NATL :NST of standards & TECH R.I.C.

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/NBS monograph „ ^ QC100 .U5^ V137;1973 C.I NBS-PUB-C 1959 NBS MONOGRAPH 131

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>c Applications of Waveguide and Circuit Theory

r>o i57to the Development of Accurate Microwave

Measurement Methods and Standards

R. W. Beatty

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued August 1973
Library of Congress Catalog Number: 73-600158

National Bureau of Standards Monograph 137

Nat. Bur. Stand. (U.S.), Monogr. 137, 322 pages (Aug. 1973)

CODEN: NBSMA6

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C 13.44: 137). Price $5.20. Stock Number 0303-01169
Contents

Page

Abstract--'^ X

Preface

Acknowledgments xii

1. Introduction 1 1.1. General 1 1.2. Theory 2 1.3. Applications 4 1.4. Conclusions and References 6

2. Basic Theory of Waveguide Junctions 7 2.1. Definitions 7

a. Waveguide Junctions 7 b. Terminal Surfaces and Terminal Variables 7 2.2. V, i, a and b for Waveguide 8

a. Introduction 8

b. Basic Derivation of v, i, e" and h" 8

c. Complex and Vector Potential Functions 9

d. Power and Impedance Normalization 10

e. Examples of Waveguide v and i 11

(1) TEM-Mode in Coaxial Waveguide 11

(2) ^^10 Rectangular Waveguide 13

f. Traveling Wave Amplitudes a and b 15

g. Other Traveling Wave Amplitudes 15

2.3. Parameter Matrices 16

a. Impedance and Admittance Matrices 16

b. The Scattering Matrix 17

c. Power 17

(1) General -- 17

(2) Realizability Conditions 18

(a) Strict Realizability 18

(b) Semi -realizability 18

(c) Losslessness 19

d. Reciprocity 19

e. Sources; Joining Equations 20

(1) General 20

(2) Sources 20

(3) Joining Equations 21

3. Introductory Network Analysis : 24

3.1. Linear Network Parameters 24

a. Introduction 24

b. Terminal Variables 24

c. Network Parameters 24

d. Complex Wave Amplitudes a and b 25

e. Parameters Associated with a and b 25

iii
Contents (Continued)

Page

f. Other Terminal Variables 26

g. Network Equivalent to a Waveguide Junction 26

3.2. The Scattering Matrix 27

a. General Remarks 2 7

b. Scattering Coefficients of a Two- Arm Waveguide Junction 2 8

c. Effects of Moving Terminal Surfaces 30

3.3. Reciprocity, Realiz ability , and Losslessness for 2-Ports 31

a. Reciprocity 31

b. Strict Realizability 32

c. Losslessness 33

3.4. Power and Efficiency 34

3.5. Representation of the Source 35

a. From Linear Relation for Source and Joining Equations- 35

b. From a Constant Voltage Generator 36

c. Summation of Wave Reflections 37

3.6. Net Power and Available Power 37

a. Net Power to a Waveguide Junction 37

b. Available Power from Generator 38

3.7. Mismatch Loss 39

a. Mismatch Loss in General 39

b. Meaning of Mismatch 39

c. Conjugate Mismatch Loss 40

d. The Zg Mismatch Loss 41

e. Difference Between Conjugate and Mismatch Losses 41

3.8. Transmission Properties of 2-Ports 41

a. Substitution Loss 42

b. Transducer Loss 43

c. Insertion Loss 44

d. Attenuation or Characteristic Insertion Loss 44

e. Components of Losses 45

3.9. Maximum Transmitted Power 47

a. Maximum Efficiency 48

(1) Gradient of Efficiency 48

(2) Maximum Efficiency from Maximum Power Considerations 50

b. Minimum Transducer Loss 51

c. Intrinsic Attenuation 52

3.10. Phase Shift 52

a. Relative Phase 52

b. Shift of Phase by a 2 -Port 52

c. Different Kinds of Phase Shift of 2-Ports 53

(1) Transmission Phase Shift 54

(2) Differential Phase Shift 54

(3) Substitution Phase Shift, Including Insertion Phase Shift-- 54

iv
Contents (Continued)

Page

3.11. Cascading 2-Ports 54

3.12. Cascading Coefficients 56

3.13. Transformation of Reflection Coefficients 57

a. Simpler Transformations 57

b. The Sliding Termination 5 8

c. Significance of the Radius of the r^-Circle 59

d. Displacement of Center from Origin 60

e. Locus of for Real 60

f. Other Circles- 61

3.14. The Linear Fractional Transformation 62

a. Constant |w|, variable ^, 62

b. Constant ii , variable Iwl 63

c. Invariance of cross-ratio 64

3.15. 3-Ports or Waveguide Junctions Having Three Arms 6 5

a. Introduction 65

b. Realizability Conditions 65

c. Conditions for Losslessness 67

d. Reciprocity 69

e. Symmetry 69

f. 3-Port, One Arm Terminated 70

g. Non-reciprocal 3-Ports 71

h. The Directional Coupler, 3 -Port 71

i. Solution of the Scattering Equations for b^ 73

i. Measurement of Reflection Coefficient V with General 3-Port 76 •' u 3.16. 4-Ports 77

a. The Directional Coupler 77

b. The Magic Tee 78

4. Power 80

4.1. Introduction 80

4.2. Mismatch Errors 80

a. Introduction 80

b. Calibration of Power Meters 81

(1) General Discussion 81

(2) Method I -- Alternate Connection to a Stable Power Source-- 82

(3) Method 2 -- Comparison Using T- Junctions 83

(a) Simultaneous Comparison , 83

(b) Alternate Connection to T-Junction 85

(4) Method 3 -- Comparison Using Magic T 85

(a) Simultaneous Comparison 85

(b) Alternate Connection to Magic T 87

c. Discussion of Calibration Methods-- 89

d. Use of Power Meters 90

(1) General Remarks 90

(2) Direct Measurement 90

V
Contents (Continued)

Page

(3) Use of Calibrated Attenuator - 91

(4) Use of Directional Couplers 92

(a) Temporarily Inserted 92

(b) Permanently Installed 94

e. Measurement of Scattering Coefficients 95

4.3. Barretter Mount Efficiency Measurement 98

a. Introduction 98

b. Impedance Method 100

c. Improved Method 102

d. Discussion of Errors 107

e. Experimental Results 111

f. Conditions for Linear r^-Locus 113

5. Impedance-Reflection Coefficient 115

5.1. Introduction 115

5.2. Adjustable Sliding Termination 116

a. Introduction 116

b. Principle of Operation 117

c. Theory 118

d. Design 120

5.3. Quarter-Wavelength Short- Circuit 123

a. Introduction 123

b. Coaxial Line 124

c. Rectangular Waveguide 129

d. Errors and Design Information 136

5.4. Squared VSWR and Magnified Responses - 137

a. Introduction 137

b. Simplified Explanations 138

(1) Squared VSWR Response 138

(2) Magnified Response 138

c. Analysis 141

d. Means of Obtaining Various Responses 143

e. Measurements Using Squared VSWR Response 145

f. Measurements Using Magnified Response 146

g. Results 149

5.5. The Tuned Reflectome ter 149

a. Three -Arm Waveguide Junctions 149

b. Tuning for Squared VSWR Response 150

c. Tuning for Magnified Response 151

d. Tuning for Infinite Directivity 152

e. Analysis of Errors 152

vi
Contents (Continued)

Page

5.6. Hybrid Reflectometer 155

a. Introduction 155

b. Basic Theory- 156

c. Experimental Equipment 15 7

(1) General 15 7

(2) Terminating Arrangements 158

(3) Standard Reflections for (Air-Dielectric) Coaxial Line 160

(4) Sliding Loads 161

d. System Performance and Applications 162

e. Results and Discussion of Errors 164

f. Conclusions 166

5.7. Connector Reflections and Losses 166

a. Introduction 166

b. Preliminary Considerations 166

c. Brief Description of Method 169

d. Review of Ref lectometer Techniques 170

e. VSWR Determination 171

f. Efficiency Determination 173

g. Supplemental Techniques in VSWR Determination 177

h. Results 184

6. Attenuation 189

6.1. Introduction 189

6.2. Definitions of Attenuation 190

a. Introduction 190

b. Broad General Meaning 191

c. Restricted Meanings 192

d. IRE Definitions 193

e. Comparisons of Definitions 195

f. Terminology- 20 3

g. Precise Definitions 205

h. Future Trends 20 8 6.3. Cascade -Connected Attenuators 209

a. Introduction 209

b. Analysis 210

(1) Two Attenuators 211

(2) Three Attenuators 212

(3) Any Number (n) of Attenuators 214

6.4. Mismatch Errors in Attenuation Measurements 214

a. Introduction 214

b. Analysis of Case of Fixed Pad 215

c. Evaluation of Error 216

d. Effects of Attenuator Characteristics 218

e. Effect of Realizability Conditions 219

vii
Contents (Continued)

Page

f. Variable Attenuators 219

(1) Introduction 219

(2) Expression for Mismatch Error 219

(3) Evaluation of the Mismatch Errors 221

g. Avoidance of Mismatch Error 22 2 h. Later Work 222

6.5. Effects of Realizability Conditions 222

6.6. Effects of Connectors and Adapters 224

a. Introduction 224

b. Previous Analyses - - 226

c. An Improved Representation 2 27

d. Substitution Loss 228

e. Standard Attenuation 230

f. Connector and Mismatch Errors 231

g. Same Fixed Attenuator in Two Systems 233

h. Adjusting Systems for Zero Reflections 237

i. Same Variable Attenuator in Two Systems 238

j. Standard Incremental Attenuation 242

k. Basic Insertion Arrangements 244

(1) Cases 2A and 3A -- Combining the Component with an Adapter- 244

(2) Cases 2B and 3B -- Substituting the Component for an Adapter 246

(3) Cases 2C and 3C -- Combining the Component with an Adapter

and Substituting for Another Adapter 247

1, Conclusions - 248

6.7. Efficiency and Attenuation of 2-Ports from Reflection Coefficient

Measurements 251

a. Introduction 251

b. 2-Port Having Terminal Pairs 253

c. 2-Port Having Waveguide Leads 255

d. Conclusions 256

6.8. Attenuation from Power Measurements 257

a. Introduction 25 7 b. The Measurement System 257

c. Theory of Measurement 259

d. Propagation of Error in Measuring d-c Power Differences 261

e. Mismatch Errors 261

f. Results 263

g. Analysis and Evaluation of Mismatch Errors 264

6.9. Two-Channel Nulling Method 266

6.10. Attenuation Divider Circuit - 272

viii
Contents (Continued)

Page

Phase Shift 277

7.1. Introduction 277

7.2. Phase Shift Equations for 2-Ports 277

a. Introduction 277

b. General 278

c. Phase Shift Equations- 2 79

(1) Phase Shift of v 280

(2) Phase Shift of i 280

(3) Phase Difference Between and 281

(4) Phase Difference Between and 281

(5) Differential Phase Shift - 282

(6) Insertion Phase Shift 283

d. Characteristic Phase Shift 284

e. An Ideal Phase Shifter 285

f. Conclusions • : 286

7.3. Standard Phase Shifter 286

a. Introduction 286

b. Tuning Errors 288

(1) Case I: S^^^ = 0, but T^^ f

0 289

(2) Case II: r^^ = 0, but S^^ f

0 290

c. Dimension Errors 291

d. Graphical Presentation of Results 292

7.4. Differential Phase Shifter 298

Conclusion 301

References 30 3

\

ix
ABSTRACT

The basic theory and analytical methods used in the development of accurate microwave measurement methods and standards are presented.

Developments at the U.S. National Bureau of Standards during 1948-1968 are described in which the above theory and analytical methods were applied.

These developments were in the fields of power , impedance , attenuation and phase

shift, and led to the establishment of National Sta.ndards and calibration methods at

frequencies from about 300 MHz to 30 GHz.

Key words: Attenuation definitions; attenuation measurement; barretter mount

efficiency; coaxial connectors; impedance measurement; microwave network

theory; mismatch errors; phase shift -measurement ; power measurement;

reflectometers ; waveguide joints; waveguid'.e theory.

X
PREFACE

The purpose bf this monograph is to show how microwave waveguide and circuit

theory was formulated and applied to the development of accurate measurement methods

and standards at the U.S. National Bureau of Standards.

The topics of power, impedance, attenuation, and phase shift standards and measurement techniques have been selected for discussion. Appropriate research papers by

the author and his associates have been partially revised and updated for the above

purposes. In addition, new material, especially on attenuation definitions, has been

included.

It is not possible here to present a complete history of NBS research in this

area, nor to accurately describe the present state-of-the-art. However, an attempt

has been made to put the work in perspective by giving references to previous and

subsequent NBS pertinent research.

It is intended that this monograph will indicate the character and extent of the

research which must be performed in order to develop accurate microwave measurement

methods and standards at the highest level. It is hoped that the collected works

and discussions will be helpful and stimulating to other workers in the same general

field.

xi
ACKNOWLEDGMENTS

This NBS monograph is a partially revised version of a doctoral thesis presented

to the Department of Electronic Engineering of the University of Tokyo, Tokyo, Japan.

It describes research done partly while the author was an employee of the U.S.

National Bureau of Standards (NBS) since 1948 and partly while a guest worker in 19 70

with the Electrotechnical Laboratory (ETL) Tanashi Branch, Tanashi , Tokyo, Japan.

Thus, the author is indebted to personnel of NBS, ETL , and the University of

Tokyo in completing the work. The work could not have been accomplished without the

approval and encouragement of Dr. Lewis M. Branscomb , Former Director of NBS, and

Dr. Fusao Mori, Director of ETL. The support at NBS of Dr. Ernest Ambler, Director

of the Institute for Basic Standards (IBS) , Mr. Bascom W. Birmingham, Deputy Director

IBS at Boulder, Colorado, and Dr. Raymond C. Sangster, Chief, Electromagnetics

Division, is gratefully acknowledged. At ETL, the strong support of Dr. Kenjiro

Sakurai , Chief of the Radio- and Opto-Electronics Division, Dr. Hideo Yamanaka,

Chief of the Opto-Electronics Section, Mr. Ryunosuke Ishige, Chief of the Radio

Electronics Section, Dr. Hiroshi Kashiwagi , of the Laser Section, and Dr. Toshio

Nemoto is most gratefully acknowledged. At the University of Tokyo, the counsel

and supervision received from Prof. Sogo Okamura, Faculty of Engineering, Department

of Electronic Engineering, was especially helpful, and essential for the completion

of the work.

The help of many co-workers and associates at NBS and elsewhere in stimulating

and carrying out the research is acknowledged as follows: Dr. David M. Kerns, Senior

Research Scientist at NBS, associate of the author since 1948 and leading NBS

specialist in electromagnetic theory for rf measurement, taught courses, wrote

tutorial and research papers laying solid foundations upon which many of the NBS

microwave standards are based, and stimulated the author's work through many helpful

discussions. His influence and assistance are particularly strong in the theoretical

portion of this monograph. Much of the material in chapter 2 is derived from his

part 1 of our book (see list of references: Kerns and Beatty, 1967), and sections

2.3c(2), 2.3d, and 2.3e are taken directly from his writings. Dr. Glenn F. Engen,

a co-worker since 1954, contributed to the research on tuned re f lectometers , attenuation from power measurements, and measurements of connector reflections and losses.

Mr. Wilbur J. Anson contributed to the work on impedance standards and the hybrid

reflectometer . Mr. William E. Little, Mr. B. C. Yates, and Mr. G. E. Fentress

contributed to the research on impedance standards, intrinsic attenuation, and the

xii
measurement o£ small losses. Dr. George E. Schafer (now at the U.S. Army Electronics

Proving Ground, Fort Huachuca, Arizona) was closely associated with the author in

carrying out the research on phase shift standards.

Mr. A. C. MacPherson (now at the U.S. Naval Research Laboratory, Anacostia, D.C.)

contributed to the work on mismatch errors in microwave power measurements and

Mr. Frank Reggia (now at the Harry Diamond Laboratories, Washington D.C.) contributed

to the research on the measurement o£ barretter mount efficiency.

Discussions with Dr. Bruno 0. Weinschel of Weinschel Engineering Co.,

Gaithersburg , Md. , stimulated and influenced the research on attenuation definitions.

Dr. Paul F. Wacker's review of the theoretical portions was very helpful.

Finally, the author is especially grateful to Mrs. Anne Y. Rumfelt who carefully

and thoroughly read the manuscript and made many helpful suggestions and corrections.

Miss Sharon Foote , Mrs. Florence M. Indorf, and Miss Janet Becker typed the manuscript

and put it into proper form for the NBS Publications Section.

xiii

APPLICATIONS OF WAVEGUIDE AND CIRCUIT THEORY TO THE DEVELOPMENT OF ACCURATE

MICROWAVE MEASUREMENT METHODS AND STANDARDS

R, W. Beatty

National Bureau of Standards

Boulder, Colorado

The basic theory and analytical methods used in the development of accurate microwave measurement methods and standards are presented.

Developments at the U.S. National Bureau of Standards during

1948-1968 are described in which the above theory and analytical

methods were applied.

These developments were in the fields of power, impedance,

attenuation and phase shift, and led to the establishment of

National Standards and calibration methods at frequencies from

about 30 0 MHz to 30 GHz

.

Key words: Attenuation definitions; attenuation measurement;

barretter mount efficiency; coaxial connectors; impedance meas- urement; microwave network theory; mismatch errors, phase shift

measurement; power measurement; reflectometers ; waveguide joints;

waveguide theory.

1. Introduction

1.1. General

This monograph presents a formulation of waveguide and circuit theory with

selected applications to the development of accurate measurement methods and

standards. A portion of the research at the U.S. National Bureau of Standards by

the author and co-workers during the period 1948-1968 is described.

Although considerable work had been done prior to 1948 in the field of microwave measurements at the M.I.T. Radiation Laboratory and other laboratories, much

remained to be accomplished before U.S. radio frequency and microwave standards

and calibration services could be established.

The theory of waveguides and of microwave circuits needed to be re-examined

and the foimdations exposed and strengthened. The quantities to be measured needed

to be precisely defined, and the conditions under which the theory remains valid

needed to be clearly stated. New and refined measurement techniques and standards

needed to be developed. Errors needed to be analyzed and limits of uncertainty

evaluated.

1
In the following, the above points are illustrated with specific examples.

In Chapters 2 and 3, respectively, a formulation of waveguide and circuit theory is

given which is slanted towards measurement applications. Selected applications are

given in Chapters 4 through 7, respectively, for the topics of power, impedance,

attenuation, and phase shift.

1.2. Theory

Waveguide and circuit theory is presented in Chapters 2 and 3, respectively.

Although this theory must be considered well-known, it has previously not been

as clearly presented in a form convenient for measurement applications. Greater

attention is paid to fundamental aspects of the theory, the assumptions made, and

the conditions required for validity.

The waveguide theory of Chapter 2 applies mainly to lossless,^ uniform cylindrical waveguides of arbitrary cross -section . A rigorous and general treatment of

this subject has been published (Kerns and Beatty, 1967). Much of the same material

is presented in this monograph, from a somewhat less general point of view, but with

specific applications in mind.

First, the basic terms "waveguide junction," "waveguide leads," "terminal

surface" and "terminal variables" are defined in section 2.1. Two sets of terminal

variables are discussed in section 2.2. One set consists of quantities v and i

which are generalizations of voltage and current. The other set are the complex

amplitudes a and b of the traveling waves which interfere to give rise to the generalized voltage and current. These terminal variables are related to the waveguide

fields corresponding to a given mode in a given waveguide.

The theory of waveguides leading to modal equations is based upon assumptions

of uniform, cylindrical waveguide geometry, and freedom from dissipative loss and

from leakage. As shown in Chapter 2 the transverse fields for a given mode can

each be resolved into two components; one in the complex plane, and one vector (space)

component. The complex components denoted by the letters v and i are regarded as

generalizations of voltage and current. Under certain conditions, they may be made

to coincide with actual voltage and current in transmission lines.

^The case of small losses is not treated in Chapter 2, but is illustrated in a later

example (sec. 5.3).

2
It further develops that these variables may be referred to a given terminal

surface in a waveguide which is part of a waveguide junction (a microwave network

or circuit). When so referred, they may be regarded as terminal variables, similar

to the familiar terminal variables "voltage" and "current" encountered in lumped

element network theory. It follows that conventional network theory may be applied

to the analysis of microwave circuits involving waveguides.

Power and impedance normalization are discussed and normalization factors are

defined. The consequences of some specific choices of these factors are explored and

a scheme for suppressing them for convenience in manipulating circuit equations is

given. The concept of "characteristic impedance" is clarified.

General power relationships for waveguide junctions are examined and the realizability conditions (including lossnessness) are precisely stated. In addition, the

reciprocity condition is given in matrix form. Finally, representation of sources by

terminal variables and impedances or reflection coefficients is discussed together

with the simple, but important, equations for joining together elements of waveguide

circuits

.

Following the presentation of the general theory, special theory for two-arm

waveguide junctions (2 -ports) is developed in Chapter 3, along lines considered

useful for analysis of measurement circuits. Some theory for 3-ports and 4-ports

is also developed.

Special emphasis is given to the use of scattering coefficients and they are

used in developing equations for many basic concepts. For example, efficiency,

mismatch loss, substitution loss, transducer loss, insertion loss, and attenuation

of a 2-port network are all carefully defined and equations given. The concepts of

phase difference and phase shifts associated with a 2-port are introduced. Transmission phase shift, insertion phase shift, and differential phase shift are defined.

Analytical tools based upon the cascading of 2-port networks and upon the transformation of reflection coefficient by a 2-port network are presented.

The realizability , reciprocity, and lossless conditions on the scattering coefficients of 3-port networks are given. Scattering matrices for circuit elements

such as directional couplers and circulators are given. Finally, the basic circuit

used in many measurement applications, consisting of a source, a 3-port network, a

detector, and load, is analyzed and equations are presented.

3
1.3. Applications

In Chapters 4 through 7, applications of the theory and analytical techniques to

the development of accurate measurement methods and standards are given.

The mismatch errors in the calibration and use of microwave power meters were

originally quite large. Application of the foregoing theory led to the reduction

of these errors and made possible the evaluation of limits of uncertainty as stated

in the calibration reports. In 4.2, the analysis and pertinent equations are

presented

.

In 4.3, the development of an improved method for measuring efficiencies of

barretter mounts is described. This work resulted in the first accurate determination

of efficiencies of coaxial barretter mounts. The analysis in terms of scattering and

reflection coefficients was found useful in later developments.

Applications to impedance or reflection coefficient measurement techniques and

standards are discussed in Chapter 5. The adjustable sliding termination described

in 5.2 is an improvement on previous designs. It employs a simple resistive strip

which can rotate and move relative to a short-circuit, in such a way as to produce

reflection coefficients ranging from zero to nearly unity. The entire termination

is designed to be slid inside a waveguide so that the phase of the reflection coefficient can be varied. This has proven to be a useful tool in measurement applications

In section 5.3, formulas, graphs, and conductivity data are presented to aid

any laboratory in designing and evaluating impedance standards consisting of quarterwavelength short-circuited sections of coaxial line or rectangular waveguide. All

standard sizes are covered over a frequency range from 200 MHz to 330 GHz.

A number of interesting circuits yielding squared VSWR response are analyzed

in 5.4. An even more useful development is "magnified response." It permits very

sensitive measurements of small reflection coefficients or of small differences

between impedances which are almost equal. The application of these techniques

to a modified phasable load method of impedance measurement is indicated.

The adjustment of tuners for some impedance measurement applications is

described in 5.5. The use of tuners led to the development of the tuned refl'ectometer, which, together with the quarter-wavelength short-circuit standard, became

the most accurate calibration technique for reflection coefficient standards.

4

/
An interesting variation of the tuned ref lectometer is described in 5.6. It

consists of constructing a tuned reflectometer using rectangular waveguide components,

except for the output waveguide, which is coaxial. One then slides loads in the

coaxial section and adjusts the tuners belonging to the rectangular waveguide instruments. Once adjusted, it is used to measure reflection coefficients of coaxial

terminations and devices.

The tuned reflectometer is also used to measure the reflections and losses of

waveguide joints or coaxial connectors by a sensitive technique described in 5.7.

There are many applications of microwave circuit theory to attenuation measurements as discussed in Chapter 6. The subject of attenuation definitions is discussed

at some length in 6.2. Circuit theory is used to clearly show different results from

different definitions. Precise definitions suitable for highly accurate measurement

applications are formulated and the conditions of measurement are tightly specified.

The error due to mismatch often causes the greatest uncertainty in attenuation

measurements. Mismatch effects in cas cade - connecte d attenuators are analyzed in 6.3

and mismatch errors in measuring fixed and variable attenuators are treated in 6.4.

Usually, only the magnitudes of the reflection coefficients of the circuits and the

attenuators are measured or estimated when evaluating mismatch errors. Of course,

the phases are also involved, but it is assumed that they can take on any value,

and the limits of error are calculated, assuming the most unfavorable conditions.

Actually, the realizab ility conditions limit the range over which reflection coefficient phases can vary. Thus the limits of error calculated by the above method

may be too conservative in some cases. In 6.5, it is shown that the effect of

realizability on error limit calculations is practically unimportant except for

low- loss attenuators, below, say 1 decibel.

In most analytical techniques, an attenuator is represented by a simple 2-port

network. However, this model gives no information regarding the effect of imperfections of connectors or adapters. A more complicated model is required, and this

is discussed fully in 6.6,

In sections 6.7 and 6.8, techniques for measuring attenuation are described.

The first makes use of the theory of linear fractional transformation of reflection

coefficient. The second technique is a simple measurement of power ratio, but is

refined to give imprecedented accuracy. Circuit theory is applied to evaluate

small mismatch errors which contribute to the uncertainty of the measurement.

5
The measurement of small attenuations , such as losses in waveguide joints and

in short sections of waveguide, by a 2 -channel nulling method is described in section

6.9. A circuit for producing known, very small changes of attenuation or phase shift

is described in section 6.10.

The topic of phase shift measurements and standards at microwave frequencies

was long neglected. In section 7.2, phase shift equations for 2-ports are presented.

The development of the tuned reflectometer made possible one form of microwave phase

shift standard. The phase of the reflectometer output signal is made to closely

track the position of a short-circuit sliding in a precision waveguide. Phase

measurements then reduce to a measurement of frequency and of mechanical displacements

and dimensions. The errors in such a standard were analyzed in 7.3.

A modification of the tuned reflectometer circuit was developed using two

short-circuits sliding in waveguides of slightly different widths in order to produce

known, small phase shifts. This is the basic principle of the differential phase

shifter described in 7.4.

Finally, the definitions of phase shift of various terminal variables are

analyzed and equations are derived in 7.2. The concept of an ideal phase shifter

is examined. This section, together with 3.10, gives a number of basic phase shift

equations useful in the analysis of phase shift measuring circuits.

1.4. Conclusions and References

In section 8, it is concluded that the foregoing applications of the theory

demonstrate its usefulness in developing accurate standards and measurement methods.

The steps in the development include precise definitions of the quantities to be

measured, development and evaluation of accurate standards and measuring techniques,

and the analysis and evaluation of errors.

A list of references, arranged alphabetically by the name of the senior author,

is given in section 9. The references not only support the work described but

indicate later work which extended or superseded the earlier work.

6
2. Basic Theory of Waveguide Junctions

2.1. Definitions

a. Waveguide Junctions

A "waveguide junction" is not simply a junction where some waveguides come

together and are joined where their walls intersect. The term has a broader meaning

but includes such simple junctions. For purposes of analysis, a waveguide junction

is considered to be an idealized representation of a given actual electromagnetic

device to which access is provided by means of waveguides.

Such a waveguide junction is linear and has uniform, lossless, cylindrical

waveguide leads which may be of arbitrary cross -section . It does not leak, or, in

other words, electromagnetic energy enters and leaves only thru waveguide leads.

In general, sources may be present inside the junction and there may be attenuated

modes present in the waveguide leads. However, in this monograph, these possibilities

are both excluded from consideration.

b. Terminal Surfaces and Terminal Variables

The outer boundaries of a waveguide junction are perfectly conducting surfaces

which prevent the flow of electromagnetic energy except inside the waveguide leads.

The waveguide junction may be considered to terminate somewhere inside the waveguide

leads at terminal surfaces which may be arbitrarily chosen. For convenience, they

are usually chosen to be planes perpendicular to the waveguide axes.

In order to provide a convenient measure of the flow of electromagnetic energy

in the waveguide leads, quantities such as v and i are defined which are derived from

the transverse electric and magnetic fields at the terminal surfaces. These quantities are called "terminal variables" and usually a set of two suffices to characterize the flow of energy in a given mode in a given waveguide lead. If more than

one mode is propagating in a given waveguide lead, there is a different set of

terminal variables associated with each mode.

7
2.2. V, i, a and b for Waveguide^

a. Introduction

It is convenient to define quantities denoted as v and i for waveguide that

behave in a similar way to voltage and current in lumped circuit element networks.

Then it becomes possible to apply conventional circuit theory to the analysis of

waveguide junctions and circuits.

The quantities called v and i are derived from the transverse components and

of the electric and magnetic complex vector field amplitudes corresponding to a

given waveguide mode. For example, is written as the product of two factors, v

and e". The factor v is complex and represents the time variation and phase of the

field. The factor e° is a vector function which gives the relative strength and

direction of the field in the waveguide (i.e., the mode pattern).

If there is more than one mode propagating in the waveguide, then a different

V and i are obtained corresponding to each mode (also a different e° and h°).

We write expressions for E^ and in terms of complex and vector potential

functions and then obtain v, i, e"* and h° in terms of them. In order to make v and

i behave like voltage and current, it is necessary to examine power and wave impedance

relationships and then define power and impedance normalization constants Wq and Z^.

The consequences of choosing these constants in different ways are discussed. For

example, by a suitable choice of Z^, v and i can be made to coincide with actual

voltage and current in a coaxial waveguide operating in the TEM-mode (transmission

line) .

b. Basic Derivation of v, i, e° and h"

For a given mode at a given terminal surface in a waveguide, we resolve E^ and

into factors as follows:

E^ = V e ° , and

= i h\ (2.1)

where v and i are complex and contain the information representing sinusoidal time

variation in the complex plane, and e° and h° contain information about the relative

magnitudes and directions of the transverse field components. We call v and i

^The principa,l concepts and conditions underlying the definitions of these symbols

are given in Kerns (1967) .

8
respectively, generalized voltage and current, and we call e° and h° the basis fields

for a given waveguide mode.

Since and may be expressed in terms of complex and vector potential

functions, it is also possible to express, v, i, e° and h° in terms of these functions as will be shown.

c. Complex and Vector Potential Functions

Let the axis of the waveguides be the z-axis and let the direction of propagation

be the +

z - direction . Waveguide field equations may be derived from Hertz potentials

of the form

n = f 4) e^, (2.2)

where f is a function only of the transverse coordinates, (f) is a function only of z,

e^ is the unit vector in the +

z -direction , and n satisfies the vector wave equation

V^n + k^n = 0, where = w^ye. (2.3)

We obtain a scalar wave equation

V^f + K^f = 0, (2.4)

where the solutions for f depend upon the boundary conditions corresponding to a given

waveguide. We also obtain a one - dimens ional wave equation in <^ whose solution is

(t> = Ae'^^ + Be^^, (2.5)

where the propagation constant y = a + j&

.

For TM (transverse magnetic) fields it can be shown (Kerns and Beatty, 1967) that

E = VxVxn = (})'Vf + K^f(j)e^

H = jweVxn = jojEct) (Vf xe^) , (2.6)

and for TE^ (transverse electric) fields ,

E = -jcjyVxn = -

j

toycj) (Vgxe^)

H = VxVxn = (!)'Vg + K^g(})e^, (2.7)

where g has been used instead of f because it is subject to different boundary

conditions

.

Some examples of f and g functions for certain waveguide cross -sections follow.

9
For rectangular waveguide of width a and height b, application of the boundary

conditions yields [Kerns and Beatty, 1967) ; For TM- modes

I mTTx mn

mn

s m sinfin^

l h

IHUE.

For TE- modes

'mn ^

= nil mn

mr

b

cos|l^

b

niT

(2.8)

^b

(2.9)

In p + C

For the TEM-mode in coaxial waveguide in which p denotes the radial coordinate,

we obtain

f =

= 0 (2.10)

In general, we can write for the transverse field components in the case of

TM-modes

E^ = (t)'Vf = ve°

= jaje<j,(Vfxep = ih° ,

where v °= ([>', e° ^ vf, i <^ jwe((), and h° <^ (Vfxe^) .

The corresponding relationships for TE-modes are

E^

(2.11)

ja)y(})(Vgxe^) = ve"

'Vg = ih°

,

(2.12)

where v -jwptj), e" ^ (Vgxe^) , i <^ (j) ' , and h° °^ Vg,

The constants of proportionality are to be determined, and will be done so in

the most convenient manner in the following.

d. Power and Impedance Normalization

Consider the relationships for power P, (integral of Poynting's vector across

the terminal surface) and wave impedance in terms of E^ and H^. We write the

following

P = Re(W) , where (2.13)

W = i

/ E^xH^'e^ ds, or (2.14)

10
W = (iv) J I e^xh^-e^ ds , or (2.15)

W = (iv) • Wq, (2.16)

where = j ! e'xh^'e^ ds . (2.17)

Wq is a power normalizing constant to be chosen for convenience. For example,

if i and v denote root-mean-square (rms) complex amplitudes, it is convenient to

choose Wq = 1, so that P = Re (Tv) . Once the value of has been chosen, it fixes

the product le^h"] for a given mode in a waveguide.

Next, we write the following expression for the wave impedance for a given

mode in a given waveguide

e xE* + e xe"

"t ^ ^

In this expression, and are the transverse components of the electric and

magnetic field waves traveling in the + z-direction , v* and i^ are the corresponding

traveling waves of "voltage" and "current," Z" is the "wave impedance" of the basis w

fields e" and h° , and Zq is an impedance normalization constant.

It can be seen that Z^ is proportional to the wave impedance Z^ , and the constant

of proportionality 1/Z^ is real (since e" and h" are not complex) and dimensionless

(since Z^ and Zq have the dimensions of ohms). Note that the dimensions of Wq, Zq ,

V and i can be chosen in several different ways , but the choice used here is considered to be the most convenient for use in circuit analysis.

e. Examples of Waveguide v and i

(1) TEM-mode in Coaxial Waveguide

The familiar coaxial waveguide operating in the TEM-mode (transmission line mode)

is an interesting example of convenient choices of normalizing constants Wq and Zq.

Depending upon how they are chosen, v and i can be made to coincide with actual voltage

and current in the transmission line. This is shown as follows. It has been shown

(Kerns and Beatty, 1967) that the potential function f for the TEM-mode in coaxial

waveguide is of the form [see eq. (2.10)]

f = Ct In p + C,

11
where the cylindrical coordinates p, e, and z apply, and the radii a and b, with

b > a, are used. Following eq . (2.11), let

f = -pipC, e

h" = C3(fxe^) = (2.19)

In the above equation, e , e , and are unit vectors and = C,C.

z ' p ' 9 2 3 The power normalization factor Wq is

^°^h°-e^ ds = ^ ^ ^

dp = TTC,C,£n -

„ 2 12 a (2.20)

The relationship involving the impedance normalization factor is

'w 1 e xe

l\i z

^0 ^o'^

(2.21)

Solving for C-j^ and C2 , we obtain

0 a

(2.22)

^^0

'2 (2.23)

' e a

Now consider the actual rms voltage V and current I in the transmission line in

terms of their line integral definitions

.

V = —

/7

• dil = V^

e^-e dp or

V rwl0_!in^, and

e 2ttZ 0

(2.24)

I = /2

2tt

• d£ = i h" 'e p -de or

I = i V^iin ^

^ e a

(2.25)

12
It is apparent from inspection of eqs . (2.24) and (2.25) that we can obtain the

convenient relationships V = v and I = i if we put W„ = 1 and

Z 0 (2.26)

This is the well-known equation for the "characteristic impedance" of the coaxial

line. Although this represents the most convenient choice for Zq , other choices

would be possible in which v ^ V and i I . (This fact is not so well-known.)

If Zq is chosen as above, it involves only the radii of the line and the

properties ]s and e of the medium inside the line. For a given line operating in

the TEM mode, there is only one value of Zq and thus, it is "characteristic" of that

line. The name "characteristic impedance" is apt in this case.

However, when modes higher than the TEM are considered, in coaxial waveguide

as well as other types of waveguide, the term "characteristic impedance" is less appropriate. In these cases, there is a v and i associated with each mode, but the line

integrals of and depend upon the path and there is no single convenient choice

of Wg and or of Zq .

If one chooses Z^ for a given mode in a given waveguide, then Zq is "characteristic"

of that mode in that waveguide for that choice. However, since different choices might

be found convenient for different purposes, Z^ is in the broad sense simply a normalization constant and is not really "characteristic" of either the waveguide or the

mode .

(2) TE^Q Mode in Rectangular Waveguide

Proceeding as in the previous section, we can obtain expressions for e° and h"

starting with eq. (2.9) and applying eqs. (2.12), (2.17), and (2.18). They are

Again, we consider some line integral definitions of "voltage" V, and "current"

I, which are arbitrary in that we choose the path of integration. Suppose that we

^It is considered that if Zg were to be characteristic of a given waveguide, then ther

could be only one value of Zg for that waveguide. Similarly, if Zq were to be characteristic of a given mode, then there could be only one value of Z_ for that mode.

e y

(2.27)

13
choose V to be the line integral of electric field along the center of the waveguide

cross -section which bisects the wide dimension. Then

V

b

0 t max

e dy = V / Ce°)

y ' i -^max

e dy, or

w

_0_

ab

(2.28)

I f we choose W

0 1 and = 4(b/a)Z , then V = v.u w

Now suppose that I is the total current in one of the wide walls . Then

a a

• e dx = -i

0

I = -/ H

0 ^ -i / h° • e dx, or

x

'

I = i '4aJYT^"

I. IT ^ Z ab w ^

(2 .29)

If we again choose and Z^ as above, we find that

(2 .30)

We would find it more convenient if I were to equal i. This can be arranged by redefining I. Suppose that the new I, or I^ represents the current in a strip of width w,

instead of the current in the entire wall. Let the strip be centered in the waveguide

wall. It can then be shown that I = i when w = 0.406a, where a is the total width

n

'

of the wall.

IT ^ b If, on the other hand, we were to choose = (-A — Z then I = i. We could ' ' 0 M-* a w

then redefine V to be the line integral over a shorter path. If the path length

were approximately 0.39 3b, then the new V would equal v.

In the above redefining of V and I, we could not take V larger than the line

integral of over the full height of the waveguide, and we could not take I larger

than the current in the total width of the widest wall. Thus, the limits between which

Zq may be chosen to obtain simultaneously V = v and I = i are

4 I

> ^ I

Z,, (2.31)

where

Z = w

377 ohms . ff

(2 . 32)

In the above expression, f is the operating frequency, f^ is the cutoff frequency,

and and are respectively the relative permeability and the relative permittivity.

14
The above example illustrates the arbitrary nature of the choice of and Z

and shows some of the consequences of various choices.

f. Traveling Wave Amplitudes a and b

Another convenient set of terminal variables are the voltage traveling wave

amplitudes a and b. They are related to v and i by

V = a + b

Zgi = a - b fCv - ZqD

The power and impedance (or reflection coefficient) relationships are

WgReCiv) = Wr,Re

0

b

a

i- Ca-F) Ca+b)

V . Zgi Z . Zq

-A (lar

^0

(2.33)

(2.34)

(2.35)

where Z V

i

Let

w

Z - Z„ (Z /ZA- 1 Z" - 1

Z + Z„ (Z /Z J +1 Z" + 1 (2.36)

Note that if Zq is in ohms, Z^ , the wave impedance of the basis fields, is a normalized

impedance and is dimensionless

.

g. Other Traveling Wave Amplitudes

Two other sets of traveling wave amplitudes will be mentioned. They are sometimes

called power waves, since they have the dimensions of the square root of power.

One set of traveling wave amplitudes is very simply related to a and b and is

used to suppress W^ and from the power and impedance relations. This can be

convenient when carrying through a complicated analysis. The suppressed constants

can be reinserted after the analysis has been completed, if desired.

If we define a new set of terminal variables

a' = a / —, and b ' = b / —

,

V 7 V 7 ^0 0

(2.37)

then the power relationship is simply

' I 2 (2.38)

15
and there is no change in the reflection coefficient

b' b

a (2.39)

Then

(a' + b')

i = a' + b'

0

^^0

(2.40)

Another set of terminal variables is used in the analysis of transistor circuits,

for example, and is defined as follows (Kerns, 1967).

11

m 2 (v + Z i ) ^ m m m"^ m

1 1 (2.41)

m 2 m

(v - Z i ) , ^ m m m-^ '

in which Z^ is the impedance terminating port m.

The existence of different sets of terminal variables each designated as a and b

is sometimes confusing and can lead to errors if the basic definitions are not clearlyunderstood.

2.3. Parameter Matrices

a. Impedance and Admittance Matrices

The sets of simultaneous linear equations relating the pairs of terminal variables

of a waveguide junction can be written compactly in matrix form. For example, for a

total number N of propagated modes, we define the column matrices

V = and i (2.42)

We can then write the set of equations relating the v's and i's in matrix form

V = Zi, or i = Yv, (2.43)

16
where

Z =

Hi hi

^21 ^22

7 7

"IN

2N

and Y is the inverse of Z. The above Z and Y matrices are called the impedance and

admittance matrices, respectively.

b. The Scattering Matrix

I£ we consider the terminal variables a and b in a similar way, we obtain

b = Sa, (2.44)

where S is the scattering matrix. It has elements similar to those of the above

impedance matrix, and the elements are called scattering coefficients.

Since there are several sets of terminal variables denoted by the letters a and

b, it happens that there are also different scattering matrices denoted by the letter

S. For example, the coefficients in the scattering matrix used to relate power waves

(Bodway, 1967) are sometimes called "S-parameters .

This situation can cause confusion. In order to distinguish between different

scattering matrices, it is necessary to examine the definitions of the terminal variables that they relate.

In this monograph, the scattering matrix will relate the complex voltage wave

amplitudes a and b as defined in eq. (2.33).

The relationships connecting Z, Y, and S, are written in matrix equations as

follows

Z = (1 + S) (1 - S)'^ Zg = y'^ (2.45)

(ZZ"^ - l)(ZZ'l + 1)"^ = S = (1 - Z„Y)(1 + Z„Y)'^ (2.46)

c. Power

(1) General

The total complex power input to a waveguide junction is by extension of

eq. (2.16), in matrix notation

W = i*WQV, (2.47)

17
where * denotes the Hermitian conjugate and

It can be shown that

^0

=

^01 °

0

0 0 w

0

0

ON

W = i*WQZi = v*Y*WqV, and

P = Re(W) = aMWgZQ^ - S*^^!'^^)^.

(2.48)

(2.49)

(2 ) Realizability Conditions

A waveguide junction is said to be "realizable" if its Z, Y, and X are such that

Re (W) > 0 for arbitrary v or i. The conditions thus placed on Z, Y, and S are the

"realizability conditions" for passive waveguide junctions.

Consider the impedance matrix Z, and define = ^(^qZ + Z*Wq) . Note that

is Hermitian (H^ = H^) . Since

Re(W) 5 Re(i*WQZi) = , (2.50)

conditions on Re (W) are equivalent to conditions on the matrix of the Hermitian form

i^H^i.

We now distinguish (Kerns and Beatty, 1967) three cases of realizability, according

to whether the dissipation in the junction is positive for every non-zero i, for only

some i, or for no i.

(a) "Strict realizability": Re (W) > 0 for every non-zero i. In this case the

Hermitian matrix H^ and the associated form are said to be "positive definite" (or,

sometimes, "strictly positive"). A useful criterion for this case is: a Hermitian

matrix is positive definite if and only if all its principal minors are positive.^

(b) "Semi-realizability" : Re (W) >_ 0 for every i and Re (W) = 0 for some non-zero

i. In this case H^ and the associated form are said to be "positive semi -definite"

;

a criterion for this case is: a Hermitian matrix is positive semi -definite if and

only if it is singular and all its principal minors are nonnegative (Mirsky, 1955).

^For a proof of this theorem see Mirsky (1955). A "principal minor" of a matrix A

is a minor whose diagonal is part of the diagonal of A. Thus a principal minor is

obtained by selecting rows and columns with the same sets of indices. Special cases

of the principal minors of A are the diagonal elements of A and the determinant of A.

18

/
(c) "Losslessness" : Re (W) = 0 for every i. For this case it is easily shown

directly that must be the (NxN) zero matrix.

For the admittance matrix Y and for the scattering matrix S, the matrices corresponding to are = ^(WqY + Y*Wq) and Hg = (WqZq"'- - S*WqZq-'-S), respectively.

Realizability conditions for Z, Y, and S may be summarized as follows: Realizability requires the matrices H^, Hy, and Hg to be positive definite, positive semidefinite, or zero, according as case a, b, or c applies.

At this point it is observed that certain simplifications may be obtained in the

above analysis and results by suitable choices of normalization. For example, if

is a scalar matrix (i.e. a scalar multiple of the unit matrix), it cancels out in

the statement of realizability conditions for Z and Y; similarly, if W^Z^''' is a scalar

matrix, it cancels out in the statements pertaining to S (see table 2-1).

Table 2-1. Realizability and reciprocity conditions under simplifying

normalizations.

Z (Wq scalar, Zq

arbitrary)

Y (Wq scalar, Zq

arbitrary)

S

(WqZq"^ SCALAR)

REALIZABILITY Z+Z*PD, Y+Y*PD, 1 -S*SPD,

PSD, OR 0. PSD, OR 0. PSD, OR 0.

RECIPROCITY Z = Z Y = Y ^ = S

NOTE: 1. PD = POSITIVE DEFINITE, PSD = POSITIVE SEM I - DEF I N I TE . 2. RECIPROCITY MAY OF COURSE HOLD SIMULTANEOUSLY WITH

ANY CASE OF REALIZABILITY.

d. Reciprocity

Provided that the parameters u, e (which may be complex) are symmetric tensors

(which may reduce to scalars) it can be shown that (Kerns, 1949a)

n

I I (E'xH" - E"xH') • n dS = 0, (2.51)

m=l S m . m

where E', H', and E", H" denote any two electromagnetic fields (of the same frequency)

that can exist in the given waveguide junction. From eqs . (2.1) and (2.17) it follows

that eq. (2.51) is equivalent to

i"WQv' - I'WqV" = 0, (2.52)

where the ~ denotes the transpose of a matrix.

19
To find the consequences of eq . (2.52) for Z, Y, we first insert v' = Zi',

v" = Zi". After taking the transpose of the second term, we obtain

which implies

i-'CW^Z - ZWQ)i' = 0,

W^Z - ZWq = 0, [2.53)

- 1

~ ~ -

1

since i', i" are arbitrary. Since Z = Y (and Z = Y ), the relation

WqY - YWq = 0 (2.54)

is an immediate consequence of eq . (2.53).

To find the conditions imposed on S we use eq . (2.33) and find from eq. (2.52)

a"Zg^Wgb' - a'ZQ^Wpb" = 0.

Hence, using b = Sa, we must have

Zq^WqS - SZ'^Wq = 0. (2.55)

The reciprocity conditions eq . (2.52) and eq . (2.55) may be simplified by appropriate

choice of normalizations, and it happens that the appropriate choices are the same

as in the case of the reali zability conditions considered above. Table (2-1)

furnishes a summary of all these relations in simplified form.

e. Sources; Joining Equations

(1) General

We have already noted that our basic expression for the power input contributed

by one waveguide mode, W = i v (for W° = 1), is of the same form as the expression

a a a a for input power at a pair of terminals in an alternating- current network. We shall

now consider two further fundamental relations that are required to establish the

basis of the application of equations of the form of network equations to waveguide

problems

.

(2 ) Sources

For simplicity consider a waveguide "junction" having just one waveguide lead,

in which just one mode propagates. We choose a terminal surface and consider the

terminal variables v^ , i^ . We assume that the junction is linear (from an external

20
point of view) but not necessarily passive. The most general linear relation connecting v-j^ and i^ can be written

or h = V^l ' \l' ^2.57)

where Z^,Y^,V-,, i-, are constants. The presence of sources may be manifested

s s s s in these equations in two ways: in the inhomogeneity of the equations (i.e. v f 0)

or in a violation of the realizability condition as applied to Z ^ and Y ^ . The

' gl gl

latter possibility means for these 1x1 matrices Re(Zg^) < 0 and Re(Yg-|^) < 0 , as

may be seen in table (2-1), It should be observed that eq. (2.56) and eq. (2.57)

respectively represent versions of Thevenin's and Norton's theorems.

Alternatively, we may describe the source in terms of the terminal variables

b^, a^ , The most general linear relation connecting these variables may be written

^1 = Sgl^l ^gl' f2.58)

where S , and b , are constants characterizing the source. This might be called Kerns'

gl gl 6 5

theorem, after D. M. Kerns. The equation states that the general emergent wave b^ is

the sum of the wave b^^ that would be emitted into a non-reflecting load and the

reflected portion of the incident wave a^. From table (2-1) we see that violation

of the realizability condition for the 1 x l scattering matrix S^-j^ means I | > 1.

(3) Joining Equations

Suppose that a waveguide lead of one system is to be connected to a waveguide

lead of another system. We assume that the terminal surfaces associated with each

system have been so located that they coincide when the connection is made (fig. 2-1).

The transverse components of E, H on the common terminal surface S are then given

by the equations

E^ = y V e°

,

t ^ a a'

= I i^ha. ' (2.59)

associated with the one system and also by the equations

= y V e° , t a a

'

=

I i^h^' , (2.60)

21
associated with the other. We assume

this implies

a a

a a

'

C2.61)

[2 .62)

Figure 2-1. Coincident terminal surfaces.

since n = -n' . For the electromagnetic fields corresponding to eqs . (2.59) and (2.60)

to be continued properly across S, it is necessary and sufficient that

= E^, H H^. This means that it is in turn necessary and sufficient that

^a = ^a

ia = t2.63)

for each mode involved. These are the joining equations of waveguide theory. They

are of exactly the same form as the equations in circuit theory that describe the

joining of two pairs of terminals. To verify this, consider that the terminal pairs

shown in figure 2-2 are to be joined.

Figure 2-2. An "equivalent circuit"

for joining.

With the sign conventions indicated in figure 2-2, circuit theory obviously requires

V = v' , i = -i'.' (The sign conventions are determined by eq . (2.62) together with

the choice of n as the inward normal on S .

)

m m

22
!i

In the equations that characterize a waveguide junction, such as the matrix

equation v = Zi, the number o£ variables is twice the number of equations. In a

waveguide junction characterized by a set of N equations, the electromagnetic

state has N degrees of freedom. However, if loads or sources'* are connected at all

terminal surfaces and the joining equations are applied, the number of equations

becomes equal to the number of unknowns in the system. Thus, except for special

cases where the equations are not all independent, the terminal variables (and hence

the electromagnetic state) become determinate.

passive waveguide junction possessing just one waveguide lead (multimode or not)

is termed a "load" or a "termination"; if not passive, it is termed a "source."

23
3. Introductory Network Analysis

3.1. Linear Network Parameters

a. Introduction

It has been shown in the previous section that network equations developed for

alternating current networks consisting of distinct circuit elements may also be used

in waveguide problems in which distinct circuit elements do not exist as lumped

elements, but are distributed in space.

These network equations are briefly reviewed as a point of departure to reintroduce scattering coefficients and other parameters used to relate wave amplitudes.

b. Terminal Variables

It is customary to provide access to a network by means of terminals, and in

most cases, these are grouped in pairs. We can have input terminal pairs to which

sources are connected to feed energy into a network, and we can have output terminal

pairs to which loads or terminations are connected to absorb or reflect energy emerging

from the network. Usually, there is a terminal pair to which a detector is connected,

especially in circuits used for measurement purposes.

The voltage v across a given pair of terminals, and the current i flowing into

one terminal (and out of the other) are the terminal variables in common use. The

relationship between the terminal variables at one terminal pair and those at another

terminal pair is determined by the characteristics of the network. If all of the

network elements are linear, the relationship is given by a set of linear equations,

having coefficients which are independent of the terminal variables. These coefficients are called network parameters.

c. Network Parameters

It is possible to obtain more than one set of parameters for a given network,

depending upon how the terminal variables are selected at the terminals. Three frequently encountered sets of parameters for a two-terminal pair network (four-pole,

or 2-port) are shown in figure 3-1.

24

/
The equations defining the parameters are given in three forms. The first form

is the usual set of two simultaneous linear equations. The second form is the corresponding matrix equation. The last form is the short form of the matrix equation,

where the matrices are indicated by single letters.

Figure 3-1. Representation of a fourpole (2-port)

^1 = ^1^1 ^ ^2^2 Hi ^12' '^l

^2 = hih " ^22^2 ^^2- ^^21

7^22-'

^1 - ^11^1 * ^12^2 ^l' •^11 ^12^ ^1

^2 = ^21^ >2. .^21 ^22'

= - Bi^ ^1^ A B' [-2^

= - Di2; >1^ C D V^2^

V = Zi. (3,1)

Yv. (3.23

(3.3)

The above matrices are called respectively the impedance matrix, the admittance

matrix, and the ABCD matrix (general circuit parameters). The ABCD matrix may also

be called the v and i cascading matrix.

d. Complex Wave Amplitudes a and b

Access to a waveguide junction is provided by means of waveguide leads. Terminal

surfaces chosen in the waveguide leads form part of the outer boundary of the waveguide

junction. The amplitude a of the voltage wave incident on the junction and the

amplitude b of the voltage wave emerging from the junction at each such terminal

surface are one type of terminal variables in common use. The terminal surfaces where

energy may enter or emerge from a waveguide junction are also called ports.

e. Parameters Associated with a and b

It is possible to define many sets of parameters relating a and b for a given

waveguide junction. Three of these sets of parameters for a 2-port network are shown

in figure 3-2, and defined in eqs . (3.4), (3.5), and (3.6).

25
1 2

j

1

1

1

U Q_

I

1

1

Figure 3-2. Representation of a 2 -port waveguide junction with

terminal surfaces and terminal variables a and b.

+ ^12^2 "^1 ^1 S 1 12

^2 =

' ^21^1 + ^22^2

^21 ^22-

^1

== Sllh + §12^2 §11

^2

== §21^1 + §22^2' §21 ^22-

^11^2 + ^12^2 ^11 ^2' '^2

^1

=

'' ^21^2 + ^22^2 1-^21 ^22J .^2.

b = Sa

gb

(3.4)

(3.S)

(3.6)

The coefficients in the above equations are called scattering, gathering, and

cascading coefficients, respectively.

f. Other Terminal Variables

Another set of terminal variables v and i can be used with waveguide junctions.

They are generalizations of voltage and current and can actually represent transmission

line voltages and currents in cases where only the TEM-mode propagates. The parameters relating v and i for waveguide junctions are called by the same names as the

corresponding ones relating voltage and current. Still other terminal variables

could be defined by forming linear combinations of the ones already mentioned. However, terminal variables other than v and i, and a and b have not been widely used.

g. Network Equivalent to a Waveguide Junction

A network which shares an identical set of parameters with a waveguide junction

is said to be equivalent to that junction. Such equivalence may hold at only one

frequency at which the parameters are defined, or in the less usual case, might

hold over a range of frequencies.

For example, the parameters relating the terminal variables v and i for a

waveguide junction may be identical with those which relate voltage and current for

a network. In this case, the impedance matrices would certainly be identical. It

follows that all of the other parameter matrices would be the same as the corresponding ones for the equivalent network. This is true because each set of

parameters relating terminal variables is linearly related to each other set.

26
Occasionally some difficulty may arise in identifying equivalent networks if,

for example, we are given a scattering matrix for a waveguide junction and find

that the impedance matrix does not exist (the elements may be infinitely large).

An example is the following, in which the equivalent circuit is a single series

impedance as shown in figure 3-3.

Figure 3-3. A fourpole having a single series

element.

By inspection, the admittance matrix is

Y = 1- ( 1 -1

-1 1

It follows from eq . (2.46) that the corresponding scattering matrix is

2Z ^1 " '^^Ol " ^02^

2Z 02

^01

If we choose Z^^ = Z^^ = Z^, then

(Z^/Zq)

(Z./Zq)

(3.7)

(3.8)

Application of eq. (2.45) in order to obtain an impedance matrix will reveal that

each of the elements is infinite. It is then said that the impedance matrix does

not exist

.

3.2. The Scattering Matrix

a. General Remarks

The scattering matrix has an appropriate name as we can see from the example

below

.

Figure 3-4. Representation of a multi-arm

waveguide junction with energy

incident in arm 1.

27
As shown in figure 3-4, a wave of amplitude a^ entering the waveguide junction

in arm 1 is scattered, and some of the energy is transmitted to each of the other

arms of the junction. (For simplicity, it has been assumed that the other arms have

nonreflecting terminations so that there are no reflected waves in these arms.) The

coefficients in the linear equations relating the amplitudes of the emergent waves

(the b's) to the amplitudes of the incident waves (the a's) are called the scattering

coefficients. The matrix of these coefficients is called the scattering matrix which

was defined more formally in eq . (2.44).

In general some scattering will also occur as a result of reflection. Since

scattering involves both transmission and reflection of energy, it is to be expected

that the scattering coefficients will be of two kinds; transmission coefficients and

reflection coefficients. This will be illustrated clearly in the case of two-arm

waveguide junctions (2-ports), to be discussed.

Much of the theory of two-arm waveguide junctions may be applied to waveguide

junctions with more than two arms, as will be shown below. Consider a waveguide

junction having n arms, all of which are terminated by non-reflecting loads except

th th the p and q arms. This requires that all incident wave amplitudes vanish except

for a^ and a^. The scattering equation b = Sa for the waveguide junction then reduces

to

b = S a + S a

p pp p pq q

(3.9)

b = S a + S a , q qp p qq q'

considering only the emergent waves in the p^^ and q^'^ arms. This is of the same

form as that for a two- arm waveguide junction, so that one can for example determine

S , S , S , and S by the same techniques developed for determining ^ , S^^,

pp ' pq ' qp ' qq ^ 11' 12'

S2-|^, and of two-arm junctions, to be described.

b. Scattering Coefficients of a Two-Arm Waveguide Junction

Consider a two- arm waveguide junction with a source connected to arm 1 and a load

connected to arm 2. The reflection coefficient of the load is designated by r^^ as

shown in figure 3-5. The reflection coefficient of a load is defined to be the

ratio of the amplitude of the wave reflected from that load to the amplitude of the

wave incident upon it.

28
SOURCE WAVEGUIDE

JUNCTION

1^02

LOAD

Figure 3-5. A two-arm waveguide junction connected

between a source and a load.

Referring to eq. (3.4) it can be seen that the input reflection coefficient

can be written

^11 ^

1 - ^22^1

"^^12^21 ^11^22^^L ^11

^22^L

(3.10)

Note that if there is no reflection from the load (Tj

reflection is within the waveguide junction, and

fb.

"

0) , the only source of

1 = S

a2 = 0

11' (3.11)

Therefore S^^ is the reflection coefficient "observed looking into" arm 1 with

arm 2 terminated in a non-reflecting load. We can say that S^^^ characterizes the

reflecting property of the waveguide junction for energy entering arm 1.

A similar argument with the above roles of arms 1 and 2 reversed will show that

is the reflection coefficient "observed looking into" arm 2 with arm 1 terminated

in a non- reflecting load. Thus, the scattering coefficients S^^^ and are reflection coefficients. We can generalize on the basis of previous remarks about eq. (3.9)

that any scattering coefficient of the form S^^ is a reflection coefficient if p = q.

It can be shown conversely that when p ^ q, the scattering coefficient S^^ is

a transmission coefficient. If the transmission coefficient is defined as the ratio

of the amplitude of the wave emerging from arm 2 to the amplitude a^ of the wave

incident in arm 1, when a non-reflecting load is connected to arm 2, then inspection

of eq. (3.4) shows that

fb.

"

21' (3.12)

a2 = 0

A similar argument applies to S^^j a-nd extension to S^^ is straightforward,

29
c. Effects of Moving Terminal Surfaces

The scattering coefficients of a given waveguide junction have been defined in

terms of wave amplitudes at certain arbitrarily selected terminal surfaces. If new

terminal surfaces are chosen, a new set of scattering coefficients will then apply.

Since only a simple change was made in the waveguide junction, one would hope for a

simple relationship between the new and old sets of scattering coefficients. That

this is the case may be shown as follows. Consider the new set of wave amplitudes

and scattering coefficients to be denoted by primes.

It is well known that a wave traveling along a uniform section of waveguide

experiences attenuation and phase delay as it progresses. If, as in figure 3-6, a

wave traveling to the right has an amplitude a^ at terminal surface 1', it will have

an amplitude a^ = a^e ' at terminal surface 1. Similarly, a wave traveling to the

left having an amplitude at terminal surface 1 will have an amplitude b^ = b-j^e'^^^

at terminal surface 1'. Here we regard £^ only as a distance between two terminal

surfaces, considered as always positive, since we have not set up any conventions of

positive and negative displacement in this case.

1 1

1 1

2 2

1 1

1

1

1

1

1

j

1—a'l 1—Q,

2*—1 Qy- 1

1

1

1

Figure 3-6. Changes of locations of terminal surfaces

from 1 and 2 to l' and Z'

.

Assuming that the waveguide leads are lossless, their propagation constants (y's)

will be j3-j^ and j&2> following relationships will hold:

a^ = a;e-j^^^\ b; = b^e'^^^^^

^2 = ^2^

' b' = b-e"JP2x^2 (3.13)

We apply the definitions of S^^ and as in eqs. C3.ll) and (3.12) to S^^ and S^-j^,

obtaining

s;^ = S^^e-j26i^i and 5'^^ = S^^e-^^^^'^^^^'^K (3.14)

Interchanging subscripts 1 and 2 yields

s' = S..e"^^^^^2 22 ^22® " ' ' ^12 " ^12^ jf^2^2 3i£i) (3.15)

No such simple relationship would be obtained with impedances and admittances, and

this is one important advantage of using scattering coefficients.

30
I£ we now designate the original set o£ terminal surfaces and scattering coefficients by primes, eqs . (3.14) and (3.15) will still hold, but we will want to solve

them for S^^, 5^^, 5^^, and S22- Upon doing this, it will be found that the algebraic

sign of the exponents will be positive instead of negative. The effect of shifting

a terminal surface to a new position in either direction should now be evident.

3.3. Reciprocity, Realizability , and Losslessness for 2-Ports

The relationships derived for waveguide junctions in general will be specialized

to apply to 2-ports. The meanings of these relationships in terms of energy flow will

be examined later.

In the following, the usual symbol will be used to designate the normalizing

impedance of the propagating mode (usually only one in each waveguide lead) , and an

additional subscript will be added to designate the particular waveguide lead of the

waveguide junction (these waveguide leads are often identical, but are sometimes quite

different from one another)

.

The power normalization matrix will be taken equal to the unit matrix so that

it will disappear from the following equations. This is felt to be justified because

it seldom happens that we need to choose otherwise, and if we do, we can refer back

to sections 2.2 and 2.3.

We will carry through the normalization factor because it is more often useful.

Cases continue to occur where the waveguide arms of a junction are not identical and

we do not wish to choose all of the Z^'s equal.

a. Reciprocity

In terms of the scattering coefficients it has been shown that the reciprocity

condition is given by eq. (2.55). Assuming that the power normalization matrix is

the unit matrix, we have

Zq^S = SZq^ • (3.16)

Performing the indicated multiplication for the 2x2 matrices, we obtain

r^i ^2' fll ^21]

hi ^01 hi h2

^21 ^22 ^12 ^22

.^02 ^02. .^01 ^02.

31
The resulting condition on the scattering coefficients is

^12^02 " ^21^01- (3.18)

It is often stated that reciprocity implies the equality of S^^ ^rid seen

that this is true only^ when Z^^ and Z^^ equal. It is not always convenient or

appropriate to choose them equal, so the correct relationship above should be kept

in mind.

b. Strict Realizability

The condition of strict realizability, discussed in section 2.3, excludes

loss lessness , and is given by the following matrix inequality

Re(W) = a*Hga > 0, for arbitrary a ^ 0,

where Hg = WqZq" C3.19)

as in eq. (2.49). This condition requires that Hg be positive definite, which in

turn requires that all of the principal minors of Hg be positive. However, it is

not necessary to show that all of the principal minors are positive, since this

follows automatically if one shows that the leading minors are positive. The leading

minors are the ones which progressively include the elements on the principal diagonal,

starting at upper left.

The equation for Hg for 2x2 matrices, assuming that is the unit matrix, is

"S

=

1

- l^lll^

'01

^2li

'02

'"^ 11^^12 ^ f 21^22

^12^11 ^ ^22^21

01 02

Thus, strict realizability requires that

01

1 - IS 22

'02

'02

IS 12

'01

(3.20)

'01

'02

'02

'01

21

1

- l^ll^

12

S I 2 ^22 I

< 1,

< 1,

^If new terminal variables are employed such that S^' ^21 ^ S., /(Z„^ /Z_-) , then reciprocity requires S

12 ^12^f^02/^01^ ^""^

12 '21' See section 2.2.g.

32
and

'01

c 12 ^22 1 C 121

02 02

2

01

> 0, (3.21]

^01 ^02

On testing a scattering matrix to see whether it corresponds to a strictly

realizable waveguide junction, only two o£ the above inequalities need to be considered;

the last one, and either of the remaining ones. It is noteworthy that the first two

inequalities involve the magnitudes but not the phases of the scattering coefficients,

while the third inequality involves both magnitudes and phases.

c. Losslessness

While a lossless waveguide junction is not actually obtainable in practice, one

can approach this condition closely, and it is therefore important. The assumption

of lossless waveguide leads which has been made, is of comparable importance. It has

been stated, in section 2.3, that losslessness requires Hg to be zero, which requires

each element of Hg to vanish. This yields the following restraints upon the scattering

coefficients

'01

'02

l^lll^

c 12 2ll ^22 I

^11^12

^21^22

^12^11

^22^21

(3.22)

It can be shown that an equivalent and more revealing set of restraints on the

scattering equations is the following:

11

= Is 22

^01 1^21 ^02 I ^12 /[Zo^Zo2(l-S^)],

and

'12 + ^21 =

'^ll ^ *22 ± ^2n - 1)^, (3.23)

where ijj is the phase of S and n is an integer.

P<^ pq

^

It is interesting to note that the condition of losslessness implies a partial

symmetry in that |S^^| = |S 22 I . and a partial reciprocity in that 2g-j^|S2-|^ ^021^12

One, also notes that this condition requires every element of Hg to vanish, while the

strict realizability condition requires that all of the principal minors be greater

than zero. Therefore, one should not in general simply replace the inequality signs

in eq. (3.21) by equals signs to convert to the lossless condition.

S

33
3.4. Power and Efficiency

A two-arm waveguide junction is often connected between a source of energy and

a load as shown in figure 3-7.

This arrangement shown in figure 3-7 is frequently encountered in measurement

systems. The rate of energy flow into the junction equals the incident power P^^

minus the reflected power Pj^-|^.

1 11

/3, ^3 1

SOURCE 1 WAVEGUIDE

JUNCTION 1 LOAD

1

1

1

Figure 3-7. Diagram of a simple waveguide system with energy

flowing from left to right, with net powers Pi and

Pl and with incident and reflected powers P

and Pp 1

.

II

The rate of energy dissipation in the load is P^^ , and the rate of energy dissipation in

the waveguide junction is P^ - Pj^ . The efficiency of the waveguide junction is

defined to be the ratio of P^ to P^. An expression for the efficiency is developed

as follows: The net power P^ crossing terminal surface 1 to the right is

Pl = Pji - Pri. (3.24)

where the incident power P^.^ = \a.^\^/l^^, and the reflected power Pj^^ = |b-j^|^/ZQ^,

and therefore

(1 r 1 M

'01

where = b^/a^,

Similarly, the net power P^^ absorbed by the load is

'2 |b. 1^

(1

'02

(3.25)

(3.26)

where V-^ = a2/b2.

The efficiency of the waveguide junction when energy is fed into arm 1 is

'01

'02 1

- (3.27)

Substituting eq. (3.10) for r^, and solving eq. (3.4) for the ratio of to

a^ , one obtains

'01

'02 II - S^^v^\^ - 1(5^2^21 - ^1^22^^! * ^lll^

(3.28)

34
The efficiency is seen to be not only a property of the waveguide junction,

ilso of the reflf

non- reflecting load,

but also of the reflection coefficient of the load. In the special case of a

r 1 01 I 2ll

['^lla, = 0 = — • : ^-TT- C3.29)

If the positions of the source and the load with respect to the waveguide junction were to be reversed, it is apparent that the expressions for the efficiency t]^

would be of the same form, but with subscripts 1 and 2 interchanged. Thus

Do = • — — C3.30)

^01 11 - ^II^lI' - 1^^2521 - hl^22^h ' ^^^^

and

r T ^02 1^12!

'

['^2]a,=o = — * rr—T (3. 31)

^ ^01 ^

' I 22I

It is interesting to compare eqs . (3.29) and (3.31) with the inequalities eq.

(3.21). The first two inequalities simply state that the efficiency of a strictly

realizable waveguide junction is less than unity for two conditions. The first

condition assumes that energy enters arm 1 and there is no reflection from the load

on arm 2. The second condition is similar except that 1 and 2 are interchanged.

3.5. Representation of the Source

If we consider a source connected to arm 1 of a waveguide junction as in figure

3-7, it is possible to show in a number of ways that

^1 = ^ ^l^G' C3.32)

where a^ is the amplitude of the wave in arm 1 incident upon the junction, b^^ is the

amplitude of the wave that the generator would emit to a non-reflecting load,^ b^

is the amplitude of the wave reflected from the junction in arm 1, and is the

reflection coefficient of the source. It is considered desirable to build confidence

in this relationship by giving alternate derivations, since it is widely used in

circuit analysis.

a. From Linear Relation for Source and Joining Equations

For convenience, the wave amplitudes referred to in eq . (2.58) will be primed,

^It is assumed here that the generator is unaffected by load changes. This is not

true in general, but is approximately true if isolation or buffering is employed

between the active source and terminal surface No. 1.

35
and and S^^ become respectively b^ and r^^. Then the diagram of figure 3-8

represents the situation when a source is joined to arm 1 of a waveguide junction.

The joining equations are evidently

and

a = b^,

^ = aj,

1 1

SOURCE o;»-j |—a,

J—^tj| b|-^

WAVEGUIDE

JUNCTION

I

Figure 3-8. Representation of a waveguide junction

connected to a source of energy.

Substituting these into the modified eq. (2.58) yields eq . (3.32). Note that the

a's and b's of eq. (3.32) are chosen in different directions than in section 2.3.e.

b. From a Constant Voltage Generator

If we postulate a constant voltage source as shown in figure 3-9 and relate a

and b to V and i in the usual manner, it is found that eq . (3.32) can again be

obtained.

I Figure 3-9. A constant voltage generator with a waveguide output

having either terminal variables v^ and i^ or a^

and bi .

The steps are as follows:

a^ + b-j^ = e

v^ = e - i-^Z^,

(a^ - b^) = e

-

1

ia.-^ - b^) , 01 " G

2aT (1 - r^r^) = e(i - r^),

^1 =

1

1 - ^iFg

7

CI - r,) (3.34)

We define b^ as the amplitude of the emergent wave from the generator (a^) when a

non-reflecting load is connected (r^ = 0). Thus

- (1 - r.) (3.35)

36

/
Then it follows that

1 - ^l^G

C3.36)

or

(3.323

c. Summation of Wave Reflections

Supposing that a wave amplitude b^ emerges from the generator, one can consider

the multiple reflections that take place as shown in figure 3-10.

Figure 3-10. Representation of multiple reflections at a terminal

surface in the waveguide connected to an energy source.

The sum of wave amplitudes reflected toward the generator is

^1 = ^G^l^l * "^^G^l^ * ^^G^l^'

or

b„r

b = H_J_, or b.

1 - ^G^l

^G^l

or

^1 = ^G " ^^G' (3.32)

3.6. Net Power and Available Power

a. Net Power to a Waveguide Junction

Suppose that a source is connected to a waveguide junction as shown in figure

3-7. The net power delivered to the junction is

P = -1— (1 - ir 1^) = . '-11—. (3.373

^01 ^01 \' - 'Gh\'

37
The components of this power associated with the incident and reflected waves are

respective ly

II

2

^01 I-"- " ^G^ll^

and

PrI = —^ • 7- C3.38)

lb 1 2 I r I ^

It is sometimes erroneously assumed that the incident power from a generator is

always independent of the load. However one can see from the above equation that this

can be true only if the generator is non-reflecting, or = 0.

b. Available Power from Generator

It is seen above that the net, incident, and reflected power from a generator all

depend upon the reflection coefficient of the effective load terminating the

generator. It is well known that the net power will be maximum when the load impedance

is the complex conjugate of the generator impedance, or when

Z-^ = Zq. (3.39)

This condition in terms of the corresponding reflection coefficients (remembering that

Zq^ is real for lossless waveguide leads) is

^1 = Tq. (3.40)

The net power output from the source under this condition is termed the available

power P^, and is obtained by substituting eq. (3.40) into eq. (3.37), as foil ows

i^gI^ 1 = • 7

77' (3.41)

z 1 - r P

The components of the available power associated with the incident and reflected waves

respectively are

^AI

lb I 2 P i°gI 1 A

and

Par = • = Pa • (3-42)

Compare these to the net power P^ delivered to a non- re fleeting load,

P = (3.43)

7'^01

Thus Pq is generally less than P^^ except when the generator is non-reflecting (T ^ = 0)

,

and then they are the same.

38

/
3.7. Mismatch Loss

a. Mismatch Loss in General

We define mismatch loss Lj^ as the ratio in decibels of the power Pj^ absorbed

by a matched load to the power Pj^j^ absorbed by a mismatched load, when these loads

are alternately connected to the same source (generator) , or

P M

L„ = 10 log,„ (3.44)

It is necessary to understand what is meant by the terms "match" and "mismatch,"

and their meanings will be discussed later.

One sees that the above mismatch loss is simply a special case of the ratio of

the powers absorbed by two different loads which are alternately connected to the same

generator. If the load initially connected has a reflection coefficient """r-j^, the load

finally connected has a reflection coefficient r^, and the generator has a reflection

coefficient r^, the ratio expessed in decibels is

= 10 log

10

1 - r ""t ^ ^ G U

1 - |^rJ2

10 log^g — (3.45)

'Pl

and is called (Beatty, 1964a) the comparison loss. An expression of this form is

widely used (Beatty and MacPherson, 19 53) in the analysis of mismatch errors in power

measurements

.

If one is given "'"P^ expressed in decibels referred to some convenient level, one

subtracts L^ in order to obtain ^P-^- Supposing that the load initially connected

were matched and the load finally connected were mismatched, then eq . (3.45) reduces to

eq. (3.44), which gives the power loss in decibels due to mismatch.

b. Meaning of Mismatch

The term "mismatch" implies that other than matched conditions exist. This is

clear enough, but there are various interpretations of the term "match." In a manner

of speaking, one impedance is said to match another when the two are identical.

Thus, a load that matches a given generator yields the condition

h = ^G °^ ^1 = ^GHowever, the concept of a conjugate match is well established and the conditions are

given by eqs . (3.39) and (3.40).

39
One could argue that a conjugate match is really a mismatch since the two

impedances involved are not exactly equal to each other. The resistive components

are equal, but the reactive components have opposite sign. However, the word

"matched" is commonly used to mean adapted, fit, or suited. For example, a married

couple are said to be well matched if they complement each other. Thus the word

"matched" can have various meanings other than "equal." In precise work, we must

be careful that we understand just what meaning is implied.

It is said that a load matches a waveguide when its impedance equals the characteristic impedance of the waveguide, or = Z^^ and = 0. If Z^-j^ is real, then

this type of non- reflecting match or match may be equivalent to a conjugate match,

providing that the impedance Z^ of the generator feeding the waveguide also equals

Zq^. However, if Z^^ is complex, a non-reflecting or Z^ match will not in general

result in maximum power absorbed in the load, although there will be no reflection of

energy back towards the generator.

One should be aware that other types of impedance matching have been defined

in addition to those above, so that the terms "match" and "mismatch" should be used

with care. A load that is matched in one sense, may be mismatched . in another.

c. Conjugate Mismatch Loss

A generator delivers its available power when terminated in a load which

provides a conjugate match as mentioned above. When a different load terminates the

generator, the net power delivered, P^^ , is less. The ratio of P^ to P-j^, expressed^

in decibels, is the conjugate mismatch loss M^.,:

p 1 1 - r r I ^ M = 10 log = 10 log

^ . C3.46)

^ ^° Pi

^° (1 - Irgl^Jd - Ir^r)

The conjugate mismatch loss M^^ cannot be negative, since P^ is either equal to

or greater than P^. In the simple case when the generator is non-reflecting (r^ = 0)

,

the conjugate mismatch loss reduces to

1 (3.47)

One sometimes finds this expression given for mismatch loss without the statement

that it requires a non- reflecting generator to be correct.

^This follows from eqs . (3.37) and (3.41).

40

/
d. The Zq Mismatch Loss

When a transmission line or waveguide is terminated in such a way that there are

no reflected waves, the impedance of the termination equals or matches the characteristic impedance of the transmission line or waveguide. If the generator were to

have the same impedance, then maximum net power would be delivered to the load,

assuming of course that there were no losses in between. If one were then to connect

a different load, there would be less power delivered, and a mismatch loss would result.

The expression for mismatch loss would be eq. (3.47).

However, the generator is not always non-reflecting so that the Zq mismatch

loss is

p 1 1 - r r I ^ M = 10 log = 10 log —rr- (3.48)

^0 ^" 1 - |r^ 1

2

It is possible for this expression to become negative (when P^ > Pq) , but this

can occur only when 7^ 0

.

e. Difference Between Conjugate and Mismatch Losses

It has been observed that the conjugate mismatch loss and the Zq mismatch loss

are the same when the generator is non-reflecting, and are given by eq . (3.47). In

general, however, these two quantities are not the same and their difference is given

by

- ^Z, = 1° l°glO r '° '°^10

, ]r 12

- (3.49)

u ^0 ' gI

This is the ratio expressed in decibels of the available power from the generator

to the power which would be absorbed by a non- reflecting load connected to that

generator.

3.8. Transmission Properties of 2-Ports

Enough theory has already been developed to enable calculation of the net power

transmitted through a 2 -port. For example, the net power input to arm 1 may be calculated from eq. (3.37) and the net power output to a load connected to arm 2 is then

obtained from eq. (3.28) for the efficiency. In terms of b^ and , the scattering

coefficients of the 2-port, Z^^, and the reflection coefficient of the load,

the net power transmitted to the load is

I^gI' l^zil'd - I^lM)

P = . il h (3.50)

^02 - ^nrcHi - s^^r^) - Si2^2irGrLl'

41
It is often desirable to compare the transmission properties on two 2-ports, and

this problem has given rise to the concept of substitution loss which will be defined.

The allied concepts of transducer loss, insertion loss, and attenuation are special

cases of substitution loss, as will be shown.

In the following, unless otherwise stated, it will be assumed that the generator

connects to arm number 1 of the 2 -port and the load connects to arm number 2. This

will allow a simpler notation. It then follows that a reversal of these connections

will require a reversal of the subscripts 1 and 2 in the equations.

a. Substitution Loss

If one 2-port is removed from between a given generator and load, and another

2-port is substituted in its place, the net powers absorbed by the load under the

initial and final conditions will have the following ratio, expressed in decibels:

h^^ia - ^s^^r,](i - ^s^^r^) - ^s^/s^.r^r^]

where the front superscripts i and f denote initial and final conditions, respectively,

and it has been assumed that the act of substituting one 2-port for the other does not

change the characteristics of either the generator or the load.

The above expression applies to cases in which the two arms of the waveguide

are dissimilar, and/or have different propagating modes, as well as to the more usual

case in which they are identical and the same mode propagates in each.

The substitution loss Lg , as defined above may range from -«> to +<» , and upon

assuming negative values, could be regarded as a gain, rather than a loss. This

continues to hold true even when we exclude "active" 2-ports such as amplifiers, for

the final 2-port might be a better transducer than the initial 2-port.

The substitution loss may be restricted to positive values by specifying certain

characteristics for the initial 2-port, or for the generator and the load, as will be

discussed later.

Actually, the substitution loss most closely corresponds to what one can measure, since even if initially no 2-port device is placed between generator and load,

one must still have a joint or connector. When very accurate measurements are to be

made, it is not permissible to neglect the reflection and dissipative loss of the

connector (Beatty, 1964), so that it must be considered as the initial waveguide

junction. As shown in figure 3-11, one always measures the substitution loss, even

when the initial waveguide junction consists only of a connector.

42

/

L,, = 10 log

10

= 20 log 10
A step attenuator, in which one attenuator is removed and another inserted in

its place, is an excellent example of the need for the concept of substitution loss.

This concept may also be applied to the case of a smoothly variable attenuator which

may be regarded as though one removes an initial attenuator (corresponding to the

initial setting) and substitutes in its place another attenuator (corresponding to

the final setting) even though one does not physically remove the variable attenuator

from the circuit. Thus it is analytically equivalent to a step attenuator, and the

substitution loss concept applies.

(0) INITIAL CONDITION

- CONNECTORS

WAVEGUIDE

JUNCTION

(b) FINAL CONDITION

Figure 3-11, Insertion of a waveguide junction into a waveguide system.

(a) Initial condition— one connector pair, (b) Final conditionwaveguide junction core and two connector pairs,

b. Transducer Loss

As a special case of substitution loss, consider that the initial waveguide

junction is a perfect transducer and transmits all of the available generator power

P

but must transform the load impedance to the complex conjugate of the generator

impedance. The substitution loss is then obtained from eqs . (3.41) and (3.50) and

is called the transducer loss:

^ to the load. In order to do this the perfect transducer must not only be lossless,

10 log^Q -ii = 10 log^Q X 1(1 '12^21^G^L

2 1

- |rj-^)(i

(3.52)

'01 I 211 I'G' ^

I L

It follows from the above equation that the transducer loss of a passive 2-port

cannot be negative. It is a measure of how closely the performance of the 2-port

approaches that of a perfect transducer connected between a given generator and load,

43
Although the transducer loss is felt to be a useful concept, any attempts to

measure transducer loss will be in error by the amount by which the initial 2-port

fails to be a perfect transducer, unless the transducer loss of the initial 2-port

can be closely estimated, and added to the observed loss.

Like substitution loss, transducer loss may apply to 2-ports having dissimilar

arms, and/or dissimilar propagating modes in each arm. It cannot be used to specify

a property of a 2-port since r^^ and are involved in eq . (3.52).

c. Insertion Loss

Another special case of substitution loss is the insertion loss."* Here it is

assumed that the initial 2-port is a perfect connector or adapter, which has no dis

sipative loss and introduces no reflection or phase shift. The initial power is given

by eq. (3.37] substituting for T^, and the final power is given by eq. (3.50), so

that the insertion loss is

^L , 1^02 I ' ^ll'^G^'^^ " ^22^L^ ^12^21^G^lI^

(3.53) L^ = 10 log^Q = 10 log^Q

It is apparent that the insertion loss is normally positive, but could be negative; for example, in the case where the load does not provide a conjugate match to

the generator, and the 2-port which is inserted is lossless and does provide a

conjugate match.

The idealized initial condition of a perfect connector may perhaps be more closely

approached in practice than the initial condition of perfect transducer, and hence

the insertion loss may be more accurately measured than the transducer loss. Neither

can be used in general to specify the characteristics of a 2-port, since generator

and load characteristics affect both. However, they both become equivalent in the

case of non-reflecting generator and load to be discussed below,

d. Attenuation^ or Characteristic Insertion Loss

A concept which is useful for specification of a characteristic of a 2-port is

its attenuation, which is the transducer loss or the insertion loss of the 2-port

'*In the IRE definition of Insertion Loss see p. 75 of IRE Dictionary (1961), the

definition is "fuzzy" because nothing is said about the initial connector which is

opened to insert the device.

^See section 6.2 for further discussion of definitions of attenuation.

44

/
when placed in a non-reflecting system. Making the substitution = r^^ = 0 in

eq. (3.52) or eq. (3.53) yields

10 log^o'"'

'02 1

'01 ^2ll

(3.54)

In attempting to measure the attenuation of a 2-port, one should note that errors

will be produced by any failure to meet the assumed initial conditions, and the condition of a non-reflecting system. The actual quantity measured will be the substitution

loss, and the error may be analyzed by comparison of eqs . (3.54) and (3.51).^

In the special but often encountered case in which arms 1 and 2 of the 2-port

are identical, it is convenient to choose Z 01 Zq2> and its attenuation is

[A] 7-7 =- 20 log^ — (3.55)

21

It is apparent that 1^21^ '"^^ ^® determined by measuring the attenuation of the

2-port, provided that we specify Z^^ and Z^^- Similarly, |S^2l "^^7 ^® determined by

connecting the generator to arm 2 and the load to arm 1.

The attenuation of a waveguide junction^ for energy traveling into arm 2 and out

of arm 1 is

^2 = 1° l°glO

'01

'02 1^12 I

(3.56)

The difference between the "forward" and "backward" attenuations is

^1 = 20 log^o

^01^21

^02^12

(3.57)

It is interesting to note that a waveguide junction which satisfies the reciprocity condition (Z^^S^^ = Zq2S-|^2) same "forward" and "backward" attenuations.

This is one test for reciprocity, but is not a sufficient condition, since it tells

nothing about the phase relationship between S^2 ^nd S2-|^.

e. Components of Losses

It is convenient and instructive to separate the substitution loss and its

derivatives into components, one associated with the dissipation of energy, and

the other with mismatch.

detailed error analysis is given later in sections 6.4, 6.5, and 6.6.

^One can specify the attenuation between any two arms of a multiport having all of

its arms terminated in some specified manner to passive loads.

45
The net power transmitted to the load equals the input power multiplied by the

efficiency of the- 2-port. If the available power from the generator is given, then

the input power to the 2-port may be obtained from the conjugate mismatch loss. Thus

two quantities, (1) the conjugate mismatch loss and (2) the efficiency, are sufficient

to determine the power absorbed by the load, if the available power from the generator

is given. (1) is associated with mismatch, and (2) is associated with the dissipation

of energy.

It is evident that substitution of another 2-port for the initial one will cause

a change of power absorbed by the load, and that the change will be equal to the

changes of (1) and (2) above. Thus the substitution loss may be written

S-'D' (3.58)

Referring to eq. [3.46) we can write

(Lg)^ = 10 logio

and referring to eq. (3.28), we can write

h-'

1

-

1 - l^r^l^

(Lg)^ = 10 log 10

'21 1

- 22'L

21 "^22^L

Si'

1 - iSl'

(3.59)

(3.60)

where is given by eq. (2.12).

The substitution loss is obtained by adding eqs . (3.59) and (3.60), which yields

. f.

Lg = 20 log10

21 r

21 "^22^L r^G 4

(3.61)

It is apparent that substitution of eq. (3.10) into eq . (3.61) will yield eq. (3.51),

verifying that eq. (3.61) does represent the substitution loss.

The components of transducer loss may be obtained by appropriate specialization

of eq. (3.58) and are

1 - r^r

T^M 10 log G' 1 10

(1 - \T^\')a - |rJ2)

(3.62)

and

[L^]p = 10 log^^ —

,

10 (3.63)

where is given by eq. (3.28)

46
Similarly the components of insertion loss are

2 1 1 - ^G^i

1 - ^G^L

and

l^D 10 log^o -

Finally, the components o£ attenuation are

^11

1

and

[A]p = 10 log10

'02

C 12

'01 ^211

(3.64)

(3,65)

(3.66)

(3.67)

Comparison o£ eq. (3.67) with eq. (3.29) shows that the component of attenuation

associated with dissipation may also be written

1 [A^]q = 10 log^o

t^l]a2=0

(3.68)

3.9. Maximum Transmitted Power

The conditions for maximum transmission of power to a load are given with

reference to figure 3-12, in which two lossless tuners are shown connected one on

each side of the waveguide junction.

First, the maximum power available from the generator must be obtained at the

input to the waveguide junction. The conjugate match = must be obtained by

adjustment of the tuner T^, and the generator will then deliver maximum power. Since

T^ is lossless, maximum power will be obtained in arm 1 of the waveguide junction.

and the conjugate match = will apply there.

WAVEGUIDE

GENERATOR JUNCTIONLOAD

Figure 3-12. Two lossless tuners attached to waveguide junction in order

to obtain niaximum power to load.

47
Second, maximum power must be obtained at the output arm o£ the wavguide junction, arm 2. This will occur when Tj^ = ^2i> where is the reflection coefficient

of the equivalent generator at that terminal surface. Under this condition, it is

evident that the efficiency of the waveguide junction is maximum. It is also evident that since is lossless, the condition = will also apply.

In practice, adjustment of tuners in the above sequence would not produce the

maximum transmission of power with only one adjustment of each, since the second

adjustment would in general upset the conditions achieved by the first adjustment.®

Rather, a series of adjustments converging upon the desired condition might be

necessary.

a. Maximum Efficiency

The efficiency of a two-arm waveguide junction was seen to depend upon Fj^, the

reflection coefficient of the load. With the load terminating arm 2 of the 2-port,

the efficiency ri-|^ is given by eq. (3.28). One expects that the efficiency would have

a maximum value ri-|^^ for a particular value Ty^ of the reflection coefficient of the

load.

The following problem is often of interest. Given the characteristics of the

waveguide junction (for example, the scattering coefficients and the characteristic

impedances of the waveguide leads) calculate Tj^ and n-|^j^^. The solutions will be given,

one based upon analysis of eq. (3.28), and the other based upon the maximum power

considerations discussed previously.

(1) Gradient of Efficiency

According to eq . (3.28), efficiency is a function of r^, which is complex, and

one can plot contours of efficiency in the Tj^-plane. Maxima and minima will occur

when the gradient of vanishes.

The gradient of n-j^ is

Vn, = e,. I —i— + e. • — (3.69)

' l' sir. I rj d^iir

^There are special cases in which the second adjustment would not upset the first,

For example, when the waveguide junction is an isolator.

48
Both components of the gradient vanish at a maximum point. In order for the -

JLj component of the gradient to vanish,

3n-, 0 = —-

2ZQ.^|S^J^(l-|rJ^)|rJ[|S^^|sin(^^3H-4;^) . IS-^^S^^-S-^^S^^I IS^JsinCy^ll^^l^L^]

^02 f I •'•^22^1 1 ^' I "^^l 2^21' ^11^22 ^^L""^!! '

^

(3.70)

where is the phase angle of (S-j^2^21 " ^11^22-' ' vanishes when

'22

'^^12^21 ' ^11^22^^11

sin((|;^^ - -

sinCt|^22

where ijjj^ is the phase of Tj^. The solution of eq. (3.71) for i^j^ gives the argument

of the reflection coefficient of the load for the maximum efficiency point. This

solution is independent of I

| .

In a similar way, it can be shown that the | Tj^ | - component of the gradient

vanishes when

(3.71)

where

AlFj^l 2 - BITj^I + A = 0,

A = IS22I cos(i|j22 + + 1(^^12^21 ' ^11^22^^11 ' cos (ijj^^ - - i}^) ,

(3.72)

and

B = (1 - |S^J^ . |S22l^ -

Note that ^-^ has been substituted for in A.

The solution for Tj^ is then

^12^21 ' ^11^22 I

1

- 2A 3^M

(3.73)

Substitution of for T.^ in eq. (3.28) will then give H-j^j^^, the maximum efficiency:

'01

~

z'^02 li - s^22^mI r P

ISzilHi - |rj^)

•^^12^21 ^11^22^^M ^lll

^ (3.74)

49
(2) Maximum Efficiency from Maximum Power Considerations

Referring to figure 3-13,

GENERATOR WAVEGUIDE

JUNCTION

Figure 3-13. Generator, tuner, and waveguide junction,

the conditions for maximum power transmission (and maximum efficiency) are

^1

= (3.75)

and

^M = ^2^, (3.76)

Rewriting eqs . (3.75) and (3.76)

= ^1

'^^IZ^Zl ^11^22^^M * ^11

^ ^22^M

(3.77)

and

'^^12^21 " ^11^22^ ^22

1 - hlh

(3.78)

We can eliminate and obtain a quadratic equation with variable rM'

^^M - Br^ + a = 0, (3.79)

where

and

^ " ^22 ^ll'^^12^21 ^11^22^'

B - 1 - * 1^22!^ 1^12^21 ^11^22!^'

The solution of eq. (3.79) is

^M

" B_

2a

1 ±

2 a (3.80)

The expression for maximum efficiency is then

Z

n IM

'01

^02 1^ ^22^mI^ " 1*^^12^21 " ^11^22^^M ^11 1

^ (3.74)

50
Since the complex Tw can be obtained from eq . (3.79), one can also separately find

the magnitude and phase of as follows. If we multiply eq . (3.79) by e , we

obtain

ae'^^lr^l^ - Blr^l . ae'^'^^ = 0. (3.81)

Taking the real part, we obtain

A|r^r - BlFj^l + A = 0.

which is of the same form as eq. (3.72), and taking the imaginary part of eq. (3.81),

we obtain

^22

^11^^12^21 ^11^22^

sin(i(iy + i^j^ - ip^^)

(3.71)

sin(ijj22 +

Thus the two solutions for Tj^ are seen to be equivalent. However, the minimum

loss method yields the complex value of Tj^ from solution of eq. (3.79) while the

vanishing of the gradient method requires first a calculation of \p^, then a calculation of iTj^l by eqs. (3.71) and (3.73).

b. Minimum Transducer Loss

The following problem is of some interest in connection with maximum transmission of power. Suppose that the lossless tuners T^ and T^ are connected as shown

in figure 3*12 and adjusted to give maximum transmission of power. Under these

conditions, the transducer loss of the resulting waveguide junction (including

the tuners) is minimum. What is the value in terms of the characteristics of the

original waveguide junction?

It is apparent that the component of transducer loss associated with mismatch

as given by eq . (3.62) is zero. Thus the minimum transducer loss is given by eq. (3.63),

where n is the efficiency of the resulting waveguide junction. However, since tuners

Ty and T^ are lossless, the efficiency is also that of the original waveguide junction.

Under the above condition of maximum transmission of power, this efficiency is a

maximum rij^^ , and hence the minimum transducer loss is

[L^]^ = 10 log^Q ^. (3.82)

It has not been assumed in this instance that and r, vanish, hence we cannot

conclude that resulting waveguide junction is non-reflecting, and in general, it will

not fulfill this condition.

51
c. Intrinsic Attenuation

The minimum attenuation of a 2 -port, obtained by adjusting tuners connected as

shown in figure 3-12, is by definition (Beatty, 1964c) the minimum transducer loss

under the conditions of non-reflecting generator and load (F^ = T^^ = 0) . It is given

by eq. (3.82) and may be called the intrinsic attenuation for reasons similar to those

given above.

The resulting composite waveguide junction including the tuners is non- reflecting

(a bilateral Zq match), since the conjugate match condition exists at both input

and output terminal surfaces, and the generator and load are non- reflecting . Rewriting eq. (3.82)

Aj = 10 log^o i-, (3.83)

3.10. Phase Shift

a. Relative Phase

A quantity such as voltage, current, or voltage wave amplitude which varies

sinusoidally at a fixed frequency f may be represented by a complex quantity

u = Aej^'^^^^^ = Ae^^ C3.84)

The phase of u at any instant of time t is

e = arg u = wt + B. (3.85)

where w = 2iTf.

Since we cannot tell absolutely when t was equal to zero, B cannot be absolutely

determined and thus phase is always relative. The phase 9 may be expressed in degrees

radians, or cycles.

We choose the convention that the angle 9 is positive when measured counterclock-

- i 9 wise in the complex plane, and we choose u as above (not u = Ae ).

b. Shift of Phase by a 2-Port'

We are interested in different kinds of relative phase or phase shitt. For

example, we may observe the phase of the output voltage^" of a 2 -port relative to the

^See Kerns and Beatty (1967), and Beatty (1964d) . In this monograph, phase shift is

positive if it is a phase advance (lead) and negative if it is a phase delay (lag)

.

However, it should be noted that the term "phase shift" is often used to denote the

absolute magnitude of a phase difference, and is then always positive.

1 0 Other quantities, such as current or electric wave amplitude may also be of interest

52

/
phase of the output from a signal source, which is padded or isolated and phase locked

to an oscillator of stable frequency. If the 2 -port changes, the relative phase of the

output voltage may be shifted to a different value. The phase shift is

Ae = - ^e, (3.86)

where front superscripts i and f denote respectively the initial and final values of

the relative phase of the output voltage.

c. Different Kinds of Phase Shift of 2-Ports

Three kinds of phase shift associated with 2-ports are illustrated in figure

3-14.

1 2

(1) TRANSMISSION PHASE SHIFT 0^

C I

2-PORT

1

2-PORT

(2) DIFFERENTIAL PHASE SHIFT AG

LOAD

LOAD

INITIAL

2-PORT

1

LOAD

1

1V

i '2

2

1 FINAL

2-PORT

1

1

1

LOAD

1 2

(3) SUBSTITUTION PHASE SHIFT 0^

Figure 3-14. Three kinds of phase shift associated with a 2-port.

53
They are

(13 Transmission Phase Shift

arg

w

; arg ; arg

^^1

(3.87)

This is similar to transmission loss

(2] Differential Phase Shift

Ae arg arg arg

f

.

"2

1

.

1,

(3.88)

This is similar to incremental attenuation.

(3) Substitution Phase Shift, Including Insertion Phase Shift

Qc = arg

'%]

; arg ; arg

i i

i ^2i

(3.89)

This is similar to substitution loss or to insertion loss.

Further discussion of this topic, including equations for various phase shifts

of 2-ports is presented later in section 7.2.

3.11. Cascading 2-Ports

In the analysis of measuring systems, it is often necessary to determine the

properties of a 2-port composed of a number of cascade-connected 2-ports. Consider

the following problem. Given the scattering coefficients of each of two individual

2-ports, determine the scattering coefficients of the composite 2-port which results

when these 2-ports are cascade-connected. This problem may be reduced to three steps

(1) conversion of the scattering matrices of each unit to cascading matrices, (2)

multiplication in sequence of these cascading matrices, and (3) conversion of the

resulting cascading matrix to the scattering matrix of the composite waveguide junction.

To illustrate and provide the basis for these procedures, consider the cascade

connection of two 2-ports as shown in figure 3-15.

1

1

M

1

1

N

1

1 Qj —

H

^1 Aj 1 .A,

1 B, 1 -B2

1

1

1

1

Figure 3-15. Cascade connection of two 2-ports.

54
The scattering matrices of M and N are

m

"^21 ^"22

and n "ll ^12

^^21 "22

(3.90)

The corresponding cascading matrices relate the wave amplitudes as follows

and

The joining equations are

Thus ,

(h 1 M A ""l = ^11 ^12 ^2

= M

^2

kl ^22- ^2-

^11 ^12' '^1

= N

'^2

.^21 ^22. ^2-

= \

and " ^2

= MN

'^2 = R

'^2

"^2. .^2.

(3.91)

(3.92)

(3.93)

where R is the cascading matrix of the composite waveguide junction. The relationship

between the scattering matrix S and the cascading matrix is as follows:

'''^^12^2l"^11^22^

21

11

22

(3.94)

and

22

'l2 *^^11^22'^12^21^

21

(3.95)

For the example above ,

^ ^ [CMiiN^^.M^^N^l^ f^l^l2^^2^225"

i(M2,N^^.M22N2,) (M2iN^2^^22^22).

In terms of the scattering coefficients,

(3.96)

11 [(m^2">2r'"ll'"22)f"l2"21 -"ll"22^ ' '"ll'"22^

1

"^21^21

12 = ["ill'^l-'"22"ll^ "'l2'"2l'^ll]

21

22

[-n22(l-m22n^^) - •"22''l2"21^

= (l-m22n]^i)

"21^21

1

'"2l"21

(3.97)

-, and

"^21^21

55
Using eq. (3.95), we obtain the scattering coefficient of the composite waveguide

jiinction in terms of those of the two cascaded units as follows:

11

12

'21

22

"^11

" "^12"'2l"ll

1 - ^22^^1

"I2'"l2

1 - n^^m22

"^21^21

1 "^22^11

(3.98)

-, and

^22 ^ "l2"2l'"22

1 - n^^m22

3.12. Cascading Coefficients

The cascading coefficients are of special interest not only in analysis of cascaded

2-ports, but are also the coefficients of the equation for the transformation of

reflection coefficient by a 2-port. Rewriting eq . (3.10) we obtain

r - ^ll^L ^ ^12

1

~ ^ll^L * ^22

Inspection of eq. (3.94) reveals that the following equation holds

(3.99)

^11^22 ^12^21

12

21

(3.100)

Referring to eq. (3.18), it is seen that the reciprocity condition on the cascading

coefficients is

Z, ^11^22 ^12^21

'01

'02

(3.101)

The lossless condition on these coefficients are as follows:

l^iz' 1^21'' l^ll' 1^221'

'^^11^22

*12 * *21

^12^21

'11

'01

and (3.102)

'02

'22 ± 2mT

,

where n is an integer and * is the argument of r pq ^ pq

56
In terms of the cascading coefficients, the efficiency can be written

'01

'02 I^Zl^L * ^22 I

' (3.103)

3.13. Transformation of Reflection Coefficients

A 2-port with arm 2 terminated in a load having a reflection coefficient

has an input (arm 1) reflection coefficient of

1 - ^22^1 cr^ + d

(3.104)

The reflection coefficient Tj^ is said to be transformed to a reflection coefficient

by the 2-port. An equation of the form of eq. (3.104) is called a linear fractional transformation. Many properties of the linear fractional transformation from

the theory of complex variables can be applied to the theory of waveguide junctions.

A few of the simpler properties will be reviewed and applications to some types of

microwave measurements will become apparent.

a. Simpler Transformations

It is helpful in visualizing transformations of a complex quantity T.^ to another

complex quantity to draw the two complex planes, and plot corresponding points.

Thus we might have the following transformations illustrated in figure 3-16.

-PLANE r^-PLANE

^12^21

Figure 3-16. Transformation of three points from

Ti -plane to -plane.

-a + b -c + d

^12^21

11

1 + S 22

2. = - - S^^; Fl = 0.

^^There is a wealth of literature on this subject and only a few examples are listed

(Deschamps, 1953), (Storer, et al . , 1953, and (Mathis , 1954).

57
3. ^1 " ~ 11 ; ^

' c + d

(3.105)

22

A well-known property of the linear fractional transformation is that circles

in the Fj^-plane transform to circles (different ones) in the r^-plane. Since three

distinct points determine a circle, one can draw the r^-circle corresponding to a

given Tj^-circle by transforming just three distinct points of the Tj^-circle. In

figure 3.16 the real axis in the Tj^-plane (a circle of infinite radius) has been

transformed into a circle in the r^^-plane.

b. The Sliding Termination

A termination sliding inside a uniform, lossless waveguide will cause the

reflection coefficient at a fixed terminal surface in the waveguide to vary in

phase, but not in magnitude. Thus a circular locus of r^^ is produced, with its

center on the origin.

The corresponding circular locus of can be found by applying eq. (3.104),

but it is simpler to use the following form of eq. (3.10):

^1

- ^12^21

(3.106)

'22

The variation of by sliding the termination changes only one term. Proceeding

one step at a time, we can determine the corresponding variation of as shown in

figure 3-17.

Figure 3-17. Steps in the transformation of a -circle to a

(Fj^ —Sii)-circle.

58
From figure 3-17 it is evident that the radius of the (r^ - S^^) -circle, and

also of the r^-circle of figure 3-18 is R^|S^2S22l) °^

^1 " ^1 " 7

+ S

1 22 22

^12^21

^12^21^lI

I^22^lI'

L '

I ' L

I

The distance from the origin to the center of the (r^ - S^^)-circle is

1 1^12^21

1 + S 22

I.

^12^21^22^l1

^22^L I

(3.107)

\\^22^L^- C3.108)

The T-j^-circle is shown in figure 3-18,

r; LOCUS ^ X

1

Figure 3-18. The -circle corresponding to a

Ti. -circle centered on the origin.

c. Significance of the Radius of the r^-Circle

Under certain conditions, the radius R^ of the F-j^-circle equals the efficiency

[n2]^ =0 of the waveguide junction. These conditions are

1. |r^| = 1 (a sliding short circuit).

2. ^01^21 ~ ^02^12 ^^^^ reciprocity condition).

When these conditions are satisfied,

Z, R. 02 1^121

^01 1 '22

2 ' '2Ja^ = 0' C3.109)

The component of attenuation associated with dissipation [^2]^ is related to the

radius of the r^-circle as follows

[A^]^ = 10 log10" (3.110)

Note that the attenuation component [^2]^ corresponds to energy flowing into

arm 2 and out of arm 1, while the r^-circle corresponds to energy flowing in the

opposite direction.

59
A similar result is obtained if the sliding short-circuit is in arm 1, and R2

is the radius of the r2-circle, as follows:

[A^]^ = 10 log^g-^. (3.111)

^^2

d. Displacement of Center from Origin

The center of the T-j^-circle is displaced from the origin by the vector representing the complex quantity = S^^ + C^, where |

| is given by eq . (3.108) and

is directed at the angle + ij;^^ - shown in figure 3-18.

It is apparent that the r^-circle will be concentric with the origin if the

waveguide junction is non-reflecting (S^^ = =0). We have then

[Ci]s^^ = S22 = 0 = and [Ri]s^^ = S22 = 0 = 1^12^21^11- (3.112)

Also, for a non - reflecting waveguide junction, for which reciprocity holds, and

|rj = 1,

[^Is, =S„ = 0 = 1° l°glO TTT—

^ • (3.113)

^11"^22""

In this case it makes no difference which is the direction of energy flow through

the waveguide junction.

The possibility of other conditions for which the r^-circle might be concentric

f with the origin exists. For example, if and S^^ are equal and opposite,

^11 * "^22 = ''^12 * ^21 - (^^n - Dtt,

and

ic; I 1^12^21^22^

l^nl = (3.114)

'22"^!

e. Locus of for V.^ Real

If is restricted to real values, the locus of is a circle. This condition

may be closely approximated in practice by varying the bias current of a barretter in

a microwave power mount, for example.

^^This technique is based upon a method originated by Kerns (1949b). See also, Beatty

and Reggia (1955)

.

60

/
Referring to eq. (3.106) we again proceed step-by-step to obtain the circular

locus o£ as shown in figure 3-19,

r- PLANE M 1

1

ISjjl sini;/22

T. LOCUS

r-s,

I

ISjjIsini/zjj

Figure 3-19- Steps in the transformation of real to a -circle.

It can be seen from inspection of figure 3-19 that

2IS22I sin if^^

(3.115)

and that the center of the F^-circle is located at

^1 = ^1 * R^e^ ^^/^^''^la + iPai) (3.116)

f. Other r^-Circles

It is possible to obtain a circular locus of by means other than those

mentioned. For example, a non-reflecting generator and variable phase shifter could

be connected so as to vary the phase of 3.^, while a fixed non-reflecting generator

operating at the same frequency is connected to arm 1. In this case,

_ ^1 _ ^2

\ ~ ~~ ~ ^ ^12 7"'

^1 ^1

(3.117)

Figure 3-20. Possible Fj^ -circles obtained by varying phase of a^

Depending upon the ratio of la^l to |a^| , we could have the loci for F^ shown in

figure 3-20.

61
Similarly

,

b a ^2 = — = + 82-^ —, (3.118)

^2 ^2

and corresponding circular loci may be obtained for

Such loci may be used in measurements of scattering coefficients of nonreciprocal and active 2-ports (Altschuler, 1962).

3.14. The Linear Fractional Transformation

Instead of the transformation of reflection coefficients by a 2-port as in

eq. (3.104), we are sometimes concerned with the transformation of a complex quantity

w to the Z-plane in which the coefficients and the variables w and Z can be assigned

any appropriate values to correspond to a given physical problem. We consider

V aw + b

(3.119)

cw + d

where Z is the dependent, and w the independent complex variable, and a = Ae-'°',

b = Be^^, c = Ce^'^, and d = De^ .

Two cases are of special interest, one in which the magnitude of w remains

constant and its phase (jj^ , varies, and the other the converse, i.e. the phase

remains constant, and |w| varies.

a. Constant w , Variable ' ' w

It is helpful to write eq. (3.119) in the following form

Z = a . (ad - bc)/cd^ (3.120)

c (c/d)w + 1

and to let e = Ee^^ = (ad - bc)/cd.

One can proceed step-by-step, as was done in a previous section, and arrive at

the diagram of figure 3-21 to represent the transformation.

Figure 3-21. Transformation of a w-circle centered on the

origin to a Z -circle.

62
The arrows on the circles correspond to increasing , the phase angle of w, w

The radius R-j^ of the Z-circle is

cdew

R =

I d I 2 - I cw I 2

and the position C-j^ of the center of the circle is at

Id^e

C = _ - ^1

(3.121)

(3.122)

b. Constant i|j . Variable |w

It is helpful to write eq. (3.119) in the following form.

„ _ a [(ad - bc)/c ]e

c

' T"^w| + (d/c)e

- i iIj W

(3.123)

and to let p = Pe-"^ = (ad - bc)/c^.

Step-by-step procedures will yield the diagram of figure 3-22 to represent

the transformation.

Figure 3-22. Transformation of a w-line thru the

origin to a Z-circle.

The arrows on the circles correspond to increasing |w|. The radius R^ of the

Z-circle is

2|d/c| sin(6 - Y

"

^1^^)

and the position of the center of the circle is

a ^ ^ Hp-1'^-^/2)

= — + R-j^e c

(3.124)

(3.125)

I

63

1
c. Invariance of Cross-Ratio

The following property of the linear transformation has measurement applications and is therefore of interest.

What is called the "cross ratio" or "anharmonic ratio" is invariant under

a linear fractional transformation.^^ This is illustrated in figure 3-23 and

eq. (3.126).

z-PLANE w-PLANE

1

Figure 3-23, Four points in the w-plane ti-ansformed into four other

points in the z -plane.

(3.126)

An application of this property is given in section 3.15j.

^^See for example, Townsend (1915).

64
3.15. 3-Ports or Waveguide Junctions Having Three Arms

a. Introduction

There are many physical forms which may be represented by a waveguide junction

having three arms, with a single mode propagating in each arm.

In some cases , a physical form having only two arms may be represented by a

3-port. For example, if one of the arms consists of cylindrical waveguide of circular cross -section , two orthogonal modes may propagate in this arm, and it may

be represented by two arms, each having a single mode.

In other cases , a physical form having more than three arms may be represented

by a 3-port. For example, a four-arm junction such as a directional coupler with

one arm terminated and not available for connection, may be represented by a 3-port.

Similarly, if the frequency of operation is below the dominant mode cutoff frequency

for all but three of the arms, then it may be represented by a 3-port (assuming that

each of the three remaining arms has single mode propagation)

.

b. Realizability Conditions

The general condition for strict realizability as given in section 2.3c(2)(a)

applies, that is the Hermitian matrices , or Hg are positive definite. Con-

,-1 sidering only Hg =

chosen as the unit matrix, we have

S*Zq S, in which the power normalizing matrix has been

where

"s

=

"ll

^H"12 ^"l3

"21 "22

Sr

"23 (3

"31 ^H"32 ^H"33>

(3.127)

'11

^ll'

'01

21

'02 '03

'12

^11^12 ^ ^21^22 ^ ^31^32

'01 '02 '03

'13

^11^13 ^ ^21^23 ^ ^31^53

'01 02 '03

65
and

H

21

'2 2

'2 3

'31

'32

'33

^12^11

+ ^22^21

+ ^ -r S ,9 ^ 32 31

^ A T 01 ^ n o 0 2 ^03 J

9

' 12 ' +

1 - |S 22

2 1 c 12 1^32 1

^01

^12^13

+ ^22^23

+ ^32^33^

. ^01U J. ^03 J

>

^1 3^11

+ ^2 3^21+ ^33^31

^ ^01 ^02 ^03 '

9

^13^12

+ ^23^22

+ ^33^32

9

i hi ^02 ^03

\ c \ 9 1^13!I C 1 ? IS23I

+

1 - 1 <^ 1 2 ^ 1^33!

hi 7^02 ^03

(3.128)

In order for Hg to be positive definite, all of the principal minors must be

positive, although it is only necessary to show that all of the leading principal

minors are positive. The latter condition is as follows:

S > 0,

"11

Sp Sj, - Sjj S„ > 0

,

"11 "22 "12 "21

and

det Hg > 0. (3. 129)

It then follows that

and

S > 0,

"22

S > 0,

"33

Sir - S„ Sj, > 0 , "22 "33 "23 32

"11 "33 "l3 "31

(3.130)

66
These conditions state that the net power input to the waveguide junction must

be positive for whatever excitation is chosen, or that for some excitation, some

energy is absorbed by the junction. For example, consider a source connected to

arm 1 and non-reflecting loads to arms 2 and 3. The power input to the junction is

P.m

(1

and the power output is

'01

S I 2 ^11 I

'01

out y^02 "'03

If power is absorbed in the junction, then

P.m

- P ^ > 0, out '

r 1 s 1

2

' ^21 I

^ ^02

^3ll

'03 J

(3.131)

(3. 132)

or

(1 S I ^ ^11 I

'01

S I 2 21 I

'02

31

'03

> 0,

or

'01

^21 I

'02

S I ^ ^31

1

'03

> 0,

or

> 0.

11

(3.133)

Each of the other conditions corresponds to a different excitation and it can

be shown that any excitation can be synthesized as a combination of those implied

by the first condition stated, i.e. the leading principal minors of Hg must be

pos itive

.

c. Conditions for Losslessness

A large class of 3-ports have very little loss, and the properties of lossless

waveguide junctions apply to a good degree of approximation in most of these cases.

'These properties are concisely stated as in section 2.3c(2)(c) by saying that

all of the elements of Hg vanish. There are nine elements, but the off-diagonal

elements at the top are the conjugates of the opposite off-diagonal elements at the

bottom and give no additional information when set equal to zero. Hence, we have just

six lossless conditions as follows:

67
1^ Is |2 |Q |2

U > ^01 ^02 ^03

^11^12 _ ^21^22 ^ ^51^52 ^

Z Z 7 •^01 ^02 ^03

S 12 l-IC 12 |c 12 ^12 I ^

^ 1^22' _ 1^32 I ^

Q

Z 7 7 ^01 ^02 ^03

^11^13 _ ^21^23 _ ^31^33 ^

q

7 Z 7 '^Ol 02 ^03

IsP131|2 Is |2 1-IS P

1^231 __l_iiL = u

,

7 7 7 ^01 ^02 '^03

^12^13 ^22^23 ^32^33

- ^ - - = 0. (3.134)

^01 ^02 ^03

These conditions state that the net power input to a lossless waveguide

junction must be zero for any combination or excitations chosen, or that no energy

is absorbed within the junction.

As in the case o£ the 2-port, the conditions obtained directly from Hg = 0

are not necessarily in the most useful form, and other ones may be derived.

For example, consider the lossless condition Hg = Zq"'" - S*Zq"'"S = 0 in a

slightly different form:

c Ay "

-'-c = 7 ^ ^0 0

'

S* = Z'^S'^Z„ = Z"^ adj_S

2 (3.135)

" ^ " det S

^

Equating corresponding elements and taking the magnitude of each side of the

equations, we obtain for a typical element, say S2-j^:

1027 l^2ll = TTTTT 7" 1^12^33 " ^13^321- (3.136)

I det S I Zq -j^

It is possible to show that | det S | = 1 as follows. We make use of the following general rules which apply to determinants:

1. det (AB) = det A det B

2. det (transpose of A) = det A

3. det A* = det (transpose of A).

68
The following steps then lead to the desired result.

det(Zg^) = detfS*) det(ZQ^) det(S),

1 = det(S*) det(S) = Cdet S) * (det S) = |det S|S

Therefore

|det S| = 1. (3.137)

The equations obtained by equating corresponding elements of the above matrices

are

ZqI

I '^lll " '^22^33 ^23^321' 1^121 " ~ 1^21^33 " ^23^3ll'

^02

^01 ^0 2 1^13! " ; 1^21^32 " ^22^3ll' l^2ll " ; 1^12^33 ' ^13^32 1' ^03 01

I ^0 2 I 1^22' " 1^11^33 ^13^3ll' '^23! " ; 1^11^32 ' ^12^3ll'

^03

l^3ll " ~ 1^12^23 ^13^22!' 1^32! " 1^11^23 ' ^13^2ll'

and

'01 02

^33! ~ 1^11^22 " ^12^2ll • (3.138)

They are useful in proving some of the theorems for lossless 3 -ports to follow,

d. Reciprocity

for 3-ports which are

application of eq. (2.55) yields the reciprocity conditions

^21^01 ^12'^02'

^31^01 ^13^03'

^32 02

= S 7 ^23 03' (3 .139)

e . Symmetry^

There are many types of symmetry possible. For example, consider symmetrically

constructed H-plane and E-plane tees. If arm 3 is the branching arm and the terminal

surfaces in arms 1 and 2 are chosen symmetrically, the conditions on the scattering

' ''Techniques for the analysis of symmetrical waveguide junctions are given in

Kerns (1951).

69
coefficients are as follows: For the H-plane tee,

^11 ^22' ^12 " ^21

^13 = ^31 = ^23 = ^32- ^'-'"'^

For the E-plane tee,

^11 22> "^12 ^21

^32 = ^23 = -^13 = -^31- ^'-'''^

The symmetrical Y structures exhibit a higher order of symmetry, and non-reciprocal

devices such as circulators may possess symmetry. Each type of symmetry yields a set

of constraints upon the scattering coefficients. It is possible to have electrical

symmetry without physical symmetry, but this is not usually encountered.

f. 3-Port, One Arm Terminated

A 3-port with one arm terminated in a load so that only two arms are available

for connection to sources and loads is essentially a 2-r)ort. The parameters of the

2-port may be found in terms of the 3-port and the reflection coefficient of the

fixed termination.

If the load of reflection coefficient r^^^ terminates arm 3, then a^ = Tj^^^'jj

and the scattering equations for the 3-port are \

-

• '^1^ ^ '^13^L3^3'

^2 = ^^22^2 * '^23^L3^3'

^3 = 3 s r h

^33^L3'^3'

(33.142)

where the front superscript identifies the scattering coefficient as that of the

3-port. Solving to obtain the scattering equations of the 2-port, we obtain

^11

3c 3 c T- ^13 ^31^L3

3s r ^33^L3

^1 " ^12

3c 3c

^13 ^32^L3

^33^L3

^2 = 3^ + ^21

3c 3C p ^23 ^31^L3

'^33^L3J

'^22 ^

3 c 3 C -n ^23 ^32^L3

33^L3

(3.143)

It is seen that if either the load is non-reflecting (T^^ = 0) or the third

arm is decoupled [^23 ^ ^31 ~ scattering coefficients of the 2-port become

those of the 3-port or

11

3c 2c

^11' ^12 12 21 '21^ and '22 22' (3.144)

70
g, Non -Reciprocal 3-Ports

An interesting non- reciprocal 3-port is the circulator, which is shown in two

forms in figure 3-24.

Figure 3-24. Schematic diagrams for lossless, non- reciprocal 3-port circulators.

The scattering matrix corresponding to the ideal forms of the two circulators

shown in figure 3-24 are

^0 0

1'

•o 1

0"

1 0 0 , s

= 0 0 1

.0 1 0 .1 0 0

(3. 145)

The ideal circulator is lossless (S*S = 1).

h. The Directional Coupler, 3-Port

Actually, directional couplers are often considered as a class of lossless 4-

ports, but if one arm is internally terminated so as to be not available for connection, it is then a 3-port. As shown in figure 3-25 the side arm (arm 3) to which

detectors may be connected couples mainly to the incident wave in arm 1. The

coupling ratio is the ratio of the incident input power to the power coupled out

to a non- reflecting detector, when the main arm of the coupler is terminated by

a non- reflecting load.

OUTPUT t X

Figure 3-Z5. Schematic diagram for a directional

coupler connected as a 3-port.

When a = a_ = 0 (non-reflecting loads) , the coupling is

C = 10 log

'03

10

or if the 3-port is a reciprocal one,

C = -

10 log

'03

10

'01

10 logio|S^3S3^

'31' "01

(3.146)

(3.147)

71
Another important property of the directional coupler is the ratio of the

emergent power coupled out for a given incident power input to arm 1, to the power

coupled out for the same incident power to arm 2, assuming non-reflecting loads on

all arms except those connected to sources. This ratio expressed in decibels is

called the directivity D. It may be written

D = 10 log

10

'01

'03

'03

\K r '02

a^=a3=0

(3.148]

f b where b^ indicates coupling to the "forward" wave, and b^ indicates coupling to

the "backward" wave, the amplitudes a.^ of the forward, and a^ of the backward wave

being equal. It follows then that the directivity is

^01 Si

CM

.^02 S2

D = 10 log^Q

or if the 3-port is a reciprocal one, the directivity can be written

D = 10 log^Q

(3.149)

^13^31

23 32

(3.150)

In case that Z 01 '02 Zq^, reciprocity implies that S^-j^ = ^-^^> ^ 32 23- or

D = 20 log^Q 31

32

(3.151)

It may be more convenient to measure the power transmitted through the main arm

than the incident power. The directivity might then be defined as the ratio of

emergent powers coupled out of the side arm for the same powers transmitted through

the main arm. This would lead to an equation for directivity similar to eq. (3.149)

except that the ratio of Z^^ to Z^^ would appear in place of the ratio of Z^^ to Z 01 01 '02'

Note that the ideal circulator is similar to a directional coupler for which

the coupling ratio is unity and the directivity infinite. Thus the ideal circulator

couples all of the energy out of the main arm into the side arm.

Two representations for ideal 3-port directional couplers are shown in

figure 3-26

.

72
Figure 3-26. Schematic diagrams for ideal 3-port directional couplers.

The scattering matrices corresponding to the 3-ports shown are

0

S

=

/(l-c^) c /(l-c^) 0 0

c 0 0

0 /(l-c^) 0 /(l-c") 0 c

0 c 0

(3.152)

(3. 153)

Observe that the scattering matrix of an ideal 3-port directional coupler is

not unitary. Although ideal in concept, it contains an internal termination and is

not lossless.

i. Solution of the Scattering Equations for

A number of measurement systems can be represented by a 3-port connected in a

system as shown in figure 3-27.

GENERATOR

DETECTOR

T

3-PORT LOAD

Figure 3-27. A 3-port connected in a system representative of

many measurement systems.

In analyzing the system, one is often interested in solving for b^, the

amplitude of the wave incident upon the detector. This is conveniently done by

writing the scattering equations for the 3-port, applying the terminal plane

relationships imposed by the generator and load, and solving the resulting equations

A solution containing determinants is conveniently obtained using Cramer's rule.

73
The scattering equations are

h = ^11^1 12^2 ^13^3'

' ^21^1 ^ ^22^2 ^23^3'

^3 =

= ^31^1 * ^2^2 ^ ^33^3- (3.154)

and

The terminal plane relationships to be applied are

a^ = + h^G'

^2 " ^2^L'

^3 " ^3^D'

(3.155)

where V^, and denote reflection coefficients of the generator, load, and

detector, respectively.

The equations to be solved are

-b^ = -(1

^21^1

^lirc^^l " hz'G^L'^Z ' hz^G^D^Z'

^31^1 " ^32^1^2 (1 ^33^D^^3- (3.156)

The solution for b^ is

(i-s,,r,)

21

31

(i-s,,r^)

21

^12^G^L

•^1-^22^1^

^32^1

31

^12^G^L

^1-^22^1^

^32^1

^13^G^D

^23^D

^1-^33^

(3.157)

or

21

31 h2\

(i-s^,r^)

21

31

^12^G^L

^^-h2'0

h2\

^13^G^D

^23^D

(i-S33r^)

(3. 158)

When one wishes to see the effect on b^ of variations in V-^, the following form

is useful:

= -b.

21 22

^31 ^32

21

31

^12^G

'22

32

^13^G^E

^23^D

^1-^33^

(3.159)

(1-s^^r,)

31

^13^G^D

(1-533^

74
Note that the above equation has the form of eq. (3.104) the linear fractional transformation of reflection coefficient. Therefore the results of sections 3.13 and 3.14

are applicable.

A further simplification of the above equation may be made as follows. Let

K =

^21 ^22

^31 ^32

(3.160)

(i-s^.r^)

21

31

31

^12^G

22

32

^13^G^D

^23^D

-(i-S33r^)

2i (i-s.^r,)

(3.161)

31

^13^G^D

(l-SjjFp)

and

31

(i-s.^r,)

'31

^13^G^D

(1-^33^0)

(3.162)

Then 1

5

b3 = -C

^21^1 - 1

1 + Kr, + r. CK (3.163)

r r

^ 2i^L 1 2i L

It can be seen by reference to eq. (3.151) that in case the 3-port represents a

directional coupler, the parameter K is approximately equal to its directivity ratio,

assuming that is small and S2-|^ - 1.

The parameter T can be shown as follows to be the internal reflection coefficient of the equivalent generator at terminal surface 2. One recalls an application

of Thevenin's theorem in which the internal impedance of a constant voltage source may

be determined by considering the inactive source and measuring the impedance "looking

back" into its output terminals. In a similar manner, the internal reflection

coefficient T^^ is obtained by "looking back" at terminal surface 2 with the

generator inactive. We then have the following conditions on terminal surfaces

1 , 2 ,

• and 3

a.

^3 " ^3^D'

^2^2i' (3.164)

^^This result appeared in a number of papers, for example: Beatty and Kerns (1958);

Beatty (1959); Engen and Beatty (1959); and Anson (1961).

75
The equations to be solved are

0 = -(1 - S

bo ^21^0*^1 ^22^2 ^23^D^3'

(3.165)

Solving for a2 we obtain

^2 = ^2

(i-s^.r^)

^31^G

^13^D

(i-S33rj,)

^21^G

^31^G

12

'22

32

^13^D

^23^D

-(i-S33rj3)

(3.166)

It can be seen that the V^^ of eqs . (3.161) and (3.166) is the same as the V

of eq. (3.164), and hence ^ 2x interpretation given above.

The short form of eq. (3.163) is useful in ref lectometer theory. It is

interesting to observe that the ratio of b3 to b^ for a 4 -port connected as a

reflectometer can also be put into this form (Engen and Beatty, 1959).

j. Measurement of Reflection Coefficient with general 3-Port

The response bj of a 3-port connected as in figure 3-27 is given by eq. (3.163),

and is a linear fractional transformation. We can use the property of invariance of

the cross-ratio in measuring the reflection coefficient of an unknown load as

follows (Beatty, 1972b).

Connect in sequence 3 different loads having known reflection coefficients

T-j^ , V^, and r3 to port 2 of the 3-port, measuring the corresponding values of

^^3 ~ ^1' ^2' ^"^^ ^3 ^measure both magnitude and phase). Then connect the unknown

load and measure corresponding to r^.

We can deduce from eq. (3.126) that

V,123ul23u - V, v_ - V r, - r, - r

• = • v„23ul23ul - V, V - V, - r, r - r/

(3.167)

Solving for T^, we obtain

where

r + rr

r = -^5 -. " 1 H- r

Z 3 ^ _1 2 ^ _3 u

r, - r, V- - v_ V - v/ 1 2 2 3 u 1

(3.168)

76
3.16. 4-Ports

a. The Directional Coupler

An ideal directional coupler is a lossless 4-port having a scattering matrix

which can be put into the form in eq. (3.169). To obtain this form, one uses a

numbering scheme as shown in figure 3-28 and chooses terminal surfaces appropriately:

/ 0 j /Cl-c^) 0 c j/(l-c2) 0 c 0

0

c

0 j/(l-c^)

j/Cl-cM 0

(3.169)

It may be shown^^ that any non-reflecting lossless reciprocal 4-port has a

scattering matrix that can be put in this form and is potentially a directional

coupler

.

3 1 X

Figure 3-28. Schematic diagram of a 4-port

directional coupler.

Suppose that arm 4 is internally terminated so that a^ = Tr^b^, and that only

the remaining three arms are available for connection. The scattering equations of

the resulting 3-port are then as follows:

[ b. c^r^a^ + Jj/(l - c2)a2 + jc/(l - c^^r^j^a^

= j/(l - c^)a.-^ + 0 + ca.

b^ = jc/(l - c^)r.pa^ + ca2 - (1 - c^)Tr^a^.

The directivity of this coupler is

D = 10 log

10

23

13

10 log

10

(1 - cMlr^l^

(3.170)

(3. 171)

It is seen that the directivity is strongly influenced by the termination. The

directivity is infinite for a non-reflecting termination but decreases as the reflection from the termination is increased.

The coupling of this coupler is

C = -10 logio|S23S32| = 20 log^^ |. (3.172)

^^See, for example, vol. 8 of MIT Rad. Lab. Series, Principles of Microwave Circuits, edited by Montgomery, Dicke, and Purcell (McGraw-Hill, New York, N.Y., 1948)"!

7 7
It is seen that the coupling is unaffected by the termination, providing that we have

an ideal 4-port directional coupler to begin with, as was assumed above. Also, the

vanishing of 'S22 is unaffected. A non-zero does affect ^S^^^, ^S^^j and the

directivity, however.

b . The Magic Tee

A magic tee is usually envisioned as an E-H tee having appropriate internal

tuning elements, adjusted to provide a non-reflecting 4-port. Its external appearance

is as shown in figure 3-29. A magic tee can be regarded as a lossless reciprocal

1^

Figure 3-29- External appearance of a magic tee.

4-port having a scattering matrix of a directional coupler with a coupling factor of

c = 1//2", or a coupling C of approximately 3 decibels. It is shown schematically

in figure 3-30.

Figure 3-30. Schematic diagram of an E-H tee.

The corresponding scattering matrix is as follows

^0 j 0

1'

S = j 0 1 0

0 1 0 j

1 0 j 0

(3.173)

This can be put into a different form by choosing different numbers for the

arms, and by moving the terminal surfaces in two of the arms. Moving the terminal

surfaces in arms 2 and 3 a quarter-wavelength outwards gives

'0 1 0

1" 10-10

0-1 0 1 10 10

S = -±- 72

C3.174)

78
Reversing the numbering of arms 2 and 3 gives

0 0 1

1"

1 0 0 -1 1 /r 1 -1 0 0

il 1 0 0

(3.175)

This is the usual form of the scattering matrix of a magic tee.

The magic tee can be used as a phase shifter and as an isolating power divider,

among its other uses. In its application as a phase shifter, sliding short-circuits

are placed in opposite arms, say arms 2 and 4 of figure 3-30. We then examine b^

relative to a^ to observe phase shifting properties. The appropriate scattering

equation is

b^ = - J- (b,e^^2 - b.e^^^), (3.176)

where and 6^ relate to the positions of the sliding short-circuits in arms 2 and 4.

But b2 = b^ = a^//2^, if arm 3 is terminated by a non-reflecting load so that a2 = 0

.

Therefore,

^ = - i (gj^z - ej^'*) . (3.177)

a^ 2

If the short circuits are initially positioned so that e-'^^ - e-'^'* = 2, they can

then have their motions ganged so that this phase addition is preserved as they

are moved to change the phase of b^ relative to a-j^ without changing the amplitude.

In order to accomplish this, both short-circuits must move in the same direction

with respect to the center of the magic tee.

In its application as an isolating power divider to form 2 channels, the magic

tee has one arm, say arm 3 terminated by a non- re fleeting load, and the source is

connected to arm 1. The power divides between arms 2 and 4 just as though two

identical generators were connected to arms 2 and 4. is the reflection coefficient of the generator connected to arm 1, then the reflection coefficients of

the equivalent generators for arms 2 and 4 are also r^-,.

79
4. Power

4.1. Introduction

Two research topics are described in this chapter. In section 4.2., the analysis

of mismatch errors (Beatty and MacPherson, 1953] in the calibration and use of microwave power meters is presented.

In section 4.3, the measurement of barretter mount efficiency by an improved

version (Beatty and Reggia, 1955) of the impedance variation method (Kerns, 1949b) is

described, errors are analyzed, and experimental results are given.

4.2. Mismatch Errors

a. Introduction

In waveguide circuits, it is often required to determine the power delivered to

a matched, or non-reflecting load, by a signal source. Assuming that the power meter

correctly indicates the power absorbed, it often will not give the desired result

because of mismatch.''

This point is illustrated by the following example, which is taken from p. 130 of

Griesheimer (1947), where the results were incorrect. A signal source delivers power

to a load according to

= T~ 11 - r r ^'-''^

^01 1^ ^GU'

If we assume that the signal source delivers 100 mW to a non-reflecting (r^ = 0)

load, and that this is the desired result, then

100 mW,

^01

If a power meter having a VSWR of 1.4, or |r^| = 1/6, reads correctly the power

absorbed, and the generator is non-reflecting (r^^ = 0) , then

1 - i- 9 7.2 mW.

36^

The power meter reads 2.81 lower than the desired value. Now if the signal

source also has a VSWR of 1.4 (but still delivers 100 mW to a matched load) the

^See section 3.7b for a discussion of the meaning of "mismatch." In this monograph

the term "mismatch" generally denotes a departure from the Z„-match, or non- reflecting condition.

80
power delivered to the power meter depends upon the phase of T^r^ according to

eq. (3.37) and lies between the limits

92.0 mW < [P, ] ,

,

, , -, /A < 102.9 mW. - 1 |rGl=|ril=i/6 -

Thus the power meter can indicate 2.91 above the desired value or 8% below

it.

Although this effect was known (Hand and Schrock , 1951), the analysis of the

error had not been published before 1953 and quoted error limits were often incorrect.

The error could be quite large when the generator mismatch was large, which was often

the case when generators were insufficiently padded or isolated. Subsequently, more

attention was paid to achieving a source match (Engen, 1958; Beatty and Fentress, 1971).

If the source is well matched, then a correction can be calculated if one knows the

VSWR of the power meter.

Application of the theory of section 3.7 and particularly of eq . (3.45) to

various circuit arrangements employed in the calibration and use of power meters is

given in the following discussion. The presentation is a slightly revised version of

Beatty and MacPherson (1953) , which first presented equations and results which

are still useful today.

b. Calibration of Power Meters

(1) General Discussion

A power meter is calibrated by comparing its indicated power with the power it

actually absorbs. Best accuracy of calibration is obtained by avoiding the use of

secondary standards, attenuators or directional couplers and comparing the meter

directly with a reference standard which may be a bolometric or calorimetric device.

This may be done by alternate connection of the meter and the standard to a stable

source or by the use of certain power splitting devices enabling simultaneous comparison, or by a combination of methods. Power splitting devices having a power

ratio of unity have the advantage that geometric symmetry is possible, permitting

precise mechanical construction which leads to a corresponding excellence of

electrical symmetry.

The end result of a power meter calibration is often a correction factor f which

is used to convert the meter reading to the power Pj^ aborbed by the meter

81
(P^ = fR^) • The correction factor £ may be obtained in terms of observed quantities:

M

M

RMR, (4.1)

where Pg represents the power absorbed by the standard power measuring device. Alternatively, a calibration factor is defined (Larson, 1962) for bolometric power

moiints employing dc substitution. It is the ratio of the substituted dc power P^^^ in

the bolometer element to the microwave power P^^ incident upon the bolometer mount.

In the event that P^^^ equals Rj^ above, then f of eq . (4.1) equals the reciprocal of

the effective efficiency rig, and

n.(l - Ml ) =

r 1

2

(4.2)

where Tj^ is the reflection coefficient of the power meter. Calibration factors have

also been defined for bolometer units (bolometer mount with bolometer element

installed) in combination with directional couplers (Desch and Larson, 1963).

All of these calibration factors are related and can be determined from measured

power ratios. In the following, it is assumed that the ratio Pg to Rj^ can be determined and that we are concerned only with the deviation from unity of the factor K

of eq. (4.1). It is normally unity except as affected by mismatches and deviations

from ideal properties of any power dividing devices that may be used.

(2) Method 1

Figure 4-1. Alternate connection to stable source.

Alternate Connection to a Stable Power Source

The power meter and the standard are alternately connected to a stable generator

as shown in figure 4-1. The generator output is padded to prevent the change in

loading from affecting its amplitude or frequency. The ratio of the powers absorbed

by the meter and the standard is^

M

1 ^G^S

^G^M

Ml

1 - l^sl^

(4.3)

^This is a direct application of the theory of comparison loss developed in

section 3.7a, eq. (3.45).

82
where T^, Tg, and T-^ are the voltage reflection coefficients respectively of the

generator, the standard, and the power meter, measured at the place of connection.^

It is convenient to measure the voltage standing wave ratio (VSWR or p)

corresponding"* to the magnitude of r. Assuming that the worst phase combinations

can exist in eq. (4.3);

> > Pg * Ps

PgPm ^ 1 (4.4)

In a specific example, if p^ = 4.0, Pg = 1.05, and pj^ = 1.25, lies between

0.84 and 1.17, a mismatch error between -16 and +17 percent if is erroneously

taken to be unity.

The range of error can be reduced by "matching back" toward the generator,

making vanish. In this case, eq. (4.3) becomes:

'Ml

Ps

+ 1

+ 1 (4.5)

With Pg = 1.05 and Pj^ = 1.25 as before, = 0.99, and the mismatch error is

-1 percent.

Caution must be used in attempts to further reduce the mismatch error by

matching the power meter input. If an adjustable transformer is used for this

purpose, the loss in the transformer itself will cause an error which cannot be

readily evaluated. Only transformers with known loss can be safely used for this

purpose

.

(3) Method 2 -- Comparison Using T-Junctions

(a) Simultaneous Comparison : The generator is connected to the center arm

(No. 3) of a symmetrical T-junction as shown in figure 4-2. The standard and the

power meter are connected to the other two arms (arms 1 and 2, respectively).

'It is evident that equals unity if Tj^ = Tg, a condition which may be recognized

by a method described by MacPherson and Kerns (1950).

r| = (p - l)/(p + 1)

83
> 2

P

s

1— M

M

M

Figure 4-2. T-junction comparison.

It can be shown that equal power will be delivered to the standard and the

meter by a symmetrical T providing that their impedances are identical. A high

degree of symmetry can be achieved by precise mechanical design and construction

and by special techniques, such as electroforming . The degree of symmetry achieved

may in some cases exceed the accuracy with which it can be measured. In the general

case, however, in which asymmetry must be taken into account, the ratio of powers

absorbed by the meter and the standard is .

K2 = ^

P

23

13

11

13 o

'22

s

) ^23 „ ^13

1

- (4.63

The coefficients of the form S are the scattering coefficients of the T. These

m ,n " scattering coefficients are either voltage reflection coefficients [m = n) , or voltage

transmission coefficients [m ^ n) , and can be measured^ with a standing-wave machine.

It is possible to obtain the magnitudes of the coefficients in eq. (4.6) from

VSWR measurements and calculate the limits of as the phases are permitted to vary.

Assuming that the T is symmetrical and lossless, lies between the limits:

PmPs i ^2 i

PmPs

(4.7)

Specifically; if pg = 1.05, and Pj^ = 1.25, lies between 0.76 and 1.31, an error

between -24 and +31 percent.

^This equation follows from the scattering equations (eq. (3.153)} of a three-arm

junction

.

^See section 4.2e,

84
(b) Alternate Connection to T- Junction : The generator is connected to the

center arm (No. 3) of a T-junction as shown in figure 4-3. An uncalibrated power

monitor is connected to one arm (No, 1) and the generator output is adjusted to

maintain a constant indication of the monitor. The meter and the standard are

alternately connected to the other arm (No. 2).

MO S MO

2

M

Figure 4-3. Alternate connection to a T junction.

The ratio of powers absorbed by the meter and the standard from these conditions,

applying eq. (4.6), is

M M

MO

MO 22

23 o

'22

23 „ M

'Ml (4.8)

Comparison of this equation with eq. (4.6) shows that the effect of asymmetry of the

T has been reduced by this method. In the case of a symmetrical lossless T, the

limits of as determined from measurements of the magnitudes of the coefficients

are the same as the limits of K^, as expressed in eq . (4.7),

PmPs ^

PmPs

(4.9)

(4) Method 3 -- Comparison Using Magic T

(a) Simultaneous Comparison : The standard and the power meter are connected to

the symmetrical arras (numbers 1 and 2, respectively) of a magic T^ as shown in figure

4-4.

'A conventional waveguide magic T may be defined as a four-arm junction having the

form shown in figure 4-4 which is symmetrical, lossless, and matched looking in each

arm.

85
Tot

/A M

M

Figure 4-4. Magic T comparison.

A generator and matched load are connected to the other arms [numbers 3 and 4,

respectively) . It can be shown that equal power will be delivered to the standard

and the meter by a symmetrical magic T provided that their impedances are identical.

But in the general case in which the asymmetry and mismatch are taken into account

the ratio of powers absorbed by the meter and the standard is,®

P

ab - cdFj^

bg - dfr, r L'M

(4.10)

where

a = ^23^1 - ^ll^s^

^ ^12^13^5

b

= S,3C1 - ''^^'^) * ^14^34^L

c = ^34(1 - ^ll^S^

^ ^13^14^5

d = ^14^23 ^13^24

£ = ^12^34 ^13^24

g

= Si3(l - S22V ^ ^12^23^M

The scattering coefficients of the magic T can be measured^ with a standing-wave

machine or the ideal ^° values can be used if the losses in the T are sufficiently

small, the internal matching is sufficiently good, and the mechanical construction

is sufficiently precise.

If the four-arm junction is an ideal magic T having properly chosen reference

planes, S^^ = = S33 = S^^ = = S34 = 0. and S^^ = -S^^ = 8^3 = S^j.

It is possible to simplify eq. (4.10) in several ways. For example, if it is

assumed that the load is perfectly matched (r^^ = 0), eq . (4.10) reduces to eq. (4.6).

^This equation follows from the scattering equations of a four-arm junction.

^See for example section 4.2e.

^"See eq. (3.172).

86
If in addition some o£ the properties of an ideal magic T are substituted in this

equation (S 11 '22 '12 0, and IS^^I ^ l^zs'-*' reduces to eq . (4.5).

nomogram representing eq. (4.5) is shown in figure 4-5

ITUJ

Ul

(E

UJO

a.

UJ

u.o

if) >

I

1.0

1.2

1.3

1.4 -

1.5 -

1.6 -

1.7

1.8 -\

1.9

2.0

E »

0.0.

q:

UJo

0.

m

1.12 -I

1. 10

1.08

1.06 -

1.04

1.02 -

1.00-

0.98 -

0.96 -

0.94 -

0.92 -

0.90 -

2.0 -I

1.9 -

•^^

I 8 UJH

Ul

OCUl5 1.7

o

Q.

O

OC<

9 1.6

^ 1.5

oc

V)

>

' 1.4

1.3 H

1.2

1.0

Figure 4-5. Mismatch effect in the calibration of power meters.

(b) Alternate Connection to Magic T : The generator and a load are connected to

the symmetrical arms (numbers 3 and 4, respectively) of a magic T as shown in

figure 4-6.

87
M

T

ZA /A L

Mr

Figure 4-6. Alternate connection to a magic T.

An. uncalibrated power monitor is connected to arm No. 1 and the meter and standard are alternately connected to arm No. 2. The generator output is adjusted to

maintain a constant indication of the monitor. The ratio of powers absorbed by the

meter and the standard is:

Kp =

where

:

M M MO

S ^MO

bg'- dfr^Fg 1

-

1

- (4.113

b = 5,3(1 - ^44^1^ ^ ^14^34^L

5,3(1 - ^22^5^ ' ^12^23^5

d = ^14^23 ^13^24

f = ^12^34 ^13^24

g

= 5,3(1 - ^12^23^MIf the load is matched (Fj^ = 0), eq. (4.11) reduces to eq. (4.8). It is evident that

the asymmetry effect is generally less in the alternate connection method than in the

simultaneous comparison method. If the magic T is very nearly ideal, substitution

of some of its properties (S

a perfect magic T.

12

M

= 0) into eq. (4.8) reduces it to eq. (4.5) For

r 1

2

Pg + 1

Pm ' 1 (4.5)

88
c. Discussion of Calibration Methods

The alternate connection of the meter and the standard to the same generator

is a simple and flexible method which can be employed with waveguide or coaxial line.

No auxiliary power dividing equipment is required and the mismatch error is relatively

small and can be easily evaluated from VSWR measurements if the generator is matched.

The generator must remain stable in power output and frequency during the calibration

and must be padded to prevent oscillator pulling caused by changes in loading.

The use of a power divider permits simultaneous comparison of the meter and

the standard. This reduces the necessity for padding the oscillator and the stability

requirements are not as great.

If the degree of symmetry is low, the mismatch and asymmetry error may be reduced

by using the alternate connection method with a power monitor. The generator padding

and stability requirements are increased and it is necessary to provide a smoothly

adjustable generator output.

Symmetrical three-arm T's are commercially available in waveguide and coaxial

line. Because of the center conductor, the coaxial T involves additional difficulties

of construction not encountered in the waveguide T. The mismatch error can be

evaluated from measurements of the parameters of the T. The range of mismatch error,

even with a perfect T, is greater than that obtained by alternate connection of the

meter and the standard to a matched generator.

Symmetrical waveguide magic T's are commercially available but magic T's or

hybrid circuits in coaxial line are not readily obtainable. Carefully constructed

waveguide magic T's make excellent power dividers for power meter calibration,

permitting simultaneous comparison with no more mismatch error than is encountered

with alternate connection to a matched generator. If asymmetry is appreciable, it

can be determined from measurements of the T parameters.

The effect of asymmetry can be reduced by using the alternate connection method

with a power monitor.

A magic T can also be used to accurately compare two impedances (MacPherson and

Kerns, 1950). If one impedance is adjustable, the two can be made equal. Applying

this principle to power meter calibration, the impedances of the meter and the standard

can sometimes be made nearly equal , permitting a reduction in the mismatch error.

89
d. Use of Power Meters

(1) General Remarks

If the power to be measured is within the range of the power meter, a direct

measurement can be made. If the power is greater, a calibrated device such as an

attenuator or directional coupler is used in such a way that a known fraction of

the power is measured by the power meter.

(2] Direct Measurement

In the direct measurement of the power that a given generator will deliver to

a given load, the power meter is simply substituted for the load. This is the same

situation encountered in the alternate connection of two power meters to a stable

source. If the meter and the load are nearly matched,^' it is often erroneously

assumed that the power measured by the meter is the same as that delivered to the

load. Assuming that the generator is well padded, the ratio of powers absorbed by

the load and the power meter is given by eq. (4.3) with an appropriate change in

subscripts

6 p

^G^M

^G^L

1 [4. 12)

Ml

In this expression the reflection coefficients of the generator, meter, and load

are designated as r^^, T^, and T^, respectively. If the power meter reading is

assumed to be correct, it is multiplied by the factor to obtain the power that the

generator will deliver to the load. It is apparent that eq. [4.12) and eq. (4.3) are

of the same form and that the limits of K, are:

0

Pg * Pm

Pm^Pg^l + 1

< < PgPm + 1

Pg ^ Pl

(4.13)

In a specific example, taking = 4.0, Pj^ = 1.05, and = 1.25, lies between 0.84

and 1.17, a mismatch error of between -16 and +17 percent.

If the generator is matched, the mismatch error in the previous example is

approximately -1 percent.

1

1

The Z. -match, or the non-reflecting condition is the desired condition.

90
(3) Use of Calibrated Attenuator

If a calibrated attenuator is used to extend the range of power meter as shown

in figure 4-7, the measured power is normally multiplied by the attenuator ratio to

obtain the power available to the load. If the error of mismatch is taken into

•^1

I -o— CALIBRATE D

ATTENUATOR

2

-o- r

M

M

M

r

Figure 4-7. Use of calibrated attenuator.

account, the previous result is multiplied by a correction factor K.^ to obtain the

power delivered to the load when it is connected directly to the generator. The

factor K.^ is given by^^

^7

=

1 - ^G^

1

- (4.14)

In eq. (4.14) the reflection coefficients of the generator, load, and meter are

denoted by r^, r^^, and Y-^, respectively. The scattering coefficients of the attenuator are denoted by ^ and represents the input voltage reflection coefficient of the attenuator with its output connected to the power meter. From the

scattering equations for a reciprocal two terminal -pair network.

hi ' '12

^22^M

(4.15)

and the attenuation in decibels is

= 10 log^Q = 10 log^Q

'12

(4.16)

"This equation follows from eq. (4.12) and the scattering equations of a two-port.

91
If only the VSWR's are measured corresponding to the reflection coefficients of

eq. (4.14), the limits of are:

(Pg + Pl) (P22 * *^Pl * i > 4 —

(Pg * Pi)CP22 Pm^

P(,Pl + 1) (P22 + 1) (Pi 1) J (4.17)

In a specific example, taking p^ = 2.0, = 1.20, p^ = 1.1, P22 = 1.20 and p^ = 1,25,

lies between 0.89 and 1.14, a mismatch error between -11 and +14 percent.

If desired, the error may be evaluated by measuring the reflection coefficients

appearing in eq. (4.14). The measured value of may be checked by measuring the

scattering coefficients of the attenuator and substituting in eq . (4.15). Inspection

of eqs . (4.15) and (4.16) shows that approximately equals S^^ if the attenuation

is large.

If the attenuator is reflection- free (S

^7

=

r r ^ G 1

11 22

^ 1 - |r

= 0), eq . (4.14) reduces to;

r r (4.18)

Ml

and eq. (4.15) reduces to;

U 12 ^MIn a specific example, taking pg = 2.0, pj^ = 1.20, 1.1, and p.

(4.19)

1.019 (10 dB

attenuator), lies between 0.970 and 1.046, a mismatch error between -3 and +5 percent .

If the generator is matched (r^ = 0), eq. (4.18) reduces to:

1 - I^L pM

1' 2

1 - I^M ' Pm [pl ' iJ

.20, k'^ equals 1.007, representing a mismatch

(4.20)

of less than 1 percent,

(4) Use of Directional Couplers

(a) Temporarily Inserted : A directional coupler is often used to extend the

range of power meters as shown in figures 4-8 and 4-9. In figure 4-8, the coupler is

temporarily inserted between the generator and load and the power is measured with

a power meter. The measured power is normally multiplied by the coupler ratio to

92
Figure 4-8. Temporary insertion of directional coupler.

obtain the power available to the load. I£ the effect of mismatch is taken into

account, the previous result is multiplied by a correction factor Kg to obtain the

power delivered to the load when it is connected directly to the generator. The

factor K„ is given by:^'

Kn = P R

^G^l c 12 I3I

^12^1 ^13^23^M

^23^^ ^11^ ^12^13

1 - I^lI^

(4.21)

In eq. (4.21) the reflection coefficients of the generator, meter, and load are

denoted by r^^, Tj^, and r^^ , respectively. The input reflection coefficient of the

directional coupler connected as shown in figure 4-8 is r^. The scattering coefficients of the coupler are designated by ^. The coupling, C, and directivity D,

are defined in the usual manner:

C = 10 log^Q R^Q = 10 lOg^Q

LO '^CO

S

13

D = 10 log 10

13

'23

(4.22)

It can be shown by a solution of the scattering equations for a three-arm junction

that the reflection coefficient is:

^1 = ^11 ' a

12

Si3(l 22'L

^12^1 ^33^M^ ^13^23^M

'13

^23^M

(1 - S33r^) - ^12(1 ^33^M^ ^13^23^M

^13^1 ^22^L^ * ^12^23^L

^23^1

(4.23)

j '^The derivation of this equation is straightforward, starting from the scattering

[equations of a three-arm junction. See eq. (3.153).

93
I£ the directional ' coupler can be considered to be perfect, having infinite directivity (S23 = 0) and being reflection- free (.^-^-^ - ~ ^33 ~ ' above equations

simplify, eq . (4.23) reducing to:

h = ^11 ^ h2"h ^ hz'^M C4.243

and eq. (4.21) reducing to eq. [4.18). If in addition the generator is matched

(Vq = 0), eq. (4.20) applies.

*-6 I

-re

D.C.

2

-oM

Figure 4-9. Permanent installation of directional coupler.

(b) Permanently Installed : A directional coupler is often permanently

installed between the generator and the load as shown in figure 4-9. The power

delivered to the load is normally obtained by multiplying the power meter reading

by the coupler ratio. If mismatch is present, it is necessary to multiply this

result by a correction factor Kg. This factor is given by: 1 h

Pl

- - I S-i -7

I ^ PR

^12*^^ ' ^33^M^ * ^13^23^M

'13(^1 ^22^0 ^ ^12^23^1 1 - r., I M

(4.25)

Note that Kg = 1 when | | = | Fj^ | = 0, and |S-|^2l 't^e directional coupler

can be considered to be perfect (S^-j^ = = = S22 = 0) and the coupling is

loose (S^2 = l)j eq. (4.25) reduces to eq. (4.20)

In a specific example let = 1.5 and = 1.25, A directional coupler is

used having a directivity of 25 decibels, a coupling of 20 decibels and reflections

in each arm producing a VSWR less than 1.1. Assuming the worst phase conditions,

the limits of error calculated from eq. (4.25) are approximately -8 and +2 percent.

If the directional coupler is assumed to be perfect in the same example, the

error calculated from eq. (4.20) is approximately -3 percent.

1 It The derivation is similar to that for eq. (4.21)

94
If these examples can be considered typical, it is evident that the simplified

equation cannot be used to evaluate the mismatch error unless the directional coupler

is very nearly perfect and the degree of mismatch is small.

e. Measurement of Scattering Coefficients

A reciprocal network having n-terminal pairs and a scattering matrix S is shown

in figure 4-10. The scattering coefficients are of the general form ^ where p and

B,

/

S =

s

s

i

I

I s

12

in

\

S,2- s

Q S ^22 ^

S2n- S

I n

2n

nn

Figure 4-10. Network having n-terminal pairs,

q are integers, each denoting a given terminal pair. It is assumed that reciprocity

holds in the form S = S . pq qp

If p = q = K, the voltage reflection coefficient S^^ is measured at the K

terminal pair with all other terminal pairs connected to reflection-free loads

th

Figure 4-11. Reflection coefficient circle.

If p ;^ q, the voltage transmission coefficient S is measured in the following

way. The q terminal pair is connected to a variable reactance. All other terminal

pairs with the exception of the p pair are connected to re flection- free loads.

95
The input reflection coefficient is measured for various reactances at q. The

locus of the measured points is a circle as shown in figure 4-11. Assuming that

reciprocity holds in the form S

ft nr R{1

1

S , the magnitude and phase of S^^ are:

is I ^

' pq'

s in

Y

pq

' Yqq) (4.26)

A short derivation follows:

If A and B denote the incident and reflected voltage waves at a pair of terminals p and q,

r B

Let

Then

P

B„ = S

S A + S A

pp p pq q

A + S A . qp p qq q

(4.27)

and:

B

= P A

qq

pp

+ spq

A

= s pq

pp - s.

(4.28)

(4.29)

p v

' qq

A variation of 9 represents a change in the reactance connected to terminal pair q.

As shown in figure 4-12, the magnitude of the vector quantity (T^ - Sp^) goes through

maximum and minimum values as 6 changes

.

96
Figure 4-12. Translated reflection coefficient circle.

At these points M and m,

pp-^M

1 - IS I

s I ^ pq' fr - s ) = P PP'm ^ ^ Is

. ^ '^pq 'qq

The radius of the circle is

R -

^ ^ -

2

qq

S P |S |2

pq' +

' pq'

1 + IS qq' ' qq'

The distance to the center of the circle is:

S I 2 pq'

1 - |s |2

' qq'

s 1

2

pq'

s I ^ pq'

1 - Is I 1 + Is I ' qq' ' qq'

s 1=^ pq' - |s

2

' qq'

R S

qq'

qq'

(4.30)

(4.31)

(4.32)

(4.33)

Denoting the phase angle i2y^^-y^^) by <)) , the diagram of figure 4-12 can be drawn.

An alternate method of measuring the scattering coefficients of an n-terminal

pair network is as follows. Referring to figure 4-10, the scattering coefficients

S S , and S are determined by terminating terminal pair q in three different

PP pq qq ^ f n

97
loads having voltage reflection coefficients r^^^, V^^, and r^^, and measuring the

corresponding input voltage reflection coefficients r^^, Y^, and at terminal pair

p with all other terminal pairs terminated in matched loads.

Solving the three equations for input voltage reflection coefficient of the

fo rm

r r = s + , (4.34)

PP 1 - s r, qq L

the following expressions are obtained for the scattering coefficients.

S

- ^l/l.^3^^ - ^2^ - hjljl^'z - ^3^ ' 'lJlJz^'z

- (4,353

^L^^La^^l - ^2^ ^ " ^3^ ^ "^lJl^'z ' ^1^

s

= - ^L.Cr^ - ^ r,^(r3 - r,] . r,jr, - r,) ^^^^^^ qq ^l/lJ^i - ^2^ " ^^^^3^^^ - r3) . T^^J^^iT^ - r,)

^'u'l^'i - ^2^ ' 'lJl^^'z - ^3) ' - '1^^'

Extension to the non- re ciprocal case is straightforward. One replaces S^^ by S^^S^^

,

and 2y is replaced by y + y 'pq ^ ^ pq qp

4.3. Barretter Mount Efficiency Measurement

a. Introduction

As mentioned in the introduction to Chapter 4, the impedance variation method

of measuring barretter mount efficiency was developed by Kerns (1949b). At first,

accuracy was poor (10-20%) because impedances could not be accurately measured

in coaxial and rectangular waveguide .

In the following section, a method is described which is based upon Kerns'

work, but avoids the measurement of impedance. The accuracy is estimated to be

better than ± 1.6% (see table 4-2). The work described here was performed in 1953-54

and was followed with further refinements,^^ The work on microwave power measurements

is well covered in a survey article (Rumfelt and Elwell, 1967).

'^Tuned reflectometer techniques (Engen , 1961) were applied to measure efficiencies

of barretter mounts in rectangular waveguide with improved accuracy (0.51). These

techniques were used at frequencies where the more accurate (0.2%) microcalorimeter

methods had not yet been applied. Finally a method was developed (Engen, 1966) to

accurately (1,4%) transfer the calibration of a bolometer mount in rectangular

waveguide to one having a coaxial output.

98
Although the method described below is presently not in use at NBS , it

illustrates the application of microwave circuit theory to the development of accurate

measurement techniques. The description of the method begins with a discussion of

the definition of bolometer mount efficiency.

The efficiency, n, of a bolometer mount is defined as the ratio of the power

dissipated in the bolometer element to the power input to the bolometer mount. If

the power dissipated in the bolometer element, , can be accurately determined, the

power input, Pj , to the bolometer mount is

P, Pj = —. (4.38)

n

P|^ is usually measured by substitution techniques in which it is customary to reduce

the audio or dc bolometer bias power (after the rf power is applied] until the

bolometer resistance returns to its original operating value. It is assumed that

the change in bolometer resistance caused by the rf power is identical to the change

in resistance caused by an equal amount of af or dc power P^. The validity of this

assumption has been treated (Carlin and Sucher, 1952), (Adams and Desch, 1968), (Jarvis

and Adams, 196 8), (Adams and Jarvis, 1969), for Wollaston wire bolometers cooled by

convection. Based upon this analysis, Carlin and Sucher concluded that "Wollaston wire

bolometers, when properly designed and mounted, afford a means of measuring cw power

over a frequency range extending to the millimeter wavelength region, with an accuracy

approaching that of low-frequency measurements." It should be noted, however, that under

less favorable conditions the substitution error for convection-cooled Wollaston wire

bolometers may be appreciable (let us say greater than 0.5 percent) at frequencies above

the estimated limit of 3,000 MHz, depending upon the length and mounting of the

bolometer element.

If the ratio of P, to P, is K , b d s

Pj = ^P^. (4.39)

n

It is possible to estimate the limits of for a specific Wollaston wire bolometer

from the calculated curves of Carlin and Sucher (1952)

.

An' impedance method of determining bolometer mount efficiency has been described

by Kerns (1949b). Unfortunately, relatively small errors in the required impedance

measurements can lead to a large error (10-20%) in the efficiency as determined by

this method.

99
A modification of Kerns' method will be described, in which the direct measurement of impedance is avoided, permitting the efficiencies of tunable bolometer mounts

to be obtained with increased accuracy (1.6%). Efficiencies of untuned bolometer

mounts can thereby be obtained with very little loss in accuracy from voltage ratio

measurements

.

b. Impedance Method

In the impedance method of determining efficiency the bolometer mount is thought

of as replaced by an equivalent two-terminal-pair network terminated in the bolometer

resistance. As shown in figure 4-13, the input impedance [of the equivalent network)

corresponding to each of three different bolometer resistances is obtained.

The normal operating resistance of the bolometer is designated as R2 . The

efficiency for this condition may be calculated from the following expression deduced

from Kerns (1949), involving the three terminating resistances w^ , m^, and w^ and

the three corresponding input impedances Z^, 1^, and Z^,

Efficiency (n)

W2 =

^2 (Z2 - Z2)(W3 - w^)

^2

' ReZ2 ^HW2 - w^) (w^ - W2)

(4.40)

s s ^11 ^22

^12 ^^1 J^l

s2 r ^1 = ^11 . ''''

1 -^21^U

"2

r2

s2 r

<- . 12' L2

11I I 1I -s r ^22' L2

To = S,, + s2 r ^1 2' L3

11 l-S22rL3

1^121' {^-|^L2l'}

'^R = R2

I ,2 2 2 l^"^22^L2l 'l^11^^"^22'-L2^'^^12'^L2l

Figure 4-13. Efficiency of a two-terminal-pair network (terminated in a resistance Rs ) determined from three measurements of

input impedance or reflection coefficient.

100
An equivalent expression for the efficiency can be obtained^ ^ in terms of the

voltage reflection coefficients corresponding to the above terminating resistances and

input impedances.

Efficiency (n)

R=R,

ri)(r3 - T^nv^^ - r^i) r I ^ (4.41)

where r denotes an input, and Tj^ a terminating reflection coefficient.

If the bolometer forms one arm of a Wheatstone bridge, it is convenient to adjust

the bolometer resistances , R^ , and R^ to predetermined values. If the factor

containing the real parameters Fj^^, '^^^2' ^^'^ ^L3 denoted by C, eq. (4.41) becomes

(T^ - r^)(r3 - r^)

n

= (4.42)

where

{1 L2l

L3

- r Ll

(r L2 (R2 - Ri)(R3 - R2)

and

Rt

Where Zq is an arbitrary real impedance.

It is generally true that the factor C can be more accurately determined than the

other factors in eq. (4.42), because C is a function of resistances determined by dc

measurement

.

The reflection coefficients V^, T^, and occur in difference terms of eq.

(4.42), with the unfortunate result that a given error in measuring individual

reflection coefficients may produce a much larger error in the calculated efficiency.

For example, if C = 19.92, = 0.0676 , = 0, and = 0.174 e^^-'-^^^^, the

efficiency is approximately 97 percent. An error of only ± 1 percent in measuring

the voltage standing -wave ratios

1 . |r|

VSWR corresponding to | and

1 -

I r|

can produce an error of approximately ± 6 percent in the calculated efficiency,

^ ^Equation (4.41) can be obtained by simultaneous solution of the equations appearing

in figure 4-13, which are based upon the scattering equations of a two -terminal -pair

network

.

See section 3.1e, eq. (3.4).

101
In order to reduce this error in efficiency to the more useful value of ± 1 percent, it would be necessary in the example to make VSWR measurements to an accuracy

better than approximately +0.2 percent. It is apparent that the determination of

efficiency by this method places rather severe requirements on the accuracy of UHF

or microwave impedance measurements.

c. Improved Method

It is possible to avoid the direct measurement of impedance of tunable bolometer

mounts having a high efficiency (above approximately 90%) and thereby increase the

accuracy of the efficiency determination.

Assuming that the bolometer mount can be made r e f 1 ec t ion - f re e

~

appropriate tuning adjustment when the bolometer is operating at its normal rated

resistance, , eq. (4.42) becomes

n = C

r r ' 1^ 3 (4.43)

1

' '3

If, in addition, the bolometer mount has a high efficiency, it can be assumed with

small error (as discussed later) that the vectors representing and terminate on

a straight line passing through the origin. The efficiency is

n

= c

r r 13

F + F ' 1

I - I ' 3

(4. 44)

The plus sign is used if the vectors representing F^^ and F^ terminate on opposite

sides of the origin, and the negative sign is used if they terminate on the same side.

Bolometer resistances R-^ and should be chosen above and below R2 in order to

obtain the greatest possible spread. In this case the vectors representing F.^^ and F^

terminate on opposite sides of the origin, and

n

= C

F F 13 C (Pi - 1)(P3 - 1)

F + F ^3! ^ Mil

(4.45)

P1P3

where p represents the VSWR corresponding to | F

|

.

Instead of measuring p-j^ and p^ , more accurate results may be obtained by measuring the relative voltage output of a loosely coupled, properly positioned fixed

probe

.

102
A simplified representation of a slotted section and probe is shown in figure

4-14. It is seen that the voltage, Ep, {in waveguide of rectangular cross section

operating in the dominant mode, Ep corresponding to the strength of the transverse

electric field) is a function of the reflection coefficients of the generator, probe,

and load referred to the probe position. From inspection of the equivalent circuit

Ep Eg

(1 - rg)(i + rp)(i + T^-)

(1 + rg}(i - Tpjd + T^) + 2(1 + rp)(i - r^r^)

in which the subscripts G, P, and L, refer to the generator, probe, and load,

respect ively

.

By means of a matching transformer following the generator, it is possible to

(4.46)

make vanish. In this case,

Ep Eg

1 + r.

(4.47)

where Xp = (1 - Fp) (1 + Tp)

GENERATOR SLOTTED

SECTION

PROBE

I

TUNABLE

BOLOMETER

MOUNT

Figure 4-14. Simplified block diagram of measuring apparatus

and an equivalent circuit representation.

If in addition, the probe is loosely coupled (yp - 0)

,

E Ep = ^ (1 ^ V- (4 .48)

If the probe is located at a position where its response is maximum when the

bolometer resistance is R-j^ , Ep-j^ is proportional to (1 + |r-|^|). With the probe fixed

in that position, the bolometer resistance is changed to R2 and then , observing

the probe response,

Epi = k(i + |rj)

'?2

^P3

k

k(l

103

(4.49)
Defining the ratios K^^ and as follows,

E

K-, PI

'P2

'?2

1 +

= 1 (4.50)

the efficiency may be written

(Ep-j^ Ep2) (Ep2 - Epj) - K3)

(4.51)

^PZ^^Pl '^PZ-' ''1 ''3

A correction to eq. (4.51), to compensate for failure of the assumption that

the vectors representing and are colinear, can be made if the other sources

of error are neglected for the moment. Let the ratio of n given by eq. (4.43) to

that given by eq. (4.51) be

r r 13 (K, - K3)

1)(1 - K3)

(4.52)

If the angular difference between and T^, as shown in figure 4-15, is

LOCUS

Figure 4-15. Diagram illustrating curvature of input reflection

coefficient locus for resistive termination of

bolometer mount,

IT + 5 , where 6 < 0.1, it can be shown that

\T^\ - (1 - K3)(l + 6V2K3),

r - r ^3 ^ il (K, - K3) 1 + 6' K^(l - Kj)^

2K3(K^ - K3)^

104
and

- 1 + 6^

1) (K, - k;)

2K3(K^ - K3)^

(4.53)

The angular difference, 6, is simply related to the curvature, K, of the locus

of the reflection coefficient. This locus may be determined by measuring the input

reflection coefficient (referred to the fixed position of the probe) as the

bolometer resistance is varied. The expression relating K and 6 is

K

6 - - (K - K ) 2

(6 < 0.1)

Equation (4.53) may be written in terms of K:

(4.54)

?1 = 1 + K

(K^ - 1)(K^ - Kp

8K,

(4.55)

A graph representing the percentage correction according to eq. (4.55) is shown

in figure 4-16

.

Another correction to eq. (4.51) is based upon the fact that there may be

appreciable losses between the fixed probe position and the bolometer mount input.

The efficiency of a length of line or waveguide having a known attenuation is shown

in figure 4-17. If the line or guide section is not uniform, the efficiency must be

K=l .333

1 .3

1 .2

1 .1

1 .0

1 .0%

LOCUS

\ \

1 1.0 0.9 0.8 0,7

1 .3

1 .2

1 .1

1 .0

K=l .000

J .0%

1.0 0.9 0.8 0.7

K=0.667

UNIT

CIRCLE

1 .3

1 .2 -

1 .0

0.5%

1.1- 0.1

1.0 0.9 0.8 0.7

Figure 4-16. Percentage correction to efficiency corresponding to curvature of reflection

coefficient locus for three values of curvature.

105
determined by other means, such as measuring the bolometer mount efficiency with

another identical slotted section inserted between the bolometer mount and the measuring slotted section. If the efficiency of that portion of the circuit between the

fixed probe position and the bolometer mount input is Ip.g, the efficiency of the

bolometer mount, applying the above corrections is

'P-B

K.

1 + K

1)(Kt - K^)

8K,

(4.56)

It is seen that both corrections increase the efficiency over the value obtained in

eq. (4.51).

,900

950

>-o

990

995

,999

n

10

ATTENUATION =

, 1 L '1 0 _ UtLlbtL

M 1 1

S

.01 .05 .01 0.5 1.0

ATTENUATION, DECIBELS

Figure 4-17. Efficiency of a symmetrical, matched attenuator

(or a length of uniform line) terminated in a reflection-free load.

The method just described is applicable to tunable bolometer mounts (adjusted

for no reflection when R = > in which the bolometer element can be represented by

a resistance terminating the bolometer mount. (Barretters are generally suitable,

but there is evidence that thermistors do not fulfill this condition).

The efficiency of tunable bolometer -mount assemblies, including matching transformers, can also be measured by this method. After the efficiency of a tunable

bolometer -mount assembly has been determined, at a specified operating frequency,

the efficiency of another tunable or untuned bolometer mount or assembly can be

obtained by comparing the power readings of the two mounts when alternately connected to a stable, well-padded generator. Assuming that the power dissipated in

106
the element can be accurately measured by dc substitution techniques (See Chapter 3

of Ginzton, 1957), and letting the subscripts A and B refer to the two mounts,

^A^A = ^Ad

Vb = ^Bd' ^4.57)

where n is the efficiency, P is the input power, and is the power dissipated in

the bolometer element.

If Tg is the input reflection coefficient of the bolometer mount whose efficiency

is to be determined, and we assume that = 0 , the ratio of powers aborbed by the two

bolometer mounts or assemblies, (assuming a matched generator) is

Pa 1 Cpb - 1)^

The efficiency of the second bolometer mount is

C4.58)

- 'a ^Bd ^Pb ^ ^Bd

rig n, n,, (4.59)

Pb ^Ad P^,

where Pg is the VSWR corresponding to | Tg |

.

An error in measuring Pg will cause an error in determining rig, but fortunately

the error is small in most practical cases. For example, if p„ is determined to

be 1.20 with an accuracy of ± 2 percent, the corresponding error in rig is approximately + 0.2 percent.

d. Discussion of Errors

An accurate knowledge of the efficiency of bolometer mounts used for microwave

power measurement is essential to accurate power measurement. For this reason, it

is felt that a detailed discussion of the error in measuring efficiency is desirable.

Certain sources of error appear to be common to most measurements at high frequency. Among these are. instability of oscillators and amplifiers, unwanted frequency modulation (FM) , spurious amplitude modulation (AM) and harmonics in the

generator output, pulling of the oscillator by changes ,in loading, erratic or unknown

detector characteristics, errors in measuring the detector output, impedance mismatches at junctions, and mechanical instability of the components. Error from

these sources is minimized by careful instrumentation and the use of recognized good

practice in measurement techniques. For example, the stability of electronic equipment is improved by using voltage -stabilized power supplies, phase locking the frequency, and by avoiding large ambient temperature variations. Oscillator pulling is

minimized by the use of nonreciprocal transmiss ion -1 ine elements or attenuator pads

107
with at least 20 -dB attenuation. Unwanted FM is reduced by careful modulation

practices or by the use of high-Q transmission cavities to attenuate undesired side

bands. Parasitic oscillations, causing spurious AM, can be eliminated by usual

procedures, e.g., damping, shielding; and minimizing feedback. Low-pass filters

are used to reduce the harmonic output of generators . Detectors can be calibrated

before use or the need for known detector characteristics may be avoided by use of

calibrated attenuators. Matching transformers can be used to reduce impedance

mismatches, and careful attention to reducing movement of the components will reduce

mechanical instability.

After the above precautions are taken, observations should be made to verify

the desired conditions. For example, the generator output can be observed with a

spectrum analyzer to verify the reduction in unwanted FM and spurious AM. The

oscillator-output amplitude and frequency can be monitored during load changes to

observe pulling, and the detector output can be monitored with a continuous recorder

to observe system stability.

Additional sources of error, which can be minimized by careful instrumentation

and experimental procedure, are instability of the bolometer bias supply, inaccuracy

of resistance measurement, mechanical irregularities in the slotted section and

traveling probe, excessive coupling, and incorrect position of the probe. The use

of heavy-duty, low- dis char ge storage batteries or solid state voltage reference

sources will generally provide a stable bias supply.

Resistances R^, R2 , and R^ are measured at direct current and assumed to be

the same at UHF or microwaves. It was pointed out by Kerns [1949), and can be

seen from eq . (4.42), that even if the dc resistances are multiplied by a constant

real factor, there will be no error in efficiency. The effect of random errors in

resistance measurement upon the efficiency is the same as the random errors in VSWR

measurements, discussed in section b. It was seen that an error in VSWR between the

limits ± 0.2 percent will produce an error in efficiency between the limits of

approximately ± 1 percent.

Resistance measurements between 100 and 30 0 ohms can be made with an accuracy o:

approximately ± 0.05 percent with a good Wheatstone bridge. The corresponding error

in efficiency would be approximately ± 0.25 percent.

The choice of a slotted section and traveling probe is important in adjusting

and V2 for minimum value, and in approximating the assumed uniform, lossless line

or waveguide

.

108
The error caused by excessive probe coupling is difficult to evaluate analytically

(see eqs. (4.46) and (4.47)). However, it is possible to determine experimentally

when the probe is sufficiently decoupled by making a series of efficiency measurements, each with a diminishing value of probe coupling. When there is no further

appreciable change in the measured efficiency, the probe has been sufficiently withdrawn. Another method of checking the effect of probe coupling consists in making two

efficiency measurements, one with the probe set to the position for maximum response

corresponding to a bolometer resistance R-j^ , and the other with the probe set to the

position for maximum response when the bolometer resistance is . If nothing else

is changed, the two probe positions are separated by approximately A/4, so that the

phase of the reflection from the load as seen at the probe position differs by

approximately 180 degrees in the two cases. An example of this method is given

in table 4-1, where it is assumed that the average of the two efficiency measurements

closely approximates the correct efficiency with the probe sufficiently decoupled.

This assumption was found to be valid for small variations in efficiency.

It is possible to evaluate the effect of certain sources of error analytically.

The error in measurement of the relative voltage output of the probe, the incorrect

positioning of the probe, the generator and load mismatch, and the curvature of the

input reflection coefficient locus can be taken into account if the resulting error

in the efficiency is small.

If e^, z^' ^"^^ ^3 3.re the errors made in measuring the probe relative voltages

^Pl ' ^P2 ' ^'^'^ ^P3' ^®spect ively , the error in efficiency from this source alone is

approximately

-1 fK.d - K3)

^1 - s i CK, - 1)

K3(K^ - 1)

(e, - £,) + (e, - e,) (4.60)

If the individual errors lie within the range indicated by |e^| = 1^2 1 ~

I £3! £ e'>

the maximum error in efficiency would be less than

2K (1 - K ) 2K (K - 1)

e = ± e' or e = ± e', (4.61)

(K^ - 1)(K^ - K3) (1 - K3)(K^ - K3)

whichever is largest. A graph of this relationship is shown in figure 4-18. Using

a 200-ohm barretter, the limiting values of Yi-^ and K3 were determined to be approximately 1.33 and 0.75. Referring to figure 4-18, with e' assumed equal to ± 0.1 percent, the error in efficiency would be less than ± 0.4 percent. As this is a random

error, improved accuracy can be obtained by averaging the results of a number of

measurements

.

109
An analysis of the error in efficiency caused by generator and load mismatch,

curvature of the reflection coefficient locus, and incorrect probe position yields,

after some manipulation, a correction factor to apply to eq. (4.51). It is

1 +

fl-K. rK,-i

2K, 2K, (K,-K3)^ (K, -1) (1-KJ

(K^-1)

KjCK^-l)

(K,-K3)

r2 1 cos(iJj2-a) + (I-K3)

T^l cos (ij;^ - 6 -a)

K^(l-K3)

r2 I cos

'gl cos((jjg+a)

'^1 cos ()J;g+6 + a)

(4.62)

where i|j and are the angular arguments of and T^, respectively, and a represents

twice the angular error (26A£) in setting the probe to its correct position. (A£ is

the distance the probe position is in error.)

In the derivation of eq. (4.62) approximations were made (very small higher-order

terms were neglected), assuming that Ir^l < 0.005, | r2 | < 0.005, 6 < 0.1, and a < 0.1.

The magnitude of the error represented by eq. (4.62) can be illustrated by considering

some of the sources of error separately. For example, if 6 = a = 0,

I cos +

I r2 I cos . (4.63)

110
I£ Ir^l = Ir^l = 0.005, the total mismatch error lies between the limits ± 1 percent

,

If = = 0.

?2 = 1 - 2(K^ - K3)

1 - K,^ i-Ki - 1

•J 2 ,

' ^ a + (K - 1)C1 - K ) (6 + a)2 + — ^

52

(K^ - K3)

(4.64)

'1 "3

A graph of the effect of changing the probe position upon the calculated efficiency is

shown in figure 4-19 for K-^ = 1.0676 , K3 = 0.826 , and 6 = 5°.

If r„ = r„ = a = 0, eq. (4.62) reduces to eq. (4.53), as represented by

figure 4-16.

0.11, DEGREES

Figure 4-19- Effect of varying probe position upon the efficiency

correction according to equation (4, 64).

e. Experimental Results

The efficiencies of two commercially available bolometer mounts were measured

at 600, 1,000, 2,000, and 3,000 MHz. The efficiency of a commercially available

tunable bolometer mount (A) was measured first, and then the efficiency of a commercially available bolometer mount (B) was determined from comparative power measurements. The data obtained in a typical measurement of the efficiency of a bolometer

mount is shown in table 4-1. It was found that the efficiency of the tunable

bolometer mount remained at approximately 96 percent over the above frequency range,

while the efficiency of mount B decreased with rising frequency, as shown in figure

4-20. Because only one of each of the two types of mounts was investigated, the

measured efficiencies are not necessarily representative of these types of bolometer

mounts

.

Ill
Table 4-1. Typical efficiency measurement at 1, 000 MHz.

K

R-| = 150 ohms R-l = 250 ohms

^a

1

2

3 ---

1 50 .0

200 .0

250.0

1 .270

1.119

1 .000

250.0

200 .0

150.0

1 .277

1 .155

1 .000

EACH CASE, THE PROBE WAS IN POSITION FOR MAXIMUM RESPONSE WHEN R

C = 16.00 (CALCULATED FROM eq (4))

0.946 (MEASURED WHEN R^ = 150 ohm)

0.916 (MEASURED WHEN R-, = 250 Ohm)

0.948 (AVERAGE OF ABOVE TWO VALUES)

0.988 (MEASURED)

1.002 (CORRECTION FOR LOCUS CURVATURE)

0.962 ((MOUNT A) CALCULATED FROM eq (18))

0.807 mw,

0.823 mw,

1 .020 (MEASURED)

0.981 ((MOUNT B) CALCULATED FROM eq (21)).

n

n

n

^P-B

^1

^Ad

^B

'B

z z

UJ u

— o

o

UJ

100

99

98

97

96

95

—

0.5 1.0 2

FREQUENCY- KMC/S

Figure 4-20. Measured efficiency of a coaxial bolometer mount B.

112
Table 4-2. Estimate of limits of error in a single efficiency measurement.

PRINCIPAL SOURCES OF ERROR

APPROXIMATE

LIMITS OF ERROR

IN EFFICIENCY

MEASUREMENT OF nP-B

MEASUREMENT OF PROBE VOLTAGE

MEASUREMENT OF RESISTANCES

GENERATOR MISMATCH

LOAD MISMATCH

ESTIMATED LIMITS OF ERROR IN SINGLE EFFICIENCY

MEASUREMENT OF TUNABLE BOLOMETER MOUNT---

MEASUREMENT OF POWER

MEASUREMENT OF VSWR

ESTIMATED LIMITS OF ERROR IN SINGLE EFFICIENCY

MEASUREMENT OF UNTUNED BOLOMETER MOUNT---

% + 0.5

± .4

± .2

± .1 ± .1

±1 .3

±0 . 2 ± .1

+ 1 .6

An approximate evaluation of the error in measuring efficiency is given in

table 4-2. It represents an estimate of the limits of error in a single measurement of efficiency. The actual error can be considerably less than this, if

the effect of random errors is reduced by averaging the results of a number of

measurements. A further reduction of error could be obtained by use of better

equipment and imrpoved measuring techniques

.

f. Conditions for Linear T-j^-Locus

It will be shown that the locus of the input voltage reflection coefficient

of a lossless, tuned, linear, passive, two-terminal -pair network terminated in

loads having real reflection coefficients is a straight line passing through

the origin.

The lossless condition requires that ^

l^ll' " 1^22' ^

1^12' ^ " ^ ' l^iil ^ ^ ^ ' 1^22! ^ = 1

-

^*12 " *11 ^ ^11 '

where \jj represents the angular argument of a scattering coefficient.

-v-

^ ^See section 3.3c, and assume Z^-j^ = Zq2-

113
The input reflection coefficient of a reciprocal two -terminal -pair network

terminated in a load having a real reflection coefficient | | , is from eq. (3.10),

^12 I '"l I r = S + i-^ ^ . (4.65) 1-S22|rj

Because the network is matched (r = 0) when terminated in a load having a

reflection coefficient ^-^2'

S 2,1r ^11

= r

^'^•^^^

^ " ^22 I ^L2

I

Combining eq. (4.65) and eq. (4.67),

Sej^^^ = S^|rL2|ej^'J^^^^^^^^ - (1 - S^) \Y ^^{e^^"^ (4.67)

or

Se^^^ii = \V^^\e^^'^'''-'^^^\ (4.68)

It is evident that S =

I I ^^'^ '^'22 ~ ^ above lossless, tuned, two -terminal

pair network. Substituting the results of eq. (4.65) and eq. (4.68) in eq. (4.66),

the input reflection coefficient is

r =

" ej"^!^ (4.69)

As I I varies, the locus of r is a straight line passing through the origin.

It should be noted that the above conditions imposed upon the network (lossless,

matched input when terminated in a load having the real reflection coefficient ^-^2^

are sufficient to produce a linear input reflection coefficient locus passing through

the origin, but are not necessary. The amount of locus curvature is not necessarily

an indication of the amount of loss, because it is possible to obtain a straight

line locus with a lossy network having = 0.

114
5. Impedance -Reflect ion Coefficient

5.1. Introduction

The accurate measurement of impedance or of reflection coefficient is easier in

rectangular waveguide of convenient dimensions (such as 0.4" x 0.9" inside, or WR-90)

than in coaxial waveguide. Rigid, high-quality components are readily available, and

the absence of a center conductor and supporting insulators makes it easy to slide

terminations inside a rectangular waveguide.

The development of an adjustable sliding termination for rectangular waveguide is

described in section 5.2. It is an improvement over previous designs and has proven

useful in adjusting tuned re f lectometers and in other measurement applications.

The quarter wavelength short-circuited section of waveguide is the most

accurately known standard of reflection coefficient. It nominally has a reflection

coefficient of unity, but due to wall losses, is a bit smaller. If the conductivity

of the metal walls is known, the magnitude of the reflection coefficient can be calculated to very good accuracy. The absence of current flow across the joint minimizes

dissipative loss in the waveguide joint. In other types of standards, this loss

causes an error which is difficult to evaluate. The short-circuited section should

be made from a single block of metal in order to eliminate any other joint losses.

In section 5.3, formulas, graphs and conductivity data are presented to aid in

designing, constructing, and evaluating the above type of reflection coefficient

standard.

Research on different kinds of waveguide junctions for use in impedance measurements is described in section 5.4. It was found that adjustment of a tuner in one

type of waveguide junction produced a squared VSWR response instead of the usual VSWR

response. The use of directional couplers was found to give magnified response.

(The ratio of maximum to minimum detector output equals k times the VSWR, where k

is the magnifying factor.) By use of a tuner, the magnification factor can be varied

and made as large as desired.

Applications of tuners in circuits for the measurement of reflection coefficient are described in section 5.5. A tuner was used to magnify the response of a

phasable load re f lectometer , and to magnify the difference between two loads alternately connected to a re f lectometer . Finally, a re f lectomet er with two tuners was

investigated and analyzed. This form of- re f lectometer , when used with a quarter

wavelength short-circuit impedance standard has provided the basis for present-day

calibrations of impedance standards at the U.S. National Bureau of Standards.

lis
As was mentioned above, accurate impedance measurements are more difficult in

coaxial line than in rectangular waveguide. Accordingly, the hybrid reflectometer

,

described in section 5.6, was developed. It consists almost completely of rectangular

waveguide components except for the output section of waveguide, (which is coaxial)

and the sliding loads, which must slide inside a section of coaxial line. Coaxial

quarter wavelength short-circuit impedance standards are used. A waveguide -tocoaxial adapter connects the rectangular and the coaxial portions together. The

hybrid principle makes it easy to adapt the existing tuned reflectometer for measurements in various types of waveguide other than those having rectangular or coaxial

cross section,

Reflectometer tuning and sliding load techniques were developed to measure small

reflections and losses in waveguide joints. The theory and technique are described

in section 5.7, and experimental results are given for WR-90 (X-band) waveguide

joints.

5.2. Adjustable Sliding Termination

a. Introduction

The adjustable sliding termination (A.S.T.) using a resistive vane which can rotate

and slide relative to a sliding short-circuit in rectangular waveguide (Beatty, 1957),

is particularly useful for fine tuning of reflectometers , as well as other applications. The design is applicable to lowest TE modes in uniconductor waveguides but

is not readily adaptable to other forms such as coaxial line.

There are many measurement situations where a sliding load, having low reflections, is useful. One having high reflections, such as a sliding short-circuit is

also useful. A sliding load which can be adjusted over a wide range of reflection coefficients,^ ranging from zero to nearly unity, is even more useful.

For example, in tuning a reflectometer, one tunes for high directivity of the

directional coupler by reducing the cyclical variations of the side arm output as

the load is slid inside the output waveguide. If the load is adjustable, a more

sensitive tuner adjustment is possible, because one can simultaneously approach a

^A resistive vane fastened to a rotating and sliding short-circuit in circular

waveguide has been described (deRonde, 1957), and proposed as a variable standard

of reflection coefficient. It is not possible to obtain complete cancellation of

reflections, however.

116
"flat" response and zero response. It is possible by this technique to obtain

effective directivities in excess of 100 decibels and thereby increase the accuracy

of reflection coefficient measurements.

The adjustable sliding termination described here is superior in performance to

previous ones. It can be adjusted for completely zero reflection as well as almost

total reflection. It is simple in design and construction.

An adjustable sliding termination for rectangular waveguide was described by

Grantham (1951). Several years later, a similar termination was produced commercially. That termination differed from its predecessor in details of construction

of the dissipative element and the reflecting antenna. Both terminations were

designed primarily to provide minimum reflection and could not be adjusted over a

wide range. The commercial termination has a range of 1.005 to 1.15 in VSWR. The

principle of the double slug tuner was used (Ellenwood and Ryan, 1953) to obtain an

adjustable sliding termination with a somewhat greater range. An adjustable sliding

termination having a wide range was described by Kato and Sakai (1955) , but it was not

possible to adjust this termination for cancellation of its reflections.

b. Principle of Operation

As shown in the diagram of figure 5-1, this termination slides inside a

rectangular waveguide and consists of a short-circuiting piston to which is attached

a dissipative strip supported by a dielectric rod, which can rotate and slide relative to the piston. The phase of the reflection from the strip can be varied by

sliding the strip, while the magnitude of the net reflection from the short circuit

can be varied by rotating^ the strip. With independent control of these two motions,

complete cancellation of reflections can be obtained. On the other hand, with the

strip surface perpendicular to the electric field, minimum losses occur in the strip

and almost perfect reflection is obtained. It is possible to adjust the termination

to any intermediate condition, then to slide the entire assembly, to obtain any

reflection coefficient desirable.

^An attenuator employing a resistive strip rotating in circular waveguide was

described by Southworth (1950).

117
Figure 5-1. Essential features of termination.

c. Theory

Apparently, the analysis of the fields inside a rectangular waveguide containing

an arbitrarily positioned dissipative strip has not been published. Lacking this

basic information, a rigorous analysis of the action of this termination has not been

attempted. An approximate study based upon microwave circuit theory is given below.

As shown in figure 5-2, the termination can be regarded as an attenuator

terminated in a short-circuited line of variable length. In terms of the scattering

coefficients S-j^^ , S-j^2 > ^22 °^ attenuator, the input voltage reflection coefficient is^

^1 = ^11

S

^ ^12

^22 ^ ^ j26£' (5.1)

where 3 = 2t\/Xq, and equals the wavelength in the waveguide. The condition for

cancellation of reflections (r^ = 0) is

S

12 11 (5.2)

This condition can be readily realized, as will be shown. Normally, |S-|^-j^| and IS22

Figure 5-2. Approximate equivalent circuit representation.

^See for example, eq. (3.10).

118
are much less than unity, and the approximate condition for cancellation of reflec

tions is

^12^ ^11®' (5.3)

Rotation of the strip will, in general, change both |S-j^2^l l^lll" ^ typical

variation of these quantities with 6 (the angle between the electric field direction

and the normal to the surface of the strip) is shown in figure 5- 3 (a). At the

angle 6c, the magnitudes of S-|^2^ ^^'^ equal, and it is possible to obtain

cancellation of reflections by sliding the strip, varying £. If the strip is too

short, |S-]^2^I ^ill i^ot decrease to equal |Sj^-j^|, as shown in figure 5-3(b). However,

it is possible to increase |S-]^-|^| by adding a reflecting object at the end of the

strip, so that cancellation of reflections can again be obtained as in figure 5-3(c).

Although this discussion of the cancellation of reflections is based upon the

approximate condition of eq. (5.3), it should be noted that exact cancellation according

to eq. (5.2) is possible under almost the identical conditions.

UNIT CIRCLE

RECTANGULAR STRIP

035"xl.5"

RECTANGULAR STRIP

035"xQ75"

METAL REFLECTOR ADDED

TO INCREASE |S|||

Figure 5-3. Dependence of [ Sn | and |Si|| upon rotation of thin rectangular strip; (a) rectangular

strip 0. 35 inch X 1. 5 inches, (b) rectangular strip 0. 35 inch X 0. 75 inch, and

(c) metal reflector added to increase | Sxi | •

119
d. Design

A number o£ considerations can influence the design (Beatty, 1960) of the termination. If the primary need is for a reflection-free termination, with no need for a

wide range of VSWR, it seems advisable to use a thicker strip of dissipative material

such as Synthane . The extra thickness can give added strength to a long, tapered

strip, which will require less rotation to achieve cancellation of reflection, and

give a smoother adjustment which is less frequency sensitive than with shorter strips.

Examples of A.S.T.'s for different sizes of waveguide are shown in figure 5-4, and a

diagram of typical variation of |S22^I 1^11^ shown in figure 5-5.

If a wide range of adjustment of VSWR is desired, a thin strip is required. IRC

resistance strip may be used to obtain fairly high ranges, but a dissipative film on

Figure 5-4. Adjustable sliding loads for different waveguide sizes.

120
f

\ \

\ VJ

^^^^^^^

Figure 5-5. Variation of | Sn | and | Si| | with rotation of thick tapered strip.

a thin mica strip is capable o£ greater range. A thin film has little strength and

it may be desirable to add a reflecting disk or rod as shown in figure 5-6. The

disk or rod permits a shorter strip to be used, while retaining the ability to cancel

reflections

.

121
Figure 5-7. Mechanical controls permitting independent adjustment

of rotation and sliding, with control locking.

122
I NCHES

2 3 4 5

Figure 5-8. Mechanical controls permitting coarse manual adjustment and locking,

followed by fine independent adjustments of rotation and sliding.

Control of the mechanical motion required may be achieved by the arrangements

shown in figures 5-7 and 5-8.

Independent control of the sliding and rotation of the strip is achieved in

both arrangements, but in figure 5-8, the strip may be rotated and slid by hand until

the clamping screw is used, which clamps the dielectric rod supporting the strip so

that it can only be moved by the limited fine adjustments A and B. One adjustment

of sliding (knob A) and an independent adjustment of rotation (knob B) are provided.

An adjustable load having satisfactory performance over the recommended frequency

range of WR-90 (X-band) waveguide was constructed, using a rectangular IRC resistance

strip 200 ohms per square, 0.35 inch x 1.50 inches. The strip was not tapered, but

mounted on a dielectric rod as was shown in figure 5-1.

5.3. Quarter -Wavelength Short -Circuit

a. Introduction

The purpose here is to derive equations and to present a graph (fig. 5-15) for

the determination of the return losses of microwave impedance standards. It applies

to impedance standards consisting of quarter-wavelength short-circuited sections of

123
rectangular waveguide or coaxial line (Beatty, 196 7a) for which the conductivity is

10'' mho/m (closely corresponding to platinum). The curves for 50-ohm coaxial line

are each labeled with the appropriate inner diameter of outer conductor. The curves

for rectangular waveguide are each labeled with the appropriate WR (USA) number except

for two WG (British) numbers and two lEC-R (International) numbers for which there

are no WR equivalents. All of the lEC standard sizes of rectangular waveguide are

covered. See Brady (1967) for cross-referenced waveguide designations. The return

loss Lg corresponding to a given conductivity a may easily be calculated by dividing

the return loss read from the graph by the square root of the given conductivity

normalized to 10^ mho/m. For example, if one reads Lg = 0.020 dB , and the normalized

conductivity is 4(4 x 10'' mho/m, closely corresponding to gold), the return loss for

an impedance standard made of gold (or gold-plated) is 0 .020 4- /T = 0.010 dB.

The bases for figure 5-15 are the following formulas. By definition, the return

loss is

Lg = 20 log-j^Q —— dB (5.4)

I Tgl

where | Tg | is the magnitude of the reflection coefficient of the impedance standard.

For a lossless quarter-wavelength short-circuit, | Tg | is ideally unity.

b. Coaxial Line

For a short-circuited quarter-wavelength section of air-dielectric coaxial line"*

operating in the TEM mode, it can be shown^ as follows that

2Rgp ( 60X

1 + 1 1 + y

^0 ^ 4bZo

where

^0

Rgp =

-j^2 0TracS ^ resistance of end plate, ^

Zq = 60 Jin — = characteristic impedance,

a = outer diameter of inner conductor,

b = inner diameter of outer conductor,

a = conductivity of metal,

a (5.5)

Tt is assumed that the same metal is used for inner and outer conductors and for the

short-circuiting end plate.

^Equation (5.5) is equivalent to eq . (2) of Beatty and Yates (1969). (Note that b

and a refer to diameters, not radii).

'See, for example, p. 70 of Jackson (1951)

124
6 = skin depth in metal

/irfya

-, and

p = permeability of metal (4it x 10 H/m if relative permeability is

unity)

.

Consider first the relationship between the magnitude of the reflection coefficient |r| and the incident, reflected, and dissipated powers Pj , Pj^, and P^,

respectively. The incident and reflected wave amplitudes a and b are indicated in

figure 5-9.

Figure 5-9- Schematic diagram of quarter-wavelength

section of short-circuited waveguide.

By definition.

= 1

where

Pj^ = reflected power,

Pj = incident power. and

D dissipated power.

It follows that

1 = 1

I 2P

I

(5.6)

(5.7)

The ratio of dissipated to incident power is calculated as follows. For a

single propagating mode in lossless waveguide of usual cross sections, the wall

current distribution is well -known. The resistance of the metal walls to the

current flow may be calculated corresponding to a given value of the skin depth 6.

One then calculates the dissipative loss by integrating the I^R losses. The expressions for P^ and Pt will both contain an I., (maximum current) term which cancels out

D I M ^

'

in the ratio. It is of course assumed that the losses are small enough so that the

perturbation of the lossless current distribution can be neglected.

125
At high frequencies, current flows in the "skin" of a metal. The resistance

R of a strip of metal is^

R = ^, (5.8]

oA

where £ is the length in the direction of current flow, a is the conductivity of the

metal, and A is the area of the cross section through which the current flows. The

case of figure 5-10 (a) corresponds to the short-circuiting end plate of a coaxial

line. The resistance between the inner and outer conductors is

n 1 H dr 1 „ b Rgp = - / = £n -. (5.9)

a a 2iTr6 2-na6 a

The loss in the metal can now be calculated, assuming the current is distributed

in the known manner for the TEM-mode as shown in figure 5-11.

^See, for example, Michels (1961), p. 982,

126
A/4

Figure 5-11. Assumed current distribution in quarter wavelength

short-circuited section of coaxial line.

The I^R loss P^p in the end plate is

P = T

2 D 2Tra6 a

The I^R loss Pq^-, in the outer conductor is calculated using the element of

figure 5-10(b).

X/4

(5.10)

"M

2ttZ1 2

\ ^'^

1 X

' 2iTba6 4 In a similar way, the loss Pj^-, in the inner conductor is

(5.11)

IC

X/4

/ |Im cos

2ttZ| ^

( dz

X ' ''TTa06 OC (5.12)

The total dissipated power is

P = P + P + P D EP OC IC

2iTa6

. b ^ X In — + —

a 4 "-a b

1 .

1 (5.13)

The incident current I is one-half Ij^. It follows that the incident power is

Pi = I^Z

(5.14)

127
The magnitude of the reflection coefficient | Tg | of the short-circuited quarter

wavelength section is obtained from eqs . (5.7), (5.13), and (5.14). It is

TTa6Z an - + - \ - +

0

^

The above equation may also be written

607ra6 J —

60

1 +—/—•— 1 + 4b

(5.15)

(5.16)

or

^EP

'0

This reduces to eq. (5.5) if y

= 1

.

1 + 60

4b

1 . b

(5.17)

r r In the following example the components of power loss are individually calculated. Suppose that f = 4 x 10^ Hz, a = 10' mho/m, = = 1 , y

= 4:7 x 10 ' H/m,

Zq = SO f^, a = 0.304 cm, b = 0.70 cm, and Ij^

6 = 2.52 X 10 cm, R^p = 0.00527 Q, P^p = 5.27 mW,

lA. We calculate the skin depth

16.9 mW, Pj(. = 38. 9 mW,

= 61.1 mW, and Pj = 12.5 W. From eq. (5.7),

= 1

- 61.2

10 = 0 . 99755,

25.0

and from eq. (5.4), Lg = 0.0213 dB. These results agree with Beatty and Yates (1969),

and show in addition the distribution of losses in end plate, inner and outer

conductors

.

In the event that the attenuation constant a of the line is more easily determined than the conductivity a, the reflection coefficient is determined as follows.

The total loss in the inner and outer conductors may be written in terms of the

attenuation constant a of the coaxial line. First consider that the end plate is

lossless, and we can write

or

-2a!i ~ ^ e =1 2ai.

Similarly

^sl^ 1 - 4a£.

(5.18)

(5.19)

(5.20)

128
Comparison with eq. (5.7) shows that

P + P

^OC ^IC

I

4a (5.21)

We can regard the losses in the end plate P^p as corresponding to an equivalent length

of line £g such that

EP

(5 . 22)

It then follows that

TgM 1 - 2a^^ -

60 ^ 1 + b/a

(5.23)

We can determine from eqs . (5.22), (5.21), (5.10), (5.11), and (5.12). It is

b

(5.24)

In the previous example, it equals 0.177 cm, and a = 0.596 x 10 Neper/cm = 15.8 dB

per 100 feet.

c. Rectangular Waveguide

For a quarter guide wavelength, short-circuited section of rectangular waveguide operating in the dominant TE-j^ ^ mode, it can be shown as follows that the magni

tude of the reflection coefficient is (Beatty and Yates, 1969),

Fgl = 1 - 2a + iEP (5.25)

where

a = attenuation constant (Np/cm)

,

= guide wavelength (cm)

,

5,gp = length of waveguide having same loss as short-circuiting plate (cm), and

^EP = ^ (5.26)

One proceeds in the same way used previously for coaxial line, calculating the

ratio of dissipative to incident power and substituting into eq. (5.7) to obtain

eq. (5.25).

129
The distribution of electric and magnetic fields in the TE^ ^ mode for a wave

incident upon a quarter guide wavelength short-circuited section of rectangular wave

guide is shown in figure 5-12 and the corresponding wall current distribution is

shown in figure 5-13.

Figure 5-12. Electric and magnetic fields in quarter wavelength section

of rectangular waveguide propagating TEi^o mode in +Z

direction and short-circuited at Z = 0.

130
DIRECTION OF

INCIDENT

TEi,o WAVE

Figure 5-13. Wall currents in short-circuited quarter wavelength

section of rectangular waveguide with incident TEi,o

mode wave.

For a TE^ q mode, incident wave in +z direction, it can be shown that

cos

TTX (1 + r^ej^^z^g-jaz

j ^

A s m TTX (1

-j

^

A

(5.27)

where

271 IT 7 T'

^ " a'

1

and = voltage reflection coefficient at z = 0 (-1 for short-circuit).

It is implicit in eq. (5.27) that sinusoidal time variation with angular frequency 0) exists, and that y is the permeability of the medium (usually air) within the

waveguide. The factor A indicates the level of the excitation and for convenience

131
the rms value of the field strengths are denoted by E and H. The incident power

Pj (at z = 0) is

>j = Re / [E X H*] • e^ dx dy, (5,28)

where e^ is the unit vector in the +z direction. Upon integration,

Pi =

2 ab

(5.29)

The waveguide walls are assumed to have high conductivity a and small skin depth

6 resulting in small losses which do not appreciably perturb the TE^ ^ mode fields.

One may then calculate the small wall losses by integrating the I^R losses in the

metal

.

In order to calculate the resistance to a current element, one uses the elements

shown in figure 5-14.

t

-dx

—

^1 l^dx

(a) R o6dx (b) R

= o6dz (c)

{^x o6dz

R

=

1 nre 5-14. Elements used to calculate I R losses in (a) shorting plate,

(b) narrow walls, and (c^ broad walls.

It is convenient to write eq. (5.27) in the form

cos

TTX

a sin 3z

s

- -2

f ttx' [6 1 A s m —

"-a '

\ [3 \

1 Z, A s m

cos Bz

I TTX

^a sin Bz,

(5,30)

132
One can now calculate the dissipated power as follows. The magnitude of the current

density in the short-circuiting plate is

J

spi 2 fB 1 A si„(E)'

[ '}

(5.31)

The power Pgp dissipated in the short-circuiting plate is

a

o a&dx a6

(5.32)

The ratio of Pgp to Pj. is

SP 1 Pj 2a6Zj^ ajya6XQ

The magnitude of the current density in the narrow walls is

(5.33)

(5.34)

The power Pj^^ dissipated in the side walls is

a6dz a6

(5.35)

The ratio of P,,,., to 2Pt is NW I

NW

2P

I 4 h

(5.36)

The magnitudes of the x and z components of the current densities in the broad

walls are

J = H = 2A

X '

' z

' cos

I ^a - J

T7X

a

s m Hz

|JJ = iHj = 2 s m TTX cos tsz. (5.37)

The power dissipated in the broad walls is

Prw = 2 / / (iJjdz)^

^. (iJjMx) I - G

o o a 6dz a6dx-' 2a6

(5.38)

The ratio of P„,., to 2Pt is

2Pj 2bZj^a6

1 + (5.39)

The total power dissipated is

P = P + P + P D SP NW BW (5.40)

133
and the magnitude of the reflection coefficient of the short-circuit is

D D

2Pn

(5.7)

'I "'I

In the event that the attenuation constant a of the waveguide is known, it is

convenient to use eqs . (5.25) and (5.26) which are derived as follows, assuming that

(1) the short-circuiting plate and the waveguide walls are one continuous piece of

metal having no lossy joints, and (2) the waveguide section is filled with a medium

such as air whose electrical properties are not significantly different from those of

a vacuum.

If we assume for the moment that there are no losses in the short-circuiting

plate

,

2a

4

= 1 = 1

P + P

2Pt

(5.41)

Thus

~ 2 P + P

2Pt

(5.42)

or from eqs. (S.36) and (5.39) it follows that

4fT a)pa6aA

G

^ —111 + 2a 2b 2b

(5.43)

or

1 . 2b, .

120iTba6 f 1 c

(5.44)

where f and f^ are respectively the operating frequency and the cutoff frequency of

the waveguide.

It is convenient to account for the losses in the short-circuiting plate by adding

an equivalent length £gp of waveguide such that the reflection coefficient |r| is

given by eq. (5.25). One derives eq. (5.26) to calculate l^^ as follows. Since

2a + a EP

P + P ^NW BW

2Pt

SP

2P,

134
it follows that

= ^ . ^= ^ . . (5.45)

2a 2Pj 4 P^^ . Pg^

Substitution of eqs . (5.32), (5.35), and (5.38) into eq. (5.45) yields eq. (5.26).

It is interesting that the expression for does not involve the conductivity

of the metal.

An example of the calculation of |r| for rectangular waveguide follows. Given

f = 9.4 GHz, b = 1.016 cm, a = 2.286 cm (WR-90, or X-band waveguide), a = 4.0 x 10^

mho/m, we calculate X = 3.189 cm, X^^ = 4.451 cm, X^ = 4.572 cm, and £gp = 0.364 cm,

a = 1.59 x 10'^ Np/cm, | Fg | = 0.99953, and Lj^ = 0.00408 dB . If we have 10.64 W incident power, then P^ = 10 mW, Pgp = 2.46 mW, P^^ = 2.2 0 mW, and Pg^^ = 5.34 mW. The

distribution of losses was calculated from the following formulas:

^SP 1 P, X,/4' ^'-'^^

^EP

and

NW BW (5.47)

EP

174

BW

1 + 2EP 2b.

2 fx 1 a

(5.48)

135
FREQUENCY -GHz

Figure 5-15. Calculated return loss of quarter wavelength short-circuit impedance

standards. Basis of normalization: a = lo' mho/m. Curves for

coaxial line are marked with the appropriate inner diameter of outer

conductor. Those for rectangular waveguide are marked with

appropriate standard waveguide designators.

d. Errors and Design Information

It is estimated that the errors due to approximations in the formulas will be

less than the errors in reading the graph of figure 5-15 over the range of values

shown. Thus, one can expect the error in Lg determined from the graph to be within

± 2 percent. To this must be added errors from other sources, such as the error in

determining conductivity, etc.

Table 5-1 gives bulk conductivities normalized to 10^ mho/m for a number of metals

which might be used to construct quarter-wavelength short-circuits. In the absence

of measured conductivities at the frequencies of interest, the values in this table

may be used with probably only a few percent error in Lg in most c^ses.

136
Table 5-1. Bulk conductivity normalized to 1 0 mho/m

of various metals at 20°C.

SILVER

COPPER (ANNEALED)

GOLD

ALUMINUM

MAGNESIUM

BRASS

NICKEL

PHOSPHOR BRONZE

PLATINUM

STAINLESS STEEL

1 .28

1 .28

1 .00

-0.110

6.29

5.80

4.10

3 .54

2.17

1.19-1.56

HANDBOOK OF CHEMISTRY AND PHYSICS. 41st ed.

CLEVELAND, OHIO: CHEMICAL RUBBER PUBL. CO.,

1959-1960. p. 2588.

5.4. Squared VSWR and Magnified Responses

a

.

Introduct ion

The following discussion of research on unusual responses obtained from various

kinds of waveguide junctions preceded and is related to the research on tuned

ref lectometers described later in section 5.5. The discussion is a revised version

of Beatty (1959)

.

In most microwave impedance measuring instruments, such as the slotted line,

the resonance line, and rotary standing-wave indicators, the ratio of the maximum

to the minimum response (magnitude of voltage output to detector) is ideally equal

to the voltage standing -wave ratio (VSWR) of the termination under test or

calibration.

Other radically different types of responses are obtainable from various kinds

of waveguide junctions. The responses to be discussed here have been called magnified

and squared VSWR responses for reasons to become apparent.

A simplified explanation will be given first, followed by a more complicated

mathematical description.

The differences among responses are shown in figure 5-16, three response curves

calculated for the same termination.

137
b. Simplified Explanations

(1) Squared VSWR Response

A simplified explanation can be given for one system yielding squared VSWR

response. Other systems which have been devised apparently do not permit simplified

explanations and have not been thoroughly analyzed. Enough theory will be given

however, to permit their use for measurement purposes.

The system shown in figure 5-17(a) consists of a straight section of uniform

lossless waveguide [which may be either coaxial line or rectangular waveguide, for

example) with oppositely located coupling probes for generator and detector. A short

circuit which may be adjusted in position terminates one end of the uniform waveguide

section while the other end is terminated in the sliding load to be measured. In this

system it is necessary to vary the phase of the load by sliding it inside the waveguide, but in other systems to be described, this is not always required.

Referring to the simplified model of figure 5-17(b),

As a varies, |E| goes through maxima and minima. The ratio of the maxima to the

minima is

Zg Z

I

(5.49)

If the short circuit is located Ap/4 from the probes, Z,, = and

1 . r^e-j^e^

(5.50)

(5.51)

where is the VSWR of the load. The meaning of other symbols used above should

become clear upon reference to figure 5-17(b).

(2) Magnified Response

A system yielding magnified response is shown in figure 5-18.

138
(a) IDEALIZED SLOHED LINE

d = c[l + Kr,e-i^^^] d = c

i8i

(b) SQUARED VSWR R{ESPONSE (c) MAGNIFIED RESPONSE

Figure 5-16. Response curve of three measurement systems with termination

having a VSWR of 1. 26.

(a) idealized slotted line,

d = c[l + Kr, -j20l]

le

(b) squared VSWR response

d = c'

{ -) magnified response,

1+r -jzpi

J-ie

1 - r -jzpi

Lie

d = c" [1 + Kr^ -j20l].

-L/e

GENERATOR

SHORT

CIRCUIT

LOAD

DETECTOR

(a) DIAGRAM OF SYSTEM (b) SIMPLIFIED MODEL Figure 5-17. Diagram and simplified model of one system yielding

squared VSWR response; (a) diagram of system,

(b) simplified model.

139
^31 '^G

^21 ^32 ''g

Figure 5-18. Diagram of simplified system to illustrate magnified response.

A directional coupler is connected to respond mainly to the wave reflected from

a phasable or sliding termination whose VSWR is to be measured. For simplicity, it

is assumed that the generator and detector do not produce reflections (r„ = r„ = 0) ,

and that no reflections are produced in arm 2 by the directional coupler (S22 = 0)

The signal coupled to the detector has two components. One is fixed and exists

because the directivity is not infinite. The other is from the load reflection and

varies in phase as the load is slid inside the waveguide. As the relative phase of

the two components vary, the magnitude of the resultant varies. If the components

are of approximately equal magnitudes, the range of variation of the resultant may

be large even though the reflection from the termination may be small.

Inspection of the diagram of figure 5-18 leads to the following equations

describing the response.^

h

~- bc^Sj, . S2iS32rLe-^26^3 ^ ^^S^^^^ ^ KFLe'^'^^. (5.52)

The response is of the same form as that of the idealized slotted line (see

fig. 5-19) excepting that r^^ is multiplied by the factor K. Since |K| may be very

large (it approximately equals the directivity ratio of the directional coupler), one

may consider T to be magnified by the factor |K| , leading to the term magnified

8 In this equation, b represents a wave amplitude, S a scattering coefficient of the

directional coupler, and the voltage reflection coefficient of the load.

140
response. It should be noted however that the response variation will increase as

one increases |K| up to the point where |Kr^| = 1, and then will decrease with

further increase in |K|.

c. Analysis

Both magnified and squared VSWR responses may be analyzed by considering the

generalized treatment of MacPherson and Kerns (1956) of a 3-arm junction measurement

system for phasable loads. Such a junction is shown in figure 5-19, where it has been

assumed for convenience that the necessary variation in phase is obtained by changing

the length I of uniform, lossless waveguide.

GENERATOR

1

1

03 tri —T,

DETECTOR

TERMINATION

Figure 5-19. Three-arm junction with phasable load.

Instead of using the gathering coefficients employed by MacPherson and Kerns,

the solution for b^ is obtained in terms of the more familiar scattering coefficients

(Schafer and Beatty, 1958) and may be expressed as follows:

1 + Kr,e-j26£

b = c = = CKr e J^p^

^ ^2i^L®

1 + J_ ej2e£

Kr,

®

1 ^Zi^L^

(5.53)

where

^G^31

(1-Sj^^r^) s^jTq

^31^G ^1-533^

K = !2l!32 . s S3I

22

141
and

(i-s,,r,)

22

32

'^13^D

"^23^D

^1-^33^

2i

^31^G

^13^D

(i-S33r,)

In the above expressions, the component o£ the emergent wave amplitude supplied

by the generator is ~ ' ^i'^g' represents the amplitude of the wave

incident on the junction in arm 1. Symbols of the form ^ are the scattering coefficients of the junction, and r^, T^, and are the voltage reflection coefficients

of the generator, detector, and load, respectively, as indicated in figure 5-19.

The reflection coefficient is that which would be obtained "looking into"

arm 2 if the generator was turned off and its impedance (as observed at T^) was

unchanged in so doing.

The variation in Ib^l as we vary the phase Ci|^^) of is defined to be the response

of the systems represented by figure 5-19, and is determined by eq. (5.53).

The properties of eq. (5.53) will be examined in an effort to classify types of

responses obtainable. It is evident that the parameter C affects only the level of

the response, while the form of the response curve (Ib^l vs ip^) is affected by the

parameters K and ^2i'

We may consider the response for the conditions = 0, |K| = 1, the usual or

normal type of response, since it leads to the form obtained in the case of an

idealized slotted line.

The response form previously referred to as magnified response is obtained when

^2i ~ ^' ^'"'^ """^ unrestricted. However values of |K| may range from zero to

infinity, and it is possible to have a magnification factor |K| greater or less than

unity. The case where |K| is less than one is of dubious interest, but the possibility of |K| greater than unity appears especially attractive for the measurement

of small reflections.

If differs from zero, it will cause distortion in the response curve, which

is considered undesirable. However, a distinctly different type of response may be

obtained if |r2il " '' ' ^^'^ phases of |K| and |r2il equal. This leads to

squared VSWR response if |K| = 1. This type of response is not only curious, but

may prove useful in some measurement applications.

142
Actually |r2il ^^^^ than unity in actual (not lossless) systems, so that the

ideal squared VSWR response may be closely approached with an actual system, but never

quite reached.

A fourth type of response is obtained if the phases of K and r^^ are the same,

|r2^| - 1, and |k| is unrestricted. The ratio of maximum to minimum detector signal

level corresponding to eq . (5.51) is

1 + I Kr

J

p = ^

Pl- (5.54) ^ 1 - iKr^l

^

It seems appropriate to call this a magnified squared VSWR response, and it may have

applications in the measurement of large VSWR.

This completes the classification of responses, since conditions other than

those mentioned may be regarded as causing distortions of the types described above.

d. Means of Obtaining Various Responses

Examples have already been given (figs. 5-17 and 5-18) of junctions permitting

magnified and squared VSWR responses. However other types of arrangements are possible and offer a variety of measurement systems, each with its possible advantages

and disadvantages.

In order to closely approach squared VSWR response (|r2^| - 1> |K| - 1) it becomes

evident that the 3-arm junction should have low loss and low coupling to the load.

(This may be shown from a consideration of the conditions imposed upon the scattering

coefficients by losslessness . ) These conditions are not sufficient however, as one may

conclude after trying junctions which satisfy only these conditions. It is necessary

for K and to have the same phase, and this is obtained by some tuning device,

such as the adjustable short-circuit in figure 5-17. There may be some difficulty

in obtaining the desired response in some cases, because it is not always possible

to obtain correct phase relationship, but the junction forms represented in figure

5-20 (a-d) have all ben found experimentally to permit a close approach to squared VSWR

response by proper adjustment of the tuner. The arrangement of figure 5-20 (e) should

also permit squared VSWR response, but has not been constructed or tested.

Magnified Response occurs upon making |K| greater than unity while T^^ = 0.

It evidently cannot be obtained with a lossless junction, for then |K| = 1. If it

is assumed that we can always make = 0, then T^^ would equal ^22' this would

vanish, so that K = S2-|^S22/S2-|^ . The directional coupler connected as shown in

figure 5-18 evidently permits magnified response since 1 822/52]^ I is the directivity

143
ratio and may be quite large while 1 52]^ I is usually between 0.7 and 1.0. The use

o£ auxiliary tuners with a directional coupler permits greater versatility since one

may adjust the directivity ratio upwards or downwards with one tuner, then adjust

the other tuner to make - 0. These adjustments are independent only if made in

the order described. Referring to figure 5-21, the tuner in arm 2 is adjusted

first in order to obtain the desired value of |K| , then the tuner in arm 1 is

adjusted to make r^. =0.

6ENERAT0R

TUNER

(a)

DETECTOR

LOADA

3

TUNER

GENERATOR

1/

-IRIS

-IRIS

(b) DETECTOR

7- LOAD

TUNER

GENERATOR

(e)

DETECTOR

IRIS

GENERATOR

DETECTOR

DETECTOR

3"

\

TUNER

GENERATOR

COUPLING

COUPLING LOOP

LOAD

LOAD

NOTE: AFTER CORRECT POSITION OF TUNER IS FOUND,

IT MOVES WITH COUPLING LOOPS INSIDE COAXIAL UNE .

Figure 5-20. Schematic drawings of junctions permitting squared

VSWR response: (a), (b), (c), (d). ,(e).

144
e . Measurements Using Squared VSWR Response

Any of the junctions of figure 5-20 or their equivalents may be used if the

unknown is phasable. This requirement is satisfied if the unknown termination slides

inside the waveguide. In principle, a phase shifter or line stretcher may also be

used to provide the phase variation, but in practice, they are less than perfect,

leading to additional errors in measurement. If the unknown termination does not

slide within the waveguide, the arrangements of figures 5-20 (d and e) may be used.

Either flexible cables must be used to couple the generator and detector to the

moving junction, or the generator and detector may be arranged to move with the

junction. If the termination is not too large, it and the waveguide section could

be moved, keeping everything else fixed. The arrangement shown for rectangular waveguide is not readily adaptable to operation with coaxial line. However the arrangement of figure 5-20 (e) should be satisfactory for operation with coaxial line.

The correct adjustment of the tuner is made with r^^ = 0 and corresponds to

maximum detector output for figures 5-17 and 5-20 (a) and to minimum detector output

for the others shown. When the correct adjustment has been made, the response curves

will be symmetrical about the maxima and minima.

A correction may be made for deviations from the ideal conditions |K| - 1,

|r2j^| - 1, using the methods indicated by MacPherson and Kerns (1956). Instead of

analyzing the response curve to obtain the parameter y required for the correction,

a shorter method is as follows. Only the real part (g) of y is needed to obtain an

intermediate VSWR Pj from the apparent VSWR p^. It can be shown that to a good

approximation (to the first order in g and b)

,

It is still necessary to determine |k| in order to obtain \^-^\ or P^- This may be

done by measuring p^ when a termination of known |r, | is connected.

(5.55)

where

Cg + 1) - IbjICFL = 0)

b,! (|rj = 1) 3'max^' L' '

145
1 TTTTT V TTTTT

1

1

1 ^ V

' > ^

' ^ V

'

DIRECTIONAL COUPLER

TUNER

TO ADJUST

KO

TUNER

TO ADJUST

DIRECTIVITY

^Figure 5-21, Directional coupler with auxiliary tuners.

£. Measurements Using Magnified Response

The arrangement of figure 5-21 may be used to measure the voltage reflection

coefficient of an unknown termination. Two basic methods will be outlined.

In the first method, the auxiliary tuners are adjusted for the conditions

= 0 and K = oo. Inspection of eq. (5.53) shows that Ib^l, the magnitude of the

detector arm wave amplitude will then be proportional to | r, | . One then measures

Li the ratio r of the |b^| values obtained when the load is first unknown Cr„) , then

a standard of known reflection coefficient magnitude | Tg | . Then

Ir^l = rlTgl. C5.56)

The adjustments of the tuners preceding the measurement is as follows. One adjusts the

tuner in arm 2 until no variation is observed in Ib^l as one slides a termination of

low reflection inside the output waveguide. Then the tuner in arm 1 is adjusted

until no variation in Ib^j is observed as one slides a termination of high reflection

inside the output waveguide. If necessary, the above operations are repeated in

sequence until no variation in Ib^l is observed as either termination is slid.

146
In the second method, the tuner in arm 2 is adjusted (with the unknown connected

to arm 2) until the detector output is zero. Then Kr^j = -1. The tuner in arm 1 is

then adjusted until T^^ =0. A reflection standard of known | Tg | is then connected

to arm 2, and the phase of Tg is varied. Substitution of the above conditions into

eq. (5.53) leads to

|b,| |r,,| + Ird ' 3 ' max ' U '

' S

' Pa " ";—; 7^—;

\—rr* (5.57)

I Til - Tn

3 ' mm '

I U ' 1311

In the event that |rg| 1, (approximately true for a sliding short-circuit) then

Note that it is unnecessary to vary the phase of r^, the reflection coefficient

of the unknown termination in either method. In the second method one needs to vary

the phase of Tg , but this is easily done if a sliding short-circuit is used.

Alternatively a fixed reflection standard may be used if a suitable line

stretcher is incorporated into arm 2 of the measuring instrument.

STANDARD

_ TOWARD WAVEGUIDE SECTION

GENERATOR ^ —

1 1

I

1

1

2

Figure 5-22. Line stretcher and scattering coefficient representation.

Because of the special condition - 0, the 1 ine ' stretcher need not be of the

constant impedance type, since reflections that it may introduce may be cancelled

by reflection from the tuner in arm 1. Also, the reference plane for arm 2 may be

located in the uniform waveguide section of the line stretcher between the source of

its reflections and the load. As shown in figure 5-22, the reference plane 1

remains fixed although the output waveguide and load move. The line stretcher must

be stable however, so that the parameters S^^ , 5.^2' ^.nd with respect to the

reference planes T^ and T2 do not vary as it operates.

147
With the addition of the line stretcher, the second method described above maybe called a magnified difference method, since the smaller the difference between

iFgl and Ir^l, the greater the variation in Ib^l as the phase is changed according

to eq. (5.57). The arrangement of apparatus employing a waveguide line stretcher

is shown in figure 5-23. Results of measurements of the VSWR of two calculable

standards (Booth, 1961) by the above techniques are shown in table 5-2.

DETECTOR

TUNER TUNER

LINE

STRETCHER u

-A. I TTTTT TTTTT

GENERATOR TERMINATION

T, V

DIRECTIONAL

COUPLER DIELECTRIC

WAVEGUIDE To

SECTION 2

HAVING

STANDARD

DIMENSIONS

Figure 5-23. Arrangement of apparatus for magnified difference method.

148
Table 5-2. Comparison of calculated and measured VSWR of half-round

obstacle impedance standards of rectangular waveguide using

a magnified response measurement technique.

RADIUS (r) 0.1000" 0.1400"

POSITION (£) 1.000" 1.500"

CALCULATED VSWR

ASSUMING NO LOSSES

1.226145 1.455465

CALCULATED VSWR

CORRECTED FOR ATTENUATION

1.2253 1.4543

MEASURED VSWR 1.2247 1.4538

DEVIATION 0.05% 0.03% —f

0.9000"

FREQUENCY = 9.3946 Km c/s ±0.015%

ESTIMATED ATTENUATION = 0.070 db per foot

-VSWR

g. Results

The techniques employing magnified response give promise of increased accuracy

in the measurement of low and intermediate VSWR. Accuracies of approximately 0.1

percent in VSWR up to 2.0 have been achieved, and perhaps an order of magnitude

better than that is possible.

5.5. The Tuned Re flectometer

a. Three-Arm Waveguide Junctions

In the following, some methods for the accurate determination of reflection

coefficient magnitude and VSWR will be discussed.

A number of measuring systems can be represented by a three-arm junction with

arms connected to a generator, detector, and the unknown.

149
The variation in detector signal (square root of detector power) as one changes

the relative phase of the reflection coefficient of the unknown is called the response

of the system. One can obtain different types of response by choosing junctions

having certain properties to be discussed. A careful analysis^ of this type of

measuring system reveals that the amplitude b^ of the output of the third arm (connected to the detector) may be expressed in a form similar to eq. (3.158)

b3 = bg

^21 ^22

^31 ^32

^31^G

^13^D

^1-333^0^

(5.58)

or similar to eq. (3.162),

31

Ci-Si,r,)

^31^G

^13^D

1 + Kr,

1 - r^.r, 2i L

(5.59)

where terms of the form ^ are the scattering coefficients of the junction, and

the voltage reflection 'coefficients of the generator, detector, and load are represented by Tg, Fp, and respectively. The term denotes the reflection coefficient corresponding to the internal impedance of the equivalent generator connected directly to the load.

In impedance measuring systems such as the slotted line, it is approximately

true that K 1 and T^^ - 0. For these systems,

= C 1 + (5.60)

and as the phase ijjj^ of varies,

' 3 ' max . 1 +

3 ' min

p^, the VSWR, (5.61)

2i

b. Tuning for Squared VSWR Response

It is possible to construct a "lossless" junction for which |K| ~ 1 and

^ 1. This can yield a squared VSWR response; for as the phase of V, varies.

3 ' max

3 ' min

1 +

1

- (5.62)

'For background material, see Macpherson and Kerns (1956) and section 3.15i,

150
It is necessary to tune the junction in order to obtain the correct phase relationship between K and ^2i' This response is not only a curiosity, but may find

applications in measuring techniques.

c. Tuning for Magnified Response

Of more promise is the class of junctions for which |K| >> 1, T^^ - 0. Inspection

of eqs . (5.58) and (5.59) reveals that |K| may be approximated closely by |S22/S2-|^|,

the directivity ratio of a directional coupler.

It is well known (Barnett, 1953) that the effective directivity of a given

directional coupler may be increased (or decreased) by adjustment of a tuner connected

to the coupler output. By this means, one can adjust |K| to any desired value. Then

another tuner connected to the coupler input may be adjusted to make = 0. This

yields a response similar to that of eq. (5.61) except that is multiplied by K.

Since |K| may be greater than unity, this can be called a magnified response.

One technique for impedance measurement consists of connecting the unknown

(Tj^ = Ty) , tuning for b^ = 0, (KT^ = -1), then tuning for = 0. If the unknown

is removed and a standard (Fj^ = Tg) is connected, the response is

(5.63)

If a variable standard were available, one could adjust it until b^ = 0 , then

Ty = Tg. If only fixed standards are available, the following procedure may be used

to obtain the magnitude of T^.

21

(i-s,,rg)

^21^G

^31^G

12

22

32

^13^D

^23^D

^--'^33^0^

(i-s,,r^)

^31^G

^13^D

[i-S33r^)

(3.160)

If one varies the phase of Tg (a line stretcher, not necessarily of the constant

impedance type, may be used), the ratio of maximum to minimum response is

^3 max

min Ir^i - (5.64)

This type of response is very sensitive to small differences in Ir^l and | TgOne

can choose a suitable half-round inductive obstacle impedance standard so that | Tg

|

is close to (It is interesting to note that the attenuation of a short section

151
of waveguide connected to an impedance standard could be determined by this technique.)

If a short circuit is used as the standard |rg| - 1, the ratio very nearly equals the

VSWR of the unknown. Variations of these techniques based upon eq. (5.64) are

possible

.

d. Tuning for Infinite Directivity

It is especially interesting to consider the response when the tuners are adjusted

for conditions T^^ = 0, and |K| = 0° (or S^^ = 0). Equation (5.58) becomes

, _ G 21^32^L ^p.. b, = , (5.65)

(1 - S^^T^)il - S^^T^)

or

|b3l = c'irj.

With this type of response it is not necessary to vary the phase of r^, but

Ly to observe Ib^l whe

the output. In this case

merely to observe Ib^l when one alternately connects the unknown and the standard to

Ibjiu Ir^l

r , or I r^l = r I Tgl . (5.66)

Any standard of known reflection coefficient l^gl, such as the ones previously

mentioned (except Tg = 0) , may be used. If a short circuit is used, the measuring

system can easily be arranged so that return loss

1 20 log^Q (5.67)

is measured directly,

e. Analysis of Errors

The basic form of the reflectometer is shown in figure 5-24.

152
DETECTOR

STANDARD

WAVEGUIDE

SECTION

1 -Ff]—

' X

1

1

^21 h

Figure 5-24. Basic form of the reflectometer.

A mathematical treatment shows (assuming linearity) that the response will in general

be of the form:

Ar^ . B

cr, . D

(5.68)

It is evident that the desired response will be realized if B = C = 0. Then

D

(5.69)

The sources of error, in determining by the methods to be discussed in this

section include: a) incorrect measurement of r, and uncertainty in value of | Tg |

,

and b) improper adjustment of the tuners, such that B and C do not vanish.

The error due to a) may be determined by inspection. If the equation for \T^\

is written in the form:

b

Tglr,

3IU

(5.70)

3

I S

it is evident that the fractional error in will equal the sum of the fractional

errors in |rg| and r if the latter are small.

With regard to the second item b) it is assumed that B and C have been reduced

to the point where the variations in the expression

|Ar^ + B|

|cr, - D|

(5.71)

are due mainly to variations either in the numerator or the denominator as the loads

of small IFj^I and large VSWR are employed respectively.

153
A first order correction to eq. (5.70) may be written as follows

3IU

1 . B ^

D

T^) . ... (5.72)

The ratios |B/A| and |C/D| may be determined from the expressions:

= 20 log B

1 + 2 (5.73)

and

= 20 log

f 1 + 2 C

D

(5.74)

where K-j^ and K2 are the ratios in decibels of the maximum to minimum outputs with

the sliding loads of small Ir^^l and large VSWR respectively, |r^| is the reflection

coefficient of the load of small VSWR, and a reflection coefficient of unity has

been assumed for the load of large VSWR.

Except for the presence of the factor r^^^, these equations for and K2 are of

the same form, and values for |B/A| and |C/D| may be obtained from figure 5-25 where

the value of |C/D| is taken from the line |r^| = 1. It will be noted that the evaluation of the right hand side of eq. (5.72) presupposes a knowledge of but for the

present purpose of assigning a limit of error, an estimated (approximate) value is

adequate.

154
As an example, i£ a sliding load of VSWR = 1.005 is employed in the first tuning

operation, and the variation in output is reduced to 1 dB , |B/A| will have a value of

approximately 1.6 x 10 ^. If the variation in the second step with the sliding short

is reduced to 0.02 dB , the value for |C/D| will be approximately 1.2 x 10

Assuming that the unknown load has a value || - 0.2, (p - 1.5), | Tg | = 1.000 [corresponding to a short-circuit) and assuming the terms in the right hand factor of

eq. (5.72) combine in the worst phase, values of 6 and 1.2 are obtained for

and for | Tg - r^| respectively, for a total error of

± (1.6 X 10'^ X 6 + 1.2 x lO"'^ X 1.2) =^ ± 2.4 x lO""^, or ± 0.24 percent.

5.6. Hybrid Ref lectometer

a. Introduction

The development and analysis of tuned reflectometers as previously mentioned

led to improved accuracy in the measurement of VSWR and magnitude of reflection

coefficient. In the following discussion, which is a revised version of Beatty

and Anson (1962), two additional developments are described.

First, it is shown in figure 5-26 how one can measure both phase and magnitude

of reflection coefficient with a tuned ref lectometer . Second, the hybrid ref lectometer

principle is introduced and illustrated. A hybrid tuned ref lectometer is one in

which the source, detector, tuners, isolators, and directional coupler are components

of a convenient (say rectangular) waveguide system, but the output waveguide, sliding

loads, and unknown are components of a different (say coaxial or stripline) waveguide

system.

Such an arrangement has obvious advantages which are illustrated by the following description of a rectangular-coaxial waveguide hybrid tuned ref lectometer

arrangement which was tested at 4 GHz.

In applications to rectangular waveguide systems an accuracy of 0.1% in VSWR was

obtained in measuring moderate and small reflection coefficients. It is difficult

to obtain corresponding accuracy in coaxial systems, especially those in which the

diameters are small.

155
In the work to be described an accuracy of 1.51 in |

| for Ir^l >_ 0.2 is obtained

for the small-diameter coaxial line (internal diameter of outer conductor, 0.276 in;

external diameter of inner conductor, 0.120 in) normally associated with the type-N

connector

.

In order to take advantage of the general superiority of rectangular-waveguide

components over coaxial components, a novel hybrid arrangement was employed. Modifications of the measurement techniques are described which overcome difficulties

inherent in the coaxial structure.

b . Basic Theory

The necessary theory has already been given in this monograph. It is well to

recall that the output of a correctly tuned reflectometer is ideally

b^ = CT^ (5.65)

where b^ is the amplitude of the wave incident upon the detector, C is a constant and

is the reflection coefficient of the load.

One normally makes a measurement by alternately connecting standard and unknown

loads having reflection coefficients Tg and r^, respectively, to the ref lectometer

and (by means of standard phase -shifters and attenuators) measuring the corresponding

change in b^. This is often obtained by noting the attenuation and phase-shift

changes required to maintain b^ constant. Often phase information is not required

and one needs to measure only the attenuation Aa required to return the detector

output to the reference level. One can then obtain Ir^l from the relationship

Aa = 20 log^Q (5.75)

It is also well to recall that one adjusts the auxiliary tuners so that cyclical

variations in the detector output [b^l (which occur when sliding the loads in the

measurement arm of the reflectometer) are minimized. In adjusting the tuner on the

load arm of the directional coupler, one slides a load having a small VSWR, say 1.05,

and in adjusting the other tuner, one slides a short-circuiting piston. The accuracy

of the measurement to be made depends in part on how well these adjustments are made.

156
c. Experimental Equipment

(1) General

In general, the arrangement of equipment is as shown in figure 5-26. The

standard phase-shifter and the reflectometer may be of the same form, and the tuning

DETECTOR

CALIBRATED PHASE SHIFTER

Tx

SIGNAL SOURCE

CALIBRATED VARIABLE ATTENUATOR

STANDARD

TUNED REFLECTOMETER

.J

Figure 5-26, Simplified arrangement of apparatus.

procedure for each would then be the same. The arrangement shown permits measurements

to be made of phase as well as magnitude of the voltage reflection coefficient. However, in order to illustrate the hybrid system, the phase -shifter was omitted and only

VSWR data were obtained.

A hybrid arrangement was used in which all the equipment consisted of rectangular

waveguide components with the exception of the measurement arm of the ref lectometer

and the loads, which were in small-diameter coaxial line. This was done in order to

take advantage of the greater availability of excellent components in rectangular

waveguide. It was also found that the mechanical stability and freedom from leakage

were greater than with coaxial systems. This arrangement permits changes in the

diameter of the coaxial line, as desired, merely by changing the waveguide/coaxial

adaptor

.

157
The discontinuity in going from rectangular waveguide to coaxial line caused no

difficulty since its reflection is tuned out in the normal adjustment of the tuners.

Commercially available WR187 (3.95-5.85 GHz) rectangular -waveguide components

were used except for the tuners, which were specially designed and constructed. In

addition to the usual stubs located in the center of the broad top wall, the tuners

had stubs for fine adjustment located in the center of the narrow side wall.

The coaxial line was made from special phosphor-bronze tubing with tolerances

of ± 0.0005 inch on the diameters.

Rods, clamps, and an H-beam were used to increase mechanical rigidity, and ferrite

isolators were used to minimize undesirable effects of impedance changes.

[2) Terminating Arrangements

Errors caused by connector discontinuities and losses were minimized by modifying

the terminating arrangements as shown in figure 5-27. A joint in the center conductor

at the plane of connection was eliminated by special design of the output terminal,

the adaptors for connecting the unknown load, and the working standard of reflection,

shown in figures 5-27(a], (b) and (c) respectively. Another adaptor, not shown,

permitted connection of loads having male type-N connectors.

158
COAXIAL LINE OUTPUT OF REFLECTOMETER

TERMINAL SURFACE FOR ARM 2

(a) OUTER END OF OUTPUT ARM OF REFLECTOMETER

(c) STANDARD TERMINATION CONNECTED TO OUTPUT ARM

Figure 5-27, Terminating ar rangennents for coaxial-line output of

reflectometer.

The working standard of reflection consisted of a A/4 short-circuit having

fingers only on the inner conductor. The current flow at its contacts with the line

terminal surfaces was thereby minimized. This working standard was calibrated against

a similar but more refined standard described later.

159
(3) Standard Reflections for (Air- Dielectric) Coaxial Line

Although it was not convenient to use as a reference in the measurement of unknown

reflection coefficients, the one-piece X/4 short-circuit as shown in figure 5-28 was used to calibrate the working standard.

ONE-PIECE

X/4 SHORT-CIRCUIT

REFERENCE STANDARD

(a)

X/4 SHORT-CIRCUIT

WORKING STANDARD

3

(b)

Figure 5-28. Arrangement for comparison of X/4 short-circuited sections

of coaxial line.

(a) One-piece \f 4 short-circuit reference standard.

(b) x/4 short-circuit working standard.

The reflection coefficient |rg| of such a short-circuit can be determined to

high accuracy from

2R.EP

'0

60X

1 + 8bZ

1 + bl

0

(5.5)

or with moderate accuracy from the graph of figure 5-15. For greatest accuracy the

conductivity is obtained from an attenuation measurement on a coaxial-line section

made of the same material used in the one-piece X/4 short-circuit.

160
(4) Sliding Loads

Unless suitable sliding loads are employed, one cannot proceed far enough in the

adjustment of the auxiliary tuners, so that eq. (5.65) applies. Hence the accuracy of

measurement depends largely upon the suitability of these components. Their reflection

coefficients should not vary appreciably as they are slid or rotated in the coaxial

line. Some variation is tolerable provided that one can still recognize when the

tuners have received adequate adjustments.

Designs for suitable s liding -short- circuits are shown in figure 5-29 and for

sliding absorbing terminations in figure 5-30.

DESIGNS FOR SLIDING

SHORT-CIRCUITING TERMINATIONS

PLASTIC

PUSH-TUBE

METAL PLUNGER

(a) STAGGERED QUARTER WAVELENGTH CONTACTING TYPE

PLASTIC

PUSH-TUBE

METAL PLUNGER

JX/4k-

(b) QUARTER WAVELENGTH CONTACTING TYPE

PLASTIC

PUSH-TUBE METAL

/////////////////////////////,

/////////////////////.'/////.')

DISSIPATIVE MATERIAL

3/4"-

(c) A NON- CONTACTING TYPE

Figure 5-29. Designs for sliding short-circuits.

161
DESIGNS FOR SLI DING, TAPERED, ABSORBING TERMINATIONS

PUSH-TUBE

-^-

DISSIPATIVE MATERIAL

\

3" 3"

(a) TAPERED SLIDING TERMINATION

PUSH-TUBE DISSIPATIVE MATERIAL

BRASS-

(b) SHORTENED ABSORBING TERMINATION

Figure 5-30. Designs for low -reflection terminations.

d . System Performance and Applications

Various tests were made at 4 GHz to evaluate both overall performance and that

of individual components. The results were as follows.

The magnitude |S^-j^| of the reflection coefficient from a joint in the outer

conductor was determined to be approximately 0.0025, using a ref lectometer technique

(Beatty et al . , 1960). If no correction is made for this reflection, it can contribute 1.25% to the error in measuring a reflection coefficient of |r^| = 0.2.

The performance of the sliding loads was obtained as shown in figure 5-31.

The conductivity of the leaded copper used in making the one-piece A/4 shortcircuit was determined to be 4.0 x 10 mho/m, from which the calculated reflection

coefficient was 0.9988. Comparison of the working standard of reflection with this

one gave its value as iTgj = 0.9931.

162
The attenuation of the uniform section of coaxial line was determined as 0.17 dB/ft.

The reflection from a joint in the centre conductor alone (its effect was avoided in

the method used) corresponded to |S,-, | = 0.01.

(b)

Figure 5-31. Performance of sliding loads,

(a) short-circuit,

(b) absorbing plunger.

163
e. Results and Discussion of Errors

The VSWR's of a number of commercially available coaxial terminations were measured

in order to demonstrate the overall performance. The results are shown in figure 5-32

and are generally within expected limits.

TERMINAL

END OF

REFLECTOMETER

TERMINATION

MANUFACTURED

BY

RETURN LOSS

IN DECIBELS

AVERAGE OF 10

MEASUREMENTS

MAXIMUM

DEVIATION

FROM

AVERAGE

MAGNITUDE

OF

REFLECTION

COEFFICIENT

VSWR

MANUFACTURERS

SPECIFICATION

VSWR AT

4 GHz

(a) COMPANY 'I^" 28.2cib O.OOdb 0.0389 1.078 < 1.05

(b) COMPANY "a" 42.1 0.35 0.0079 1.016 < 1.05

(b) COMPANY "B" 9.22 0.02 0346 2.058 NOMINAL 2.0

(a) COMPANY "C" 20.1 0.15 0.0989 1.220 <l.20 at 3 GHz

(a) COMPANY "D" 172 0.05 0.1380 1.320 <I20 from

7-10 GHz

TEFLON TEFLON

0.272

Fig. (0)

0.215

0.360

0.272

Fig. (b)

Figure 5-32. Results of measurements.

164
13 O

l-H o

o

0 Z

01 O

OS l-H

O UJ

>-i Q£ I—O Li- ce O

a:

0.0002-

0.0001

0.001 0.02 0.05 0.1 0.2 0.5 1 0.002 0.01

ERROR IN RETURN LOSS-DECIBELS

Figure 5-33. Attenuation error.

The principle sources of error which have an appreciable effect, together with

their maximum contribution to the overall error in measuring Ir^l - 0.2, are as

follows

(a) Uncorrected reflection coefficient of the joint in the outer conductors:

1.25%.

(b) Error in determining | Tg | : 0.1% or less.

(c) Error in measuring Aa: 0.3% or less, provided that Aa is less than 20 dB

(see fig . 5-33)

.

(d) Error in adjusting the auxiliary tuners: 0.51 or less, provided that |r^|

is less than 0.2.

The overall r.m.s. error is calculated as 1.4%. Thus an estimate of 1.5% is

conservative for Ir.,] - 0.2. For |r„| less than 0.2, the error would be higher.

165
£. Conclusions

The results of this investigation of the application of improved reflactometer

techniques to the measurement of VSWR in coaxial systems indicates that an accuracy of

1.51 or better is immediately possible. This accuracy can be obtained over a limited

but useful range of |ry|. It is felt that techniques can readily be devised to extend

the range without serious loss of accuracy. For example, one needs to reduce the

reflection from the joint in the outer conductor, and to make a correction for its

effect. Increased accuracy can be expected with the refinement of components such

as the sliding loads. Also one would expect less difficulty in working with coaxial

line of larger diameters.

The hybrid principle works well in this example, and would be expected to work

well with other combinations of different waveguide components.

5.7. Connector Reflections and Losses

a. Introduction

A knowledge of the reflections and losses of waveguide joints or connectors is

important in evaluating certain errors occurring in nearly all types of microwave

measurements. In some cases, these errors are the limiting ones, and further

improvement in the state of the art depends upon improvements in joints or connectors.

Sensitive and accurate techniques to measure the characteristics of joints and

connectors are vital to such an improvement.

The improvement and refinement of microwave reflectometer techniques has led to

sensitive means for determining the small losses and reflections normally associated

with good waveguide joints and connectors.

In the following, based upon Beatty, et al . (1960), we describe and discuss

tuned ref lectometer techniques which provide a powerful tool for the investigation

of the properties of waveguide joints or connectors.

b. Preliminary Considerations

As shown in figure 5-34, a waveguide joint or connector may be represented by a

2-arm waveguide junction characterized by four scattering coefficients, S^^, S^2>

S2-|^, and S22' One seldom needs to know all four in order to predict the behavior of

the junction. For example, if nonreciprocal behavior is excluded and Yt ~ >

166
I

01

c c

1

1

S2I

L

02

T

r. = S..+

^12 ^21

r,

^22

r. = -2 72*2

Figure 5-34. 2-arm waveguide junction representation of

waveguide joint.

^01 ~ ^02' ^^^"^ ^12 ~ ^21' ^"^^ only three are needed. If, in addition, the junction

is symmetrical, S-^.^ = ^22' ^^'^ only two coefficients are required. If the junction

is lossless or nearly so, then for practical purposes |S-j^2l " 1^21^ ^^'^ l^lll ~ 1^221*

It is sufficient for many purposes to determine only the VSWR corresponding to

|S^-|^| or IS22I and/or the efficiency n (for energy flowing into arm 1 and out at

arm 2 ,

^21

and for the reverse direction,

^12

of a 2-arm junction terminated in a nonref lecting load. In practice, the direction

of energy flow will make little difference in the efficiency of a low-loss connector

[even if it is not physically symmetrical), for the reasons mentioned above.

Similarly, the VSWR will be essentially independent of the direction of energy flow.

^ 1 2 ^21 I

1 - 1 S I 2

1 - is^^r

(5.76)

(5.77)

167
ISOLATOR

— TO

OUTPUT

INDICATOR

OUTPUT LEVEL vs POSITION

OF

SLIDING TERMINATION

TUNER Y

DIRECTIONAL

COUPLER

TUNER X

UNIFORM

WAVEGUIDE

n r

TO SOURCE

OF

MICROWAVE ENERGY

SLIDING TERMINATION

HAVING SMALL REFLECTION

WAVEGUIDE JOINT OR CONNECTORUNDER INVESTIGATION

Figure 5-35. Reflectometer arrangement for measuring VSWR of

waveguide joint.

168
ISOLATOR

— TO

OUTPUT

INDICATOR

OUTPUT LEVEL vs POSITION

OF

SHORT CIRCUIT

JL

TUNER Y

JL

DIRECTIONAL

COUPLER

TUNER X

UNIFORM

WAVEGUIDE

T r

TO SOURCE

OF

MICROWAVE ENERGY

SHORT-CIRCUITING

PLUNGER

WAVEGUIDE JOINT OR CONNECTOR

UNDER INVESTIGATION

Figure 5-36. Reflectometer arrangement for measuring efficiency of

waveguide joint.

c. Brief Description of Method

The measurement techniques used to obtain the VSWR and efficiency are illustrated

in figures 5-35 and 5-36, respectively. In both cases, one uses a single directional

coupler reflectometer employing two auxiliary tuners, X and Y, which are adjusted in

turn in the following way. Tuner X is first adjusted so that the cyclical variations

in the sidearm (arm 3) output as one slides a low-reflection termination in waveguide

section A (fig. 5-35) are essentially eliminated. Then tuner Y is adjusted to achieve

the same condition, as one slides a highly reflecting termination in waveguide

section A (fig . 5-36) .

The joint or connector under investigation is between the identical waveguide

sections A and B. It is intuitively evident that if the joint or connector were

perfect (S 11 22

= 0, and |S 12

= S

21 1) , there would be no cyclical variations

169
in the sidearm output as one slid either termination from waveguide section A into

section B. However, i£ the joint or connector were not perfect, it seems reasonable

to expect that the adjustments made with termination sliding in section A would not

hold when these terminations were transferred to section B.

This is actually the case, and the VSWR and efficiency of the joint can be

obtained from the observed data in such an experiment. The presence of very small

reflections is sensitively determined by the arrangement of figure 5-35 and the

presence of very small losses by the arrangement of figure 5-36.

d. Review of Reflectometer Techniques

In order to interpret the above experiments and obtain quantitative results,

we briefly review some reflectometer theory.

The amplitude b^ of the wave emerging from the sidearm (arm 3 in figures 5-35

and 5-36) is related to the reflection coefficient by eq. (3.162), or

i + r K A b = kb„ (5.78)

^ ^2i^A

where b^ is the component of the incident wave amplitude furnished by the generator,

K is a function of the scattering coefficients of the reflectometer, V^^ and k are

also functions of these scattering coefficients and the reflection coefficients of

the generator and detector, and finally is the reflection coefficient terminating

arm 2 of the reflectometer. The terminal plane of arm 2 is located in waveguide

section A and may have any arbitrary position sufficiently removed from the joints

so as to avoid higher modes. The term V^^ is the reflection coefficient of the

equivalent generator at this reference plane, while K is practically equal to the

directivity ratio of the reflectometer.

It is possible to adjust tuners X and Y to make both 1/K and T^^^ vanish. Under

these conditions eq. (5.78) becomes

= cr^, (5.79)

where c = kb^. This factor will remain constant if the generator is stable and well

isolated, the detector impedance terminating arm 3 is constant, and the tuner adjustments are stable. The detector output power P^^ is proportional to or

= p|r^r. (5.80)

170
The constant of proportionality can be determined if a reflection standard, such as

a quarter wavelength short circuit, is connected to a waveguide section A. However,

it can be eliminated from consideration by measuring only power ratios in which it

cancels out.

e. VSWR Determination

The cyclical variations obtained as one slides the low-reflection termination in

waveguide section B (fig. 5- 35) occur as the reflection from the termination goes in

and out of phase with the reflection from the joint. Assuming that the reflectometer

has been adjusted so that eq. (5.79) applies, the behavior of is of interest. As

indicated in figure 5-34, it may be written in the form of eq. (3.10),

^A = ^11 - / ' ' ' (5.81)

where the ^are the scattering coefficients of the 2 -arm junction representing

the waveguide joint and short sections of waveguide on either side of the joint.

Equation (5.81) may in practice be simplified to

^A " ^11 r^e'^Y^^z, (;5.82)

since one uses a termination with small |r^|(|r^| < 0.005) and the loss and reflection

of the joint are small. Excluding nonreciprocal behavior of the joint, these considerations lead to the conditions |S22-^lI ' ' ^^'^ I ^12^21 ^ ~ ' ^^ich are necessary

for the above simplification of eq . (5.81).

An illustration of eq. (5.82), neglecting attenuation of the waveguide, is given

by the diagram of figure 5-37. The upper diagrams show the circular loci of as

varies in phase, while the lower graphs show the corresponding variations in the

output level of the sidearm. In each case, it is seen that

Ib3lmax = I^KISiJ - |r^|), (5.83)

where c is the same as in eq. (5.79).

The determination of |S^^| proceeds as follows. Let Ib^l^^ - |c| represent the

output level of arm 3, when a low- loss circuit terminates waveguide section A, and

let [bjlrj, =^ |cr^| represent the corresponding output level, when the short-circuit

is removed and the sliding termination inserted in waveguide section A, First |r^|

may be determined by measuring the ratio of l^^lj to Ib^lg^, with a calibrated variable

attenuator, maintaining the detector output as a fixed reference level so as to

make unnecessary any knowledge of the detector law. Second, the quantity

171
is similarly determined from the ratio o£ lb, I to lb,! . Finally. |S I is ' 3 ' max ' 3 ' sc ' 11

determined by subtraction. There are many refinements and variations of this basic

technique, some of which are discussed later.

FOR EACH CASE •• |b3L,x= c(|S„| + iFrl)

Figure 5-37. Reflection coefficient diagrams and detector response

curves arising from eq (5. 82).

172
f. Efficiency Determination

In the 2-arm junction representation of a waveguide joint or connector, it is

necessary to select terminal surfaces on either side of the Joint to avoid appreciable

higher modes at these surfaces. The length ij of waveguide between the two terminal

surfaces will introduce some wall loss or attenuation in addition to the loss in the

joint. The measured efficiency may be written

n = n^Hj, (5.84)

- ZaUr where = e , the efficiency of a section of waveguide of length 1^ having an

attenuation constant ct, and rij is the efficiency of the joint alone. (This separation

of the effect of losses is not rigorous, but should hold closely in practice.)

The determination of ri follows from previous work^ ° in which one measures the

radius of the circular locus of the input reflection coefficient as one slides a

short circuit in the output waveguide. The situation may be represented by figure

5.34, in which now denotes the reflection coefficient of the sliding short circuit.

In a uniform waveguide section having finite attenuation, the locus of is a

logarithmic spiral converging toward the origin. However, this is transformed by

the 2-arm junction, and the corresponding locus of V-^ is, in general, a distorted

logarithmic spiral converging toward S^^. It is distorted because the spiral l/T^

is displaced or translated an amount before being inverted. The output level

Ibjl of the sidearm of the reflectometer in this general case exhibits variations

as shown in figure 5.38. Analysis of the data in this case is not covered here.

However, recent papers by Almassy (1971) and Engen (1972) are pertinent.

Fortunately, in many cases the reflection coefficient is small and has

negligible effect in distorting the spiral locus of r^. In this event, one obtains

data similar to that shown in figure 5-39. On each side of the transition length

, it is appropriate to consider the waveguide lossless and apply the theory developed

under this assumption. This theory is briefly as follows.

^°See section 6.7, Cullen (1949) and Beatty (1950).

173
ro

POSITION OF SHORT-CIRCUIT

Figure 5-38. Detector output vs position of sliding short when reflection from

joint is moderately large.

It was shown in section 3.13b, eq. (3.107), that the radius o£ the F^-circle

as the phase of T,^ varies is

21

^12^21^tI

I^22^tI^

(5 .85)

The efficiency of a 2-arm junction with energy flowing into arm 2 and with arm 1

terminated in a nonreflecting load is

Is,'121

'12

^221^

(5.77)

It is evident that to the extent that 1 - |S22l^ equals 1 - \S22'^j\'^ > and

1^12^21^ = IS^^I^j then R^^ - n^^lVrj.]. This approximation will be quite good if a lowloss short circuit is used. If the connector has low loss, it will very nearly be

true that |S-|^2l ~ 1^21^' nonre ciprocal behavior were permitted.

174
POSITION OF SHORT-CIRCUIT

Figure 5-39, Detector output vs position of sliding short when reflection

from joint is small.

175
SHORT-CIRCUIT SLIDING

ro

WG

SECTION A

WG

SECTION B

Na lb.

31 MAX

31MIN

POSITION OF SHORT-CIRCUIT

lb.3'A Ic Tscl

Figure 5-40. Reflection coefficient diagrams and detector response curves for a

sliding short in lossless waveguide.

Referring to figure 5-40, it is apparent that

1 lb,! +|b,| . nlr

1 ' 3 ' max ' 3 ' mm ' sc

'^3lA I'scI

since nlr | is the radius of the r. -circle and the constant |c| cancels. It is

' sc ' A

= n, (5.86)

noted that |

| has been replaced by |Tg^|.

It is convenient to use instrumentation developed to measure power differences

directly, when observing the changes in the sidearm output as one slides the shortcircuit. In order to obtain data as shown in figure 5-39, it would be necessary

to plot the square root of the observed relative power. However, it is somewhat

easier to analyze the data as obtained and take square roots of only the three

points needed in the calculation of efficiency.

176
The attenuation o£ the waveguide sections is obtained from the slope of the

square of the curve to the left of £^ in figure 5-39. If and the side arm

powers corresponding to two positions of the short circuit spaced a distance i, the

attenuation constant a may be calculated from the expression

-AaSi ^2

e = —, (5.87)

Pi

where ?2 smaller than P^.

g. Supplemental Techniques in VSWR Determination

In the technique described for measuring the VSWR corresponding to |S-^^-|^| of the

2-arm junction representing the waveguide joint or connector, it was assumed that

|1/K| was much smaller than I S^-j^ ] . When dealing with very small reflections, the

reduction of |l/K| to even smaller values may become difficult, and a number of

supplemental techniques have been developed for the solution to this problem.

The behavior of the detector output as one slides a termination having a small

reflection ( |

| << 1) in waveguide section B may be described by

C5.88) bj ^ kh^

In general', the previous technique for obtaining |S-|^^| will yield | (1/K) +

instead. If |l/K| << | S^^ | , there is no difficulty, but if this is not true, we

cannot determine |S-|^^| even if we knew |l/K|, because the phase difference between

the two terms is unknown. It then becomes important either to reduce |l/K| until '.t

is much less than |S-j^-|^| and to know when this is the case, or to employ a technique

in which the relative phase of 1/K and S^^ can be varied so as to separate their

magnitudes

.

In adjusting tuner X to reduce ll/K| to a small value, it is necessary to avoid

the possible false adjustment which could result in approximately constant output if

|l/K| >> \Tj\ . One way to do this is to begin the adjustment of tuner X with all

stubs out of the waveguide, and then adjust for a detector null with the sliding

termination in an arbitrary fixed position in waveguide section A. Under this

condition, 1/K = -T^, and if the termination is then displaced for maximum detector

output, the level will be Ib^l = 2|cr^|. The adjustment of tuner X is then continued

until an essentially constant output level is obtained as one slides the termination,

for which the detector level should be approximately half the above level, or 6 dB

down.

177
One should know when to stop trying to improve the adjustment of tuner X so as

to avoid needless tedium. This may be done with the aid of figure 5-41, as illustrated

in the following example. Suppose that |S^^| approximately equals 0.00025 and that

it is considered sufficient to reduce |l/K| to one tenth of this value or 0.000025.

(The corresponding error in determining |Sj^^| by the previously described technique

would then be less than 10 percent.) If the sliding termination has a VSWR of 1.003,

the graph shows that the adjustment of tuner X can cease when the total variation of

the detector output is within 0.3 dB . This example is representative of what can be

done with commercially available components.

The supplemental techniques to be described are included as alternate ways to

reduce |l/K| or to prevent error in measuring |S-|^-|^| because of finite |l/K|.

1) The method which is potentially the most powerful in reducing |l/K| is similar

to that described above, except that in place of a sliding termination of constant

|r^|, an adjustable sliding termination [Beatty, 1957) is used. One alternately

adjusts both tuner X and the termination, so that the sidearm output is reduced to a

lower and lower constant value as the termination is slid. By doing this, both |l/K|

and Ir^pl are reduced together. It is often found that the adjustment is limited, not

by the sensitivity of the detector to respond to the small reflected signal, but by

the ultimate failure of |r^| to remain constant as the termination is slid. However,

a return loss of 100 dB has been obtained with suitable adjustable sliding terminations.

This corresponds to |l/K| certainly less than 0.00001. It could be considerably less,

depending upon the observed variations in detector output.

Apart from the reduction of |l/K|, an adjustable sliding termination may be used

in the measurement of |S.j^-|^| as follows. Referring to figure 5-37, it is apparent that

the detector output will vanish if = "S-j^^. This condition can be easily achieved

and recognized if an adjustable sliding termination is used. Subsequently, without

changing the adjustment, the termination is slid until the detector output is a maximum and proportional to the sum of |S^-|^| and |r^| or 2|S^-|^|. Comparing this output

level to that obtained when the termination is removed and replaced by a short circuit

will eliminate the constant of proportionality and permit determination of IS^j^l^

2) An auxiliary waveguide channel^ ^ arranged as in figure 5-42 permits introduction of a signal to the detector of such a phase and magnitude so as to cancel the

^^The circuit is quite similar to one used in the determination of barretter mount

efficiencies by an impedance technique.

178
RETURN LOSS OF TERMINATION - DECIBELS

Figure 5-41. Graph for determining |1/k| from variation in detector

output as a low-reflection termination is slid in the

waveguide section A.

signal component due to finite 1/K. As is indicated by the vector diagrams, the procedure begins with the adjustment of tuner X for a detector null, whereupon 1/K = -T^.

The sliding termination is then moved to a position where the detector output is

maximum, changing the phase of by 180°, so that the signal components from and

1/K add. The switch in the auxiliary arm is now opened, introducing a signal component

"A" to the detector. The amplitude and phase of this component are adjusted using the

179
AUXILIARY

CHANNEL A

3 \

TO

OUTPUT

INDICATOR

J L ^ISOLATOR

TUNER Y

'/k

0

0

0

TUNER X

T

A

A.

n r n r ir *g^l [)

r-T

Figure 5-42. Auxiliary channel technique.

phase shifter and variable attenuator to null the detector output. It is apparent

from the diagram that "A" is in phase opposition to the 1/K signal component and will

cancel it if reduced in amplitude by one half without changing its phase. This is

done by adding 6.02-dB attenuation in the auxiliary arm. The measurement of |S-|^j^|

can now proceed in the manner discussed previously.

Additional tuning (not shown) could be employed to prevent possible interaction

between the two channels; however, the isolator shown in the figure should prove

adequate for this purpose. The resulting adjustment can be checked by means of the

procedure associated with figure 5-41, and it is possible that |l/K| will not be

small enough. This could easily be the case if the attenuator did not accurately

produce the 6.02-dB change required, or produced some phase shift. Further adjustment of either tuner X or the phase shifter and attenuator would then be necessary

180
to reduce |l/K| to the desired value. In case of difficulties in obtaining fine

adjustments of the tuner, the latter procedure is quite convenient and can provide

good resolution.

3) An extension of the previous technique in which a second auxiliary channel

is employed permits cancellation of the 1/K signal component without the use of a

calibrated attenuator. The arrangement is similar to that of figure 5-42, except for

the addition of another similar channel. Vectors representing detector signal components from 1/K, and channels A and B are shown in figure 5-43, corresponding to

ADJUST TUNER FOR NULL

SLIDE TERMINATION TOWARD LOAD

SWITCH IN CHANNEL A, ADJUST IT FOR NULL

SWITCH OUT A, SLIDE TERMINATION TOWARD GENERATOR >

SWITCH IN CHANNEL B, ADJUST IT FOR NULL

SWITCH IN A AND B TO CANCEL '/k

K

I/,K

K

Figure 5-43. Steps in a technique employing two auxiliary channels.

the steps in the procedure mentioned in the figure. Although a calibrated attenuator

is not required, it is necessary to vary the phase of Tr^ by prescribed amounts, so that

a micrometer drive for the sliding termination is convenient.

4) The relative phase of 1/K and S^-j^ may be varied by means of the line

stretcher arrangement shown in figure 5-44. Alternate manipulation of the line

181
stretcher and the sliding termination to obtain maximum detector output results in

alignment of the vectors representing 1/K, S^^ , and r.^ , as shown in figure 5-45.

The remaining steps in -the procedure and the corresponding vector diagrams and

pertinent equations are shown in the figure. It is apparent that the ratio of jS^^

to I I equals the ratio of Ibjl^^^^^ to Ibjl^^-j^.- If | | is known or determined

independently, then |S-|^^| may be calculated. In order to deal with conveniently

measured ratios, one should use a termination having a |

| not greatly different

from I S-j^-j^

I .

5) Another technique employing the same arrangement as in figure 5-44 is as

follows. Tuner X is adjusted for minimum variations in detector output as a lowreflection termination slides in the waveguide section in which terminal surface 2

is located. This reduces |l/K| to a small value; then, the three vectors 1/K, S^^^^,

and are lined up as described in the previous technique. One adjusts tuner Y

DETECTOR

TP

TO OUTPUT INDICATOR

MOVABLE WAVEGUIDE SECTION

DIRECTIONAL

COUPLER

TUNER X

SLIDING TERMINATION

HAVING SMALL REFLECTIONnr ir

FIXED WAVEGUIDE SECTION

WAVEGUIDE JOINT

OR CONNECTOR UNDER INVESTIGATION

Figure 5-44. Arrangement for varying relative phase of 1/K, Sn, and F,-

.

182
LINE THEM UP.

ADJUST '/k

FOR NULL.

SLIDE

TERMINATION

l/iK

FOR MAXIMUM. Yj I/k

LINE THEM UP

AGAIN.

SLIDE TERMINATION

FOR MINIMUM.

'/k S| T

r-T

lb

w = |s,MrTl

IKI 2|crT

NMAx=l^l(H-ls.^lr,|)

3'MAX

'/k S,

lb

3'MIN |cl(ii||+IS„l - irrl)

Figure 5-45. Steps in a technique for determining | Sn | in terms of | [

.

until the detector output variations are minimized as a short circuit is slid in the

waveguide section. The detector output level Ib^l will now be proportional to the

sum of |l/K|, |S.j^-|^|, and | | . The constant of proportionality may be eliminated,

in the usual way, by taking the ratio of this output to that obtained when the

waveguide section is terminated in a high-quality short circuit. One then determines

I r,p

I and |l/K| independently by methods previously described and finally calculates

183
h. Results

The techniques described above are applicable in principle to waveguide systems

employing waveguide of rectangular, coaxial, or any kind of cross section. However,

experimental results have been obtained in a WR90 (X-band) rectangular waveguide system

operating at a frequency of approximately 9.39 GHz.

The effects of lateral displacement upon the reflection and efficiency of a plane

butt joint in rectangular waveguide were investigated; a few measurements were

performed on other types of joints, and the attenuation constants of some short

sections of waveguide were determined.

Figure 5-46 is a photograph of the arrangement used to obtain prescribed repeatable

lateral displacements of the waveguide at a simple butt joint. Clamps were used for

alignment and to insure repeatability, and strips of shim stock of various thicknesses

were used to obtain the prescribed displacements. The heavy brass flanges shown were

originally one piece. It was soldered in the center of a section of uniform waveguide

and the edges were machined flat and square; then it was cut so as to separate into

two sections of waveguide, each with its own flange. The mating surfaces were then

ground so as to be flat and square.

The initial alignment was checked by visual inspection. A flashlight was used

to illuminate the interior of the waveguide, and reflections of light from any

visible edges at the joint proved to be a sensitive indication of misalignment or

mechanical imperfections, such as burred edges. In spite of care taken to obtain

good alignment, the reflection coefficient of the joint was never below 0.00015, and

in the results shown, was approximately 0.00071.

184
rSHIM STOCK

Figure 5-46. Photograph of alignment and clamping apparatus for

waveguide joint.

185
A comparison of experimental and calculated results is shown in figure 5-47.

The calculated curve is based upon the equation shown, which differs from that given

by Kienlin and Kiirzl (1958) by a factor of two, but agrees (when corrected for the

different ratio of f to f^) with the appropriate curve given in their figure 5. The

agreement is quite good over a limited range, but it is apparent that the residual

reflection that one obtains at zero displacement prevents agreement at the low end.

It is probable that the approximations made in deriving the equation contribute to

the disagreement at the other end.

DISPLACEMENT "d", INCHES

Figure 5-47. Measured and calculated reflections from junction of

displaced waveguide sections.

186
0 10 15 20 25 30 35

DISPLACEMENT IN INCHES/ 1000

Figure 5-48. Measured efficiency of junction of displaced waveguide sections.

A plot o£ observed data obtained in an efficiency measurement is shown in figure

5-39. The attenuation constant of the brass waveguide as determined from the slope

of the squared curve is 0.056 dB per foot. Measurements were made of the efficiency

of the same butt joint described above for the same lateral displacements and the

results are shown in figure 5-48. It is not known whether or not the results are

representative of this type of joint, since the loss would be expected to depend on

the surface finish and cleanliness of the metal at the contacting surfaces, as well as

other factors .

187
The results of additional measurements on commonly used types of joints are given

in table 5-3. Again the results may or may not be representative, but were obtained

with careful alignment of the waveguides and cleaning of the joint surfaces. It should

be noted that the waveguide sections united by a joint were originally a single section

of waveguide which was sawed in half. Thus, there is very little, if any, change in

the waveguide cross section at any of the joints.

Table 5-3. Reflections and losses of some joints, in WR-90 (X-band)

rectangular waveguide measured at 9. 39 GHz.

TYPE OF FLANGES USED

WITH BUTT JOINT

REFLECTION COEFFICIENT EFFICIENCY

CHOKE-COVER

CHOKE-COVER

0.00064

0.001

5

0 .9996

0.9993

188
6 . Attenuation

6.1. Introduction

In attempting to set up National standards of attenuation in the U.S.A. and to

compare these with the standards of other countries, the first concern is for a

precise definition of what is to be measured and for tight specifications of the

measurement conditions . A quantity cannot be accurately measured unless it is sharply

defined.

Unfortunately, the term "attenuation" has been defined by many people in many

ways so that there is not perfect agreem.ent on what the term really means. Also,

the conditions under which it is to be measured have not been carefully specified.

In 6.2, the general meaning of the term is examined, as well as various specialized

meanings that have been used. Although the term "attenuation" has been used in various

fields and scientific disciplines, its use in electrical and electronics engineering

is of main concern in this monograph. Various definitions, such as those published

by the American Institute of Electrical Engineers (AIEE) , the Institute of Radio

Engineers (IRE) , and other are discussed and compared.

Using a simple 2 -port network representation of a two-port device, equations are

given based on the various definitions. By means of the equations, quantitative

differences in various attenuation concepts are clearly shown. Examples are given

for typical cases .

It is concluded that existing definitions are in some cases contradictory, and

in all cases imprecise. Therefore they are inadequate to form a basis for accurate

attenuation measurements. A set of definitions is proposed which is based upon

specific measurement procedures and precisely specifies the conditions under which

the measurements are to be made. Errors in making attenuation measurements due to

not perfectly satisfying the specified conditions are discussed. The use of models

representing the attenuator or 2 -port device and the measurement system is discussed

and more complicated models are proposed. Finally, the concept of representing the

behaviour of 2 -port devices with terminal-invariant parameters is discussed.

Following the discussion of attenuation definitions, specific research is

described in developing methods of attenuation measurement and in analyzing errors.

For example, in 6.3, the representation of a 2 -port device by an idealized linear

189
2 -port network having uniform lossless waveguide leads is used to analyze the

cascading of attenuators, and equations are derived for the error one can make in

adding individual attenuations to obtain the total attenuation of a cascaded pair

or group

.

Similarly, in 6.4, the mismatch errors one can make in the calibration and use

of fixed and variable attenuators are analyzed. The limits of uncertainty due to

mismatch are usually determined by assuming that the phases of the interacting

reflections can have any value. However, it is known that the realizability conditions on linear, passive 2-ports restrict the range over which the phases can vary

in particular cases. In 6.5, the effect of applying the realizability conditions

to the estimation of mismatch error limits is analyzed and discussed.

In most cases where particularly high accuracy in attenuation measurement is

not required, the representation of an actual 2 -port device by an idealized linear

2-port network having uniform lossless waveguide leads is satisfactory. However,

if the accuracy is to be improved in the future, more complicated models, such as

that in 6.6, will be needed.

The development of a method for measuring attenuation by measuring the radius

of a reflection coefficient circle is described in 6.7. The achievement in 1960 of

unprecedented accuracies in attenuation measurement by the use of a very stable power

measurement system is described in 6.8.

A method for measuring very small attenuations of low-loss components, such as

short sections of waveguide and waveguide joints, was developed using a 2-channel

RF nulling technique. The method is described in 6.9 and it is shown how to avoid

errors due to losses in waveguide joints and due to changes in losses in variable

phase shifters

.

Finally, in 6.10, the development of an attenuation divider circuit is

described. This circuit makes possible the production of very small accurately

known attenuations, such as 0.0001 decibel.

6.2. Definition of Attenuation

a. Introduction

The accuracy with which attenuation can be measured depends upon the limitations

of the measuring equipment and methods, the stability and other characteristics of the

devices to be measured, and ultimately upon how sharply the quantity to be measured

and the conditions of measurement have been defined and specified.

190
The latter point is discussed in some detail in the following sections. A number

of existing definitions, particularly those published by the two institutes which

merged to form the IEEE (Institute of Electrical and Electronics Engineers) are

examined and compared. A quantitative comparison is made possible by deriving equations

based upon a simple 2 -port waveguide junction model.

The terminology used in writing and talking about the attenuations of attenuators

is discussed, and some concurrence is noted to exist in a confused situation. However, it is recognized that the actual definitions are more important to the achievement of higher accuracy than the terminology. Rather than suggest changes in

terminology, improved definitions are proposed, which sharply define the quantity

to be measured and the measurement conditions.

Finally, possible future trends, such as the use of more complicated models

for analysis, or the use of terminal -invariant parameter definitions are discussed.

b. Broad General Meaning

The word "attenuation" comes from the Latin "attenuatio" which is built from

the simpler words "ad" meaning "to," and "tennuis," which means "thin." It

generally refers to a decrease of something, especially a gradual weakening or

drawing out. It had been used in this sense for a long time before man's use of

electricity for signaling and his invention of radio. In Webster (1806), attenuation

was defined as the act of making thin or slender. It is still used in the same

sense today, to denote a decrease, weakening, emaciation, or rarification (Crowell,

1962) .

For example, the following definitions illustrate the wide use of the word.

Biological Sciences: Used specifically of the gradual reduction

in virulence of a microorganism (Gray, 1967).

Textiles: The process of making a roving or sliver progressively smaller

by doubling and drafting (Fairchild, 1967).

Medicine: The process of preparing homeopathic medicines by repeated dilutions

(Funk and Wagnall, 1965).

Distilling and Brewing: The clarification and thinning of saccharine worts

incident to the conversion of sugar into alcohol and carbon dioxide by fermentation

(Funk and Wagnall, 1965).

191
It is seen that in each field, a restricted meaning exists which is in agreement

with the broad meaning. In the field of electrical and electronic engineering, there

are many definitions involving the words "attenuation," and "loss" which are in

agreement with the broad general meaning, but are very specialized in application.

The bases for these restricted meanings are discussed next.

c. Restricted Meanings

As used in electrical and electronic engineering, the term "attenuation" is used

in the most general sense to denote a decrease of the amplitude or magnitude of

coherent or incoherent electromagnetic waves or electrical impulses without specifying

(1) what quantity shall be used to measure the decrease, (2) whether the decrease is

space-dependent, time dependent,^ or both, (3) the cause of the decrease, or (4) the

conditions under which the decrease is to be measured.

However, in various restricted meanings of the term, each of the above has

been specified, sometimes in contradictory ways. For example, in the case of an

electromagnetic wave radiated by an antenna and traveling over the earth (ground

wave) , the wave amplitude decreases as one goes farther from the antenna due to

(1) spreading out of the radiation -- the inverse distance effect, (2) dissipation

or conversion of some energy to heat in the earth, and (3) scattering by irregularities

or by objects in the transmission path. In general, "attenuation" denotes the

decrease of wave amplitude due to any and all of the above causes, but some definitions exclude the decrease due to spreading.

For example in Michels (1961) on p. 113, one finds the following: "In the most

common usage, attenuation does not include the inverse -square decrease of intensity

of radiation with distance from the source." However, in the IRE Dictionary of

Electronics terms and symbols" (1961), one finds on pp. 8-9 the following: "Note:

In a diverging wave, attenuation includes the effect of divergence,"

According to the cause of the attenuation or reduction, one finds different

terms. In Tweney and Hughes (1961), pp. 3, 373, there is the following: Absorption

(Radio) Reduction in the intensity of an electromagnetic wave, due to eddy currents

and dielectric losses in the earth. Also called attenuation. Geometrical Attenuation

(Radio) The reduction in field strength of an electromagnetic wave as it progresses

from the source on account of spreading out.

'A time -dependent decrease might be caused by a variable attenuator and observed

by a detector at a fixed point in space. A space -dependent decrease might be

caused by a lossy transmission path and observed by two detectors located at

different points along the path.

192
The American Institute of Electrical Engineers (AIBE, 1942) published the

following definition: "Attenuation of a Wave" -- The attenuation of a periodic

wave is the decrease in amplitude with distance in the direction of wave propagation

when the amplitude at any given place is constant in time, or the decrease in

amplitude with time at a given place."

This definition is somewhat restrictive because: 1) It mentions only periodic

waves. However, it is possible to decrease the energy of a transient or aperiodic

wave, or even electrical noise, by placing some dissipative, reflective, or scattering

objects in its path. 2) It ignores the possibility that a wave might simultaneously

be decreasing in amplitude with time as it decreases with distance in the direction

of propagation.

Consider the following three cases: 1] A detector is placed at a certain

position in a transmission path or circuit and observes a decrease with time of

the wave amplitude, 2) two detectors are placed at different points along a transmission path or in a circuit and the wave amplitudes are observed simultaneously,

giving information about the decrease of wave amplitude with distance along the

propagation path, 3) a detector is moved continuously along a wave transmission path,

and detects a decrease in wave amplitude in going away from the source. Such a

decrease could be both time and space dependent, although quite often, the source

is amplitude -stabilized to remove any time-dependence, or the amplitude is monitored

at some fixed point and corrections are made to eliminate any time-dependent

component .

)

It should be clear from the above examples that it is important when using

restricted definitions of attenuation that the basis for the restrictions is clearly

understood.

d. IRE Definitions

The definitions of attenuation published by the Institute of Radio Engineers

(IRE) in 1961, also appear in the IEEE Standard Dictionary of Electrical and Electronics

Terms (1972). For this reason, they deserve special study. It appears in IRE (1961)

on p. 8 and states: "Attenuation. General transmission term used to denote a decrease

of Signal magnitude."

193
There are two apparent restrictions. First, the phrase "Signal magnitude" would

seem to exclude waves^ which were not "signals." And second, the phrase "transmission

term" implies that the decrease takes place as a result of transmission, and thus is

space -dependent but perhaps not time -dependent

.

In practice, the use of the term "signal" has been expanded to include any

electromagnetic wave or electrical impulse whether or not it contains any message

to communicate information. Also the phrase "transmission term" does not really rule

out time -dependent attenuation. Thus the apparent restrictions have largely been

ignored or circumvented.

One notes that the IRE definitions of terms which include the word "attenuation"

refer to space -dependent attenuations with one exception. Thus, "Current Attenuation,"

"Voltage Attenuation," "Power Attenuation," "Attenuation (of Radio Waves)," and

"Attenuation (in a Waveguide)" are restricted to space dependent attenuation. However, "Direct -Coupled Attenuation" (TR, Pre-TR, and Attenuator Tubes). The Insertion

Loss measured with the resonant Gaps, or their functional equivalents, shortcircuited" is a time -dependent attenuation.

On the other hand, the IRE definitions which include the word "loss" consistently

employ power as the observed quantity although they are divided among space -dependent

and time -dependent attenuations. For example, time -dependent losses include Bridging

Loss, Insertion Loss, Transition Loss, Reflection Loss, Return Loss, and Transformer

Loss. Among the space -dependent losses are Absorption Loss, Divergence Loss, Heat

Loss, Power Loss, Radiation Loss, Refraction Loss, and Transmission Loss.

One can conclude that the IRE recognized both space -dependent and time - dependent

attenuations and considered "Loss" concepts as special kinds of attenuation involving

power as the observed quantity.

It is noted in some of the definitions (but not all) that attenuation or loss can

be expressed either as the amount of the decrease of a quantity (such as power) , or the

ratio of the large to the small value of the quantity which is decreased. In case

the ratio is used, it is often expressed in Decibels or Nepers. On p. 38 of IRE

^On p. 132 of IRE (1961), we find " Signal . 1) the physical embodiment of a Message . 2) a) A visible, audible, or other indicat ion used to convey information. b] The

intelligence, message, or effect to be conveyed over a communication system. c) A

Signal wave." On p . 90, we find " Message . 1) An ordered selection from an agreed

set of symbols, intended to communicate information. 2) The original modulating wave

in a communication system. Note: Definition 1) is the sense in which the term is

used in communication theory; definition 2) is the sense in which the term is often

used in engineering practice."

194
(1961) one is cautioned against confusion which may result i£ ratios of currents or

of voltages are expressed in Decibels. The point may be illustrated by the following

example. Consider two points in a linear circuit excited by steady-state, sinusoidal,

single-frequency electromagnetic waves. If the impedances at the two points are Z-^

and and the corresponding powers, voltages, and currents are respectively P^,

, and I^, and P2 , V2 , and I2, the decibel relationships for the power, voltage,

and current ratios are the following:

p |vr

Rp = 10 log^Q —, where P = |l|^ ReZ = -—— ReZ

,

^2

Ry = 10 log

10

= 10 log

10

Rj = 10 log^o 10 log^Q

2

ReZ.

ReZ.

|Z| 2

ReZ2

ReZ,

, and

(6.1)

(6.2)

(6.3)

Note that if the impedances are equal (Z^ = ^2) , all three ratios expressed in

decibels are equal. In the special case where only the real parts of the impedances

are equal (ReZ-j^ = ReZ2) , then the power and current ratios expressed in decibels are

equal and the voltage ratio gives a different number of decibels (except for ImZ-j^ =

-ImZ2) . In general, if the impedances are not equal, then the three ratios above

expressed in decibels can be different numbers.

The above example applies to the comparison of IRE definitions of Voltage

Attenuation, Current Attenuation, and Power Attenuation of Transducers, where the

input impedance of the transducer may in general be different from the impedance

of the load connected to its output port.

e. Comparisons of Definitions

In this section, comparisons will be made of various IRE definitions of

attenuation (including "loss") and of a few definitions derived from those given

by the IRE. Practically all of the IRE attenuation definitions pertain to a

"transducer" which can be represented by an ideal linear passive 2-port having

uniform, lossless waveguide leads. Equations may then be derived for the different

attenuations and quantitative comparisons are then possible. In the following,

the equations are given without derivation. The derivation follows from the

principles given in section 3.8 and the equations use the same notation and

convent ions

.

195

I
First, simple equations are given which define the attenuations in terms of

the ratios of the appropriate quantities. Then, the corresponding equations are given

in terms of the scattering coefficients S-j^^, ^-^2' ^21' ^^'^ ^22 ^'^ two-port

model of figure 6-1 assumed to represent the transducer, the characteristic impedances

Zq^ and of the principal modes in the waveguide leads, and the reflection

coefficients and of the equivalent generator and load "seen by" the

transducer. Finally, simplified equations are given for the case of Zq^ =

and = = 0, or simply r^^ 0 .

Consider first space -dependent attenuations which can be defined in terms of the

terminal variables v^ , i^, a^^, b^, v^ , i2> a2 , and b2 which are indicated on figure 6-1,

Note that for generality, the generator system is a Z^^ system and the load system

is a Zq2 system.

1

2-PORT ^02

1

Figure 6-1. Model assumed to represent a two-port device.

Voltage Attenuation = 20 Log

10 (6.4)

Current Attenuation A^-, = 20 log 10 (6.53

Power Attenuation or ^ =10 Log Transmission Loss P ^10 10 Log^Q -, (6.6)

where

Pi = —(|aj^ - |bj^),

^01

'2 - 7-^1^2!

196
and r\ is the efficiency. Note that and ?2 ^^'^ (transmitted or delivered) powers

1 Intrinsic Attenuation^ Aj = 10 Log^^

n (6.7) max

(See section 3.9.)

Voltage Wave Amplitude . = ?n }na

Attenuation"* ^VWA " ^^10 10 Log^Q II '01

'02

(6.8)

where denotes incident power and Pg denotes emergent power.

TP Available Power Attenuation^ A^ = 10 Log-j^^

Al

^^A2

(6.9)

where P^ = Available Power.

The following relationships between the available powers and the terminal

variables shown on the above figure 6-1 can be derived from inspection of section 3.6,

1 - r^r,

I

" b.

Al

(1

^G^il^ , , -1

where T, = —

,

(6.10)

and

P = P ^A2 ^2

2i'Ll

(1 2i ^)(1

' ^h^^^ = ^22 ^ ^12^21^G

1 - ^ll^G

(6.11)

and

^2

The above eqs . (6.4) through (6.9) all define an attenuation associated with the

ratio of a quantity at port 1 to the corresponding quantity at port 2. It would

also be possible to form a ratio of a quantity at port 1 to a different quantity

at port 2, but this would not be in agreement with the general definition of

attenuation

.

'This is an extension of Power Attenuation to the case in which the 2-port is

terminated in a load impedance for which it has maximum efficiency n^^^^-

''This is not given in the IRE Dictionary, but follows from application to waveguide

circuits of the IRE definition: Attenuation (in a Waveguide). Of a quantity

associated with a traveling waveguide wave, the decrease with distance in the

direction of propagation (53 IRE 2. SI).

^This is the inverse of the IRE definition of the "Available Power Gain" (of a linear transducer). (51 IRE 20. S2.)

197
The ratio of a power at port 1 to a different kind of power at port 2 is used in

some attenuation definitions, for example in the definition of "transducer loss."

However, this is not regarded as a space -dependent attenuation unless the two

different kinds of powers can be simultaneously observed. In the case of "transducer

loss," the available power might be measured at port 1 by removing the 2-port and

connecting to the source a detector whose impedance is adjusted to conjugately

match that of the source. Once P^^ has been measured, the 2-port is reconnected to

the source and (the power delivered to a specified load) is measured.

For the purpose of making comparisons, equations for the above space -dependent

attenuations are given as follows (the equation for Aj is not given because it is

rather large, and does not reduce to a simple form when Z„, = and r„ = 0).

A.Y

= 20 Log 10

(i.s^^jd-s^^rL) ^12^2l'^L

S2i(i-rL)

20 Log^Q

1+S

11

21

(6.12)

for = 0;

'02

'01

(i-s^^)(i-S22rL) - s,2S2irL

s^id-r^)

20 Log^Q

1-S

11

21

(6.13)

for = 0 and Z^-^ = Z 02

= 10 Log^Q-

'02

'01

1-S 22' Ll •^^12^21 "11^22 s..s..)rL - 10 Log^Q

isiir

^21 I

(6.14)

for r. 0 and Z 01 '02 '

VWA 20 Log^Q

21

20 Log

10

'21

(6. IS)

when = 0 ; and

\ = 10 Log 10

'02

'01

1-S 11' Gi

for = 0 and Z^^ = Z^^'

'^^12^2l'^11^22-'' G 22

21

(1-

10 Log^Q

^22 I ^

^2ll^

(6.16)

It is significant that the above equations do not become equal when the conditions

0 and Z, Zq2 are imposed. However, if, in addition, the transducer is ' G L " "01

non-reflecting (S^^ = S22 = 0), then they are all equal.

198
The differences are due to reflection interactions, and if the reflections are

small, the differences will be small. However, in the case of highly accurate measurements, even small differences may be important.

For example, if the input and output VSWR's of the 2-port^ are 1.05, corresponding

to 22 0.0244 , and we assume that = Tj^ = 0 and Z 01 > then the

11 1 attenuations Ap and will be about 0.026 decibel lower than A^^, and A^ and A^

can be about 0.4 decibel higher or lower than A^^^, depending upon the phases of

These differences are significant and illustrate the importance of clearly

defining the quantity to be measured.^

Consider next the time dependent attenuations "insertion loss" (Lj) , "transducer

loss" (L^) , and "substitution loss" (Lg) . The first two can be defined in terms of

the third, as will be demonstrated. In figure 6—2, the substitution of a final 2-port

for an initial 2-port is shown.

GENERATOR

^G' ^G

INITIAL

2-PORT

LOAD

1 ^L—*^ ^02

11 1U'p^ 1 GENERATOR

'^G' ^G

FINAL

2-PORT ^L—^ ^02 LOAD

1

Figure 6-2. Model representing substitution of final 2-port for

initial 2-port in a simple system.

The defining equation for all of the three losses is:

L = 10 Lo£ 10 (6.17)

where Pj^ is the power absorbed by the load, and the front superscripts i and f

refer to initial and final conditions, respectively.

^VSWR's higher than 1.05 are commonly encountered, even in high-quality components.

Thus, the example given is conservative.

^These differences also verify the advisability of avoiding the use of current and

voltage attenuation concepts in situations where the impedance varies along the

transmission path.

199
In the case of "substitution loss," there are no restrictions on the properties

of the initial 2-port. If the initial 2-port is a perfect connector pair having no

dissipative loss or leakage, no reflection, and no phase shift, then the substitution

loss equals L-j. , the insertion loss. (The case where the initial connector pair is

not perfect is discussed later in section 6.6.) The insertion loss under non-reflect

conditions [r^ = Tj^ = 0) is characteristic of the 2-port and is called the insertion

loss of the 2-port, the word "characteristic" being understood.

If the initial 2-port is a perfect transducer having no dissipative loss or

leakage and provides a conjugate match to the generator, then """P^ is the available

power, and the substitution loss equals Lnp , the transducer loss.

Note that the definitions of "insertion loss" and "transducer loss" are

equivalent to those given in the IRE Dictionary on pages 75 and 152, respectively.

In the equations to follow, it is assumed that the initial and final characteristics of the generator and load are unchanged by substitution of the 2-ports.

However, in making an analysis of the errors due to these assumptions failing to

hold, the front superscripts i and f would be retained on and r^^.

From eqs . (3.51), (3.53), and (3.52), respectively, we have the following:

Lg = 20 Log^Q

^S2,[(i - ^s^^rg)(i - ^s^^Fl) ^^12^^21^G^l]

--s^^Ui - ^s^^r^)(i - 's^^T^) ^^12^^21^G^L^

20 Log^Q

'21

21

(6.18)

Lj = 10 Log^Q

= 20 Log-^Q

= 10 Log^Q

'02

>-^01

21

'02

^^01

(1 - s^^r,)(i - s^^r^) - s,2S2ir,rJ

if = = 0. and = Z^^.

1(1 - s^ir^)(i - s^^Fl) - s,2S2irGrLl

(6.19)

= 20 Log 10

21

, if r. Tl = °' ^01 = ^02' (6.20)

It is seen that when the system is non-reflecting (T^ = = 0) and Z^^ = '^q2>

the transducer loss, insertion loss, and the voltage wave amplitude attenuation (a

space -dependent attenuation) are identical. It is also noteworthy that if the

200
condition r, = 0 is removed, the insertion loss and according to eq. (6.15)

continue to be identical for any value of V^.

Consider now another set of concepts which are defined by the IRE in terms of

..i.i>_n can De i.ioa<,„x <=.a simultaneously. These are tlie transition loss L^j^^

the reflection loss Lj^^ » and the return loss Lj^. These are defined with reference

to figure 6-3. It will be shown that these losses are equivalent to time-dependent

attenuat ions

,

GENERATOR LOAD

Figure 6-3. Model representing generator connected to load

by a uniform, lossless waveguide lead.

4n = ^° ^°ho (6.21)

where is the available power, and Pj^ is the net power absorbed by the load,

10 Log^Q —, where Pj

Lj^ = 10 Log^Q —, where Pj^

''O

bl 2

-, the incident power.

, the reflected power,

(6.22)

(6.23)

R o

Additional insight is obtained about the meaning of these losses when they are

expressed as time -dependent attenuations as follows. Consider the situation in

figure 6-4 where two loads are alternately connected to the same generator.

201
GENERATOR INITIAL

L n A U

1

GENERATOR FINAL

LOAD

Figure 6-4. Model representing two loads alternately

connected to the same generator.

The attenuation of the reflected wave is:

= 20 Log 10

10 Log

10 (6.24)

This can be shown to be equal to the return loss Lj^ when two conditions are imposed

1) non-reflecting generator, and 2) perfectly reflecting initial load. Considering

section 3.6, the above equation can be written as follows:

i,

when Tj, = 0 and

Ar = 20 Log^Q

TJ = 1,

T,

1 - r r ^ L

r.

20 Log^Q [6. 25)

Initially, the incident wave is totally reflected and returned to the generator,

where it is absorbed. When a different load is finally connected, the reflected wave

is reduced or attenuated. The return loss L„ is thus equal to the attenuation of

K

the wave returned to a non-reflecting generator when a perfectly reflecting load is

replaced by a different load.

From the defining equation for return loss given above.

10 Log

10 20 Log

10 (6.26)

Thus the return loss is equal to A„ for the two assumed conditions

202
Next, the transition loss can be seen by inspection of its definition to be

equal to the attenuation of the net power delivered by a generator when a conjugately

matched load is replaced by a different load. This follows because a conjugately

matched load absorbs maximum power and a different load then must absorb less power.

Finally, the rerioction loss can De sKo^^n to be ec^udj. ^^<3 the attcu^aLiun or

the net power delivered by a non- reflecting generator when a non-reflecting load

is replaced by a different load. This follows because a non-reflecting load absorbs

maximum power from a non- re fleeting generator and a different load must then absorb

less power.

In addition to the IRE definitions of attenuation, many others have been proposed

and used. However, it is not considered necessary or useful to discuss them in this

monograph.

f. Terminology

There has been a long-standing lack of agreement about which names are most

suitable for which attenuation concepts as well as which concepts are most useful to

characterize a device for measurement purposes. Although a complete discussion of

this situation would be interesting, it is o^ly briefly summarized in order to relate

the terminology used in this monograph to what is in common use today. Particular

attention will be given to the terminology of attenuation concepts used to characterize attenuators.

Attenuating devices may be classified as iither 2 -ports or multiports. In the

case of multiports, two of the ports may be sejected as available for connecting

to a transmission system and the remaining are Vot connected to the system, but to

loads or independent circuits. The system may generally be a complicated one

involving many paths between the source and thefletector . However, for the purpose

^of defining attenuations, it is usually tacit ly Issumed that there is only one path

ifrom the source to the detector. One deals onlyWith the attenuation in a single

path. Thus, attenuation measurements are basicaly concerned with 2 -ports, which

nay be either fixed or variable. The attenuationwhich characterizes a variable

dWice is different than that which characterizesW fixed device.

I In the case of non-variable devices such as itenuator pads, space -dependent

al^tenuations such as Ap and A^^^j as well as time -Upendent attenuations such as

Lg' L-^ , and which involve power ratios might be^sed to characterize the device.

I' T

203
It has been noted that in the case of a non-reflecting system, A^^^, L^, and

give the same simple equation in terms of the scattering coefficient 82;^, assuming

the 2 -port model has ideal waveguide leads. Thus, differences between these three

would be very small and would depend upon how much the actual situation differed from

tKc ideaX jnoUt-1. - ' -

It has been common practice to measure the insertion loss in a non- reflecting

system and to call this either 1) the insertion loss of the device, or 2) the attenuation of the device. It is understood that the word "characteristic" should

precede the word "of," but for brevity, it is usually omitted. The use of either

1) or 2) is optional and has been a matter of individual preference.

In discussing mismatch errors in attenuation measurements (see 6.4b) one defines

the error as the difference between what is actually measured (insertion loss or substi

tution loss) and the desired quantity (attenuation or characteristic insertion loss).

In this usage, the word "attenuation" is understood to imply a non- reflecting system

at the insertion point.

When the device is an attenuator, it is natural to speak of "its attenuation."

However, since Lj is usually measured, there is some preference for the term "its

insertion loss." It would be equally valid to measure

^ywA' ^^^^^ insertion

loss. The term "attenuation of a 2-port device" is a more general term than "insertion loss," and includes both and L^.

There is a class of two-port devices which cannot be simply inserted into a

system, but must be substituted for someihing already in the system. For example,

an attenuator pad having female connectos on both ends can be substituted for a

female-to- female adapter which is originilly in the system. In this case, the

substitution loss in a non- reflecting system may be used to characterize the attenuator

providing that the characteristics of te adapter initially in the system are

completely specified. Other items in tiis category are devices such as waveguideto-coaxial adapters and connector pair themselves.

Although the term "insertion lossof a connector pair" has been used, what is

usually measured is a substitution los . For example, a section of cable having the

connector pair under test in its centr is substituted for an identical section of

cable which has no connector pair ints center. Since a connector pair cannot be

simply inserted into a system and reoved again, the term "insertion loss" is not so

applicable as "substitution loss of connector pair" or the "attenuation of a con- nector pair." Other alternatives a3 available such as the relative efficiency of

a connector pair. In addition to te efficiency, one would also require the VSWR or

204
return loss. The phase shift might also be important. This topic is considered

further in section 6.6.

In the case of a variable 2 -port device such as a variable attenuator, two terms

are commonly used to denote its attenuation. They are 1) "its insertion loss" and 2)

"its attenuation." The first term refers to the initial attenuation when the

variable attenuator is set to "zero" or is switched to the state of minimum attenuation.

It could be either space - dependent or time-dependent, according to how it would be

defined and measured. Usually, the time - dependent attenuation is measured and called

the insertion loss." The second term is used to denote the time -dependent attenuation

due to its adjustment or its switching from one position to another. Usually the

reference point is the "zero" setting or the minimum attenuation switching state,

but any reference may be used. For example, in a step attenuator, one may allude

to the attenuation between the 25 dB step and the 30 dB step. In all cases, it is

assumed that the system has been adjusted for the non-reflecting condition prior

to the measurement.

The variable attenuator may then be characterized by its insertion loss and

its attenuation versus dial setting or step position data. If the variable attenuator

has only two states, then its insertion loss and its attenuation are all that is

required to characterize it.

If one considers a fixed pad as a special case of a variable attenuator having

only one state, then by extension, it would be characterized by its insertion loss

and its attenuation would be zero. This is one argument for the use of the term

"insertion loss" rather than "attenuation" to characterize a pad. As previously

mentioned, there are valid arguments for the opposite preference. It really makes

no difference which is used, as long as the concept is clearly defined.

In other words, for precision measurement applications, the terminology is not

as important as the tightness of the definition of the concept. In the next section,

an attempt is made to tighten up the definitions of insertion loss and attenuation.

g. Precise Definitions.

In the following section, precise definitions of attenuation will be presented.

All of the things which were mentioned in the general definition of attenuation in

section 6.2c as not being specified will be specified here. The definition will be

based upon the basic measurement method and will tightly specify the conditions of

measurement. It will be independent, as far as possible, of any idealized model,

although strangely enough, it has been found difficult to completely avoid any

idealized conditions in definitions.

205
In all o£ the definitions, it will be assumed that the source of electrical

energy supplies a .sinusoidal wave of single frequency and that we are considering

only a single mode propagating in each waveguide of the system. It is assumed that

the net power to the detector excludes any leakage (energy coming by a path other than

directly through the device in question) . If the detector responds to leakage power

and it cannot be separated from the direct path energy, this represents a source

of error.

The scope of the definitions will be limited to those which involve net power

or delivered power. Only time -dependent attenuations will be defined, and only attenuations due to dissipation or to reflection of energy will be considered.

The general definition of attenuat ion given in the first paragraph of section

6.2c is considered to be an improvement on the AIEE and IRE definitions in that it

is less restrictive. It is used as a reference in formulating the specific,

restrictive definitions to follow.

The " insertion loss (characteristic) of an insertable , fixed 2 -port device " is

defined to be in decibels, 10 times the logarithm to the base 10 of the ratio of

the initial to final net powers delivered to a non-reflecting detector by a nonreflecting source initially connected directly to the detector and finally connected

so as to feed energy only through the two-port device to the detector. (In order

for a device to be insertable, the connectors at its two ports must be of the same

type as the system connectors which mate at the insertion point, and be either

sexless, or of the opposite sex.)

The conditions under which the detector powers are measured are as follows:

1) The source is initially connected to the detector by means of a connector

pair constructed so as to tightly adhere to standard specifications for such connector types as are used. Each connector of the pair is attached to a uniform section

of waveguide constructed so as to adhere as closely as possible to standard dimensions

and of sufficient length to effectively eliminate any higher modes which may have been

excited by the connector pair.

2) The system is adjusted so that ideally there are no reflections looking

towards the source or towards the detector in the waveguide sections belonging to

the system. (Actually the adjustment is usually made in practice so that no reflections are observed in an auxiliary waveguide which is part of the reflection measuring instrument. Thus the system will then have reflections equal to those caused

by a connector pair plus any residual reflection of the measurement instrument.

This will represent a source of error.)

206
An improved technique to adjust for no system reflections is illustated in

figure 6-5. Only the case of adjustment for no reflections looking towards the

detector is shown. One can easily deduce how to use this technique to obtain no

reflections on the generator side of the system.

TUNER

TO GENERATOR

STANDARD

CONNECTOR

PAIR

STANDARD

-WAVEGUIDESECTION

TUNER

J,

TO DETECTOR

a) INSERTION POINT

RECEIVER

TUNER

STANDARD

CONNECTOR

PAIR ADJUSTABLESLIDING

V t TERMINATION—

7

SIGNAL

SOURCE

T

' / STANDARD

^ WAVEGUIDE

SECTION

b} ARRANGEMENT TO ADJUST REFLECTOMETER TO INDICATE NON-REFLECTING

CONDITION ON SYSTEM DETECTOR SIDE OF CONNECTOR PAIR. TUNER AND ADJUSTABLE SLIDING TERMINATION ARE ALTERNATELY ADJUSTED TO REDUCE

AND MAINTAIN OUTPUT TO RECEIVER BELOW NOISE LEVEL AS TERMINATION

SLIDES AT LEAST HALF WAVELENGTH

.

SIGNAL

SOURCE

RECEIVER

TUNER

STANDARD

CONNECTOR

PAIR TUNER

T T TO DETECTOR

/ STANDARD

^ WAVEGUIDE

SECTION

o) ARRANGEMENT TO ADJUST SYSTEM DETECTOR SIDE OF CONNECTOR PAIR FOR

NON-REFLECTING CONDITION. AFTER ADJUSTING TUNER ON SIGNAL SOURCE

SIDE OF DETECTOR AS ABOVE, ADJUST TUNER NEAR DETECTOR FOR

RECEIVER NULL.

Figure 6-5. Improved technique to adjust for no system reflections.

3) The generator wave amplitude and internal reflection coefficient r^,

and the detector sensitivity or gain and reflection coefficient r^, are initially

the same as when the final relative net power is observed.

4) It is assumed that the power level is either low enough to avoid appreciable

non-linear effects, or is set at a specified level before insertion of the device.

207
The " attenuation (characteristic) of an insertable , fixed 2 -port device " is

synonymous with the above definition of the insertion loss. However, one could also

define a "voltage wave amplitude attenuation (characteristic) of a fixed 2-port

device," an "intrinsic attenuation of a fixed 2-port device," or any attenuation

regarded to be characteristic of the device. Such definitions are not presently

required for precise measurement purposes, and will be left for future research.

In the case of a fixed 2-port device which is not insertable, the following

definition may be useful. The " substitution loss of a fixed 2 -port device " is in

decibels, 10 times the logarithm to the base 10 of the ratio of the initial to

final net powers delivered to a non-reflecting detector by a non-reflecting source,

connected initially to feed energy only through a removable fixed value 2-port

device (having specified properties) and finally the fixed 2-port device under consideration. The conditions of measurement are the same as those listed for "the

insertion loss of an insertable fixed 2-port device," with the exceptions indicated

above

.

The " attenuation (corresponding to a particular adjustment or change of state)

of a variable 2 -port device " is in decibels , 10 times the logarithm to the base 10

of the ratio of the higher to the lower net powers delivered to a non-reflecting

detector by a non-reflecting source connected so as to feed energy only through the

device, which is adjusted or switched from an initial setting or state to a final

setting or state. The conditions of measurement are the same as those specified

above with exceptions which are evident upon inspection.

The foregoing definitions are understood to apply also to any pair of ports of

a multiport device, whose other ports which are not involved in the definition, being

connected in some specified way, such as, but not necessarily, all connected to nonreflecting loads.

h. Future Trends

One can predict future trends on the basis of certain assumptions. The

probability that they will actually occur depends in part upon the validity of the

assumptions. It seems reasonable to assume that higher accuracy will be required

in the future in attenuation measurements. This may lead to improvements in waveguides, connectors, and in attenuators themselves. If these improvements are substantial enough, then errors will be reduced. The more that errors are reduced in

this way, the less need there will be for improved analytical techniques to evaluate

208
errors, as discussed in section 6.6. Also there will be less need to consider

replacing conventional insertion loss and attenuation concepts by a "terminal

invariant parameter" (Engen, 1969) such as intrinsic attenuation.

If, however, improvements in waveguides, connectors, and attenuators do not

keep pace with accuracy requirements, these analytical techniques and special

concepts should prove more useful.

It is also conceivable that new methods will be developed to make more accurate

attenuation measurements, and that they might make popular the measurement of some

attenuation other than insertion loss. For example, the voltage wave amplitude

attenuation appears promising as an attenuation to characterize a 2 -port device.

The increased use of computer-controlled automatic measurement systems will

make feasible new measurement techniques that would have formerly been impractical

because of tedious adjustments or calculations that are required. These techniques

are likely to place a higher demand on connector repeatability and make greater use

of substitution, as opposed to insertion techniques.

6.3. Cascade -Connected Attenuators

a. Introduction

It is well known that if one connects two fixed attenuators, say exactly

5 dB each, together and measures without error the attenuation of the combination,

the result may not exactly equal the sum of the attenuations (say 10 dB) . A

difference can occur due to reflection interactions in the waveguides or transmission

lines both where the attenuators are connected together and where they are connected

to the measurement system. Depending upon the relative phases of the reflections,

the difference in attenuation may be positive, negative, or (in relatively rare

cases) zero.

This effect becomes important when one is testing the accuracy and repeatability

of a measurement system by measuring attenuators individually, then in combination;

and comparing the results (Beatty, 19 71). Such a test will give useful information

about the accuracy of the measuring system only if the uncertainty due to attenuator

reflections is considerably less than the uncertainty or inaccuracy of the measuring

system under investigation. The above effect is also important when attenuators

209
are used as "gage blocks" to extend ranges of power meters, or when fixed and

variable attenuators are cascade -connected in order to extend the range. In turrettype attenuators, this effect may limit the useful frequency range because reflections

usually increase with frequency.

An early analysis of this effect has been published (Beatty, 1950). The

analysis was performed before the use of scattering coefficients had been widely

accepted and wave matrices were employed. The results are still valid and the

nomogram giving limits of error in terms of VSWR's is still useful.

An extension of this analysis was made by Hashimoto (1968), in which the

connector pairs joining the pads were represented by cascaded 2-ports in a manner

similar to Beatty (1964b), and the 2-port representing the contacting portion of

the joint was represented by a single series impedance in a manner similar to

Harris (1965) .

The analysis was extended to the case of cascade -connected variable attenuators

by Schafer and Rumfelt (1959).

In the present discussion, the results of the original analysis are presented

in a more convenient and slightly more general form. Scattering coefficients are

employed and non-reciprocal elements are not excluded.

b. Analysis

In the following analysis, UHF or microwave attenuators are considered. It

is assumed that the individual attenuators have been calibrated in a transmissionline system having the same characteristic impedance (Zq) and critical dimensions

as the system in which the attenuators are to be used. A further requirement is

that the attenuators are passive linear 2-port devices having connectors or waveguide

flanges that permit connection to the waveguide or transmission-line system without

discontinuity. Although attenuators can usually be considered as reciprocal devices,

there is little need to assume reciprocity in the analysis, and it is not assumed.

First the case of two attenuators is considered. Then the cases of 3 or more

cascaded 2-ports are analyzed.

210
(1) Two Attenuators

The analysis of cascade two-ports have already been described in section 3.11

and 3.12. The results of those sections are directly applicable. First, the

attenuation of a single 2-port for energy entering port 1, is

1 A = 10 Log^Q (3.55)

For the case of two attenuators designated as M and N connected in cascade as in

figure 3-15, the scattering coefficient S^.^ of the composite 2-port is written in

terms of the scattering coefficients ^2-1^, ^22' ^21' ^11 '^^ individual

attenuators as follows.

^21

The attenuation of the combination is

1

"^21^21

'^ll'"22

(3. 98)

+ 20 Lot m

10

+ 20 Log^Jl "ll'^22 (6.27)

21 ^21

°^ \ = " ^2-

The last term represents the error that must be added to the sum of the

attenuations of the individual attenuators to obtain the correct attenuation of the

combination.

If we have (as is often the case) knowledge of the VSWR's ^P-^-^ and ^P22

responding to n^^ and ''^22' knowledge of the phases, then we can calculate

the following limits between which £2 ""ust lie

20 Log^Q 1 + '22 ^11 m

P22 " 1 Pll ' 1

>^ £2 > 20 Log^Q

P22 - ^

'P22 ^ 1

Pll - 1

Pll ' 1 (6.28)

These limits are shown in the nomogram of figure 6-6

211
/ o^z-l o:''-l \ \ V (,

°2~' \

OB

10 —

1.5-

CM

<». I.I —

u.o

llJ 3

5

1.05-

i.oi1.005—

1.002—

(-) (+)

10 —

— I

0.1-^—0

u>O

13 si

ui £

> o

!3 °

.01-

.0001-

.00001-

u.o

.01 -.

UJ o

> 111

o

a. .001 —— .001

.0001

.00001

-10

-1.5

—1.05

<

>

I.OI

-1.005

1.002

Figure 6-6. Limits of error according to eq (6. 28).

(2) Three Attenuators

The analysis of 3 cascaded attenuators can be based upon that for 2 attenuators

as follows. Suppose we designate the 3 attenuators as L, M, and N connected as shown

in figure 6-7, and their scattering coefficients by the lower case letters, t, m, and

n with the appropriate subscripts. We can consider that attenuator L is cascaded

with one other attenuator (which is formed by combining M and N) . Thus the 3

attenuator case is a simple extension of that for 2 attenuators.

212
It can be shown (Kerns and Beatty, 1967) that for the cascade connection of

3 attenuators.

^2l'"2l"21

'21 (6.29)

[(1 - ^22"^lini - m22nii) - £22'"l2'^21^11^

The attenuation of the combination is

\ = \ * ^ \ * 20 Log^J (1-^22'^11^^^-'^22^11^ " ^22'^12'^2l''ll I ' ^^'^^^

where the last term is e^, the error that one would make in assuming that the attenuation of the combination equals the sum of the attenuations of the individual

attenuators

.

Note that the error term may be written

= 20 Log^pl [1 J!,22 (mn) -^-^] [1 - '^22'^11-' ' ' (6.31)

where (mn)^^ is the scattering coefficient of the combination of M and N and is

given by eq. (3.98) if S-^^ there is replaced by (mn)j^-|^.

1 2

L M N

^11 '^12

•^22

""ll "^12 "ll "12

"^21 "^2 2 "^21 ^^22

(mn)

1

1

(mn)

(mn)

12

22 (mn)2i

Figure 6-7. Three cascaded 2-ports, L, M, and N.

Assuming that we can obtain the VSWR's corresponding to ^22' '^"'^^n' "^22' ^'^'^

'"ll

' ^® then obtain the limits of by referring twice to the nomogram of figure

6-6. In the special case where the term I ^-2 2'^12'"21''^11 I ^R- (6.30) is small,

we can obtain the limits of ^ly referring to the nomogram twice if we can obtain

the VSWR's corresponding to ^-22' '"ll ' "^22' ^^'^ ^11' ^'^^ limits of are obtained

by adding the limits obtained for each term, 20 log^^ 1 1 - ^22"'ll ^ '

20 Log^gll - in22n-L]_| .

213
(3) Any Number n of Attenuators

The attenuation of any number n of cascaded attenuators is

= ^ * ^n' (6.32)

where K denotes the position of the attenuator, numbering from the input side.

In evaluating e^, one can proceed as in the cases of 2 or 3 attenuators,

deriving expressions which are extensions of eq. (6.27) and eq. (6.29). These

involve transmission coefficients of the interior located attenuators both in the

Ai, and the e terms

.

K n One can avoid deriving complicated expressions by the procedure used in the case

of 3 attenuators. Consider first two attenuators at one end of the chain, using

eq. (6.27) to find the attenuation of the combination. Then one considers these

two attenuators as a unit, and determines S-j^^ and for the combination. These

are combined with the next attenuator in the chain and again an equation of the form

of eq. (6.27) will give the attenuation of the new combination. One repeats this

process until one reaches the end of the chain. It can be deduced that the term

€^ in eq. (6.32) is

^n

= J, 20 Log^gll - '^^322 , (6.33)

K -

1

where S22 denotes the scattering coefficient of the combined attenuators to the

th K left (direction of decreasing numbers) of the K attenuator, and S^-^ denotes the

scattering coefficient of the K^^ attenuator.

6.4. Mismatch Errors in Attenuation Measurements

a. Introduction

Presently, the most significant source of error in microwave attenuation

measurements is the mismatch error. It is not only significant in calibrating

attenuators, but largely determines how much one must degrade the accuracy of measurements subsequently made with a calibrated attenuator in a different system than

that in which it was calibrated.

In the following, a simple analysis and evaluation of mismatch errors dating

back to 1947 will be reviewed, and subsequent work mainly by the author will then be

described. The cases of measurement errors for fixed pads and for variable attenuators

will be discussed.

214
b. Analysis o£ Case of Fixed Pad

Ernst Weber (1947) analyzed the mismatch error in the measurement of the

attenuation of a fixed pad. He assumed that a pad could be represented by a 2-port

network which could be inserted into a system in the manner shown in figure 6-8.

He expressed the mismatch error e as

e = - A, (6.34)

where Lj is the insertion loss one actually measures, and A (the attenuation) is the

insertion loss that one would measure if the system were matched, or non-reflecting

(^G

=

We will derive an expression for z using modern notation that is equivalent to

Weber's expression. The insertion loss Lj is

= 0).

4 = 10 Log^p D,

D

where the front superscripts i and f refer to initial and final conditions,

respect ively

.

(6. 35)

a)

r *T* r G T D

Signal

Source

Detector

1 1

1

1

b) Signal

Source

2-Port

Network

^2i T

Detector

Figure 6-8. Model illustrating a) a measurement system, and

(b) insertion of a 2-port network into the measurement

system.

Before insertion of the 2-port, the power "'"Pj^ delivered to the detector is

(Kerns and Beatty, 1967)

1 - r r P

(6.36)

where P. is the available power from the generator,

215
After insertion of the 2-port network, the power Pj^ delivered to the detector

IS

-p = p

1 - I r P 1 - r r I ^ (6.37)

where r\ is the efficiency of the 2-port network as terminated in figure 6-8Cb).

It follows that

Lj = 10 Log^Q

1 - I r I 2 1 - r r

1 - r r ^ ^ G^D

(6.38)

However

,

21

1 - hzh

' 1 - i^pr

1 - ir^r

(6.39)

Hence ,

Lj = 20 Log

10

(1 - r^r^)(i - s^^Tl)

^21^1 - ^gV

(6.40)

If the system is non-reflecting (T ^ = = 0) , the insertion loss is the

attenuat ion

A = 20 Lo£ 10 (3.55)

'21

The mismatch error e is

e = Lj - A = 20 Log^Q

(1 - r^r^)(i - S^^T^)

1 - r r (6.41)

This is equivalent to Weber's eq. (24), p. 826 where = , = S22» ~

'^D'

R' = r-^, and R^ = r^.

c. Evaluation of Error

Weber noted that his. eq. (24) could be decomposed into three terms, each of the

same form. In a similar way, eq. (6.41) equals

e = 20 Log^gll - r^rj + 20 Log^pll - S^^T^\ - 20 Log^Jl - r^^r^l. (6.42)

If the complex coefficients belonging to these terms were determined, one could then

calculate each term and combine them to obtain e.

216
It has been the practice to determine only the magnitudes of these coefficients

(not their phases) and calculate the corresponding limits of e, assuming that the

phases might take on any value. Weber presented a graph (fig. 13*18 on his p. 827)

which enables one to determine the limits of the components when one is given the

VSWR corresponding to the magnitude of the complex coefficient.

Suppose that the VSWR's corresponding to \^q\ , |r-j^| , IS22I > I^qI equal

respectively 2.0, 1.15, 1.1, and 1.4. (These are the numbers in Weber's example.)

The limits of the individual terms are:

- 0.21 dB <_ 20 log^pll - r^r^l < 0.20 dB

- 0.07 < 20 log^pll - S22rj3l 1 0.07

- 0.48 < 20 log^gll - T^T^l <_ 0.51

- 0.76 dB < e < 0.78 dB

The mismatch error e must lie somewhere between the above limits, depending upon

the relative phases of the reflection coefficients.

It would seem desirable to determine both phases and magnitudes of the reflection

coefficients in order to determine e and make a correction so as to get closer to the

correct value

.

It would also seem desirable to reduce r„ and r_ to extremely low values, because

then e would lie between very small limits and might even be negligible.

In practice, one can adjust and to have very small magnitudes indeed, but

there remains an uncertainty about their residual value. For example, if a slotted

line is used to indicate the achievement of r„ = 0, the best one can normally do is to

obtain a flat response of the slotted line. The residual VSWR of the slotted line then

represents the uncertainty in the assumption that Ir^l = 0. This might be 1.02, for

example. If the same slotted line is used to indicate the achievement of |rj^| = 0,

the uncertainty would be the same, or 1.02. One seldom has any corresponding phase

information. Hence the mismatch error limits would be obtained by the above procedure.

Assuming the same VSVifR's corresponding to |r-j^| and to IS22I > the limits of error

would be as follows:

- 0.0064 dB _< 20 log^^ | 1 - r^T-^ | _< 0.0064 dB

- 0.0047 < 20 log^gll - S22rj)| < 0.0047

- 0.0008 <_ 20 logj^pll - T^T^l < 0.0008

- 0.0119 dB £ e <_ 0.0119 dB

The importance of adjusting both generator and detector for low reflection is

clearly shown by the large reduction in the limits of e.

217
d. Effects of Attenuator Characteristics

The 2-port representing the attenuator is characterized by its scattering

They can appear explicitly in the equation

for E by substituting from eq. (3.10)

coefficients S-j^^^-, S^2> ^21' ^^'^ ^22

^^^12^21 ^11^22^^D * ^ 11

into eq. (6.41). One obtains

e = 20 log-i^Q

^22^D

' ^ll^G^*^-*- ^22^D^ ^12^21^G^D

(6.43)

(6.44)

This can be used to evaluate the mismatch error when a given attenuator is inserted

into a known system. One notes in particular that the magnitude of the term

^12^21^G^D smaller as the attenuation becomes larger. In the practical case,

it is negligible for attenuations of 20 dB or larger. If the attenuator is then

reciprocal and nearly symmetrical, the simple graph of figure 6-9 (Beatty, 1967b) can

be used to rapidly e'stimate approximate limits of mismatch error.

3 0.002

0.001

1.001 1.002 I.OI 1.02 I.IO 1.20

VSWR OF SYSTEM

Figure 6-9- Graph for rapid estimates of limits of mismatch error.

218
As the limits of error become smaller, the magnitudes of the positive and

negative limits become nearly equal.

e. Effect of Real izabil ity Conditions

In calculating the limits of mismatch error, it has been assumed that the

phases of the complex coefficients involved can have any value. In theory,

real izab il ity conditions for 2-ports impose limits on the permissible variations

of phase. However, it is shown in section 6.5 that this is not important for most

attenuators and only becomes significant when the attenuation is small (say less

than 3 dB)

.

f. Variable Attenuators

The above analysis is extended to variable attenuators as follows. The

discussion is a modified version of Beatty (1954).

(1) Introduct ion

The error in the measurement of changes in attenuation is important in the

calibration of variable attenuators and in the calibration of large attenuators,

using known pads as gage blocks or fixed attenuation standards.

The calibration of a variable attenuator consists in measuring the change in

the insertion loss as the attenuator dial moves from a zero or reference position

to another position that is marked or can be read on a scale. The change in the

insertion loss equals the change in attenuation if the attenuator is placed in a

ref lectionless , or matched, system. There is always a degree of uncertainty regarding

the match, depending upon the accuracy of the instruments used to indicate or

recognize matched conditions and upon the reflections from connectors. For this

reason the change in the insertion loss cannot be considered to be exactly equal

to the change in attenuation, and the difference is called the mismatch error.

(2) Expression for Mismatch Error

Variable attenuators are of two types; (1) continuously variable, and (2)

variable in steps . The mismatch errors may be analyzed in the same way for both

types, since a change in a continuously variable attenuator from the reference

position to another position is equivalent to removing one attenuator and inserting

219
another attenuator in the circuit. The insertion loss, in decibels, of the

attenuator corresponding to the initial [reference) position of the variable

attenuator is given by eq. (3.53) assuming that Z^^ = Zq2> and is

(1 - ^^T^ni - ^S^^T^) - ^S^/S^^

20 log

10

^21^1

(6.45)

The voltage-reflection coefficients and refer to the generator and load,

respectively, and are measured at the terminals where the attenuator is inserted.

The scattering coefficients S^^, S-|^2' ^21. and S22 refer to the attenuator, corresponding to the reference or zero position of the variable attenuator. The

corresponding insertion loss for a different (final) setting of the variable

attenuator is

20 log

10

(1 - ^s^^v^ni - ^^22\^

£ f

^12 ^21-^G^L

^S2,(i - r^r^)

(6 .46)

The change in the insertion loss is

'21

- 20 log

10

+ 20 lo! 10

20 lOg-j^g

(1 - --^S^^V^)

'21

£ £

^12 ^21^G^L

(1 - ^s^irg)(i - ^8221^) - ^^u^2i^G^^

(6.47)

or

AL A + e. (6.48)

where "^A, ^A, and e in eq . (6.48) correspond to the three terms in eq. (6.47). The

error e must be subtracted from the change in the insertion loss to obtain the change

in attenuation, '^A - ^A.

The error term may also be written

(1 - ^r,rp)(i - ^s„r,)

e = 20 log^„

^

22_L_

^ (6.49)

(1 - ^r,rc)(i - ^522^

where ^r^^ and "^T-j^ are the input -vo Itage reflection coefficients of the attenuator

terminated in a load having a voltage reflection coefficient V-^.

The mismatch error in the measurement of a single attenuator can be obtained as

a special case of eq. (6.49). The reference attenuator vanishes in this case,

220
changing to r^, S^^ unity, and S^^ and S^^ ^° zero. Substituting these values

for T-^ and into eq. (6.49) yields

e = 20 log^^Q

(1 - ^r^r,)(i - ^S^^V^)

1 - 'g'l

(6.50)

which corresponds to our eq . (6.41) and to eq. (24), page 826 of Weber (1947).

(3) Evaluation of the Mismatch Errors

It is possible in principle to evaluate the mismatch error by measuring the voltage

reflection coefficients r^, r^, r^, T-^, ^22' ^^'^ ^22' ^^'^ substituting them into

eq. (6.49).

In many cases, the magnitudes of the reflection coefficients can be determined,

but their phases cannot be conveniently determined because of the limitations of a

particular measuring apparatus. Equation (6.49) can then be used to find the limits

of the mismatch error, permitting the phases of the reflection coefficients to have

all possible values. The limit of error can be expressed in the form

(1 ± l^r r I) (1 ± l^s r |)

e..^.^ = 20 log ^J-^ . ^

, (6.51) ^° (1 + l'r^rj)(i + |'s22rLl)

and the corresponding limit of error for single attenuators is

(1 ± |r Tpl) (1 ± |s r I)

^limit = 20 log^Q '^

^ I C6.52)

For example, in order to reduce mismatch errors in the calibration of attenuators,

the magnitudes of r„ and r, are made as small as possible, and their probable amplitude

is estimated from the accuracy of the apparatus used to recognize matched conditions

and from the known connector characteristics. It is difficult to accurately determine

the phases of these small reflection coefficients, and the mismatch error can generally

be determined not exactly, but within limits.

An example will illustrate the determination of mismatch error. If the voltage

standing-wave ratios p^, p^^, , p^, P22> and P22 [P = (1 + |r|)/(l - |r|)] corresponding to Ir^l, |r^|, |r^|, |r^|, |S22l> and IS22I are 1.1, 1.1, 1.2, 1.5, 1.2,

and 1.5, the limits of error for the initial attenuator calculated from eq. (6.52) are

approximately ± 0.095 decibel. The corresponding limits of error for the final attenuator are approximately ± 0.185 decibel. The limits of error for the change in

221
attenuation calculated from eq. (6.51) are approximately + 0.242 decibel. It is seen

that the mismatch error for the change in attenuation is less than the sum of the

mismatch errors in measuring each attenuator individually.

g. Avoidance of Mismatch Error

It has been mentioned that it is possible in principle to measure both amplitudes

and phases of reflection coefficients and make a correction to an attenuation measurement. (Although not often done in the past, it may be done more often now

when using computer-controlled measurement systems.)

Another method for avoiding mismatch error has been proposed (Rabinovitch , 1962),

but its value has been questioned (Leber, 1964).

Additional analysis of mismatch errors when using rotary-vane type variable

attenuators has been published (Engen and Beatty, 1960), (Holm et al . , 1967). This

type of attenuator causes little change in phase shift and hence the mismatch errors

are small.

The effects of reflections and dissipative losses in connectors and adapters

are not adequately taken into account in the previous analysis because the attenuator

is represented by a single 2-port. Later, in section 6.6, the attenuator and the

connector pairs are represented by three cascaded 2-ports (Beatty, 1964b), so as to

involve the connector parameters explicitly in the analysis of mismatch error.

The purpose here is to discuss the effect of the realizability conditions for

2-ports upon the estimation of limits of mismatch error in high-frequency and microwave attenuation measurements. Only the case of fixed attenuators is considered.

It will be shown that the realizability conditions need be considered only for

certain ranges of attenuation and VSWR, and that the conventional method of estimating

mismatch error limits is satisfactory for most attenuators.

As shown in the previous section 6.4c, the mismatch error is

h. Later Work

6 .5 . Effects of Realizability Conditions

e = 20 log

(1 - S^irg)(l - 522^ - 5^2221^0^0

(6.44) 10

1

-

222
I£ only the magnitudes of the individual factors in this expression are known, and

it is assumed that the phases can have any values, e will have limits determined by

(1 ± |s,,r,|)(i ± |s,,rp|)± |s,,s,^r,rp|

^limit = 20 log^g - .

,

• (6.53)

The limits of mismatch error in measuring the attenuation of a fixed attenuator are

usually determined from eq. (6.53), or its equivalent. This is a valid procedure

if the above assumption about the phases is valid. However, within certain ranges

of attenuation and VSWR, realizability conditions limit the values which may be

reached by both magnitudes and phases of the scattering coefficients. Within these

ranges, the limits of error estimated from the above equations may be too conservative.

Therefore it is of interest to determine these ranges of attenuation and VSWR.

In order to simplify the analysis and presentation of results, only symmetrical

reciprocal attenuators are considered. This is felt to be a useful approach, since most

attenuator pads approximate these assumed conditions.

The realizability conditions (Kerns and Beatty, 1967) for a symmetrical reciprocal

2-port network can be written as follows. (From symmetry, the characteristic

impedances Z^^ = Zq2> and = = |S^^|e-'^^\ and also from reciprocity, ^

^21 l^2ll^ •-'

^2l' ^ 1 ' l^lll ' ^6.54)

and

^lll l^2l'

|cos(ii;^ < ~ — • (6.55)

The above equations define the ranges of attenuation and VSWR (regions) over which a

passive, symmetrical, reciprocal 2-port is 1) not realizable; 2) realizable providing

(.^21 ' ^11^ limited to certain values, when the right side of eq. (6.55) is less

than unity; and 3) realizable with no limits on (.^21 ' when the right side of

eq. (6.55) is greater than or equal to unity. These three cases are shown in figure

6.10. When the attenuator characteristics lie in region 3), the conventional method

of estimating mismatch error limits is satisfactory. In region 2) the conventional

methods would give too -conservative limits, and a different method (Youla and Paterno,

1964) would be required.

223
Attenuation, Decibels

Figure 6-10. Ranges of attenuations and reflection coefficients of

symmetrical reciprocal attenuators for which

realizability conditions restrict the relative phases

of the scattering coefficients.

It can be seen by inspection of figure 6-10 that the conventional method of

estimating limits of mismatch error is satisfactory for attenuators having an

attenuation greater than 2 dB and a VSWR less than 1.50. Since most fixed attenuators

have characteristics which fall within these limits, it is not often necessary to

consider the effect of realizability conditions upon mismatch error limits.

6.6. Effects of Connectors and Adapters

a. Introduction

This section is a modified version of Beatty (1964b). Suppose that we wish to

answer the following question: (If the attenuation of a stable fixed attenuator is

measured at the same operating frequency in two different systems, to what extent is

224
the difference of results attributable to differences in the waveguide^ joints or

connectors^ used at the insertion points?^" (Effects of connector reflections and

dissipative losses are taken into account, but leakage is not considered.)

In seeking an answer to this question, the representation ordinarily used for

the insertion of an attenuator into a waveguide system, and the concept of insertion

loss itself were found to be inadequate, and a new analysis was developed. The

essential feature of the new approach is that the waveguide joints or connectors at

insertion points are not assumed to be perfect,' i.e., having no loss, no reflection,

and effectively no characteristic phase shift, but are represented by two-ports

having appropriate characteristics. In addition, the attenuator when installed in

a circuit is no longer represented by a single two-port, but by three cascaded twoports, the outer ones representing the connector pairs, and the inner one representing

the core or kernel of the attenuator. The quantity of interest is the loss in power

delivered to the load when the above three cascaded two-ports (representing the

attenuator and its associated connector pairs) are substituted for the single twoport representing the connector pair at the insertion point. This loss in power,

expressed in decibels, is called the substitution loss (Beatty, 1964a).

Since the result of an attenuation measurement does depend upon the characteristics of the connectors at the insertion point, one must specify these characteristics if such a measurement is to be precisely defined. This leads to a slightly

modified definition of the quantity of interest in the measurement of a quantity

characteristic of the attenuator, and it is called the standard attenuation. (In

section 6.2g, it is called "characteristic insertion loss" or "characteristic

attenuation . ")

The analytical methods developed here are applicable to other situations in which

a waveguide component is inserted into a waveguide system. They might be applied

^The term "waveguide" is used here in a broad sense to include, for example, both

uniconductor waveguide having a rectangular cross section and two -conductor waveguide

having a concentric -circular or coaxial cross section.,

'The term "connector" is used here in a broad sense to designate the devices designed

to join together two sections of waveguide having the same cross section. A "perfect

connector pair" is one which would have no leakage, no loss, no reflection, and

effectively-zero-characteristic phase shift.

^°The term "insertion point" is used to designate the place where a waveguide component

such as an attenuator is inserted into a waveguide system. It is thus not a geometrical

point, but may be the region where a connector pair belonging to the system is disconnected, or where an adapter belonging to the system is removed, in order to insert a waveguide component.

225
for example to analyze the effects of connectors or adapters on accurate measurements

of the characteristic phase shift of phase shifters, or on the frequency of transmission-type cavity wavemeters. However, at present they will be applied only to microwave attenuation measurements.

Previous analytical methods used to obtain equations for insertion loss,

attenuation, and mismatch error for the case of a single fixed attenuator have been

given in sections 3.8 and 6.4. Corresponding equations will be obtained using different methods and the results will be discussed. The specified question mentioned

earlier will be answered and a calculated limit given for the effect. The techniques

developed will then be applied to the case of variable attenuators, and to fixed

attenuators having nonmating^^ connectors. Some of the cases having immediate

interest will be discussed in some detail and calculated examples given to illustrate

the use of the error equations. Useful formulas and graphs supplemental to the

analysis will be given.

b. Previous Analyses

It has been customary to represent a waveguide component such as an attenuator

by a two-arm waveguide junction (two-port), as shown in figure 6-11 where """P and

I

Generator Lood

Insertion Point-

- Waveguide

Generotor 2 -Port Load —1

1

1

Attenuator

Figure 6-11. Simple representation of an attenuator by

a two-port inserted into a waveguide circuit.

^^The term "adapter" is used to designate a device designed to join together two

sections of waveguide which have already been fitted with connectors. A perfect

adapter -connector combination would have no leakage, no loss, no reflection, and

effectively-zero-characteristic phase shift.

^^The term "nonmating connectors" as applied to a waveguide components such as an attenuator is meant to imply that the connectors on each end of the components are of such a type or sex that they could not be joined together even if it were possible

to move them physically into a favorable position for such joining.

226
P designate the powers dissipated in the load under the initial and final conditions,

respectively. The reflection coefficients of the generator and the load are

designated as and T^, respectively. It is usually assumed that they are the same

at the times that and "^P are observed. (Since the load in attenuation measuring

may also be a detector, one finds r^^ instead of r^^ in many previous equations.

It is also apparent from the diagram that the connector pair or adapter at the

insertion point is assumed to be perfect.

It has been customary to assume that one actually measures the insertion loss

and that it can be written

10 log^Q— = 10 log 10

02

^^01

(1 - s^^r^)(i - S^^T^) - S^^S^^T^T^

'21 (1

(3.53)

It has also been assumed that the desired quantity is the attenuation, which may

be written

fz,'02

7 I <; I 2 \^0l l^2ll

(3.54)

The difference between A and Lj is due mainly to system reflections or mismatch and

has been called the mismatch error. It is written

= Lj - A = 20 log^Q

(1 - s^^r^)(i - 5^2^ - hz^ii^Gh

1 - ^G^L

(6.44)

It is seen that the connector or adapter used at the insertion point is not

shown in the diagram, and its characteristics do not appear in the equations. Hence

there is no way to calculate its effect on the measurement.

c. An Improved Representation

A two-port, or two-arm waveguide junction or transducer has associated with it

two waveguide leads through which energy may enter and leave. The terminal surfaces

of the two-port are cross -sect ional surfaces within the waveguide leads. Usually

only .one propagating mode is associated with each lead and each terminal surface.

A waveguide component such as an attenuator evidently cannot itself be represented

by a two-port unless the connectors are perfect and all connectors mate at coplanar

butt joints. Since this requirement is often not very closely approximated by actual

connectors, the representation of figure 6-12 is more realistic. The attenuator

installed in the circuit is represented by a composite two-port composed of three

cascaded two-ports, A, B, and C.

227
The central core or kernel of the attenuator is represented by B. The connectors

and at each end of the attenuator mate with the connectors and , respectively, of the system to form connector pairs represented by two-ports A and C.

When the attenuator is removed, the system may be closed as shown in figure 6-13 by

joining connectors and Dj^

.

The precise manner in which the connectors join and separate is not specified,

as this would call for further more complicated analysis of specific discontinuities.

Further work has been done in characterizing connectors (Harris, 1965), (Hashimoto, 1968)

The analysis of the situation represented in figure 6-13 proceeds as follows.

Waveguide Sections

Bg

DgIBc

Kernel

or Core

Woveguide Component

(1) (2)

Bl

I

B,iD,

Connector

Poir —1

Kernel

or Core

1 1 Connector

Pair

1 —1

1

1 A B C I

Composite Waveguide Junction

or Composite 2 -Port

Figure 6-12. Representation of a waveguide component.

t 2

Load

I

Initial 2-Port

Attenuator

"I " 1

1

Generator A B C —1

1 —1— Load

1 r 1

Final 2-Port

Figure 6-13. Improved representation of attenuator insertion

into a waveguide system (Case 1).

d. Substitution Loss

When inserting an attenuator into a measuring system, as shown in figure 6-13,

one actually substitutes the attenuator and two sets of connector pairs A and B,

for the connector pair D which initially connects the system together. Thus one

228
measures the substitution loss, or the ratio of to '^P. It is similar to

eq. (3.53), but the characteristics of the connectors are now implicitly involved,

as they influence the scattering coefficients of the initial and final two-ports.

The substitution loss Lg is written as follows:

21 (1

f f

^12 ^21^G L

= 21 - SiirG)(l - ^22^^ ^^12^^21^G^L

(3.51)

where the front superscripts i and f on the scattering coefficients refer to the

initial and final two-ports, respectively.

Additional insight is obtained by writing eq. (3.51) in two additional forms.

First, it is written as the difference between the insertion losses of the initial

and final two-ports as follows:

L = 20 log^Q

20 log-^Q

(1 -s^^.r^)(i ^22^1^

f f

^12 ^21^G^L

(1 - ^S^^r^)(l - 'S^^T^) ^^12^^21^G^L

(6.56)

It is apparent that, analytically, substitution loss is equivalent to a difference

between two insertion losses. One could thus avoid the use of the concept of

substitution loss, if desired. However the assumptions made in the definition of

substitution loss are more easily realized in practice. In addition, it is more

convenient to use when analyzing variable attenuators, as will be seen later.

Another form of eq. (3.51) is the following

A) + 20 loj 10

(1 - 's^^r^)(i - 's^^v^) f f

^12 ^21^G^L

^G^

20 log-Lo

(1 - s^^rg)(i - S22rL) ^12 ^21^G^L

^G^L

(6.57)

The first two terms above on the right are attenuations as defined in eq. (3.53) and

last two terms are similar in form to eq. (6.44), the mismatch error of the previous

analys is

.

More complex expressions, containing the scattering coefficients of two-ports,

A, B, C, and D could be obtained by making the appropriate substitutions for the

scatterinp coefficients of the composite final two-port (Hashimoto, 1968).

229
The substitution loss which would occur in a non-reflecting system is of

interest and can be deduced from eq. (6.57) when = r^^ = 0 . It is

21

21

(6. 58)

It is simply the difference in the attenuations between the initial and final twoports .

e. Standard Attenuation

£

In practice, one is not interested in the attenuation A of the final two-port

for two reasons. First, it is difficult to measure since the initial two-port would

need to be a perfect connector which cannot be actually realized. And second, it is

not characteristic of the attenuator itself but of the attenuator kernel B plus

two associated connector pairs A and C. The system connectors D„ and D. which do

u. L not belong to the attenuator are parts of A and C.

The quantity of interest is the standard attenuation which is the above difference

in attenuation when the initial two-port represents not a perfect connector pair,

but a standard connector pair.^' An expression for standard attenuation A is

obtained as follows: Equation (6.58) is written in terms of the scattering equations

of two-ports A, B, C, and D as

A„

'^^s^r(,=rL=o \ "^B "^C

+ 20 lo£ 10

(1 a22b^^)(l ^22*^11-' " ^22'^12^2l'^ll

^22*^11

or

as)r^=rL=o = 20 log 10

20 log

21

10

^21^21^^21

[[1

^22*^11

a22b^^)(l ^22^11^ ^22^^12^21*^11 (6.59)

If the two-port D represents a standard connector pair, and d^-j^ is replaced by ^21'

the standard attenuation is

So

A = 20 log 10

21

^21^21^21

[(1 - a22b^i)(l - b22C^i ^22^12^21*^11^ (6.60)

^ ^A "standard connector" is one which is made precisely to standard specifications for

the particular type of connector under consideration. Standard connector pairs usually

have low but appreciable dissipative loss, reflection, and leakage.

230
It is of interest to examine the form of eq. (6.58) when the connector pairs are

all identical and nonreflect ing . One obtains

b L AeChD

Under this assumption, eq. (6.61) expresses a quantity characteristic of the attenuator

itself as represented by the kernel B and connectors and B^^.

To the extent that all connectors are identical and nonref lect ing , eq. (6.60)

will also express a quantity characteristic of the attenuator itself. The standard

attenuation as defined above is thus to a good approximation characteristic of the

attenuator itself, and will be considered the desired quantity in an attenuation

measurement.

f. Connector and Mismatch Errors

The error Eg in the measurement of standard attenuation cannot be obtained

simply by subtracting eq. (6.60) from eq. (3.51), eq. (6.56), or eq. (6.57). This

is evident when we consider figure 6-14 which represents the substitution loss of

an attenuator measured in two different systems, M and N. Even though the attenuator

is the same in both cases, it is associated with connector pairs A and C in system

M and with P and Q in system N. Thus the final two-ports are different in the

two systems

.

Initial 2-Port *:M Initial 2-Port *"N

1 Attenuator

!M " Dg|Bg B, iD

M

B

Final 2-Port

1 Attenuator

IjM , . HgjBg

B

Bl|Hl

Q

Final 2-Port

(a) (b)

Figure 6-14. Representation of same attenuator installed alternately

into two different systems, (a) System "M. "

(b) System "N. "

231
In order to obtain an expression for Eg, let system M be nonref lecting and

two-port D represent a standard connector pair having an attenuation Aj, . Then the

error Eg is the difference between the substitution losses in systems M and N, and

is written

where

= CAc

" ^11 ^ ^III

Ajj) + (Ap - A^) + (Aq - A^)

^11 = 20 log^Q

20 log^Q

(1 P22^1l) " b??"^ 22^11 ) P22^12^21^11

^ - P22^11

(1 a22b^^)(l ^22^11^ ^22^12^21"^!!

^22^^11

+ 20 lof 10

P22^11

^22^^11

and

Ejjj = 20 log

- 20 log

10

10

CI fN fN

^22^L^

fN„ fN^ „ „ ^12 ^21^G^L

(1 - h^^T^)il - h^^T^) - \2^^^^Gh

(6.62)

The error component e-j- will vanish if corresponding connectors at the insertion

points are identical, since it is seen from figure 6-14 that the resulting condition

will be that D e H, A e P, and C E Q. This is a sufficient, but not a necessary,

condition since e-j- will vanish for any condition for which Ag + Ap + Ag = A^

+ A^ + Aj-,. Usually the corresponding connectors at the insertion points will be

similar, and it is evidently worthwhile to make them as nearly identical as possible.

The error component e-j-j will also vanish if corresponding connectors at the

insertion points in the two systems are identical, and this is again a sufficient

but not a necessary condition. It will also approach zero if the attenuator kernel

is nonreflect ing and its attenuation becomes arbitrarily large. The individual terms

are of the same familiar form as eq. (6.44) for mismatch error in the simpler analysis.

The error component Gjjj is very similar in form to eq. (6.44) and will vanish

if the system reflection coefficients r„ and vanish. Again this is a sufficient

but not a necessary condition, since the relative phases of the reflection coefficients involved might possibly be such as to make the two terms vanish or cancel

232
each other. It is clear that Ejjj differs from the other two components in that the

condition of identical corresponding connectors at the insertion points does not

make it vanish. The limits of terms in e^j and Gj-jj similar to eq. (6.44) may be

evaluated by methods previously described in section 6.4, and they are typically

each of several tenths of a decibel, or less. Since each such positive term is

paired with a similar negative term, it is the differences which are important

in these errors .

Since the various terms in eq. (6.62) may not all be of the same sign, some of

them may tend to cancel others, and the overall error Eg may be lower than some of

the individual terms. In careful measurements however, one cannot afford to take

this for granted, but should either make a thorough investigation, or quote a

conservative limit of error, assuming the most unfavorable phase relationships of

the coefficients involved.

g. Same Fixed Attenuator in Two Systems

The motivating question asked earlier can now be answered using the representation of figure 6-14 letting two-port D represent a connector pair which is not

necessarily a standard one, and letting r^^j^ and T^^^ be representative of an actual

system, and not an idealized one having no reflections.

Assume that in attempting to measure the standard attenuation of a fixed

attenuator in two different systems, we actually measure the substitution loss.

We are interested in the difference ALg in substitution loss as measured in two

systems. As in the previous situation, a part of this difference is primarily due

to differences in connectors D^^ and H^^, and Dj^ and H^, and the other part of the

difference is due primarily to differences in system generator and load reflection

coefficients .

233
Referring to figure 6-14 and eq . (6.57), we can write

iN,A)

+ 20 log

- 20 log

10

10

- 20 log

+ 20 log

10

10

(1 11 ^GN^fl

£No

J- Li fNg

i< J. UlN JjlN

^ ' ^GN^LN

CI iNc

11

- ^^s r 1

- 2 2 LN"^ iNc

1 7 ^^s r r

71 CM 1

M

1 - r r GN LN

CI - fM„

^11 ^GM^ ^1 ^22^LM-'

fM„

^12

fMg

21^ GM LM

1 - r r ^GM^LM

CI iM„

^11 ^GM^^l - ^^s r 1

- iMc

^12 21^ GM LM

1 - r r ^

' GM' LM

C6.63)

The sufficient conditions under which ALg will vanish are M ^N, % N,

^GM

~ '"gN' ^ \M ~

^ \M' practice, one might try to achieve these conditions, but

some uncertainty will always exist. In order to evaluate ALg using eq. (6.63), information would be needed on the characteristics of the two-ports ^M, "^M, ^N , and '^N , as

well as on the system generator and load reflection coefficients T^j^^, r^M' '^GN' ^LN

effective at terminal surface 1.,, 2,,, 1.,, and 2... M' M' N' N

It is possible to reduce the magnitudes of the system reflection coefficients to

very low values (say 0.001) by the use of tuners, in which case the last four terms

of eq. (6.63) would be negligible, e.g., less than 0.001 dB . Thus the case of ALg

for r„.. = r,., GM LM GN ^LN 0 is of interest and is given by

[rGN=rLN=01

= AA (Aj3 - Aj^) + (Ap - A^) + (Aq Ar)

+ 20 log

10

CI ^22^11^ ' ^22^11 ^ii) - P22^12^21^11

CI ^22^11^ CI ^^22^11^ ^22^12^21*^11

C6.64)

The above result equals Sj + z-^^ of eq. (6.62) if we replace Aj^ by Ag

,

and

vanishes if connectors = H„ and D, «L234
The use of eq . (6.64) in the evaluation of errors due to differences in the system

connectors is illustrated by the following calculated examples. Consider the case in

which all connector pairs in system M are identical, or A = C = D, but the left-hand

connector at the insertion point is modified and becomes H^, but and H^^ remain

the same. Under this supposition, connector pairs H and P will be the same, and

connector pairs A, C, D, and Q will be alike.

If connector were a male type-N connector, for example, it might be modified

by removing the compensating step in the outer conductor and moving the step in the

male center conductor outward so as to close the gap or notch normally present. In

the following examples certain values are assumed for the reflection coefficients

appearing in eq. (6.64) which are thought to be realistic in view of the measured

results obtained at NBS for certain type-N connectors shown in figure 6-15. Since

large variations among connectors of the same type are possible, these results

cannot be regarded as typical of all type-N connectors.

INCHES SEPARATION FROM

FEMALE CENTER PIN TIP

TO SHOULDER ON MALE PIN

OF TYPE N CONNECTOR

Figure 6-15. Experimental data for VSWR of type-M connector pair.

235
Example 1

Given a 3 dB fixed attenuator having a kernel VSWR of 1.22, connector pairs A,

C, D, and Q having VSWR's of 1.22, and connector pairs H and P having VSWR's of 1.00.

Thus Ip^^I = IP22I = |h-^^| = |h22l = 0, |b^-^| = = \c^-^\ =

| q-j^^ | = 0.10, and

|b-^2b2il = 0.5.

The attenuation terms in eq. (6.64) have components due to dissipation and

reflection, and those due to reflection cancel in the above example. The dissipative

components should also nearly cancel, leaving only the final term of eq. (6.64).

It can be written to a good approximation as

20 log-^pll + b^^(a22 - P22) + ^22(0^^^ - q^^^) + b^2b21 ^^22^11 " P22^11^l. (6.65)

if the connector reflections are small, corresponding to reflection coefficients of

magnitude less than 0.11. Limits of AA, assuming the worst phase combinations are

-0 .131 dB £ AA £ 0 .129 dB

.

The limits of error in the above example are significant ones, well above

the usual precision of a good attenuation measurement. However, many attenuators

which are presently commercially available have better characteristics than were

assumed above. Hence, another example follows.

Example 2

Given a 3 dB fixed attenuator having a kernel VSWR of 1.15, connector pairs A,

C, D, and Q having VSWR's of 1.15, and connector pairs H and P having VSWR's of 1.00.

Thus |p-j^]^| = IP22I ~ l^ll^ ~ 1^22^ ~ ^' l^lll ^ 1^22^ ~ I ""ll I ~ 1*^111 ~ 0.07, and

1^2^21! =

Proceeding as in Example 1, the limits of AA are

-0.064 dB < AA £ 0.064 dB

.

The limits of error in this example are smaller, but still significant, since a

precision of 0.03 dB is often obtained in a measurement of 3 dB at frequencies of

4 GHz and above.

If the fixed attenuators in Examples 1 and 2 above have attenuations of 20 dB or

more (instead of 3 dB) , the calculated limits of AA will be reduced to approximately

± 0.09 dB and ± 0.04 dB , respectively.

236
System Slotted Line

Slotted Line System

Figure 6-16. Representation of arrangement often used

to adjust system reflections prior to attenuation measurement.

h. Adjusting Systems for Zero Reflections

In adjusting an attenuation measurement system in an effort to make = Tj^ = 0,

the arrangement represented in figure 6-16 is often used. A slotted line is connected

in turn to system connectors and Dj^, and tuners are adjusted until the reflection

t I coefficients r„ and r, observed in the slotted section effectively vanish. This does

b L not necessarily make the reflection coefficients and of the system vanish

because, in general, the two-ports U and V may have reflections. The two-ports U

and V are composite two-ports representing a connector pair and the taper or transition

section at the end of the slotted line.

Alternatively, a tuning stub may be included, and is used to adjust for the

condition = - 0 (Mathis , 1955). This condition, together with the condition

I t = Tj^ = 0 , will make the reflection coefficients and V of the measurement

system vanish. The measured substitution loss will then equal the standard attenuation,

provided that connector pair D is standard. Experimental arrangements are indicated

in figure 6-5, section 6.2.

A tuning circuit using directional couplers can be used to achieve source or load

Zg-match (Beatty and Fentress, 1971). It is worthwhile to tune for the condition

U22 = ~ ^ (tuning out of the residual VSWR) because one can then adjust for system

VSWR's of 1.01 or less instead of having to "add" to this the residual VSWR of 1.04

to 1.10 that may be present.

237
If the condition = v^-j^ = 0 is not obtained and and are not made to

vanish, the measured substitution loss then is given by eq. (3.51) where

^"12^21 - "11^22) " "ll^G

- ^11

^^2^21 - ^11^22^ " ^22^1

^11 *^11' ^12 ^12' ^21 ^21' ^22 ^^22

£„ . , _ ^11 ^^^12^21 ' ^11^22^^11

^11 ^11 ^12^21

(1 - ^22^11^ ^'^ ' ^

fc _ ^12'^12'^12

^12

^'^ ' ^22'^11^^^ " b„c,T) - a^b^^b^^c^

£<, _ ^21^21^21

^21

and

(1 - a22b^^)(l - b22C^i) - a22b;L2^21*=ll

-

c . c c ^22 " ^^2^21 - '^ll'^22^^22

^22 "^22 '^12'^21 ~ 7 ~ T ; T

• C6.66)

(1 - a22b^p(l - b22C^^) - a22bi2b2iCii

i. Same Variable Attenuator in Two Systems

Both continuously variable and step attenuators can be analyzed by the same

method, which is an extension of the previous analysis. A continuously variable

attenuator can be regarded as though one removed an initial attenuator corresponding

to the initial setting, and substituted in its place a final attenuator corresponding

to the final setting. This point of view is valid, even though the continuously

variable attenuator remains in the circuit at all times.

A representation which applies to the measurement of a given change in a variable

attenuator first in one system and then in a different system is shown in figure 6-17.

The system connectors D„ and D, , H„ and are not shown as joined together because

in some cases it is not possible to do so. For example, if the step attenuators have

female connectors on both ends, the system connectors will all be male and will not

mate together.

Although the initial attenuator, represented by - J - Jj^ and the final

attenuator, represented by B^^ - B - B^^ are the same in two systems, the initial

and final composite two-ports are not.

238
The difference in the substitution loss measured in the two systems M and N is

written down directly from inspection of figure 6.17 and a knowledge of eq. (3.51).

i

,

'SM N

+ 20 log 10

20 log^Q

+ 20 lo! 10

20 log^Q

(1 Si^gmHi - "^12 '"21^GM^LM

1 ^GM^LM

CI • Sl^GM^I^l - f f

P P "^12 "^21^ GM^ LM

1 ^GM^LM

(1 - ^^ll^GNni

f , ^22^LN-'f f

^^12 "21^GN^LN

1 - r r GN LN

(1 '"ii^gnHi '^22^LN^ ^^12 ^2l'-GN^LN

1 - r r

' GN LN

(6.67)

In order for eq. (6.67) to vanish we not only would need identical connectors at

the insertion points (D^^ = H^^ and = H^) but corresponding system reflection coefficients would need to be equal (F^j^ = T ^-^ and r^^j^ = • Even if these (sufficient)

conditions are not obtained, it is possible for eq. (6.67) to vanish under other less

easily described conditions, although it is not very probable. In general, there will

be a difference in the substitution losses measured in the two systems.

If both systems are nonre fleet ing , eq. (6.67) reduces to

SN ^GM ^LM °

^^GN^^LN=°

= (A, Aj^) = 20 log^Q

f i "^21 ^21

1 • T

'11 '21

(6 . 68)

239
Del

2m

jDL ^-^^LM "gN^ HrJ

Initial Attenuator 2

Initial 2-Port

Initial Attenuator

I ^^6_|Jg

I K

Initial 2-Port N.

Final Attenuator

Dg|B^ BuDlI

B

Final 2-Port

Final Attenuator 2N

I HqiBg B|_;Hl

]

1

i i I V J V Find 2-Port

Figure 6-17. Representation of substitution of same final and initial

attenuators in two different systenns.

The above equation is written in terms of the scattering coefficients of the

individual two-ports which make up the initial and final composite two-ports as

follows

r = r = 0

l^GN^^LN^°

+ 20 lo 10

20 log^Q

+ 20 log 10

- 20 log-^Q

+ 20 log^Q

+ (A^ - Aq) - (A

E

" ^G^

- (Ap - Aj^)

(1 - ^22^1^ - b c 1 2 2 11'' ^22^12^21*^11

1 ^22^^11

(1 - ^22^11^^! - ^22^11^ ®22^12^21^11

1 - e f

22 11

(1 - §22^11)^1 - ^22^11^ §22^12^21^11

1 §22^^11

CI - P22bilHl - ^22^^11^ " P22^12^21^11

1 - P22^11

(1 - a22Ci;L)(l - 822^^11^

(1 - C22fii)(l - P22^1l)

(6.69)

240
If corresponding connectors at the insertion points in the two systems are

identical (D^ = and = H^^) , eq. (6.69) will vanish. Even if this condition

does not hold, it could vanish if the positive and negative terms canceled each other,

It is interesting to consider the case in which corresponding connectors at the

insertion points in the two systems are not identical, but corresponding connectors

on the two attenuators are. (J^ = B^^ and = Bj^ . ) In this case we have A e E,

F E C, G E P, and K e Q. Many terms cancel and eq. (6.69) reduces to

CL SN r =r =0

^ GM LM

r = r =0 ' GN ^ LN ^

^G^^G

20 log^Q

- 20 log 10

+ 20 lo£ 10

20 log^Q

(1 - a22bi^)(l - b 22^11^ ^22^12'^2l'^ll

1 ^22^^11

(1 - a22ju)(l - j 22^^11^

1 ^22*^11

(1 §22^11^^!

- j 22^11^ §22^12^21^11

1 §22^^11

(1 - §22^1^1 - b 22^11^ §22^12^21^11

1 §22^^11

(6.70)

The conditions = B^^ and Jj^ = Bj^ would apply in the case of a continuously

variable attenuator that was not physically removed from the circuit. They would also

apply to step attenuators provided their connectors were sufficiently identical.

The additional conditions J^^ ^

"'^L ^ ^G ~ ^L' ^G ~ ' ^G ~ ^^"-''^ reduce

the size of eq. (6.70), but would reduce the uncertainty in the measurements of the

two systems.

It is clear from the foregoing that in order to insure the same change in loss

in two systems from the same change in settings of a variable attenuator, 1) the

corresponding system reflection coefficients must be the same and 2) the corresponding

system connectors at the insertion points must be the same. Even when these conditions

are not realized, some reduction in the difference is obtainable if all other connectors are as nearly alike as possible. It is also clear that reduction of connector

reflections and dissipative losses will also reduce the difference. A fortuitous

relationship of the phases of the reflection coefficients may also make the difference

vanish, although the probability of this happening is low.

241
j. Standard Incremental Attenuation

The desired quantity in a measurement of a variable attenuator is the change in

attenuation or incremental attenuation (Weinschel, 1960) from an initial setting to

a final setting. This equals the substitution loss when the system is nonreflecting

and is given by eq . (6.58). However it must be recognized that the characteristics

of the connectors D^^ and Dj^ at the insertion point are implicitly involved, even if

only to small degree. This is evident from inspection of figure 6-17, where it is

seen that connectors D^^ and are involved in a slightly different way in the initial

and final two-ports.

In order to define a precisely-repeatable incremental attenuation, the connectors

at the insertion point should always be the same, and should be standardized. With

standard connectors at the insertion point, we then can measure "standard incremental

attenuat ion .

This quantity may be expressed by reference to system M in figure 6-17 as

i

A^'A = 20 log^Q

'21

'21

or

A A = A^ + Ag + A(. Aj - Ap

+ 20 log^pl (1 - a22b-^^) (1 ^22*^11 ^22^12^21^11

- 20 log^LolCl - 622311^1 - 122^11^ " ^22^12^21^111- C6.71)

where it is understood that and , which form parts of two-ports A, E, C, and F,

are standard connectors.

The error in measuring standard incremental attenuation (due to system reflections and differences from standard conditions in the actual system) can be evaluated

by comparing eq. (6.71) with eq. (6.67), letting M represent the idealized system with

standard conditions (r^j^ = r^j^ = 0 , and D^^ standard connectors) and letting N

represent the actual system. The procedure in writing equations and calculating

examples follows along lines of the analysis pertaining to fixed attenuators. The

error is written

-SI ^SN

- A A = 20 log 10

'21 n

21

'21 '21

+ 20 log^Q

(1 n..r..j)(l + n22rLN) '11' GN^

f f

^12 '^21^GN^LN

(1 ^ii^gnHi '^22^LN^ ^^12 "21^GN^LN

(6.72)

242
where the m's and n's are the scattering coefficients of the composite two-ports in

figure 6-rl7 and it is understood that system M represents the idealized, and system

N the actual system. The first term in eq. (6.72) vanishes if connectors H^^ and Hj^

are standard, for then '''M = , and "^M = The second term vanishes if the reflection coefficients r^j^ and r^j^ of the actual system vanish.

In order to calculate a simple example, let the second term above vanish and

assume that only the connector is nonstandard. Then two-ports A, C, E, F, K, and

Q are alike and represent standard connector pairs and two-ports G and P are alike

and represent nonstandard connector pairs. The first term in eq. (6.72) can be

written

f_

20 log^Q

m

'21 "21

• F—

21 '21

Ap + Aq - - Aj^ + Ag

+ 20 log 10

(1 '22-'ll 22^11^ ^22^12^21^11

+ 20 log 10

(1

(1

a22b^^)(l

P22hl^tl

^22'^11-

"^22^11^

^22^12^21^11

P22^12^21^11

(1 . 22^11-'^" ^22^11^ " ^22^12^21^^11

Applying the above assumptions, the right side of eq. (6.73) reduces to just its last

two terms. Assuming that reflections are small and neglecting the smaller terms, as

in obtaining eq. (6.65), the last two terms of eq. (6.73) become

(6.73)

20 log-^gll + 11^^22 ( '22-^ ^ ^22^^11 ^tt) 11'

* ^12^2l'^§22^11 " ®22^11^

+ 20 log^Q I 1 + (a

f.

11 ^'"22 P22^ " ^22^^=11 '11-

^ ^2^21^^22^11 P22^11^ (6.74)

Example 5

Suppose that the initial attenuator is a 3-dB pad having a kernel VSWR = 1.222,

the final attenuator is a 10 dB pad having a kernel VSWR of 1.15, the standard connector pair VSWR's are 1.15, and the other connector pair VSWR ' s are 1.00. Thus

^12^2ll 1^11 3 22 0.1, Ib^^b^i

22

= c 11 '22 '11 = Ik 11

= |q 11

0.1, Ib^^l = Ib^^l =

0.07, and | g22 1 ~ I

P

22

0.07,

= 0

243

I
The resulting error limits are

-0.13 dB < < 0.13 dB.

The error limits in this example are appreciable and it is possible to make

such an error in actual measurements. However, it is easy to reduce this error bycareful control of the connectors used in the measuring systems.

k. Basic Insertion Arrangements

The representation schemes and analyses presented can be extended to other

insertion arrangements for waveguide components such as attenuators.

The basic insertion arrangements considered are classified into three groups

as follows, depending upon whether the waveguide component has 1) sexless or mating

connectors, 2) nonmating connectors of the same type, and 3) nonmating connectors of

different types

.

An example of the first case was shown in figure 6-12, and examples of the other

cases are shown in figures 6-18 to 6-20. In the last two cases, adapters are

employed. Adapters for Case 2 are simpler than for Case 3, since a transition

between waveguide of different cross sections is not employed.

Case 1 has already been considered, and some of the analysis applying to figure

6-17 will find applications in Cases 2 and 3. Insertion arrangements other than

those shown are possible, and may also be of some interest, but will not be

specifically considered. For example, the case of two or more cascade -connected

variable attenuators is of interest and has been analyzed (Schafer and Rumfelt, 1959)

by simpler techniques, but will not be presently considered.

(1) Cases 2A and 3A - - Combining the Component with an Adapter

In the arrangement of figure 6-18, an adapter is connected to the waveguide

component having nonmating connectors to form a composite waveguide component having

mating connectors. The insertion arrangement for the composite component is then

the same as for Case 1, and the previous analysis applies.

However, if we are primarily interested in the attenuation of the waveguide

component itself without the adapter attached to it, this insertion arrangement will

not give a direct answer.

244
Neglecting reflections, it is seen that the attenuation of the cascaded twoports A, B, C, and P would be obtained, provided that D e Q. The attenuation of A and

B together would be nearly characteristic of the waveguide component itself, so that

the attenuation of C and P together would need to be determined and subtracted. If

reflections were taken into account, it would be even more difficult to obtain the

desired characteristic loss from the measured substitution loss.

It is concluded that unless the adapter is to be permanently attached to the

waveguide component, this insertion arrangement does not directly yield the desired

informat ion

.

Example of Case 2A,

Connectors Same Type

Dg Dl

Example of Case 3A,

Connectors of Different Types

Bl Pg

I DgId

2

L I

Initial 2 -Port

Attenuotor Adapter

Dg|Bg e, !Pr,

Finol 2 -Port

P[ ID,

A B C P 0

1

1

1

1

1

1

1

1

1

Figure 6-18. Representation of insertion of a waveguide component (such as an attenuator)

having nonmating connectors oy connecting it to an adapter (Cases 2A and 3A).

245
Example of Case 2B,

Connectors Same Type

—

B

1 r

Bg Bl

J

Example of Case 38, Connectors of Different Types

Dg Jg Jl Dl

I L_

Dg|Jg

I

Dg Dl

I

I IP

I I

Adapter

AJi ID

Initial 2-Port

Attenuator

BliDl

I

Final 2-Port

Figure 6-19. Representation of insertion of a waveguide component (such as an attenuator) having

nonmating connectors by substituting it for an adapter (Cases 2B and 3B),

(2) Cases 2B and 3B -- Substituting the Component for an Adapter

The same basic representation shown in figure 6-19 applies analytically to either

Case 2B or 3B as one can see from the examples given. The example of Case 2B is

especially familiar as the drum or turret-type step attenuator. The attenuation steps

are usually referred to a "zero dB attenuator" which is an adapter designed to have

nominally no loss. This example has already been analyzed in the section on the

variable attenuator in two systems as represented by figure 6-17. It is apparent

that calibrations of such an attenuator in different systems may not agree if the

connectors at the insertion points are different. To avoid such a possibility, it

is advisable to make certain that they are identical. This requires standardizing the

design of various types of connectors and then adhering to that standard in their

construction. (In addition, tolerances of construction should be extremely small.)

If one is interested in the change of attenuation of such an attenuator relative

to an adapter, then the design of the adapter should also be standardized and the

adapters should be constructed according to this standard design.

246
If one is interested in the attenuation characteristic of the attenuator itself,

this insertion arrangement will not give direct information. Instead, additional

calculations would be necessary after having first determined the characteristics

of the adapter and connector pairs used. It will be found that Case 2C is a better

arrangement for the above purpose.

Case 3B is similar to Case 2B except that the waveguide component such as an

attenuator has nonmating connectors of different types, and the adapter for which

it is substituted must also have corresponding types of connectors. This case occurs

in practice for example if one desires to measure the coupling of a directional

coupler by measuring the attenuation between two arms, the other arms terminated.

The side arm in some cases may have a different waveguide than the main arm so that

the connectors are of different types.

In using the substitution arrangement of figure 6-19, one can determine the

attenuation relative to a given adapter, which must be standardized if the measurement is to be repeatable and significant. No direct information is obtained concerning the waveguide component such as the directional coupler itself, and additional

calculations would be necessary, given the characteristics of the adapter. It will

be found that the arrangement of Case 3C gives better direction information, but

still may not be completely satisfactory.

(3) Cases 2C and 3C -- Combining the Component with an Adapter and Substituting f or Another Adapter

These cases are of interest because the quantity directly measured is to a good

approximation characteristic of the waveguide component alone, and no additional

calculations are required to take into account the adapter. This is true at least for

Case 2C, if not for 3C, for which additional measurements are required.

In the examples of Case 2C, it is seen that if the kernels J and P of the two

adapters are the same, and all of the connector pairs are standard, the measured

substitution loss to a good approximation can equal the attenuation of the kernel B

plus one connector pair. This is characteristic of the waveguide component which

consists of the kernel B plus two connectors. The argument is similar to that

following eq. (6.61), and is subject to the additional assumption here that the

connector loss splits equally between the male and female connectors. A detailed

analysis will not be given, but would follow along lines of those already presented.

247
In the examples of Case 3C, shown in figure 6-20, it is seen that the measured

quantity is not likely to be a good approximation to a characteristic of the waveguide component itself as represented by the kernel B and connectors and Bj^.

We would have to assume not only that connector pairs E e A and F = Q [which is

quite reasonable) , but also that adapter kernels J e P, and that the losses in

connector pair C equal those in connectors B^ and Bj^. The latter two assumptions

could be quite unrealistic and not correct to a good approximation.

A combination of Case 3A and Case 3B substitution measurements as shown in

figure 6-21 could be used to obtain more or less directly a quantity characteristic

of the waveguide component itself. It is evident that waveguide components such as

are shown in these examples are troublesome, and require extra effort in their

evaluation. In case that the waveguide component under consideration is itself an

adapter, this technique is of particular interest and deserves further study. A

detailed analysis would follow along the lines already presented.

1. Conclusions

A more rigorous representation and analysis have been presented to enable calculation of the effects of connectors and adapters on accurate attenuation measurements. The measured substitution loss replaces the insertion loss, and the former

mismatch error is replaced by an error having three components. One condition under

which the error vanishes is that the system is nonreflecting and has standard connectors at the insertion point.

A method of obtaining the nonref lecting condition using a tuning stub and

slotted line is discussed.

The need for use of standard connectors is emphasized by some calculated example;

in which error limits up to 0.13 decibel are obtained when such a connector is

nonstandard. Examples are calculated for both fixed and variable attenuators and

are based upon measured data on type N connectors

,

Situations in which adapters are used in different insertion arrangements are

discussed, and it is concluded that adapters also need to be standardized when used

in precision measurement techniques.

248
Example 1 of Case 3C, Connectors of Different Types

Bg Bl Pg

Example 3 of Case 3C, Connectors of Different Types

Dg Jg Jl Ol

Example 1 of Cose 2C,

Connectors Some Type

Dg Jg Jl Dl

Bg Bl Pg Pl

Example 2 of Case 2C,

Connectors Same Type

D,

Adapter

Dr,iJ gi^g JmDl

Initial 2 - Port Attenuator Adopter

DgiBg

i

Bi iP, L ^G

A

-rFinal 2-Port

PliDl

Figure 6-20. - Representation of insertion of a waveguide component (such as an attenuator)

having nonmating connectors by connecting it to an adapter, and substituting

the composite component thus formed for another adapter (Cases 2C and 3C).

249
Neglecting Reflections'

LsrAe + Ap + Ap + Ac. AND Ls2 = AB-Ap.

i^2(Ls,^Ls2) = A3-^^^

Figure 6-21. Example of arrangement for measuring attenuation

approximately characteristic of waveguide component

of Case 3C.

20 logio(i-|a22 bii|)>£> 20 log^o d + hazbul)

(-)(+)

C0-|

101

2

1.5

o

C\J <M O

I

<

1 1.05

>

1.01

1.005

1.002 J

oo-r-6

10

1-rl

JD* 0.1

C\i CM O

i 0.01

o

_o

o 0.001

CM

= E-0.1

CMCM

O —0.01

^

o

0.001 o

0.0001 -r 0.0001

0.00001-^0.00001

rco

no

r2

-'1.5

-1.1 CD on

1.01

1.005

L 1,002

Figure 6-22. Nomogram of "Error Limits. "

250
The form 20 Log^gCl -

I ^22^11 I frequently observed in the error equations.

In order to facilitate calculations, the nomogram of figure 6-22 may be used. The

nomogram gives only error limits assuming that there is some possibility that the

phases of the reflections might combine to produce the greatest effect. In some cases,

the error limits given by this procedure are overly conservative because the variation of the phases is limited by realizability conditions (previously mentioned in

section 6.5).

Some basic insertion arrangements are described for waveguide components having

nonmating connectors. The ones giving a measured loss most nearly characteristic

of the waveguide component are singled out for special mention, although a complete

analysis is not presented,

6.7. Efficiency and Attenuation of 2-Ports from Reflection Coefficient Measurements

a. Introduction

It was shown in section 3.13c that under certain conditions,^"* the efficiency of

a two-port terminated in a nonref lect ing load equals the radius R2 of the r2-circle

obtained when short-circuit terminations of various lengths are connected to arm 1.

The dissipative component of attenuation equals

[A^]p = 10 log^Q ^. (3.111)

^2

An additional measurement of the VSWR p-^^ corresponding to IS^-j^| of the attenuator

yields the component of attenuation due to reflection

f^ilR = 10 log^o = 10 log^o i—-. (6.75)

4Pll 1 - ISiil'

This is similar to eq. (3.66). The total attenuation [A^]rj, is the sum of [A-j^J^ and

[A-|^]j^. Some measurements using this method are shown in figure 6-23 and results

of some measurements are given in table 6-1.

^ '*The 'conditions are (1) short-circuit terminations are lossless, and (2) the

reciprocity condition CZq2S-j^2 ^ ^01^21-* 2-port.

251
252
Table 6-1. Observed attenuation data.

MEASURED ATTENUATION

Nominal

Value of

ATTenuaT ion

Input

VSWR

A^ From

Power

Rnt in

Method

Difference

3 dB 1.070 .005 dB 2.87dB 2.88 dB 2.85 dB .03 dB

6 dB 1.235 .048 5.74 5.79 5.75 dB .04 dB

10 dB 1.180 .030 9.50 9.53 9.53 dB .00 dB

20 dB 1.240

1 .050 19.74 19.79 19.81 dB .02 dB

In the case of a reciprocal 2-port, the attenuation [A^J.^, for energy incident

upon arm 2 equals [A^]^. However it does not follow that [A-^J^ = [A2]q and

[A^]j^ = [A2]|^ unless the 2-port is also symmetrical.

The above method for measuring the efficiency of a 2-port terminated in a non-reflecting load has been described (Beatty, 1950). It has also been shown how

one can obtain from the same data, the efficiency of a 2-port terminated in any

arbitrarily chosen load [Beatty, 1972a). In the following, it will be shown how to

extend this method to the case where one measures impedances rather than reflection

coefficients, and the 2-port may consist of lumped circuit elements and have terminal

pairs rather than waveguide (or transmission line) leads.

b. 2-Port having Terminal Pairs

Consider such a 2-port as shown in figure 6-24.

253
"1 = ^11^^ ^12^2

^2=^21^+^22^2 -^2^L

Figure 6-24. Diagram of 2-port having 2 sets of

terminal pairs.

For energy incident upon port no. 1 and a load of impedance connected to port

no. 2, the efficiency r\.^ is

P, |i,P ReZ,

Hi

ReZ,

'21

or

'21

^22 ^ h

Zj^ I cos (jij^

Z^ I cos

ReZ,

ReZ-

(6.76)

where is the phase of and (j)^ is the phase of Z^.

Similarly, for energy incident upon port 2 and a load of impedance Z^ connected

to port 1,

'12

^11 ' h

Z^\ cos

Z^ I cos (f)2 (6.77)

The method of measuring is described as follows. We connect reactive loads

to port 2 having an impedance |x2|e-'^ . For each load so connected we measure the

corresponding input impedance Z^ at port 1. We measure at least three values of Z^,

and possibly 5 or 10 values in order to reduce the effect of random errors.

Suppose that we wish to determine the efficiency rx^ when a load of impedance Z-^

is connected to port 1. Let

where

^1 - \

Z Ix \e^'"^^ + Z Z

(6.78)

^12^21

x^|e +

254
We can write eq. (6.78) in the form

a|xJeJ'^/2 ^ ^

1

" ~i I iiT/2 ^ c I I e-' + d

(6.79)

where a

'11 ^12^21' = ^11 * ^L' ^'^'^

^^1 '

We note that eq. (6.79) is a linear fractional transformation and hence, as |x-|^

varies, r^^ has a circular locus of radius

^1

= ad - be

2|5- sin (6 - Y

- ^/2)|

(6.80)

where 6 is the phase of d, y is the phase of c, and the derivation of eq. (6.80) is the

same as for eq. (3.124) of section 3.14b. We can also write

^1

= ad - be 1

2 Re 4

It can be shown as follows that is closely related to R^^ . Consider that

ad - be = 2 ZyZ^^Z-j^ = 2 '^\J'\2

~ ^9i)

= Z 22

L 12 21

2

12 21'

'12

2ll

^

and c = ^L'

(6.81)

(6.82)

It follows from eq. (6.77) that

ad - be

n.

2 Re[|l

or

^1 ~ ^\ "^L'

We note that when Z^ is real, = R-j^

.

In a similar way, it may be shown that

cos

R2 cos (jjj^ ,

where '(j)^^ is the phase of Z^^, the impedance of the load connected to port 1

(6.83)

(6.84)

(6. 85)

c. 2-Port Having Waveguide Leads

If we wish to measure in a similar way the efficiency of a 2-port having waveguide leads, it may be more convenient to measure the reflection coefficients

255
and at ports 1 and 2, respectively. Suppose that we have measured at

port 2 for reactive loads on port 1, and have determined

2r'01

'02

S I ^ ^2ll

1

- ^11'

C3.29)

we can process the same data to obtain (Beatty, 19 72) n-|^ for any value of (load

connected to port 2).

The procedure is to convert each measured value of to a new reflection coefficient T by means of the transformation

2N

1 - '2^1

(6 .86)

We plot r2j^ and observe it has a circular locus. We measure the radius R

Then we calculate

2N'

2N

1 +

2 I r sin

1

-

(6.87)

where is the phase of T-^.

In a similar manner, it follows that

'IN

1 +

2

I Ft sin

1

-

(6.88)

Although it is assumed that the short-circuited waveguide terminations are lossless,

one could in principle measure the losses and make a correction taking into account

eq . (3.10 7), for example.

d. Conclusions

The method of determining 2 -port efficiencies from the radius of an impedance

circle or reflection coefficient circle may be applied to reciprocal 2-port networks

having either terminal pairs or waveguide (or transmission line) leads. The method

is not recommended for determining attenuation if conventional attenuation measuring

equipment is available, but is useful for determining efficiencies. If more than

3 measurements are made to determine a circle, then the accuracy can be better than

that for 3-point methods, at the cost of additional measurement time.

256
6.8. Attenuation from Power Measurements

a. Introduction

In the measurement o£ microwave attenuation, the stability or resolution of the

measuring system places a limit on the accuracy. Reduction of the other sources of

error (such as mismatch) will tend to make the error limit approach the system

instability limit. In practice, this limit has been typically of the order of

±0.01 dB , for systems in which the generator is not frequency-stabilized.

In the following, the application of amplitude stabilization and accurate power

measurement techniques to the problem of attenuation measurement is described. A

measurement system having a stability and resolution of the order of ±0.0001 dB was

obtained. This system was used to calibrate a rotary vane type of variable microwave attenuator, which has a high degree of resolution for small values of attenuation.

In order to take maximum advantage of the improved stability and resolution,

refined techniques were used in the evaluation and reduction of mismatch error. The

capability of the system is indicated by the tabulated results of the attenuator

calibration, and the estimate of the limits of error is supported by an analytic and

experimental treatment. The following discussion is a modified version of Engen and

Beatty (1960)

.

b. The Measurement System

A simplified diagram of the measurement system is shown in figure 6-25. The

attenuator under test is placed between an amplitude-stabilized microwave signal

source (Engen, 1958) and a bolometer mount-power meter. The power meter consists

of a self -balancing d-c bolometer bridge (Engen, 1957) having provisions for measuring

and recording the d-c bias power required to maintain the bolometer at its operating

resistance of 200 ohms. The two bolometer mounts M^ and M2 shown in the temperature

stabilized water bath are for power measurement and signal source stabilization,

respectively. A reasonable amount of care was exercised in order to obtain good

performance from each item of equipment, with a resultant system performance as shown

in figures 6-26 and 6-27. Figure 6-26 shows the stability and repeatability with

the attenuator alternately set at the 0.00 and 0.01 dB positions. It will be noted

that the stability and repeatability are better than 0.0001 dB (10 microbels) . A

recording of the long term stability is given in figure 6-27 where the maximum

variation is of the order of ±10 yB.

257
The accuracy of measurement of small changes in d-c power at this level is

estimated to be of the order of 0.02 yW. This indicates that further improvement

could be expected in the results if the system stability were improved, perhaps

by the use of a frequency-stabilized signal source.

SELF-BALANCING DC BOLOMETER BRIDGE

AND DC POWER

MEASURING EQUIPMENT

RECORDER

STABILIZED

TEMPERATURE

WATER- BATH

DC

AMPLIFIER

SELF-BALANCING

BOLOMETER

BRIDGE

TUNABLE BOLOMETER MOUNTS

Figure 6-25. Simplified diagram of measurement system.

Figure 6-26. System response to a 0. 01 dB attenuation.

258
c. Theory o£ Measurement

One can make an attenuation measurement by measuring the microwave powers P-j^ and

P2 absorbed by the bolometer mount M-j^ when the attenuator is set first on zero, then

to some other setting. The attenuation is

A = 10 log 10 (6.89)

It is of course necessary that the interaction of reflections between the

attenuator and the measuring system is negligible.

In making such a measurement, one assumes that the microwave power P absorbed by

the bolometer element is proportional^^ to the d-c power required to bias the bolometer

at its operating resistance when P = 0. Then,

A = 10 log^Q

^ - ^1

'^O - ^2^

(6. 90)

where and W2 are the d-c bias powers corresponding to P^ and P^- (For the bolometer

used, Wq = 15 mW and P^ = 10 mW.)

0.002 db

T

3 HOURS

Figure 6-27- Long-term system stability.

The apparatus employed permitted direct measurements of differences in d-c power,

a procedure permitting greater accuracy and convenience than calculation of differences from separate measurements. The changes in d-c power level during an

attenuation measurement are shown in figure 6-28.

^^The constant of proportionality is determined by the substitution error of the

bolometer which is known from previously obtained experimental data to be independent

of power level

.

259
oa.o

a

(Wo-W|)

oa.o

TIME

ATTENUATOR SET TO

LESS THAN 3DB

'-ATTENUATOR SET TO ZERO

•MICROWAVE POWER TURNED OFF

Wo

W2 1

*l

1

1 1 TIME

ATTENUATOR SET TO GREATER THAN 3 DB

ATTENUATOR SET TO ZERO

MICROWAVE POWER TURNED OFF

Figure 6-28. Changes in d-c power level during an attenuation measurement.

For attenuations less than approximately 3 dB , the power difference ' '^^^

measured directly. If we let Vl^^ = - and W^^ ^ " ^1' ^^^-^^^ becomes

1 10 log

10 W

21

W01

(6.91)

For attenuations greater than approximately 3 dB , the power differences W„t = W

and W

02

becomes

Wq - were measured directly. Then the attenuation according to eq. [6.90)

W

A = 10 loj 10

01

02

(6.92)

260
d. Propagation of Error in Measuring d-c Power Differences

It is estimated that the error in measuring d-c power differences is within 0.1

percent +0.1 yW. When - is measured, it can be shown that the limit of error in

determining the attenuation is

1 e = 10 log^Q

W

21

01

0.001

(6.93)

and when W2 - is measured,

e = 10 log^Q

0.1 yW + 0.001 W

02

W

02

(6.94)

The calculated limits of error are shown in figure 6.27.

e. Mismatch Errors

The mismatch error (Beatty, 1954) in calibrating a variable attenuator depends upon

the reflections from the system in which the attenuator is placed and upon the changes

in characteristics of the attenuator as its dial is moved from the reference position.

The graph of figure 6-29 shows calculated limits of error for the attenuator used,

based upon measurements of the magnitude of the changes in the scattering coefficients

S^^ and of the attenuator. It was assumed that there was negligible phase change

in ^2\' "^^^ mismatch error is below 0.0001 dB for attenuator settings up to 0.1 dB

,

and remains below 0.001 dB for higher settings.

261
262
f. Results

The calibration data taken at 9.3897 GHz is shown in table 6-2. Three sets o

data are shown in order to give an idea o£ the res ettability of the attenuator, an

more significant figures are given than one can normally use when interpolating

between the marked dial divisions

.

Table 6-2. Results of three sets of measurements and calculated limits of error in single measurements.

itt 61X113.1 or Measured Attenuations Maxinium Calculated Limit

Corresponding to Dial Reading Deviation of Error in Single

Reading \ 5C 3 Average From Average Measurement

dB A D dB dB dB rIFt U D d B

01 01077 01 083 0 1 044 0107 0. 0003 . 000055

02 . 02 123 02 145 n? 1 4. 0. 0002 . 000068

03 . 03015 0. 0001 . 000080

04 . 04043 . U^Uoo UrtU 1 0. 0003 . 000091

05 05181 . iO 0. 0003 . 00010

06 06 102 , UO UD f . UDUOO 0609 0. 0002 . 00011

07 06992 . 07 027 . 06985 0700 0. 0003 . 00013

08 . 0799

1

. 08056 . 08017 0802 0. 0004 . 00014

. 09 . 09075 . 09104 . 09078 0909 0. 0001 . 00015

. 1 . 10226 . 10203 . 10210 1021 0. 0002 . 00016

. 12 . 11886 . 11884 .11951 1191 0. 0004 . 00019

. 14 . 13681 . 13760 . 13819 1375 0. 0007 . 00021

. 16 . 15737 . 15765 . 15703 1573 0. 0003 . 00024

. 18 . 17806 . 17792 . 17888 1783 0. 0006 . 00026

. 2 . 20071 . 20089 . 20041 2007 0. 0003 . 00029

. . 25 .24702 . 24709 . 24724 24:1 \ 0. 0001 . 00035

. 5 . 49799 . 49762 . 49795 4979 0. 0003 . 00065

1 1. 0037 1. 0037 1. 0037 1 004 0. 0000 . 0013

2 1. 9954 1.9972 1. 9948 1 996 0. 0010 . 0029

3 2. 9968 2. 9975 2. 9993 2. 998 0. 0015 . 0047

5 4. 9841 4. 9923 4. 9927 4 990 0. 006 . 0048

10 9. 9624 9. 9671 9. 9647 9 965 0. 003 . 0053

15 14.991 14. 988 15. 001 14 99 0. 000 . 0063

20 19.963 19. 956 19. 945 19 95 0. 01 . 0093

25 24. 999 25. 031 24. 987 25 01 0. 02 . 014 30 30. 080 30. 049 30. 074 30 07 0. 02 . 015 40 40. 354 40. 281 40. 367 40 33 0. 05 . 02 50 52. 338 52. 041 52. 336 52 24 0. 20 . 06

The estimated limits of error for the complete range of the attenuator, as

determined from figure 6-29, are also shown in table 6-2. Above 20 dB , the calibration was made in two parts: Measurement of the 20 dB step, and measurement of

the additional attenuation referred to this step. For these values, the quoted

limit of error is the sum of the errors in the individual steps. The accuracy

of carefully setting the attenuator dial on the marks is not as good as the

accuracy of the measurements.

263
It is noted that even at the low end where the resolution is approximately

0.0001 dB , the estimated accuracy of the measurement is also better than the

repeatability of setting the attenuator on the mark. As a result of this work,

it was evident that attenuators needed improvement if full advantage was to be

realized of the accuracy available. Such improvements have been recently reported

(Little, et al . , 1971, Warner, et al . , 1972).

g. Analysis and Evaluation of Mismatch Errors

The analysis of mismatch errors in the calibration of variable attenuators

(Beatty, 1954), yielded an equation for the error in terms of the scattering

coefficients of two fourpoles corresponding to two settings of the attenuator dial,

and the reflection coefficients of the system in which the attenuator was placed

(see section 6.4e). The measurement of all of these quantities may be tedious or

difficult, and to avoid this, an approximate method has been developed. One obtains

reasonably close limits within which the error lies from a fairly simple experimental

procedure

.

The complete expression for the mismatch error is adapted from eq. (6.47) and

is

(1 - s;,rc)(i - s;2rL) - (S2i)'rcrL

e = 20 log 10 (6.95)

(1 - S^^V^ni - S^^T^) - iS^^)-T^T^

where scattering coefficients are denoted by S^-j^, 5^2' ^21' ^22' ^G'

represent, respectively, the reflection coefficients of the system "looking towards"

the generator and load. Primes are used to designate a setting of the attenuator

other than the zero or reference setting.

For small reflections, the following expression was derived from eq. (6.95)

e ~- 20 logroll - - CS;2-S22)rL ^ [S;^S;2-SllS2 2 ' ^ ^21^ '"^21 ' ^ ^G^L I ' ^6.96)

I I If the attenuator VSWR is not much greater than unity, the products S^-j^S22 and

^11^22 neglected. Then eq. (6.96) becomes

e ^ 20 logroll + (Sii-S;^)r(, ^ (S22-S22)rL ^ [S21' " (^21) ^ r^rL I . (6.97)

It is convenient to determine the magnitudes of the individual terms but not

their phases, so that the limit of error, allowing the phases of S^^^ and to take

on any values (but the phase of S2-|^ is assumed constant) is

e ~- 20 log,o[l - |s;^-S^J|rc| - |s;2-S22l|rLl - ( | S21 ^ -

| ^ ) I T^rL | ] . (6.98)

264
The quantity - is determined as follows. With the attenuator under

test connected as shown in figure 6-30, and set to its zero or reference position,

tuner A is adjusted for a detector null, and tuner B is adjusted until the reflection

coefficient T2j^ of the equivalent generator at terminal plane 2 vanishes. (This

condition may be recognized by means of an auxiliary re f lectometer)

.

DETECTOR

TUNER

"r"

1 B V

TUNER

"a" 2 /

GENERATOR l-uiiH X ATT.

1

mil

/

NON-REFLECTING

TERMINATION

Figure 6-30. Schematic diagram of system for measuring | - S^n |

.

Movement of the attenuator dial to some other setting will then give an

observable output. Using the theory of a directional coupler having auxiliary

tuners (Beatty and Kerns, 1958) one can obtain IS-j^-j^ - S^^| in the following way.

The magnitude of the wave emerging from the side arm of the directional coupler

in figure 6-29 is

1 + KTt

= k

2i L

(6. 99)

where k is a constant for a given stable generator operating level.

With the attenuator set first on zero, r^^ = S^^. Adjusting tuner A for b^ = 0

changes K so that KS^^^ = -1. Adjusting tuner B for = 0 makes the dependence of

Ibjl on Tj^ simply

11

(6.100)

Suppose that the attenuator dial is moved to a new position such that now equals

S^^ ; then

'11

'11

^11 S]_]_ (6. 101)

'11

265
The factor k/|S-j^^| is obtained by replacing the attenuator with a waveguide section

containing a sliding short-circuit. Upon sliding the short, [b^l goes through small

variations so that one may observe lb, I and Ib-I - . It is easily shown that ' 3 ' max ' 3 ' mm '

= ^^bjl^a^ + Ibjl^ij,) . (6.102)

One can assume that l^^l^^-j^^ = 1^3 ^min employ a fixed short-circuit with negligible

error if the VSWR corresponding to |S-j^^| is less than 1.15. It may be that this error

is also tolerable for higher VSWR's since it is not important to know |S^^ - S^^ | to

great accuracy.

The quantity IS22 " ^22 ^ found in the same way as above with the attenuator

turned end for end.

6.9. Two-Channel Nulling Method

We introduce a new technique to measure the attenuation constants of short sections

of waveguide and the losses of waveguide joints.'^

The measurement method is described with reference to figure 6-31 which shows a

two-channel system. The test section of waveguide is placed in the lower channel

in such a way that the microwave energy traverses it twice, being reflected from a

short circuit. The energy then passes through a level-set attenuator and is combined

with energy from the upper channel which contains a direct - reading attenuator and

phase shifter. Upon adjusting the upper channel to obtain a detector null, the

losses in the two channels are equal. It is apparent that one can measure the loss

in the test section of waveguide by observing the difference in attenuator settings

to obtain nulls with the test section inserted and then removed from the circuit.

The main difference from previous methods (Altschuler, 1963) is that no special

items of equipment are needed such as precision-sliding short circuits or special

amplifiers. Only stock items of commercially available equipment are used, such as

are encountered in most measurement laboratories. The main disadvantage of the method

is the necessity for making four to seven measurements at slightly different frequencies each time an attenuation constant is desired. This procedure reduces errors

from several troublesome sources, leaving the uncertainty of the reference attenuator

as the major source of error. If one uses a good commercially available rotary-vane

^^This discussion is a modified version of Beatty (1965).

266
attenuator with 0.01-dB divisions at the low end o£ the scale, and uses a

superheterodyne receiver as a null detector, one can expect a resolution and

repeatability of ± 0.001 dB , and an accuracy of perhaps ± 0.002 dB without calibrating the attenuator.

Details of the measurement procedure are as follows:

1) Compute an operating frequency such that the length of the test section of

waveguide is an integral number of half -guide -wavelengths . Calculated frequencies

for a 10.537" test section of WR-90 (X-band) waveguide are shown in figure 6-32.

2) Set the signal source and the receiver to one of these frequencies, and

set the adjustable short circuit to be a quarter-guide-wavelength (X^/4). This is

conveniently done by using the direct -reading phase shifter to obtain nulls when first

a flat plate, and then the adjustable short circuit, are placed, in turn, at terminal

surface "T . " The short circuit is adjusted until it requires a change of 180° of

the phase shifter to restore the null.

3) With the >^q/4 short circuit at terminal surface "T , " and the direct -reading

attenuator set on zero, adjust the level-set attenuator and the phase shifter to

obtain a detector null, or "initial balance."

4) Insert the test section of waveguide as shown and adjust the direct - reading

attenuator and phase shifter to restore the detector null or to obtain a "final

balance .

Note that if the frequency selected were exactly right, insertion of the test

section of waveguide would have introduced no phase shift, and it would have been

unnecessary to change the phase shifter in going from initial balance to final balance.

The measurement of attenuation constant would then be completed by dividing the reading

of the attenuator by two, and then dividing this result by the length of the test

section of the waveguide.

In practice, one cannot predict the required frequency exactly, and must approach

it by trial and error, in the manner indicated by figure 6-33. The phase shift to

restore the null is plotted vs. the frequency. At the crossover frequency, this

phase shift is zero and the attenuator reading is then the correct one to use. If

the foregoing procedure is not followed, it will usually be found that serious errors

are introduced by the change in signal level caused by the change in the phase shifter.

The reduction in phase shift in this manner also reduces error due to changes of

reflection interactions between the short circuit and the system. These changes

267
occur when the test section of waveguide is inserted. The use of a quarter - guide

wavelength short circuit and an N(Xg/2) test section of waveguide makes the losses

in the waveguide joints negligible, since current flow across the joining planes is

reduced to zero. (N is an integer.)

The change of the direct -reading attenuator causes a corresponding change in

phase-shift which affects the above results. However, in a typical commercially

available rotary-vane type attenuator, such phase shifts are certainly less than one

degree and can be safely neglected.

ISOLATOR

lOdB

lOdB

AW

X

XAW

DIRECTREADING

VARIABLE

ATTENUATOR

LEVEL-SET

ATTENUATOR

NULL

DETECTOR

DIRECTREADING

VARIABLE

PHASE

SHIFTER

X

A/yv

3dB

TERMINAL SURFACE "T"

TEST SECTION OF WAVEGUIDE

Figure 6-31, Simplified diagram of measurement system.

268
/-WAVEGUIDE JOINT / UNDER TEST

I II I

_ll

1

2

1

NUMBER OF

Xq IN L

XG (INCHES)

FOR

L = 10.537"

FREQUENCY

GHz

4.5 2.340 8.27

5 2.107 8.62

5.5 1.915 9.00

6 1.756 9.39

6.5 1.621 9.80

7 1.505 10.22

7.5 1.405 10.65

8 1.317 11.10

8.5 1.239 11.56

9 1. 171 12. 02

Figure 6-32. Arrangement of waveguide joint for measurement

of its loss, and calculated crossover frequencies

for X-band (WR-90) rectangular waveguide.

269
CO

COO (f)

0.080

1^ 0.070

I UJ O Q

? 0.060

LU

< L±J

CD

LU

<

X

CL

UJ

2

-4

-6

CROSSOVER

|_ FREQUENCY

10.660 10.662 10.664 10.666 10.668 10.670 10.672

FREQUENCY, GHz

Figure 6-33. Data illustrating trial and error method of obtaining two-way los

at crossover frequency.

270
0.12

e 0.10

0.08

0.06

0.04

0.02

0

+ MEASURED POINTS

8.0

MAX. LOSS IN WAVEGUIDE JOINT

•CURVE OF TWO-WAY DISSIPATIVE LOSS

VS FREQUENCY FOR THE WAVEGUIDE

9.0 10.0 1 1.0

FREQUENCY, GHz

12.0

Figure 6-34. Plot of data to obtain maximum loss vs. frequency of waveguide joint.

The loss in the waveguide joint can be measured by the same method. Instead of

using a continuous section of waveguide as the test section, one uses two shorter

sections which are joined by the waveguide joint under test. The overall length of

the resultant waveguide is equal to the length of the single continuous section

which it replaces. Assuming that the waveguides have the same attenuation constant,

the increase in loss of the joined sections over the continuous section of the same

length is caused by the loss in the waveguide joint. Maximum loss occurs when the

test joint is an integral number of half -guide-wavelengths from the short-circuiting

plate of the short circuit. One divides the maximum loss by four in order to predict

the loss of this joint when used in a nonreflecting system.

A number of variations of this technique have been devised. One that is

convenient is illustrated in figures 6-32 and 6-34. Note that the test section of

waveguide is an integral multiple of half -guide -wavelengths at a number of

frequencies, as shown in figure 6-32. At some of these frequencies the

271
test joint is located an odd number of from "T." Thus, its loss is negligible

at some frequencies and maximum at others. A plot of the two-way loss through the

test section of waveguide at the various crossover frequencies is shown in figure

6-34. The lower points lie on a curve of attenuation vs. frequency for the waveguide.

The vertical distance between this curve and each of the upper points is the

"maximum" loss in the waveguide joint. One divides these by four to obtain the

losses of the waveguide joint when used in a nonref lecting system.

6.10. Attenuation Divider Circuit

A circuit has been investigated (Beatty and Fentress, 1968) which divides by

a known ratio small attenuation and phase shift changes of variable attenuators and

^SideArm

Input

/

9 Af

/

/

/

/

Ac Als

Output

7 7

Main Arm

Figure 6-35. Basic attenuation divider circuit.

phase shifters. For example, attenuation changes of 0.01 dB of an attenuator can be

made to produce accurate changes of 0.0001 dB in the circuit output. The attenuator

application will be described first.

The circuit employs variable attenuators and a phase shifter as shown in

figure 6-35. It is assumed that adjustment of the attenuators produces negligible

differential phase shift and that adjustment of the phase shifter produces negligible

change in attenuation. These requirements are closely met by commercially available

attenuators and phase shifters of the rotary vane type. The basic principle is as

follows: A fraction of the energy in the main arm is routed around a side arm and

recombined in phase with the energy in the main arm. An attenuator Ap in the side

272
arm then produces a change in signal level which has a reduced effect on the circuit

output. A phase shifter is necessary to produce the correct phases for addition of

the signals from the two channels. A level-set attenuator A^^ is employed to obtain

the correct ratio between the signals.

The basic theory is as follows. Consider the wave amplitude components bj^

and b^ from main arm and side arm, which combine at terminal surface 2 (the

output). Assume that [b^l = (1/K)|bj^|, where K is the ratio of division which

results mainly from the decoupling of the side arm from the main arm.^^ Initially,

before any change in Ap , the circuit output level is

I^b2| = |bj^| + l^bj , (6.103)

assuming that bj^ and """b^ are in phase. A change of AAp dB in Ap produces a change

in |bg| according to the relationship

AA

F

= 20 log^o

^b

20 log^Q(l + 6), (6.104)

where the front superscripts denote initial and final conditions, respectively,

For small AA„ , we can write the following approximate expression:

r AA, 8.686 6 dB, (6.105)

The change in decibels of the output circuit level is

AA = 20 log

^b.

10

This can be written

AA = 20 log^Q

= 20 log

'Ml 10

Ml

20 log^Q K + 1

K +

1 + 6

(k + 1) (1 + 6)

20 log^Q K + 1

K + 10

•AAp/20'

(6.106)

(6.107)

When AA„ is small, 6 is small, and the following approximatat ion gives accurate

r results

AA = 20 log^Q 1 + K + 1

A A,

8.686

K + 1 K + 1

(6. 108)

^ 'The ratio K depends upon both the coupling ratios of the two directional couplers

and the difference in loss between the main arm and the side arm. The latter is

strongly influenced by the amount of level-set attenuation remaining after adjustment. It is desirable to keep the loss in the main arm, and hence the transmission

loss through the circuit, low by 1) choosing the coupling ratio to give nearly the

desired ratio A, as recommended, or 2) using say 3-dB couplers and installing

attenuators in both arms, increasing the one in the side arm, in order to obtain

the desired ratio.

j 273
The percent error in calculating AA from eq. (6.108) rather than from eq. (6.107)

is shown in figure 6-36. It is seen that the change in AAp is divided by the factor

K + 1, which can be controlled as follows.

Change AAp of Vernier Attenuator -Decibels

Figure 6-36. Percent error in calculating desired attenuation AA

using approximate formula AA = AAp/ (K ± 1).

Suppose that we wish to make K + 1 = 100. This requires the side arm contribution |bg| to be i-g of |bj^|, or 39.91 dB down. We choose two 20-dB couplers^' to

connect the side arm to the main arm and get a bit more decoupling than we need.

The exact ratio of |b^| to |bj^| is obtained with the level-set attenuator by the

following steps: 1) Adjust A^g and the phase shifter to obtain a null output. Then

bj^ = - '^b^. 2) Change the phase shift by 180. Then bj^ = ^b^, 3) Reduce Aj^g by

39.91 dB , reducing l^^b^l to I'^b^l = |bj^|/99. The circuit is now adjusted to give

the required division ratio.

' ^The circuit will operate a bit more accurately without this step, but eq. (6.108) then

becomes AA = -AAp(K -1). An increase in A, then causes a decrease in the circuit

output, which is awkward in some applications.

274
This circuit has been found useful for producing and measuring small attenuation

changes when available attenuators do not have sufficiently low ranges. (Many variable

attenuators have a minimum dial marking of 0.05 dB or 0.01 dB.) The circuit gives

fine control which is often important in adjusting null circuits. The circuit can

function as an attenuation vernier. (One can set to an exact dial division much

more accurately than one can estimate between divisions of . ) The circuit can also

function as a step attenuator by switching off the side channel after adjustment. The

attenuation change is 20 log^g(l + 1/K) . This procedure may be used to check the

circuit adjustment. Another check consists of balancing a change of Ap against a

corresponding opposite change of A^.

In order to obtain good accuracy, rotary vane attenuators are recommended because

they produce little or no change in phase shift. When adjusting the phase shift^^

180°, any change in signal level which it produces will cause error in setting the

desired ratio. This is normally small for rotary phase changers and can be reduced

by determining the change and readjusting Aj^g accordingly. Typically, changes in

signal level will be less than - 0.2 dB , corresponding to changes of less than

2.4 percent in the dividing factor. It is good practice to employ isolators having

low VSWR on each side of the attenuators and the phase shifter. With calibrated

attenuators, the accuracy of this attenuation divider circuit can be very good.

The application to division of small changes of phase shift is similar. The

circuit adjustment is the same, and the desired differential phase shift cj) is obtained

in terms of the change 9 in the vernier phase shifter by the following expression:

where one uses the + or - sign according to whether the signals from the two channels

are in or out of phase, respectively. For small changes in phase, one can use the

approximate formula

<)) = tan -If s in e (6.109)

K + cos 9

9 (6.110)

K ± 1

The error in using the approximate formula is shown in figure 6-37.

275
100

£

£

a>o

6

x

o

Q.

Q.

<

c

CO

o

LU

c

a>uV- CL

K + l = 100

K + I = 10

0.001 J I I I I I I I I I I I

100

10

0.1

0.01

10 100

Change 9 of Vernier Phase Shifter -Degrees

P.OOl

Figure 6-37. Percent error in calculating desired phase shift

§ using approximate formula $ — 9/ (K ± 1).

276
7. Phase Shift

7.1. Introduction

Prior to 1963, little attention was devoted to the subject of phase shift

measurements and standards. However, the increased use of phased array antennas

led to greater interest and basic microwave phase shift equations were published

(Beatty, 1964d) . In section 3.10, fundamental definit ions of phase shift were

briefly given. These are applied to a 2-port model in section 7.2, which is

an extended version of the above referenced work.

A standard phase shifter was devised by loading a tuned ref lectometer with a

sliding short-circuit in a precision section of waveguide. The phase of the

reflectometer output tracks the position of the short-circuit and can be accurately

calculated for any frequency of operation. Errors due to imperfect tuning,

dimensional variations of the waveguide, etc. were analyzed (Schafer and Beatty,

1960). The analysis of errors is described in section 7.3.

The final topic is the development of a standard differential phase shifter

(Beatty, 1964e) which consists of two tuned ref lectometer phase shift standards

having ganged short circuits sliding inside waveguides of different widths.

This topic is discussed in section 7.4.

7.2. Phase Shift Equations

a Introduction

In section 3.10, the phase of a sinusoidally varying quantity was defined

and the phase shift of voltage, current, or some other quantity associated with

a 2-port was discussed. Three kinds of phase shift (1) transmission, (2) substitution, and (3) differential phase shift were defined.

In the following, circuit theory developed in chapter 3 is applied to the

development of phase shift equations for 2-ports. Impedances, admittances, and

scattering coefficients are used in the equations. By means of these equations,

one clearly defines the quantity to be measured and can evaluate some errors due

to assumed conditions not being completely satisfied. In addition, the concept

of an ideal phase shifter is clearly explained.

277
The equations are based upon the representation of a phase shifter by a 2-port

waveguide junction. As mentioned in section 6.6, the effects of connectors

are neglected when using such a simple model. If extremely high accuracy in

phase measurements is required in the future, it may be necessary to use a better

model.

b General

Several types of phase shift of 2-ports may be considered, as will be shown

with reference to figure 7-1.

The usefulness of the different types of phase shift considered will depend upon

what types of detectors are used in a measurement of phase difference and whether

they respond to wave amplitudes or generalized voltage and current. (An electric

field probe in a slotted line would respond to v, for example.)

In figure 7-1, the terminal surfaces 1-1, and 2-2, in the waveguide leads

of the 2-port, are the places where the complex amplitudes a and b of the incident

and emergent voltage waves, and v and i, the generalized voltage and current,

(Kerns, 1967) are to be considered. The assumptions inherent in this representation

of a 2-port, such as single-mode propagation in the lossless waveguide leads have

been set forth by Kerns, (1967).

1 2

1 r -'^

il '2

a,

b2

S21 S22

^2

bi-

-Ol 02-

» b2

1 2

Figure 7-1. Representation of a 2-port, showing two sets of

terminal variables.

278
The relationship between the two sets of terminal variables is as follows

V = a + bl

= a - bj

(2.33)

The amplitude b^ of the generator wave, a^ , b^, and are related by the

following equation:

One can consider the phase shift of the 2-port to be the difference in phase

between V2 and v-^, between 12 and i-j^, between b2 and a-^, or between b2 and b^,

for example. "The phase shift of a 2-port" may, therefore, be a misleading and

ambiguous expression, since it seems to imply either that there is only one,

or that all phase shifts of a given 2-port are the same. In the following,

equations will be given to show how these various phase shifts will differ in

general, and under what conditions some of them may be the same.

The equations will be given for the case of a linear 2-port (which may be

nonreciprocal) inserted into a system in such a direction that a generator feeds

arm 1 and the load terminates arm 2. For simplicity, the symbols such as for

phase shift do not indicate that this direction has been chosen. One can easily

obtain the corresponding phase shift for the opposite direction of energy flow thru

the 2-port by interchanging subscripts 1 and 2 in the equations.

If the phase shifter is a reciprocal one, the condition Zq^S2-]^= ^92^12 ^^^^

hold, where Z^^ and Zq2 are the reference impedances chosen for waveguide leads

1 and 2 of the 2-port. Very often, Zq^= Z^^) and the reciprocity condition is

written S2-|^= ^12' This condition may be substituted into the phase shift equations

in order to reduce the number of variables by one for a reciprocal 2-port.

c Phase Shift Equations

The derivation of the following expressions for phase shift will not be given,

as they follow from straightforward algebraic manipulation of eqs . (2.33) and

(3.32)- and the scattering equations of the 2-port.

^1 = ^^G (3.32)

(3.4)

279
(1) Phase Shift of v

The transmission phase shift of v, denoted by may be written as follows

^2 ^21'^^

^ ^ = arg

^ = arg ^ 1 (7.1)

(1 . S^^)(l - S^^r^) - 5^2521^1

This phase shift depends upon the reflection coefficient of the load as well

as upon the characteristics of the 2-port. When the load is nonref lecting

,

^21

[ij^^Jp _o

= arg (7.2) ^ " ^ ^11

If one employs the impedance matrix instead of the scattering matrix of the 2-port,

the equations corresponding to eq. (7.1) and to eq. (7.2) are

Z Z = arg (7.3)

and

^21

=1 = arg ^ , (7.4)

^ ^11 '•^22 " ^12^21

where normalized impedances are used. For an open-circuited phase shifter,

^21

L Z^^

(2) Phase Shift of i

Proceeding in a similar way, the transmission phase shift of i is

ii. = arg = arg . (7.6)

When the load is nonref lect ing ,

[^.] = arg (7.7)

If one employs the admittance matrix instead of the scattering matrix of the

2-port, the equations corresponding to eq. (7.6) and to eq. (7.7) are

-Y Y

^. = arg

^ , (7.8)

^11*^^22 ^L^ ' ^12^21

280
and

-Y

[^ily =1 = — ' (7.9)

^ ^11^^22 ^ 1^ - ^2^21

where normalized admittances are used. For a short-circuited phase shifter,

Y,

L

21

^11

(7.10)

(3) Phase Difference Between b2 and a^

One can see by inspection of the scattering equations and figure 7-1 that

^2 ^21

'''b, ai D2,ai " — " • (7.11) v^

^1 ^22^L

When the load is nonreflecting

,

[*b2,aJrL = 0 = ^21 = *21- (7.12)

This phase shift is particularly interesting because it is simply the phase ii^^ of

the scattering coefficient S2-|^, a fundamental characteristic of the 2-port.

(4) Phase Difference between and b^

It is clear from eq. (3.32) that the generator wave b^^ does not, in general,

have the same amplitude as the wave incident upon the 2-port in arm 1. The phase

difference between b2 and b^ is therefore of interest and may be written

^2 ^21 \ h " — " , (7.13)

b^ (1 - s^^r^)(i - s^^r^) - s,2S2ir,rL

or

^21

*b b

" • (7.14)

This phase shift depends upon the reflection coefficients of generator and load as

well as upon the scattering coefficients of the 2-port. When only the load is nonrefleeting

,

^21

°2'°G L

" 1 - s^^r^

281
When only the generator is nonreflecting

,

When both generator and load are nonref lecting,

[*b2.bG^r^=rL=0 = ^21 = *21- (7.17)

It is observed that this is the same result as eq. (7.12) and is of special

interest for the same reason that eq. (7.12) is of interest.

The phase difference between b2 and b^ may be expressed in terms of two components,

(1) the phase difference between a-j^ and b^, and (2) the phase difference between

b2 and a-j^ as follows.

1^2 [\ K ] = arg — = arg

fb^ a^

^2'^G b^

"

"''b a ''^a b • (7.18)

The first of these components is given by eq . (7.11), and the second follows from

subsequent inspection of eq. (7.14).

*a,,b„ = arg = arg \

(7.19)

1' G 1 - r^r^

Thus , one obtains

\2>^G

^ "^21 - ^^g^l - ^22^^ - ^^gfl - ^G^l^- (7.20)

This latter expression shows clearly how reflections from the generator and load

can affect tl^, , . b2'^G

Other phase differences such as between b2 and b^^, a2 and a^, b2 and v-j^ , etc.,

could be considered, but are perhaps of less interest than the above examples. Writing

of equations for these phase differences would not be difficult, if they were desired.

(5) Differential Phase Shift

When variable phase shifters are adjusted, one is interested in a change produced i

the phase of V2 or b2 at the output waveguide lead of the 2-port as shown in figure 7-1.

For the purposes of analysis, it is convenient to regard a variable phase shifter

adjustment as though one removed an initial phase shifter and substituted a final

phase shifter, even though the variable phase shifter remains in the system at all

times. Using front superscripts i and f to denote initial and final conditions,

respectively, it can be shown as follows that the change in phase of V2 produced by

282
a given change of a variable phase shifter is the same as the corresponding change

in phase of Writing the change in phase

V2 arg

J— = arg j

^2 ^2

arg Y

1 + ^r.

1 + (7.21)

If there is no change in the load, ^T^ = and

V2 arg arg Y (7.22)

"2 "2

One can now obtain an expression for the differential phase shift, using^

^21^G

(1 - S^^r^)(l - S^^T^) - S^^S^^T^T^

and eq. (7.22), and assuming that there is no change in b^^, r^^, or r^^,

The differential phase shift is

-f

A-\p = arg

^21 ' ^^ll^G^ " ^^22^L^ ^^12^^21'- G^L

^^^21 (1 "SiirG)(i - ^22h^

f f

^12 ^21^G L

If only the generator is nonref lecting

'f arg

If only the load is nonreflecting

21 ^22^L

^ ^21 1

-

(7.23)

(7.24)

(7.25)

[A<i^] 21

'21

^ll'^G

"^ll'^G

If both generator and load are nonreflecting

,

fc- arg

21

21 '21

21

(7.26)

(7.27)

(6) Insertion Phase Shift

.When a (final) 2-port is substituted for another (initial) 2-port, a phase shift

of V2 or b2 occurs at the load. The general expression for the phase shift of b2 is

eq. (7.24). This was shown to be the same for V2 as for b2. When both generator and

^This follows from manipulation of eqs . (3.32) and (3.4) substituting = ^2^L'

283
load are nonreflecting the substitution phase shift reduces to the difference

between the characteristic phase shifts of the initial and final 2-ports.

Consider that the insertion phase shift is a special case of the substitution

phase shift, just as it was considered in section 3.8c that the insertion loss is a special case of substitution loss. The special condition is that the initial 2-port

is lossless, nonref lecting , and has zero characteristic phase shift. These conditions

on the scattering coefficients are

^^11 = ^^22 = °' ^"'^ ^^12^^21 = 1-

Substitution of these conditions into eq. (7.24) yields

r

I arg

r r G L 21

' ^ll^G^ ' ^22^0 ' ^12^21^G^L

(7.28)

(7.29)

where the front superscript f is no longer needed and has been omitted,

system is nonreflecting [T ^

= F^^ = 0) ,

arg S^^,

When the

(7.30)

the characteristic phase shift of the 2-port,

d. Characteristic Phase Shift

The foregoing equations have served to illustrate that the phase differences

considered will in general depend not only upon the characteristics of the 2-port,

but also upon the characteristics of the load, and in some cases also upon the

characteristics of the generator.

If one is interested in a phase difference which depends only upon characteristics

of the 2-port, the eqs . (7.2), (7.7), (7.12), and (7.17) can be considered. Of these

eqs. (7.12) and (7.17) are simplest. Thus it would seem desirable to select

one of the characteristic phase shifts of a 2-port. (The other would be ^-^2'^

would be defined^ as the phase difference between b2 and when nonreflecting

^This definition is in harmony with that given in IRE Standards on Antennas and

Waveguides: Waveguide and Waveguide Component Measurements, 19 59, Proc. IRE 4_7 No. 4, 568 582.

284
generator and load are connected to arms 1 and 2, respectively, of the 2-port.

The differential phase shift of a 2-port in a nonref lecting system as given by

eq. (7.27) is then simply the differential characteristic phase difference, as defined

above

.

When phase differences of 2-ports are measured under different source or load

conditions, different results will be expected. The discrepancies can be called

mismatch errors, which can be evaluated by reference to the foregoing equations.

e. An Ideal Phase Shifter

The concept of an ideal phase shifter is useful for comparison purposes in

evaluating the performance of actual phase shifters. Such a phase shifter is nonreflecting, lossless when terminated by a nonreflecting load, and nonreciprocal

.

The conditions on its scattering coefficients are

^11 " ^22 " ^12 " ^' ^"'^ ^21 " e^^'^^S (7.31)

assuming that energy is incident upon arm 1 and there is no reflected wave in arm 2.

One notes that the phase shifts of an ideal phase shifter more closely approach

^21' than one which is not ideal, even if generator and load reflections are present.

For example, compare eqs . (7.1), (7.6), (7.13) and (7.11), respectively with

= *21 * * ^L^ ' (7.32)

4^. = i>2^ + arg (1 - T^) , (7.33)

*b2,bQ = "^21' (7.34)

and

K = "l^oi • (^ • 35) ^b2,ai ^21

The differential phase shift of an ideal phase shifter is from eq (7.31) and eq. (7,

Ai); = (^4^21 - ^'''21) • (7.36)

It is seen that the use of an ideal phase shifter obviates the need for a

nonreflecting system, if the characteristic phase shift is to be produced, in the

cases of ^^^^^^^ "^b2,a^, and Ai|j. However in the cases of and i)^, a nonreflecting load is also required.

In an ideal situation, it can be seen from inspection of eqs. (7.23) and (7.31)

that the output level is not changed when the phase shifter is adjusted. However, it

285
can be seen that if the lossless, nonreflecting phase shifter is reciprocal (S

^21 ~ e-'^^^), the change in level expressed in decibels of either or b^ is

h 1

' °2 1 - r^Fj^e^^

(7.37)

The change in level is due to the interaction of reflections between generator

and load which would be suppressed by a nonreciprocal phase shifter.

Of course there would be no change in level from a lossless, nonreflecting

reciprocal phase shifter in a nonref lecting system (r„ = r, = 0)

.

f

.

Conclus ions

It has been shown that one can reasonably choose

4'2i>

the argument of S22,

and i>-^2' argument of ^-^2' characteristic phase shifts of a 2-port. For

a variable phase shifter, the characteristic differential phase shift Atjj equals

the change in

^>2-i between initial and final settings.

In measuring these quantities, it is important to insert the phase shifter into

nonreflecting systems {V ^

= Tj^ = 0) , and in using the phase shifter thereafter, it

is no less important to duplicate these conditions. Any deviation from these conditions

will result in a mismatch error, and these have been analyzed (Schafer, 1960) for some

types of measurement systems.

The use of an ideal phase shifter which is nonreciprocal, lossless, and nonreflecting

obviates the need for a nonref lecting system in phase shift measurements of

4;, , , and Atjj. However, a nonref lecting load would be required in the cases

°2' G' °2'^1

of ipy and ijj-j- . It was shown that there will be no change in the output level of an

ideal variable phase shifter. However, a reciprocal, but otherwise ideal variable

phase shifter in general requires either or to vanish if level changes are to

be eliminated.

a. Introduction

A standard microwave phase shifter was proposed (Magid, 1958) which utilizes an

adjustable short circuit attached to a tunable three-arm waveguide junction. This

7.3. Standard Phase Shifter

286
standard is illustrated in figure 7-2. It can be shown (Beatty and Kerns, 1958) that

the change of phase of the emergent wave from the third arm can be ideally made to

b3

J-^ai=b|rG+b6

' TTTTT

°3

TTTTT

I

TUNER TO

ADJUST

r2j«o

DIRECTIONAL

COUPLER TUNER TO

ADJUST

S3,«0

r2i-

'2

A PHASABLE

SHORT

CIRCUIT

^Figure 7-2, A standard microwave phase shifter.

equal the change of phase of the equivalent load attached to the second arm, whether

or not = = 0. Practical limitations in tuning the junction result in departures

from this ideal behavior which leaves an error to be evaluated.

The difference between the change of phase of the emergent wave from arm 3 and

the change of phase of the load attached to arm 2 because of imperfect tuning is termed

the tuning error. This analysis relates the tuning error to amplitude changes which

are observed at the detector attached to arm 3 during the tuning procedure. Graphs

are presented for determining parameters needed to estimate the limits of tuning

error from observations of amplitude variations during the tuning procedure.

Other sources of error which are considered are those which enter in determining

the change of phase of the reflection coefficient of the load attached to arm 2.

This load is a short circuit whose axial motion in the waveguide can be measured.

The change of phase produced depends on the waveguide wavelength and the length of

travel of the short circuit. For the dominant mode of propagation in lossless rectangular waveguide, the only dimension which affects the waveguide wavelength is the

broad dimension. The difference between the phase shift of the reflection coefficient

287
o£ the short-circuit and the calculated phase shift of the load caused by broad dimension

nonuniformity , and by inaccuracy in determining the motion of the short circuit is

termed dimension error. Limits of dimension errors are calculated for WR-90 waveguide

in the recommended frequency range of 8.2 to 12.4 GHz, and presented in graphical

form.

b . Tuning Errors

It has been shown (Beatty and Kerns, 1958), (see also section 3.15i) that the

amplitude of b^, the emergent wave from arm 3 (connected to the detector), may

be expressed in the form

S. 21

'31

22

32

(i-s,,r^)

s r ^31 G

(7.38)

^13^D

(l-SjsFp)

The phase of b^ with respect to an arbitrary reference, b^,, may be defined as

3G'

Adjustments of the junction are made to render as nearly as possible S^-j^ = 0

and r 2i 0. Under these ideal conditions, eq . (7.38) reduces to

^21^32^1

= cr,

(1 Si,r^)(i

(7 . 39)

f i where C is a constant. The change of phase of the emergent wave, 6^^ - ^3q» when

the load is changed from intial to final settings, to '^Tj^, may be determined from

the ratio of the final to initial values of b^ as obtained from eq. (7.39)

(7.40)

^3 L

This may be written to show the changes of phase explicitly as

j(fe3^-i(

3G ) (7.41)

where "^ijjj^ and ^ii-^ are the phases of the reflection coefficients of the equivalent

load at final and initial settings, respectively. From which it is apparent that

the change of phase of the emergent wave is equal to the change of phase of the

load attached to arm 2, ^ii^ - ^^-^ = ^^ii-^-

If the tuning errors are small, then departures from this ideal response because

of S2-|^ ^ 0 and T^^ f 0 can be considered separately and the contributions added. The

following analysis uses this first order approximation.

288
CI) Case I

S_T = 0, but r-- ^0. The ratio %_/''"b,, for these conditions may be derived

from eq. (7.38) and written as

1—

'

from which it is apparent that the change of phase of the emergent wave, e^g

differs from the change in phase of by j

, where

1

-J- = argument of 2 1 L

1 - Ft r,

2 1 L

(7.42)

'3G

(7.43)

= argument of (1 - ^2i^^L^ ~ ^^STM^'^'^ °^ (1 " '^2i^^L-^ ^ 2mT , where n is an integer.

In order to evaluate j from eq (7.43), one would need to know

'^2i> ^^L'

It is more convenient to calculate a limit of error assuming that one knows the

magnitudes of these quantities and the change in phase of T^, which one controls

during the measurement. The phases are then assumed to have the values which would

give maximum Cj j

. Referring to figure 7-3, in which the phase of r^^ and the

initial phase of are chosen to give the maximum tuning error (lim j) for a

given one obtains

sm

lim e T.I

sin [

fi

+ niT

1

- 2

1 '

' L

' — ' 2

1

(7.44)

>-

or<t

o

REAL

Figure 7-3. A representation of (1 — T^.^Fl )/(1 — T^/r,. ), to show

maximum e for a given — i0|. .

289
Since - 1, and since the errors are assumed small, eq (7.44) yields

fi

lim e~ J - 2 r.

.

T.I ' 2

1

sm (7.45)

It is noted that the tuning error from this source cannot exceed 2 jl^^l radians for

any phase measurement.

One can determine 1^2^^! as follows. In the tuning procedure for setting r^^ - 0

the reflection coefficient of the phasable load which is attached to arm 2, ^,

is nearly of unit magnitude and S^^ - 0; therefore, the ratio of the maximum to

minimum response of the detector as the phase of -j. is varied may be obtained

from eq (7.38) as approximately.

b, 1 + I r„.

3 ' max ' 2

1

b J . 1 - I r,

.

3 ' mm ' 2

1

1 + 2|r2.| (7.46)

or,

20 log^Q - 20 log^pd + 2|r2. I) - 17.4 112 -

I . (7.47)

I b V I

' 3 ' mm

(2) Case II

T^^ = 0, but S^^ f 0. For this case, the ratio of the final emergent wave to

the initial emergent wave may be derived from eq. (7.38) as

Si

f f

1 ^ ^

^5

^ _^ . '^^32^21 " ^31^22^

(7.48)

^^3 '^L ^ ^ Si

^^^32^21 ' ^31^22^^^L

from which it is apparent that the change of phase at the emergent wave differs from

the change of phase of the load by where.

Si 1 + —

^^32^21 ~ ^31^22"^

II

~ argument of (7.49)

^31

1 + j_

*^^32^21 ' ^31^22^

290
which for |S2-]^S22l l^32^2ll written as

II

~ argument of 1 + '31

^32^21 '^L

Since

- argument of 1 +

1, one may write l/r^^ - e

'31

^32^21

+ 2mT

.

(7.50)

From a derivation similar to that

used for j, it can be shown that, for small errors.

lim = 2 'T.II

^31

• ^32^21

fi

sm

(7.51)

Since |S2-|^|is of the order of unity, lim e^p is, to the same approximation, proportional

to the inverse of the directivity ratio. One can determine this ratio as follows.

The observed amplitude variation of the side arm output when tuning adjustments are

made to set S 31

0 can be shown to be

20 log

3 'max ~ 10

3 ' min

20 log^Q 1 + 2 ^31 1 = 17.4

r ^32^21 T . II

31

s s r 32^21^ T. II (7.52)

where r T. II

is the reflection coefficient of the phasable load, which is attached to

arm 2 when tuning for the condition S^^ = 0. The magnitude

| II I small for this

adjustment and an estimate of its value must be made in order to evaluate the error.

c. Dimension Errors

Ideally, the change of phase of the standard phase shifter at a single frequency

is

fi 4^(£^ - £.)

ipj^ = radians, (7.53)

where - is the distance between the final and initial positions of the short

circuit within a waveguide in which the waveguide wavelength is A .

The error in the change of phase of r^^ due to the uncertainty in measurement of

the axial motion of the sliding short circuit is termed the motional error, e^. A

291
small motional error is readily evaluated by considering the partial derivative of

^^^l^

with respect to I. In terms of this partial derivative.

£(, = — )A£ = — {Mr - A£.) radians,

3<i ^ (7.54)

where the A£'s are the errors in setting the positions of the load. If the uncertainty in

setting the initial and final positions of the load is | A£ | , then the limit of motional

error, lim is

lim en = ' radians = 1440 l degrees. .

In general, the waveguide wavelength will not be uniform over a particular path

between and 1^ because of variations in the dimensions. A limit of this error may

be established by calculating the difference between the change of phase of T.^ in a

uniform waveguide with the maximum (or minimum) dimension and the change of phase in a

uniform waveguide with the nominal dimension. Let this difference be termed the limit

of tolerance error. If the tolerance of the waveguide dimension (maximum variation from

the nominal value) is given by Aa, then a small limit of tolerance error, lim e , can

3. be obtained from

9 f i ^ ^ lim e = ( """tjj, ) —^ Aa radians

^ 8X ^ 3a (7.57)

This error limit is proportional to the total change in phase of r^, and therefore

is presented as a fractional error, e /^^ii-., as

e A lim r-^ = —^

Aa.

^^^'l

4a-^ (7.58)

d. Graphical Presentation of Results

It was assumed that the errors in the change of phase were small and therefore the

individual contributions to the error could be summed. Two graphs, figures 7-4 and

7-5, present values of |r2j^l, and respectively, which are used to

estimate the limits of error from the two tuning errors given by eqs . (7.45) and (7.51)

respectively. Two more graphs, figures 7-6 and 7-7 present limits of dimensional

errors. The graphs of |r2^|and I S3]^/S22S22^ I are applicable for any frequency range or

waveguide size, while the graphs of limits of dimensional error are only applicable to

WR-90 waveguide over the operating range of frequencies noted on the graphs. The

equations used to construct these graphs, however, may be used for any size waveguide.

292
Figure 7-4 is a graph o£ the value of \^2i\ V^otted against the ratio of the

maximum to the minimum response of the detector attached to arm 3, in decibels, as

the tuning load (a short circuit) is moved along the waveguide. This value of |r2j^|

is to be used in eq. (7.45) to estimate lim j

.

Figure 7-5 is a graph of the value of 1532/522^22^1 plotted against the ratio

of the maximum to the minimum response of the detector attached to arm 3, in decibels,

as the tuning load (having small reflection) is moved along the waveguide. This value

of |522/52252il -"-^ ^° ^® used in eq . (7.51) to estimate lim . In this portion

of the tuning procedure, the magnitude of reflection coefficient of the tuning load

usually lies within the range 0.001 to 0.1. Therefore, several curves are plotted

for different i T^^- j j I • It is only necessary to determine an upper limit to the

magnitude of r,j, j ^ to estimate limits of error from this source.

Figure 7-6 is a graph of the limit of motional error plotted against the

maximum uncertainty of motion imparted by the drive mechanism to the short circuit.

Several curves are plotted for various frequencies throughout the recommended frequency

range of WR-90 waveguide.

Figure 7-7 is a graph of the limit of tolerance error per degree of change of

phase, in degrees error per degree of change of phase, applicable for any value of the

change of phase of r^. Several curves are plotted for different frequencies throughout

the recommended frequency range of WR-90 waveguide.

As an example of the use of the graphs, assume that a standard phase shifter

was made and used as follows. The load attached to arm 2 is made with a short-circuit

adjustable with a micrometer of 0.0005-in. maximum uncertainty placed in a WR-90 waveguide of standard tolerance (± 0.003 in.). The tuning procedure for was carried

out to 0.01-dB variation in the maximum to minimum response. The tuning for S^^ was

carried out to 1.0-dB variation in the maximum to minimum response with a tuning load

of maximum VSWR of 1.01. The operating frequency is 9,000 MHz. The change of phase

is 60°.

293
IO''f 1 1—I—I—I I I I

I

1 1—I—I—I I I I

Figure 7-4. Graph for the determination of |r |.

294
From figure 7-4, \^2i^ ^ 0.01-dB variation is 0.00058. From eq. (7.45),

the limit of tuning error lim e„ -r is therefore 0.00058 radians or 0.033 deg. From

295
Figure 7-6. Limit of motional error.

296
0.1

'i 0.01

<uo»

a> o

« 0.001

o>

a> -o

" 0.0001

0.00001

0.000001

- 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1:

xf-- 8.2 GHz : rf=9.0 GHz

-

-f= 10.0 GHz

\f= II.O GHz

Vf= 12.0 GHz

f= 12.4 GHz

-

-

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.00001 0.0001 0.001 0.01 0.1

A a (inches)

Figure 7-7. Limit of tolerance error.

figure 7-5, \^ -t^iI 2\\ ^ 1.0-dB variation with a | ^ ^

| of 0.005 (VSWR = 1.01)

is 0.00029. From eq , (7.51), the limit of tuning error, lim jj, is 0.000 29 radians

or 0.018 deg. The total limit of tuning error is then 0.051 deg. From figure 7-6,

for a tolerance of the micrometer of 0.0005 in., the limit of motional error at

9,000 Mc/s is 0.38 deg. From figure 7-7, for a tolerance of 0.003 in. in the dimension of the waveguide at 9,000 MHz, the limit of waveguide dimension error per

degree of change of phase is 0.0038 deg/deg. For 60°, this is a limit of waveguide

dimension error of 0.228°. The total limit of error. from these sources is then

0.66 deg.

297
The above example is considered to be typical of readily constructed phase

shifters since the,- tolerances were typical (WR-90) commercial tolerances and the

tuning variations used can be attained in reasonably stable systems. However,

precision waveguide sections, and tuning loads of very small reflection coefficients

permit constructing standard phase shifters of extremely high accuracy. For example,

the motional error may easily be reduced to 0.038 deg while precision waveguides

have been constructed which have 0.00013°/deg limit of dimensional error per degree

of change of phase. With such improvement in the dimensional errors, the total limit of

error in the phase measurement described in the above example would be only 0.097 deg.

It was mentioned in section 7.3 that a standard phase shifter can be constructed

by combining a short-circuited section of uniform waveguide and a tuned, single

directional coupler type of reflectometer , as shown in figure 7-2. In operation,

the phase of the side arm output tracks the position of the short-circuiting plunger,

once the tuning adjustments have been correctly made. The phase change cf) corresponding

to a displacement i of the short circuit is

where 3 = 2t\/Xq and is the "guide wavelength."

In this section (a modified version of Beatty, 1964e) we suggest a method of

obtaining a differential phase shifter by combining two phase shifters of the above

type with ganged short circuits, as shown in figure 7-8. The phase shifters are

7.4. Differential Phase Shifter

(7.59)

OUTPUT

Figure 7-8. Differential phase shifter.

298
arranged so that the phase shift ^ o£ the output is the difference of the phase

shifts produced by the two units, or ^ = - (l>^. If the uniform waveguide sections

in which the short circuits lie have identical cross sections, then = (j)^ and

the phase shift, i|j = 0 . But if one waveguide section has a different cutoff wavelength \^ from the other, then the phase shift \l) is not equal to zero; instead,

(7.60)

where the guide wavelength is related to the cutoff wavelength by

1 ^ 1_ _1_

X^'

~ X'

' X C7.61)

G c

The cutoff wavelengths of the two waveguides can be chosen to produce any phase

shift \p between zero and- the limiting case when one waveguide is operating below

cutoff and the phase shift \p is that of a single phase shifter alone.

Such a differential phase shifter has a number of potential applications such as

the following. As the above standard phase shifter is extended to higher frequencies,

say above 30 GHz, it takes a smaller displacement to produce the same phase shift;

hence, errors in determining this displacement produce correspondingly larger errors

in the phase shift. This situation can be avoided by using a differential phase

shifter as described above, with the waveguide cutoff frequencies chosen so that the

phase of the output varies more slowly than it would if it were tracking the position

of one short circuit.

For example, at an operating frequency of 75 GHz, if one waveguide section is

WR-15 and the other is WR-12, X„ = 0.5230 cm, and A„ = 0.4719 cm. A displacement

of 0.2615 cm will produce a phase shift of 39 degrees, which is approximatly one

ninth of the phase shift that would be produced by a single phase shifter using WR-12

waveguide.

Another application is in the investigation of uniformity of waveguide sections

and the suitability of short circuits for phase shift standards. If the arrangement

of figure 7-8 is used and the two waveguide sections are nominally identical, there

will ideally be zero phase shift of the output as the short circuits are moved. Any

phase shift which actually does occur is due to deficiencies in the short circuits

or the waveguide sections, or in both.

Another application might consist of the determination of relative displacement

from the measurement of phase shift. This would require that the motion of the

299
two short circuits be independent rather than ganged. The sensitivity of phase

shift to relative displacement could be preselected by choosing the cutoff frequencies

of the individual waveguides as desired.

If the tuners are dispensed with, the differential phase shifter will still

function but with reduced accuracy, due to the finite directivities of the directional

couplers and the reflections in the system. Errors in such a phase shifter were

investigated (Ellerbruch, 1964) to find that tuning and dimensional errors were greater

than with a single phase shifter, while the short-circuit position error was less.

Thus, the device works best for small phase changes (less than 60 degrees). However,

errors were further reduced in a different design based upon the same principle (Keys,

1968) , in which two waveguide line stretchers were used.

300
8 . Conclusions

In precisely defining the quantity to be measured, in developing accurate measuring

techniques and standards, and in analyzing errors and evaluating limits of uncertainty,

waveguide and circuit theory have been indispensable tools.

Theory deals with idealized concepts and models which approximately represent

actual devices, circuits, and systems. The development of accurate measurement

methods and standards is often a compromise in which the situation in the real world

is controlled and adapted to fit a manageable theory. For example, uniform, cylindrical waveguide is used which has very low loss so that the theory of lossless waveguides

will closely represent the actual situation. If losses or non -uni f ormities had to be

rigorously taken into account, a more complicated model would need to be used, and the

mathematics might become unwieldy or unmanageable. The use of "terminal invariant

parameters," as proposed by Engen, would relieve the requirement of uniformity, but

would still require losslessness

.

The above point is also illustrated in section 6.6, where a more complicated

model is used to represent an attenuator and the connector pairs which connect it into

a waveguide circuit. The additional complexity is not justified unless one requires

a reduction in the uncertainties in measuring attenuation.

In section 5.3, an impedance standard is described which can be closely approximated by a lossless section of short-circuited waveguide. The effect of small losses

is calculated without assuming any change in the wall current distribution due to

the losses. Also, the standard has in theory no current flowing across the waveguide

joint connecting it to the measurement system. Thus the normal loss in the joint can

be safely neglected.

In devising precise definitions of quantitites to be accurately measured, it is

necessary to consider all of the assumptions and approximations which are made, then

try to specify and control the measurement system and the conditions of measurement to

fit the theory. There is always some "fuzziness" in .the evaluation of measurement

uncertainties. The reduction of this fuzziness begins with the sharpening of the

definition of the quantity to be measured and tightly specifying the conditions of

measurement.^ This point is illustrated in section 6.2g with regard to definitions

of attenuation. The same principle might also be applied to other quantities if

accuracy requirements increase.

^ In Eisenhart (1963) the concept of a "true value" is discussed.

301
Circuit theory can sometimes be used to explain the operation of a device such

as the adjustable sliding termination of section 5.2. The termination is regarded

as an attenuator terminated in a short-circuited waveguide of variable length,

and design criteria are developed. Instead of using complicated field theory, simpler

circuit theory suffices in this instance.

The use of tuners with ref lectometers was well known, but the analytical techniques which were developed in section 3.15f, section 5.4, and section 5.5 permitted

the analysis and evaluation of limits of uncertainty due to imperfectly adjusted

tuners. There is always a limit to how well one can adjust a tuner due to limitations

of the equipment used to recognize when the adjustment has been correctly made.

The reduction of errors depends upon how well they are understood. The analysis

of mismatch errors in power measurements in section 4.2 led to a large reduction in

the errors due to mismatch. It became clear that it was important to reduce the

source mismatch, which could be controlled and minimized by the calibrator. Once

the source was well -matched , the limit of uncertainty could be calculated from a

knowledge of the VSWR's of the power meters. The calibrator could then quote a

conservative limit of uncertainty for this measurement.

The examples of applications given here deal only with power, impedance,

attenuation and phase shift. They represent only a small portion of the applications

of waveguide and circuit theory to the development of accurate measurement methods

and standards. There are numerous other applications in the measurement of frequency,

noise temperature, antenna and horn gain, etc.

After an accurate measurement method and standard has been developed for a given

quantity in a frequency range where none existed before, the need for increased

accuracies and ranges of measurement continues, but sometimes at decreased urgency.

The theory and techniques described should prove useful in extending the frequency

range higher into the millimeter, sub -millimeter , infrared and optical regions.

302
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307
FORM NBS-114A (1-71)

U.S. DEPT. OF COMM. "1 . PUBLICATION OR REPORT NO. 2. Gov't Accession

BIBLIOGRAPHIC DATA NBS-MN-137

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

AoDlications of Waveeuide and Circuit Thporv to tVip

Development of Accurate Microwave Measurement Methods

and Standards

5. Publication Date August 1973

6. Performing Organization Code

7. AUT!l0R(S)

R. W. Beatty

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NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

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16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The basic theory and analytical methods used in the development of accurate

microwave measurement methods and standards are presented.

Developments at the U. S. National Bureau of Standards during 1948-1968 are

described in which the above theory and analytical methods were applied.

These developments were in the fields of power, impedance, attenuation and

phase shift, and led to the establishment of National Standards and calibration

methods at frequencies from about 300 MHz to 30 GHz.

17. KEY WORD3^ (Alphabetical order, separated by siemicolons) Attenuation definitions; attenuation measurement; barretter mount efficiency; coaxial connectors; impedance measurement;

microwave network theory; mismatch errors; phase shift-measurement; power

measurement; reflectometers; waveguide loints; waveguide theory.

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