editing from J. A. Flemming [2]and other sources.

(b) Short-circuited lines

If the receiving end of any uniform transmission line is short-circuited, E

r= 0, and

(A-8)

I r coshPI (A-9)

1. General Relationships for Any Finitelength Transmission Line

and

(A-I) Zo tanhPI (A-lO)

(A-2)

and

2. Relationships for any Finite-length Lossless Transmission Line

E r cosh PI + IrZO sinhPI(A-3)

I r coshPI + (E/Zo) sinhPI

Zo cosh PI + (Zo 2IZ) sinhPI

(ZoIZr) coshPI + sinhPI(A-4)

The following equations for lossless linesare obtained from Eqs. (A-I) through (A-lO)by setting the attenuation constant a equal tozero, in which case P = a + jf3 = + jf3 (seeEq. (2-7)), and by employing the followingrelationships between hyperbolic and circulartrigonometric functions:

coshjf3 cos f3

Two special cases of finite line lengths arepresented by lines which are terminated in (a)open circuits and (b) short circuits, viz.,

sinhjf3 j sin f3

(a) Open-circuited lines and

(A-II)

(A-12)E

I r cosf31 + j _r sinf31Zo

(a) Lines terminated in an impedance

If the receiving end of any uniform losslesstransmission line is terminated in an impedance Zr

tanhjf3 = j tanf3

(A-7)

(A-6)

(A-5)

Zo cothPI

Er coshPI

If the receiving end of any uniform transmission line is open circuited, Ir = 0, and

and

APPENDIX A 195

and

E r cosf31 + jZo IT sinf31

IT cosf31 + j(E/ZO) sinf31(A-13)

or

- jZo cotf31 (A-18)

where Ys and Yo are, respectively, the sendingend admittance and the characteristic admittance of the line.

From Eq. (A-16) it can be seen that inlossless open-circuited lines the receiving-endvoltage Er varies between the sending-endvoltage Es and infinity as the length of line isvaried. The receiving-end voltage is equal tothe sending-end voltage when I is 1/21.., 3/21..,etc., and is infinite when I is 1/41..,3/41.., S/4A,etc.

(A-I 9)jYo tanf31

(A-IS)

(A-14)Z, + jZO tanf31

ZO-----Zo + jZT tanf31

(Z/Zo) (l + tan2 f3l)

1 + (Zr2/Zo2) tan2 f31

Rationalizing Eq. (A-14)

or

(c) Short-circuited lines

(b) Open-circuited lines

If the receiving end of any uniform losslesstransmission line is open circuited, IT = 0, and

By substituting real values for Z/Zo in Eq.(A-IS), the loci of the normalized inputimpedance components along a transmissionline, as represented graphically along a circleof constant standing wave ratio on a SMITHCHART, can readily be obtained.

If the receiving end of any uniform losslesstransmission line is short-circuited, E r = 0, and

(A-23)

(A-22)

(A-2l)

(A-20)

Since the tangent and the cotangent of agiven angle always have the same sign, it ispossible to change the input impedance of a

Es jZo IT sin f31

Is ITcos f31

and

Z =

E sjZo tanf31s

Is

or

Ys

Is- jYo cotf31

E s

(A-16)

(A-I?)

Es

Er

sec f31

EIs j_r sinf31

Zo

and


line which is capacitive when open circuited atits receiving end into an inductance by shortcircuiting its receiving end. It is also possible

to change an inductive open-circuited line intoa capacitive line by short-circuiting its receivingend.

APPENDIX BCoordinate

Transformation

BILINEAR TRAI\ISFORMATION where

and

Rv =

Zo

Similarly let each point on the circular chartbe denoted by

(B-1)

As will be shown herein, a conformal transformation can be applied to the curves on

the rectangular transmission line chart in Fig.1.2 in order to obtain the more convenientcircular form shown in Fig. 1.3. When thelatter figure is rotated 90° counterclockwisefrom the orientation shown (see Sec. 1.4), thetransformation whose general form is

aZ + b

cZ + dw =

will be found to give the desired result.By assigning the proper values to the constants a, b, c, and d, the axis of X/Zo may betransformed into a circle of any convenientradius, and the entire chart will then lie withinthis circle. Each of the circles correspondingto a particular value of D will become adiameter of the new boundary circle and allof the other circles or straight lines in therectangular chart will become circles or arcsof circles in the circular chart.

In order to perform such a transformationlet each point on the rectangular chart bedenoted by a complex number

w = u + jv

Z = x + jy

Then the following conditions may be set up:The u axis is to be transformed into a circleof radius A whose center lies on the y axis adistance A above the origin. At the same time,the point (u = 0, v = 1) is to be transformedinto the center of this circle. These conditionsfix the transformation of the following points:when

w = 0 + jO Z o + jO

w = 0 + j1.0 Z o + jA (B-2)

w = +00 + jO Z o + j2A

197


Substituting in Eq. (B-1), the transformationbecomes

x2 + r _2 Ay (1 + 2 v)

(1 + v)

_4A2 v(1 + v)

w-jZ

Z - j2A(B-3)

When v is constant, this is the equation of acircle in the Z plane. By adding the quantity(1 + 2 v)2 A2 / (1 + v)2 to each side, the following more useful form is obtained:

a. The Lines v =a Constant

Either one of these equations makes it possible to determine the location of the pointin one chart which corresponds to any givenpoint in the other. However, it would bemore convenient to have the equations of thecurves in the circular chart which correspondto the circles and the coordinate system inthe rectangular chart. These equations willtherefore be derived.

This may also be written

(B-6)x2 + Y _ (1 + 2 v) A2

(1 + v)

From Eq. (B-6) it is evident that the radiusof the circle is A/(1 + v) while its center islocated on the y axis at a distance[(1 + 2 v)/(l + v)] A from the origin. Sincev = R/Zo' the circles corresponding to variousvalues of R/Zo can now be drawn. It will benoticed that the circle for R/Zo = 0 hasradius A and center at (x = 0, y = A). Forhigher values of R/Zo' the radius of the circlebecomes smaller and its center rises until forR/Zo = 00 the circle is of zero radius and itscenter is at (x = 0, y = 2A).

b. The Lines u =a Constant

(B-4)j2Aw

w + jZ

The lines v = a constant comprise thosestraight lines parallel to the u axis in theoriginal chart. To find what these linesbecome in the circular chart it will benecessary to substitute for wand Z in Eq.(B-3) the complex expression (u + iv) and(x + iy). Then the reals and imaginaries maybe separated to give

The ordinates of the rectangular coordinatesystem correspond to various constant valuesof u. Consequently, the equation of thecurves to which they are transformed canbe obtained by eliminating v from Eqs. (B-S).When this is done, there results

ux - vy - y + 2 Av

+ j (uy + x + vx - 2 Au) = 0

To satisfy this equation, the real and imaginary parts must be separately zero. Thus,

xu - (y - 2 A) v y

For each value of u this represents a circleof radius 2A/u, whose center is at(x = A/u, y = 2 A). Thus these circles are alltangent to the y axis and their centers are allon the line y = 2 A which is parallel to thex axis and tangent to the circle for v = O.Since u = X/Zo the circle for each value ofX/Zo may readily be drawn, its radius andcenter now being determined.

(B-7)x2 + (y _ 2 A)2 = 2 ~ xw

(B-S)-x

Eliminating u,

(y - 2A) u + xv

APPENDIX B 199

c. The Circles of ConstantElectrical Line Angle

Each of the circles, on the rectangularchart, corresponding to a particular valueof D will transform into a diameter of thecircle whose center is at (x = 0, y = A). Thesestraight lines may be represented by

This equation is identical to the equation forthe lines of force about two equal and oppositeparallel line changes. Consequently <(J is proportional to the parameter D and thus isdirectly proportional to the electrical angleof the line.

y=mx+A (B-8)d. The Circles of Constant Standing

Wave Ratio

x (mu + v + 1) = Au

x(u - mv - m) = A (1 - v)

where m is the slope. Substituting Eq. (B-8)in Eqs. (B-5):

xu - (mx - A) v

(mx - A) u + xv

or

Dividing,

mx + A

-x

Since the transformation is conformal, andsince the circles of constant standing waveratio T are orthogonal to those of constant D,these two systems must be orthogonal in thecircular chart. But the circles of constant Dturned out to be straight lines, all passingthrough one point. Hence the circles of constant T must transform into circles having acommon center at the intersection of the setof straight lines. Also, the value of T may bedetermined by determining the scale of R/Zowhich is to be plotted along the y axis. Thisis obtained directly from Eq. (B-4) by puttingx = °and u = O. Then

(u - mv - m) u (mu + v + 1) (l - v)

ory

2Av

v + 1

(B-9) or, putting v = R/Zo

1

Thus for

o

2A

A

o

y

y

y

2A<R/Zo)

<R/Zo) + 1

R- = 00

Zo

ywhich, when m is a constant, represents acircle in the rectangular chart. The radiusis vI + m2 and the center is on the u axis adistance m from the origin. When u = 0,this equation gives v = ±1, which shows thatall the circles pass through the required point(u=O,v=+l).

Now the slope is the tangent of the anglewith the horizontal or is the cot cp where <{J isthe angle between the y axis and the diameterin question. Thus Eq. (B-9) becomes

r


and so on. Also, the point XIZo = 0,RIZo = 1/2 and becomes y = 2/3 and thepoint XIZo = 0, RIZo = 2 becomes y = 4/3.

Hence, these two points lie on a circle ofradius 1/3 on the circular chart. On the

rectangular chart they were also on the samecircle, that for r = 0.5. Thus having determined the scale for RIZo in the circular ~hart,

the radii of the circles of constant r are automatically determined.

Symbols

The following is a list of symbols used throughout this book. Where more than a singledefinition is given for the same symbol the proper choice will be evident from its usage in the

text. Voltage and current symbols, except where specifically stated, indicate the root meansquare value of the alternating sinusoid.

B

-Bp

c

D

dB

Susceptance, mhos

Susceptance of an equivalentparallel circuit element

Susceptance of an equivalentseries circuit element

Capacitive susceptance of anequivalent parallel circuit element

Inductive susceptance of anequivalent parallel circuit element

Capacitive susceptance of anequivalent series circuit element

Inductive susceptance of anequivalent series circuit element

Normalized susceptance component of admittance

Capacitance, farads

Distance from a voltage maximum or minirimm

dBS

E

e

e

(

f

Standing wave ratio in decibels=20 loglOS

Voltage (rms sinusoid) in volts

Voltage at a point along astanding wave

Voltage at receiving end of awaveguide

Voltage at sending end of awaveguide

Voltage at a standing wavemaximum

Voltage at a standing waveminimum

Potential at any distance x

from the sending end of awaveguide at time t

Base of natural logarithms

Base of natural logarithms

Normalized waveguide voltage(rms)

Normalized waveguide voltage(rms)

Frequency, hertz201


G

Gmax

g

Hz

I

Midband frequency of a resonant (or antiresonant) circuitor waveguide

Conductance, mhos

Conductance at avoltage standing wave minimum

Conductance of a resonantwaveguide

Conductance at a voltage standing wave maximum

Conductance of an antiresonant waveguide

Conductance of an equivalentparallel circuit element

Conductance of an equivalentseries circuit element

Equivalent circuit conductanceof an active device

Equivalent circuit conductanceof an active device at resistancecutoff frequency

Equivalent circuit conductanceof an active device at zerooperating frequency

Equivalent circuit conductanceof an active device at its selfresonant frequency

Normalized conductance component of admittance

Cycles/second

Current (rms sinusoid), amperes

Current at receiving end of awaveguide

Current at sending end of awaveguide

Current at a standing wavemaximum

Current at a standing waveminimum

(s

iI<PY )112o

L

L

lore

n

P

Current at a point along awaveguide

Current at a point one-quarter 'wavelength removed from thevoltage along a waveguide

Current at any distance x fromthe sending end of a waveguideat time t

Amplitude of incident travelingwave (voltage or current)

Normalized waveguide current(rms)

Normalized waveguide current(rms)

Inductance, henries

Electrical length of a waveguidematching stub of characteristicimpedance Zo

Electrical length of a waveguidematching stub of characteristicimpedance Zos

Electrical length of an individual waveguide slug

Total electrical length of amultiple waveguide slug transformer

Electrical length of a sectionof waveguide

Number of integral quarterwavelengths in a resonant (orantiresonant) waveguide

Complex propagation constantof a waveguide Cl + jf3

Power, watts

Relative power, watts

Reference power, watts

In a pair of voltage (or current)sampling probes along a waveguide, the probe nearest thegenerator

Q

R

R

Rz or Re

Rr

T

T

Ts

s

In a pair of voltage (or current)sampling probes along a waveguide, the probe nearest theload

Ratio of midband frequencyand half-power bandwidth

Resistance, ohms

Resultant amplitude of incident and reflected travelingwaves (voltage or current)

Resistive component of loadimpedance

Resistance at a voltage standingwave maximum

Resistance of an antiresonantwaveguide

Resistance at a voltage standingwave minimum

Resistance of a resonant waveguide

Resistive component of characteristic impedance

Resistance of an equivalentparallel circuit element

Resistive component of receiving-end impedance

Resistance of an equivalentseries circuit element

Normalized resistance component of impedance

Amplitude of reflected traveling wave (voltage or current)

Standing wave ratio when lessthan unity

Series resistance of small-signalequivalent circuit of an activedevice

Electrical spacing between twovoltage (or current) samplingprobes along a waveguide

s

SWR

vV

V.min

VSWR

v

wX

X

X

x

X'

SYMBOLS 203

Standing wave ratio (voltageor current)

Standing wave ratio (voltageor current)

Maximum value of the standingwave ratio

Time, seconds

Voltage (rms of sinusoid)

DC bias voltage on an activedevice

Relative voltage

Reference voltage

Maximum voltage along a waveguide

Minimum voltage along a waveguide

Voltage standing wave ratio

Phase velocity of propagationin a uniform waveguide,meters/second

Power, watts

Reactance, ohms

In-phase component of angleof power factor

In-phase component of complex voltage (or current) transmission or reflection coefficient

Input reactance of waveguidestub of characteristic impedance Zo

Input reactance of waveguidestub of characteristic impedance ZOB

Capacitive reactance, ohms

Inductive reactance, ohms

Reactive component of loadimpedance

Reactive component of characteristic impedance


reactance of anseries circuit ele-

reactance of anseries circuit ele-

+xp

-xp

x

y

y

y

Yz or Y£

Ymax

Ymin

Reactive component of anequivalent parallel circuitInductive reactance of anequivalent parallel circuit element

Capacitive reactance of anequivalent parallel circuit element

Reactive component of receiving-end impedance

Reactive component of sending-end impedance

Reactance of an equivalentseries circuit element

Inductiveequivalentment

Capacitiveequivalentment

Normalized reactance component of impedance

Distance along a waveguidefrom the sending end

Admittance, mhos

Quadrature-phase componentof complex voltage (or current)transmission or reflection coefficient

Quadrature-phase componentof angle of power factor

Load admittance of waveguide

Maximum admittance along awaveguide

Minimum admittance along awaveguide

Admittance of an equivalentparallel circuit

Admittance of an equivalentseries circuit

Z

Zz or Z~

Zmin

Characteristic admittance of auniform waveguide, mhos

Normalized admittance of auniform waveguide

Impedance, ohms

Image impedance at the inputport of a two-port symmetricalpassive network

Image impedance at the outputport of a two-port symmetricalpassive network

Load impedance of waveguide

Maximum impedance along awaveguide

Minimum impedance along awaveguide

Characteristic impedance of auniform waveguide, ohms

Input impedance of a two-portsymmetrical passive networkwith output port open-circuited

Characteristic impedance of awaveguide stub

Impedance of an equivalentparallel circuit

Receiving-end impedance of awaveguide

Sending-end impedance of awaveguide

Input impedance of a two-portsymmetrical passive networkwith output port short-circuited

Characteristic impedance of atransforming section of waveguide (slug)

Normalized impedance of auniform waveguide

a

f3

f3

f

P

PI

Px

Attenuation constant of awaveguide, nepers/unit lengthAngle of voltage (or current)reflection coefficient

Phase constant of a waveguide,radians/unit length

Angle of voltage (or current)transmission coefficient

Dielectric constant, relative toair

Wavelength, units of electricallength

Phase of voltage (or current) ata point along a standing waverelative to that at nearest minimum

Voltage insertion phase

Current insertion phase

Voltage (or current) reflectioncoefficient

Voltage reflection coefficient

Voltage reflection coefficientvector

In-phase component of voltage

Py

T

()

SYMBOLS 205

(or current) reflection coefficientQuadrature-phase componentof voltage (or current) reflection coefficient

Voltage (or current) transmission coefficient

Current transmission coefficient

Voltage transmission coefficient

In-phase component of voltage(or current) transmission coefficient

Quadrature-phase componentof voltage (or current) transmission coefficient

Image transfer constant in atwo-port passive symmetricalnetwork

Angle of the impedance vector(angle of the power factor)

217 times the frequency f

References

References cited in the text which are not directly related to theSMITH CHART are included in numbers I through 20, below. Thenumbering system for the "Bibliography of SMITH CHART Terms,"which follows, starts with reference number 21 to avoid duplication,and continues with some omissions through number 219.

1. Encyclopaedia Britannica, Inc., vol. 2, pp. 639-640, William Benton,Publisher, University of Chicago, 1966 (since 1768).

2. Flemming, J. A.: "The Propagation of Electric Currents in Telephone and Telegraph Conductors," D. Van Nostrand Co., Inc.,New York, 1911.

3. Kennelly, A. E.: "Tables of Hyperbolic and Circular Functions,"2nd ed., Harvard University Press, 1921.

4. Bell System Tech. J., January, 1929.5. Sterba, E. J., and C. B. Feldman: Transmission Lines for Short

Wave Radio Systems, Proc. IEEE, July, 1932, pp. 1163-1202.6. Gunther, R. T.: "Astrolabes of the World," vol. II, Oxford Uni

versity Press, Inc., New York, 1932.7. Hqrold, E. W.: Negative Resistance and Devices for Obtaining It,

Proc. IRE, October, 1935, pp. 1201-1223.8. Smith, P. H.: High Frequency Transmission System, U.S. Patent

2,041,378, May 19, 1936.9. Carter, P. S.: Charts for Transmission Line Measurements and

Computations, RCA Rev., vol. 3, pp. 355-368, January, 1939.10. Terman, F. E.: "Radio Engineers Handbook," 1st ed., McGraw

Hill Book Company, New York, 1943.II. Standards on Waveguides - Definitions of Terms, 53 IRE 2.SI,

1953; also, "IRE Dictionary of Electronic Terms and Symbols,"1961.

12. Kyhl, R. L.: Plotting Impedances with Negative Resistance Components, IEEE Trans. G-MTT (Correspondence), vol. MTT-8, no.3, p. 377, May, 1960.

207


13. Trombaru1a, R. F.: Bell Telephone Laboratories technical memorandumMM 63-1347-6, February, 1963.

14. Kay Electric Company, 14 Maple Avenue, Pine Brook, N.J. 07058.15. General Radio Company, 22 Baker Avenue, West Concord, Mass.

01781.16. The Eme10id Company, 1239 Central Avenue, Hillside, N.J.17. Bell Telephone Laboratories, Whippany Road, Whippany, N.J.

07981.18. Bell System Tech. J., January, 1929.19. Smith, P. H.: L-Type Impedance Transforming Circuits, Elec

tronics, vol. 15, pp. 48-52, 54, 125, March, 1942 (also BellTelephone Laboratories Monograph B-1335).

20. Analog Instruments Company, P. O. Box 808, New Providence,N.J. 07974.

Bibliography forSmith ChartPublications

Much has been written on the theory, construction, and applicationof the SMITH CHART. However, the scope of individual publicationsis generally limited and considerable redundancy exists, but taken collectively, it constitutes an impressive mass of technical literature on asingle subject.

For convenience, this bibliography is divided into three principalparts. In all three parts references are made only to sources dealing insome significant measure with one or more aspects of the chart.

Part I lists books, including encyclopedias, reference handbooks, andtextbooks. These are numbered, respectively, 21 through 26, 31through 40, and 41 through 83. Part II lists articles published indomestic and foreign periodicals, numbered 101 through 150. PartIII lists nonperiodical-type bulletins, reports, etc., which are numbered201 through 219.

With a few exceptions, all references are listed chronologically. Thereader can estimate the extent of material presented from the referencednumber of pages.

PART I-BOOKS

Encyclopedias

21. Freiberg, W. F. (ed.): "The International Dictionary of AppliedMathematics," p. 851, D. Van Nostrand Co., Inc., Princeton,N.J., 1960.

22. Michels, W. G. (ed.): "The International Dictionary of Physicsand Electronics," 2nd ed., p. 1094, D. Van Nostrand Co., Inc.,Princeton, N.J., 1961.

209


23. Ramsey, H. B., in J. Thewlis (ed.): "Encyclopedia Dictionary ofPhysics," vol. 6, pp. 511-513, Pergamon Press, Inc., New York,1962.

24. Ska1nik, J. G., in C. Susskind (ed.): "Encyclopedia of Electronics," pp. 728-729, Reinhold Publishing Corp., New York, 1962.

25. Smith, P. H., ibid., pp. 760-763.26. Markus, J.: "Electronics and Nucleonics Dictionary," 3d ed.,

p. 605, McGraw-Hill Book Company, New York, 1966.

Handbooks

31. Moreno, T..: "Microwave Transmission Design Data," pp. 41-51,McGraw-Hill Book Company, New York, 1948; reprinted byDover Publications, Inc., New York, 1958.

32. Marcuvitz, N. (ed.): "Waveguide Handbook," M.LT. RadiationLaboratory Series, vol. 10, pp. 10-11, McGraw-Hill Book Company, New York, 1951; reprinted by Boston Technical Publications, Cambridge, Mass., 1964.

33. Pender, H. A., and K. McIlwain (eds.): "Electrical Engineers'Handbook," 4th ed., pp. 4-56, John Wiley & Sons, Inc., NewYork, 1950.

34. Starr, A. T.: "Radio and Radar Techniques," pp. 120-122, SirIsaac Pitman & Sons, Ltd., London, 1953.

35. Westman, H. P. (ed.): "Reference Data for Radio Engineers,"4th ed., International Telephone and Telegraph Corporation,pp. 586, 587, 652, printed by American Book-Stratford Press,New York, 1956.

36. Fink, D. G. (ed.): "Television Engineering Handbook," pp. 1149 to 11-51, McGraw-Hill Book Company, New York, 1957.

37. Walker, A. P. (ed.): "NAB Engineering Handbook," 5th ed.,pp. 9-26 to 9-30, McGraw-Hill Book Company, New York,1960.

38. Jasik, H. (ed.): "Antenna Engineering Handbook," 1st ed., pp.31-4,34-8,34-9, McGraw-Hill Book Company, New York, 1961.

39. Birenbaum, L., in M. Sucher (ed.): "Handbook of MicrowaveMeasurements," vol. 3, pp. 1103-1132, Polytechnic Press, Polytechnic Institute of Brooklyn, Brooklyn, New York, 1963.

40. Kellejian, R., and C. L. Jones: "Microwave MeasurementsManual," pp. 95-122, McGraw-Hill Book Company, New York,1965.

Textbooks

41. Jackson, W.: "High Frequency Transmission Lines," pp. 120143, Methuen & Co., Ltd., London, 1945.

BIBLIOGRAPHY 211

42. Smith, P. H., in J. Markus and V. Zeluff (eds.): "Electronics forEngineers," 1st ed., pp. 326-331, McGraw-Hill Book Company,New York, 1945.

43. M.LT. Radar School Staff, in J. F. Reintjes (ed.): "Principles ofRadar," 2nd ed. (1946), pp. 8-66 to 8-68, 3d ed. (1952), pp.500-502, McGraw-Hill Book Company, New York.

44. Harvard University Radio Research Laboratory Staff, in H. J.Reich (ed.): "Very High Frequency Techniques," vol. II, pp.26-29,37,59,68,511, McGraw-Hill Book Company, New York,1947.

45. Terman, F. E.: "Radio Engineering," 3d ed., pp. 94-98, McGrawHill Book Company, New York, 1947.

46. Marchand, N.: "Ultra High Frequency Transmission and Radiation," pp. 44-45, John Wiley & Sons, Inc., New York, 1947.

47. Bronwell, A. B., and R. E. Bean: "Theory and Application ofMicrowaves," pp. 164-175, McGraw-Hill Book Company, NewYork, 1947.

48. Wheeler, H. A.: Wheeler Monograph 4, Wheeler Laboratories,Great Neck, N. Y., 1948 (also ibid. "Wheeler Monographs," vol.1,1953).

49. Bell Telephone Laboratories, Murrary Hill, N.J.: "Radar Systemsand Components," pp. 341, 342, 542-545, D. Van NostrandCo., Inc., Princeton, N.J., 1949.

50. Karakash, J. J.: "Transmission Lines and Filter Networks," 1stpr., pp. 101-104, The Macmillan Co., New York, 1950.

51. Johnson, W. C.: "Transmission Lines and Networks," 1st ed.,pp. 126-143, McGraw-Hill Book Company, New York, 1950.

52. Southworth, G. C.: "Principles and Applications of WaveguideTransmission," pp. 62-73, D. Van Nostrand Co., Inc., Princeton,N.J., 1950.

53. Jordan, E. C.: "Electromagnetic Waves and Radiating Systems,"1st pr. (1950), pp. 255-258,469,471, 5th pr. (1958), PrenticeHall, Inc., Englewood Cliffs, N.J.

54. Kraus, J. D.: "Antennas," 1st ed., pp. 457, 484, McGraw-HillBook Company, New York, 1950.

55. Slater, J. c.: "Microwave Electronics," pp. 27-30, D. VanNostrand Co., Inc., Princeton, N.J., 1950.

56. Breazeale, W. M., and L. R. Quarles: "Lines, Networks, Filters,"pp. 109-114, International Textbook Co., Scranton, Pa., 1951.

57. Terman, F. E., and J. M. Pettit: "Electronic Measurements,"2d ed., pp. 152-157, 183, McGraw-Hill Book Company, NewYork,1952.

58. Ridenour, L. N. (ed.): M.LT. Radiation Laboratory Series, vol.6, pp. 318,707, vol. 8, pp. 73,226, vol. 9, pp. 60-67, vol. 10,see ref. 32, vol. 12, p. 29, vol. 14, p. 283, McGraw-Hill BookCompany, New York, 1948.


59. Radio Physics Laboratory Staff, C.S.I.R.O., Australia: in E. G.Brown, "Textbook of Radar," 2d ed., pp. 133-135, 604-605,Cambridge University Press, New York, 1954.

60. Terman, F. E.: "Electronic and Radio Engineering," 4th ed.,pp. 100-103, McGraw-Hill Book Company, New York, 1955.

61. King, R. W. P.: "Transmission Line Theory," pp. 108-112,McGraw-Hill Book Company, New York, 1955; reprinted byDover Publications, Inc., New York, 1965.

62. Ryder, J. D.: "Networks, Lines, and Fields," pp. 324-327,Prentice-Hall, Inc., Englewood Cliffs, N.J., 1955.

63. Ginzton, E. L.: "Microwave Measurements," pp. 229-234,McGraw-Hill Book Company, New York, 1957.

64. Reich, H. J., J. G. Skalnik, P. F. Ordung, and H. L. Krauss:"Microwave Principles," pp. 9-13, D. Van Nostrand Co., Inc.,Princeton, N.J., 1957.

65. Walsh, J. B.: "Electromagnetic Theory and Engineering Applications," pp. 242-253, The Ronald Press Company, New York,1960.

66. Thourel, L.: "The Antenna," p. 401 (appendix), John Wiley &Sons, Inc., New York, 1960.

67. Gledhill, C. S.: "VHF Line Techniques," pp. 1-57, EdwardArnold (Publishers) Ltd., London, 1960.

68. Goubau, G.: "Electromagnetic Waveguides and Cavities," vol. 5,p. 267, Pergamon Press, Inc., New York, 1961.

69. Brown, R. G., R. A. Sharpe, and W. L. Hughes: "Lines, Waves,and Antennas," pp. 43-56,271-272, The Ronald Press Company,New York, 1961.

70. Atwater, H. A.: "Introduction to Microwave Theory," pp. 12-18,McGraw-Hill Book Company, New York, 1962.

71. Engineering Staff of Microwave Division, Hewlett Packard Co.,in Kosow, I. L. (ed.): "Microwave Theory and.Measurements,"pp. 93-145, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.

72. Boulding, R. S. H.: "Principles and Practice of Radar," 7th ed.,pp. 782-784, D. Van Nostrand Co., Inc., Princeton, N.J., 1963.

73. Wheeler, G. J.: "Introduction to Microwaves," pp. 89-105,Prentice-Hall, Inc., Englewood Cliffs, N. J., 1963.

74. Ghose, R. W.: "Microwave Circuit Theory and Analysis," pp.97-99, McGraw-Hill Book Company, New York, 1963.

75. Harvey, A. F.: "Microwave Engineering," pp. 41-43, AcademicPress, New York, 1963.

76. Lance, A. F.: "Introduction to Microwave Theory and Measurements," pp. 38-44, 49-61, 72, McGraw-Hill Book Company,New York, 1964.

77. Ware, L. A., and H. R. Reed: "Communication Circuits," 3d ed.,p. 239, John Wiley & Sons, Inc., New York, 1964.

BIBLIOGRAPHY 213

78. Kosow, I. L. (ed.): "Electronic Precision Measurement Techniques and Experiments," Appendix D, pp. 322-329, PrenticeHall, Inc., Englewood Cliffs, N.J., 1962, 1964.

79. Ramo, S., J. R. Whinnery, and T. Van Duzer: "Fields and Wavesin Communication Electronics," pp. 35-39, John Wiley & Sons,Inc., New York, 1965.

80. Magnusson, P. C.: "Transmission Lines and Wave Propagation,"pp. 68-97, Allyn & Bacon, Inc., Boston, Mass., 1965.

81. Ishii, T. Koryu: "Microwave Engineering," pp. 183-188, TheRonald Press Company, New York, 1966.

82. Berkowitz, B.: "Basic Microwaves," pp. 123-125, Hayden BookCo., Inc., New York, 1966.

83. Collin, R. E.: "Foundations for Microwave Engineering," pp.203-217, McGraw-Hill Book Company, New York, 1966.

PART II-PERIODICALS

101. Smith, P. H.: Transmission Line Calculator, Electronics, vol. 12,p. 29, January, 1939.

102. Smith, P. H.: An Improved Transmission Line Calculator, Electronics, vol. 17, no. 1, pp. 130-133,318-325, January, 1944.

103. Jackson, W., and F. G. H. Huxley: The Solution of Transmission-Line Problems by Use of the Circle Diagram of Impedance,I.E. E. J., vol. 91, pt. III, pp. 105-116 (Discussions, pp. 117-127),March, 1944.

104. Dees, B. C.: Graphical Methods of Solving Transmission LineProblems, Radio-Electron. Eng., pt. I, pp. 16-18, 41-44, August,1945, pt. II, pp. 18-20,24-26,34-35, September, 1945.

105. Baum, R.: Solving 4-Termina1 Network Problems Graphically,Communications, pt. I, pp. 51-53, March, 1946; pt. II, pp. 4043,53, May, 1946.

106. Fellers, R. G., and R. T. Weidner: Broad Band Wave GuideAdmittance Matching by Use of Irises, Proc. IRE, October,1947, pp. 1080-1085.

107. Altar, W., and J. W. Coltman: Microwave Impedance PlottingDevice, ibid., July, 1947, pp. 734-737.

108. Linton, R. L., Jr.: Graph for Smith Chart, Electronics, January,1950, pp. 123, 158.

109. Krauss, H. L.: Transmission Line Charts, Elec. Eng., vol. 68,p. 767 , September, 1949.

110. Chodorow, M., and E. L. Ginzton: A Microwave ImpedanceBridge, Proc. IRE, June, 1949, pp. 634-639.

111. A1sberg, D. A.: Universal Equalizer Charts, Electronics, November, 1951, pp. 132, 134.

BIBLIOGRAPHY 213

78. Kosow, I. L. (ed.): "Electronic Precision Measurement Techniques and Experiments," Appendix D, pp. 322-329, PrenticeHall, Inc., Englewood Cliffs, N.J., 1962, 1964.

79. Ramo, S., J. R. Whinnery, and T. Van Duzer: "Fields and Wavesin Communication Electronics," pp. 35-39, John Wiley & Sons,Inc., New York, 1965.

80. Magnusson, P. c.: "Transmission Lines and Wave Propagation,"pp. 68-97, Allyn & Bacon, Inc., Boston, Mass., 1965.

81. Ishii, T. Koryu: "Microwave Engineering," pp. 183-188, TheRonald Press Company, New York, 1966.

82. Berkowitz, R: "Basic Microwaves," pp. 123-125, Hayden BookCo., Inc., New York, 1966.

83. Collin, R. E.: "Foundations for Microwave Engineering," pp.203-217, McGraw-Hill Book Company, New York, 1966.

PART II-PERIODICALS

101. Smith, P. H.: Transmission Line Calculator, Electronics, vol. 12,p. 29, January, 1939.

102. Smith, P. H.: An Improved Transmission Line Calculator, Electronics, voI.17,no. l,pp.130-133,318-325,January, 1944.

103. Jackson, W., and F. G. H. Huxley: The Solution of Transmission-Line Problems by Use of the Circle Diagram of Impedance,IE.E. J., vol. 91, pt. III, pp. 105-116 (Discussions, pp. 117-127),March, 1944.

104. Dees, B. C.: Graphical Methods of Solving Transmission LineProblems, Radio-Electron. Eng., pt. I, pp. 16-18,41-44, August,1945, pt. II, pp. 18-20,24-26,34-35, September, 1945.

105. Baum, R.: Solving 4-Terminal Network Problems Graphically,Communications, pt. I, pp. 51-53, March, 1946; pt. II, pp. 4043,53, May, 1946.

106. Fellers, R. G., and R. T. Weidner: Broad Band Wave GuideAdmittance Matching by Use of Irises, Proc. IRE, October,1947, pp. 1080-1085.

107. Altar, W., and J. W. Coltman: Microwave Impedance PlottingDevice, ibid., July, 1947, pp. 734-737.

108. Linton, R. L., Jr.: Graph for Smith Chart, Electronics, January,1950, pp. 123, 158.

109. Krauss, H. L.: Transmission Line Charts, Elec. Eng., vol. 68,p. 767, September, 1949.

110. Chodorow, M., and E. L. Ginzton: A Microwave ImpedanceBridge, Proc. IRE, June, 1949, pp. 634-639.

111. A1sberg, D. A.: Universal Equalizer Charts, Electronics, November, 1951, pp. 132, 134.


112. Smith, P. H.: Charts for Coaxial Line Measurements, Proc. Natl.Electron. Con!, Oct. 22-24, 1951, vol. 7, pp. 191-203.

113. Bryan, H. E.: Impedance Circle Diagrams, Radio-Electron. Eng.,February, 1951, pp. 15A-18A.

114. Markin, J.: Smith Chart Applications, Tele-Tech., vol. 12,no. 5,pp. 85-87, 116-118, May, 1953.

115. Packard, K. S.: Automatic Smith Chart Plotter, Tele-Tech., April,1953, pp. 65-67,181-183.

116. Gabriel, W. F.: An Automatic Impedance Recorder for X-Band,Proc. IRE, vol. 42, no. 9, pp. 1410-1421, September, 1954.

117. Sutherland, J. W.: Electrical Measurements at Centimeter Wavelengths, Electron. Eng., February, 1954, pp. 46-51.

118. Knowles, J. E.: A Mechanical Version of the Smith Chart, J.Sci. Instr., vol. 35, pp. 233-237, July, 1958.

119. Southworth, G. C.: Graphical Analysis of Waveguide Components, Microwave J., pt. I, pp. 25-32, June, 1959; pt. II, pp.33-39, July, 1959.

120. Ibid., Using the Smith Diagram, January, 1959, p. 25.121. Waltz, M. c.: A Technique for the Measurement of Microwave

Impedance in the Junction Region of a Semiconductor Device,ibid., May, 1959, pp. 23-27.

122. Simonsen, 0.: Impedance Transfer Ring, Electronics, July 31,1959, pp. 82, 84.

123. Steere, R. M.: Novel Applications of the Smith Chart, Microwave J., March, 1960, p. 97.

124. Hickson, R. A.: The Smith Chart, Wireless World, pt. I, pp. 2-9,January, 1960; pt. II, pp. 82-85, February, 1960; pt. III, pp.141-146, March, 1960.

125. Mathis, H. F.: Logarithmic Transmission Line Charts, Electronics,vol. 34, no. 48, pp. 48-54, Dec. 21, 1961.

126. Evans, G. E.: Eliminating Trial and Error in Broadband Impedance Matching, ibid., September 28, 1962, pp. 68-72.

127. Dawirs, H. N.: How to Design Impedance Matching Transformers, Microwaves, August, 1962, pp. 22-24.

128. Glaswer, A.: Distributed Parameter Networks, Electro-Technol.,October, 1962, pp. 105-124.

129. Deschamps, G. A., and W. L. Weeks: Charts for Computing theRefraction Indexes of a Magneto-Ionic Medium, IRE Trans.G-AP, vol. AP-IO, no. 3, pp. 305-317, May, 1962.

130. Mathis, H. F.: Conformal Transformation Charts Used byElectrical Engineers, Math. Mag. , January-February, 1963, pp.25-30.

131. Ibid., Graphical Solutions of Scattering Coefficient Problems,Microwaves, June, 1963, pp. 28-33.

132. Ibid., Still Another Method for Transforming Impedances

BIBLIOGRAPHY 215

through Lossless Networks, IEEE Trans. G-MTT (Letters), vol.MTT-II, no. 3, pp. 213-214, May, 1963.

133. Johnson, G. D., P. Norris, and F. Opp: High Frequency Amplifier Design Using Admittance Parameters, Electro-Technol., pt.II, pp. 66-72, 140, December, 1963.

134. Roberts, D. A. E.: Measurement ofVaractor Diode Impedances,IEEE Trans. G-MTT, July, 1964, pp. 471-475.

135. Baum, R. F.: Sink-Pole Chart Solves for Transmission LineVoltage and Current, Microwaves, August, 1964, pp. 18-21.

136. Weill, V.: Insertion VSWR and Insertion Loss Measurementof Connector Pairs in Coaxial Cable, Microwave J., January,1964, pp. 68-73.

137. Smith, P. H.: A New Negative Resistance Chart, ibid., June,1965, pp. 83-92.

138. Ibid., How to Use Negative Resistance Charts, Intern. Electron.,vol. 10, no. 4, pp. 32-36, October, 1965.

139. Ibid., Graphicas Smith Negativas, Intern. Electron., vol. 10, no.4, pp. 34-38, October, 1965.

140. Mathis, H. F.: Graphical Procedures for Finding Matrix Elements for a Lattice Network and a Section of TransmissionLine, IEEE Trans. G-MTT (Correspondence), vol. MTT-13, no.I, pp. 130-312, January, 1965.

141. Houlding, N.: Varactor Measurements and Equivalent Circuits,ibid., November, 1965, pp. 872-873.

142. Gallagher, R. D.: A Microwave Tunnel Diode Amplifier, Microwave J., February, 1965, pp. 62-68.

143. Hall, G. L.: Smith Chart Calculations for the Radio Amateur,QST, pt. I, pp. 22-26, January, 1966; pt. II, pp. 30-33,February, 1966.

144. Mathis, H. F.: Fermat's Last Theorem and the Smith Chart,Proc. IEEE (Letters), vol. 54, no. I, p. 65, January, 1966.

145. Clavier, P. A.: Pierre Fermat and the Smith Chart (and author'sreply), ibid., no. 9, July, 1966.

146. Bandler, J. M.: Frequency Responses in the Reflection Coefficient Plane Plotted by Digital Computer, IEEE Trans. G-MTT,(Correspondence), vol. MTT-14, no. 8, pp. 399-400, August, 1966.

147. Blanchard, W. C.: New and More Accurate Inside Out SmithChart, Microwaves, vol. 6, no. II, pp. 38-41, November, 1967.

148. Blune, Hans-Juergen C.: Input Impedance Display of One-portReflection-type Amplifiers, Microwave J., November, 1967, pp.65-69.

149. Phillips, E. N.: Ell-Network Charts Simplify Impedance Matching, Microwaves, May, 1968, pp. 44-58.

ISO. Verzino, J. W.: Computer Programs for Smith-Chart Solutions,pt. I,Microwaves, July, 1968, pp. 57-61.


PART III-BULLETINS, REPORTS, ETC.

201. Seeley, S. W.: The Theory of Impedance and AdmittanceDiagrams and Allied Subjects, M.I. T. Rad. Lab. Rept. T-lO,Feb. 18, 1943.

202. Goudsmit, S. A.: Reflection Coefficients and Impedance Charts,M.I.T. Rad. Lab. Rept. T-II, 1943.

203. Seeley, S. W.: Impedan'ce and Admittance Diagrams for Transmission Lines and Waveguides, reprint of (201), BuShips NavyDept., U.S. Government Printing Office, Washington, D.C.(NAVSHIPS 900,038), 1944.

204. Hopper, A. L.: Plotting Board for 65 mc Impedance Measurements, Bell Lab. Record, June, 1948, pp. 258, 259.

205. Smith, P. H.: Transmission Line Chart, ibid., February, 1950,p.75.

206. General Radio Co. Staff, Impedance Measurement-Smith Chartwith Example Plotted, G. R. Experimenter, November, 1950,pp.5-7.

207. Soderman, R. A., and W. M. Hague: U.H.F. Measurements withthe Type 874-LB Slotted Line, ibid., vol. 25, no. 6, pp. 1-11,November, 1950.

208. Smith, P. H.: Charts for Coaxial Line Probe Measurements,reprint of periodical ref. 112, Bell Lab. Monograph 1955,1952.

209. Deschamps, G. A.: Hyperbolic Protractor for Microwave Impedance Measurements and Other Purposes, Federal Telecommunication Labs., 1953.

210. Seeley, S. W.: An Immittance Chart, R.C.A., Lab. Div., IndustryService Lab., LB-976, April, 1955.

211. Deschamps, G. A., and W. L. Weeks: Use of the Smith Chartand Other Graphical Constructions in Magneto-Ionic Theory,University of Illinois, Scientific Rept. I, Contract AF 19(606)5565; April 26, 1960.

212. Hewlett Packard Co. Staff, Introduction to Microwave Measurements, Hewlett Packard Co. Application Note 46, sec. 4,1960, pp. 11-24.

213. Simonsen, 0.: The Impedance Transfer Ring, U.S.N. ElectronicsLab., Research Rept. 1107, April 9, 1962.

214. Ballard, R. C.: Practical Applications of the Smith Chart, Phi/coTech. Rep. Division Bull., vol. 13, no. I, pp. 17-19, JanuaryFebruary, 1963.

215. Smith, P. H.: Smith Charts-Their Development and Use, KayElectric Co., Tech. Bulls. 1-9, 1962-1966.

216. Beatty, R. W.: Microwave Impedance Measurements and Standards, Natl. Bureau of Standards, Monograph 82, August 12,1965, pp. 3-5.

BIBLIOGRAPHY 217

217. Smith, P. H.: Instructions for Use of Smith Charts, Kay ElectricCo. Data Sheet, 1966.

218. Chu, T. S.: Geometrical Representation of Gaussian BeamPropagation, Bell Syst. Tech. J., vol. 45, no. 2, pp. 287-299,February, 1966.

219. Anderson, R. W.: S-Parameter Techniques for Faster, MoreAccurate Network Design, Hewlett-Packard J., February, 1967,pp. 13-22.

Admittance:characteristic (see Characteristic ad-

mittance)final input, 18, 19input, of a waveguide, 18, 19matching of, 97-99(See also Impedance, and admittance)

Admittance matching (see Matching, impedance and admittance)

Angle of reflection coefficient:description of, 21, 25scale of, 21(See also SMITH CHART, peripheral

scales of)Astrolabe (see Charts, in navigation)Attenuation:

scale of, 34, 92, 186of traveling waves, 3,4,34,35(See also Standing wave loss factor;

Transmission loss)Attenuation constant, description of, 14,

15(See also Phase constant; Propagation

constant)

Balanced L-type matching circuits, 128Bandwidth:

representation on SMITH CHART, 77,78, 188

of a waveguide stub, 81, 82(See also Expanded SMITH CHART,

pole regions of)Building-out section (see Transformers,

stub)

Carter, P.S., Ref. 9Carter chart:

coordinates of, 63, 65-67coordinates with transmission and re

flection coefficient scales, 68Characteristic admittance:

relation to characteristic impedance, 19use as normalizing constant, 14, 19

Characteristic impedance:definition of, 3, 12

Characteristic impedance (cont'd):relation to primary circuit elements, 12of transmission lines, 12, 13

nomographs for, 13of uniconductor waveguides, 13, 14use as normalizing constant, 19of a waveguide, 1, 3, 13(See also Initial sending-end impedance;

Surge impedance)Charts:

early transmission line, xiii, xv-xviii,82, 83

for guided waves, xiii, xv-xviiiin navigation, xiii, xiv(See also Carter chart; Expanded

SMITH CHART; SMITH CHART)Circuit elements, primary, 11Coefficient of reflection (see Reflection

coefficient of)Conformal transformation ofcoordinates,

xv, 197-200Coordinates of SMITH CHART, con

struction of (see SMITH CHART,construction of coordinates)

Current reflection coefficient (see Reflection, coefficient of)

Current transmission coefficient (seeTransmission, coefficient of)

Current-voltage overlay for SMITHCHART, 33, 38-41

(See also Normalized current; Normalized voltage)

Electrical length :discussion of, 24relation to physical length, 24, 25scale of, 23, 24(See also SMITH CHART, peripheral

scales of; Wavelength scales onSMITH CHART)

Equivalent circuit representations:parallel impedance and series admit

tance,60,61,63,65alternate SMITH CHART coordi

nates for, 62

Index

Equivalent circuit representations(cont'd):

series impedance and parallel admittance, 58, 61

vector relationships for, 58, 59Expanded SMITH CHART:

central regions of, 71-75by coordinate distortion, 71, 73, 82

83by coordinate inversion, 71-73, 82,

84-86pole regions of, 71-73,76-78

for bandwidth of stub lines, 82forlocus of impedance, 79-81for Qof stub lines, 81, 82

Flemming, J.A., xv, 194(See also Telephone equation)

Glossary of SMITH CHART terms, 185191

Graphical representations (see Charts)

Hyperbolic functions, tables of, xiii

Impedance:and admittance:

alternative representations of, 60-63series to parallel conversion, 62-64series to polar conversion, 63, 65,

66coordinates, 187

characteristic (see Characteristic im-pedance)

[mal sending-end, 16initial sending-end, 3input of a waveguide, 16-18

conversion to admittance, 19,20formula for, 17normalization of, 19properties of, 17, 18representation on SMITH CHART,

17

219

220 ELECTRONIC APPLICATIONS OF THE SM ITH CHART

Impedance matching (see Matching, impedance and admittance)

Initial sending-end impedance, 3(See also Surge impedance)

Inverted SMITH CHART (see ExpandedSMITH CHART, by coordinateinversion)

Kennelly, A.E., xiii

L-type matching circuits:analysis of, 116-118capabilities of, 115-117circuit arrangements of, 115-116forbidden areas on SMITH CHART,

116, 117, 120-127losses in, 115, 116overlays for SMITH CHART, 118,

120-127examples of use, 118-119

selection of, 116,117transformable impedances, 115, 116(See also Balanced L-type matching

circuits; T-type matching circuits)

Matching, impedance and admittance,97-128

(See also Balanced L-type matchingcircuits; L-type matching circuits;T-type matching circuits; Transformers)

Matching circuits (see Balanced L-typematching circuits; L-type matching circuits; T-type matching circuits)

Matching stubs (see Transformers, stub)MIT Radiation Laboratory, xvModes (see Waveguides, modes of propa

gation in)

Negative resistance:definition of, 137, 138power flow in waveguide, 142properties of, 137, 138representation on SMITH CHART, 138,

139, 187advantages of, 141applications for, 139effect on conventional scales, 138,

139Negative SMITH CHART:

examples of use: reflection amplifierdesign, 150-156

equivalent circuit for, 150-152evaluation of circuit constants, 155

Negative SMITH CHART (cont'd):representation of operating param

eters, 152-155radial scales for: power reflection coef

ficient, 145reflection coefficient, 141, 142, 144,

145return gain, 145, 189standing wave ratio, 149

dB, 149transmission loss, 150transmission loss coefficient, 150

reflection coefficient of: angle of, 141,142, 147

magnitude of, 141, 142, 144, 145,147

transmission coefficient of: angle of,142, 144, 148

discussion of, 142, 144magnitude of, 142, 144, 148

Negative SMITH CHART coordinatesand scales, 143, 145, 146

Normalization:of admittances, 18, 19of impedances, xv, 16, 17

example of, 19Normalized current, defmition of, 187,

188(See also Current-voltage overlay for

SMITH CHART)Normalized voltage, defmition of, 188

(See also Voltage-current overlay forSMITH CHART)

Numerical alignment chart, 166-168

Overlays for SMITH CHART:amplitude of E or I, relative to mini-

mum, 53angle of E or I, relative to minimum, 51L-type circuit design, 120-127matching stub design, 104, 105negative impedance coordinates, 143normalized E or I, 39parallel impedance coordinates, 62polar impedance coordinates, 67probe ratio loci, 131-134reflection coefficient, 27

for negative SMITH CHART, 147transmission coefficient, 49

for negative SMITH CHART, 148or reflection coefficient, rectangular

coordinate, 90

Parallel impedance SMITH CHART, 6064

coordinates, 62and scales for, 64

Peripheral scales of the SMITH CHART(see SMITH CHART, peripheralscales ot)

Phase:absolute, 44example of, 44-46of incident wave, 44, 45insertion, voltage and current, example

of,50-55of reflected wave, 44-46relative lag and lead, 44, 45of resultant wave, 44-46of standing wave, 46, 50-52unit of, 44

Phase angle:of current reflection coefficient, 43,44,

46, 185, 186of power factor, 39, 40, 43of transmission coefficient, 47-50, 186of voltage reflection coefficient, 43,

44, 46, 185, 186Phase constant, 14, 15

(See also Attenuation constant; Propa-gation constant)

Phase conventions, 44-46Polar impedance chart (see Carter chart)Polar impedance vectors, chart for com-

bining, 66, 69-70Pole region charts (see Expanded SMITH

CHART, pole regions ot)Power factor:

angle of, 39-43magnitude of, 43

Power reflection coefficient:definition of, 26, 189representation on SMITH CHART, 35scale of, 26(See also Negative SMITH CHART,

. radial scales for)Power transmission coefficient:

defmition of, 46, 47scales for, 47

Primary circuit elements, 11Probe measurements of current or volt

age (see Standing waves, probemeasurements ot)

Propagation constant, 14,15

Q of resonant and antiresonant lines (seeExpanded SMITH CHART, poleregions of, for Q of stub lines)

Radial scales of the SMITH CHART (seeNegative SMITH CHART, radialscales for; SMITH CHART, radialscales ot)

Rectangular chart, history and description of, xiii, xv, xvi, xviii, xx

Rectangular representations of reflectionand transmission coefficients, 88,89

(See also Reflection, coefficient of,representations of; Transmission,coefficient of, representation ot)

Reflection:coefficient of: angle of, 4, 25, 27, 46,

48,185,186conversion to impedance, 91magnitude of, 4, 26, 27, 48,189for negative SMITH CHART (see

Negative SMITH CHART, reflec·tion coefficient of)

overlay for SMITH CHART, 27, 90relation to transmission coefficient,

47,48representations of, 25-27scale of, 25, 26, 88, 89, 95voltage or current, defmition of, 4,

26, 189X or Y component of: representation

of, 88-91, 189plot of components, example of,

94,95(See also Phase angle, of voltage

reflection coefficient; Power reflection coefficient)

of traveling waves, 4, 5Reflection gain, 38Reflection loss:

definition and evaluation of, 37, 38,189

scale for, 34Reflection-transmission coefficients:

composite X-Y representation of, 8991

conversion to impedances, 91Resonance and antiresonance:

definition of, 76in waveguide stubs, 71,76·79

dependence on electrical length of,72,76

dependence on loss in, 76-79equivalent circuits for, 76

Return gain, 145, 189(See also Return loss)

Return loss:equivalence with power reflection coef·

ficient, 26, 189evaluation of, 38scale for, 34(See also Return gain)

Scales for SMITH CHART (see SMITltCHART, peripheral scales of;SMITH CHART, radial scales of)

Slide screw tuner, 98SMITH CHART:

conformal mapping of, 139-14 Ibilinear transformation of coordi-

nates, 139-141construction of coordinates, 21, 22coordinates of, 15, 16, 186, 187coordinates with polar and rectangular

transmission and reflection coefficient scales, 93-94

SMITH CHART (cont'd):coordinates with transmission and re

flection coefficient scales, 91, 92,94,95

description of, 1, 189, 190history of, xiii, xv-xxother names for, xviiiperipheral scales of, 21,23-25,33,188

(See also Angle of reflection coeffi·cient; Transmission, coefficientof, angle of; Wavelength scales onSMITH CHART)

radial scales of: loss scales, 33, 34, 92,94,95,186,189-191

reflection scales, 25, 26, 92, 94, 95,189,

translucent overlay, Chart "A" (see

envelope inside back cover)uses of: basic, 157

solution ofvector triangles, 166-168special: data plotting, 159-161

circuit design, 163·166netwoIk applications, 158, 159numerical calculations with, 166-

168Rieke diagrams, 161scatter plots, 161, 163

specific, 157-158SMITH CHART instruments:

blackboard charts, 183, 184calculator, 169-171

improved version, 171·173with spiral cursor, 173·174

computer-plotter, 182-183impedance transfer ring, 174, 175laminated charts, 184large paper charts, 182, 183Mega-eharts (8'12 x 11 paper charts),

184Mega·Plotter, 176-178Mega-Rule, 179-182plotting board, 175, 176types of, 169

Standing waves:loci of voltage (or current) ratios, 129,

131-134construction of overlays for SMITH

CHART, 135, 136overlays for impedance evaluation,

129-136peak value of, 190position of minima, 29, 30

relation to reflection coefficient, 30probe measurements of, 129, 130

impedance evaluation by, 129·136probe separation, 129, 130

relative amplitude along, 52, 53relative phase along, 5D-5 2shape of, 5-9,28-30,48,94,95, 149

construction of spacial shape of, 79

(See also Phase, of standing waves)

INDEX 221

Standing wave loss factor (coefficient):discussion of, 36, 37,190scale for, 34(See also Transmission loss)

Standing wave ratio:in decibels, 26, 30, 31, 190effect of attenuation on, 30effect of generator impedance on, 17,

18maximum to minimum, 26, 27,190minimum to maximum, 73, 82, 83with negative resistance terminations,

142, 144, 149relation to normalized resistance, 30relation to traveling waves, 5-7representation on SMITH CHART, 28scale of, 26

Stub lines (see Resonance and antiresonance, in waveguide stubs)

Surge impedance, 3(See also Characteristic impedance;

Initial sending-end impedance)

T-type matching circuits, 119, 128Telephone equation, xv, 17, 194Transformers:

slug: definition of, 97dual, 110·114

analysis with SMITH CHART, lID112

matchable boundary, 112, 113overall length of, 113-114

single, 97,107,109,110operation of, 107, 109, 110quarter-wave, 107, 109, 110

stub: definition of, 97dual, 102, 103, 106, 107

forbidden areas on SMITH CHART,103, 106

matching capability, 102,103,106,107

single, 97-102design of, 99, 100design overlays for SMITH CHART,

102, 104, 105length versus impedance, 100-102matching capability, 97,98mismatch, length, and location of,

98-100operation of, 97-99

Transmission, coefficient of: angle of,46·50, 87-89, 91, 92, 186

magnitude of, 46·50, 88·94, 190for negative SMITH CHART (see

Negative SMITH CHART, transmission coefficients of)

overlay for SMITH CHART, 47-4989, 90

power, definition of, 46, 47, 190, 191


Transmission, coefficient of (cont'd):

relation to reflection coefficient, 47,48

representation of, 47-49, 87-91scale of, 47. 92voltage or current, definition of, 46,

47, 190X or Y component, 88-90, 191

application of, 94, 95(See also Phase angle, of transmission

coefficient; Power transmissioncoefficient)

Transmission loss:in coaxial and open wire lines, 34coefficient, 191in cylindrical waveguides, 34definition of, 34, 191determination of, 41, 42ratio in decibels, 35scale for, 34, 95scale direction on SMITH CHART, 35types of, 34

Transmission loss, types of (cont'd):

conductor loss, 3, 34dielectric loss, 3, 34radiation loss, 34

units of, 34, 35(See also Attenuation; Negative SMITH

CHART, radial scales for; Standing wave loss factor)

Transmission loss coefficient (see Stand-ing wave loss factor)

Transmission-reflection coefficients (seeReflection-transmission coefficients)

Traveling waves:discussion of, 3-5, 35graphical representation of, 7relation to standing waves, 5-7

Voltage-current overlay for SMITHCHART, 33, 38-42

Voltage reflection coefficient (see Reflection, coefficient of)

Voltage transmission coefficient (see 0

Transmission, coefficient of)

IWaveguide admittance (see Admittance)Waveguide reflection coefficient (see Re

flection, coefficient of)Waveguide transmission coefficient (see

Transmission, coefficient of)Waveguide impedance (see Impedance)Waveguides:

electrical constants of, 11-15losses in, 3, 33-38modes of propagation in, 2, 3, 18reflection in, 4,5,25,26types of, 1

Wavelength scales on SMITH CHART,21, 24, 25, 191

(See also Electrical length)

Documents

78897620 Electronic Applications of the Smith Chart SMITH P 1969