A UNITED STATES DEPARTMENT OF COMMERCE

PUBLICATION

WatTona^ Bureau of Standaras

Library, g-Ol Mmin. Bldg.

FEB 2 4 1971

NBS MONOGRAPH 1 1 2

A111Q3 Dfi4S5fi

An Introduction to the Description And Evaluation of

Microwave Systems Using

Terminal Invariant Parameters

National Bureau ot Standards

JUL 9. R 1972

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
NATIONAL BUREAU OF STANDARDS

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Errata Sheet For

NBS Monograph 112

An Introduction to the Description and Evaluation of Microwave System

Using Terminal Invariant Parameters

by

Glenn F. Engen

Page 6 Equation (16), last factor should read:

1 - r r

I g

Page 11 Equation (32)

Insert equality sign

Page 14 Third paragraph, third line,

Delete asterisk on N „

Page 21 Equation (A- 3 0)

Insert equality sign after first line

Equation (A- 3 5), last factor should read:

e

"

UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

National Bureau of Standards • Lewis M. Branscomb

An Introduction to the Description and

Evaluation of Microwave Systems

Usina Terminal Invariant Parameters

Glenn F. Engen

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

National Bureau of Standards Monograph 112

Nat. Bur. Stand. (U.S.) Monogr. 112, 27 pages (October 1969) CODEN: NBSMA

Issued October 1969

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 — Price 35 cents
NATIONAL BUREAU OF STAN

f

Library of Congress Catalog Card Number: 77-601820
Preface

This monograph gives the technical background and basis for a new method of evaluating

certain performance parameters which are common to all microwave systems. A unique feature

of the new measurement concept is its elimination of the precision waveguide and connector

requirements for an important class of measurement problems. This feature results from a reformulation in terms of "terminal invariant" parameters.

The text of this monograph was previously submitted to the faculty of the Graduate School

of the University of Colorado under the title, "A New Concept in Microwave Measurement TechI niques," in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Dej partment of Electrical Engineering, 1969.

iii
Abstract

The description and evaluation of microwave systems is usually by means of microwave circuit analysis, which may be regarded as an extension of the practice at lower frequencies. In order to insure its validity, it is necessary to postulate that the different components, which comprise the

microwave system, are interconnected via uniform and lossless waveguide, and which is usually

(but not necessarily) restricted to single mode operation. As a consequence, precision (uniform)

waveguide and connectors are usually considered necessary elements for an accurate experimental

evaluation of a microwave system.

It is possible to avoid this requirement in an alternative formulation where the description is based upon net power, instead of the complex traveling wave amplitudes. In this reformulation

the basic parameters include available power, maximum efficiency (or intrinsic attenuation), and

several different "mismatch factors." The important feature of these parameters is their "terminal

invariant" property, i.e., their invariance to an arbitrary shift in the terminal reference surface

(in an assumed lossless region). In this way the precision waveguide and connector requirement is avoided for an important

class of measurement problems. In addition the physical model, upon which the description is based, is a simple one which provides improved insight into mismatch errors and corrections.

Key words: Attenuation; impedance; microwave power; precision connector; terminal

invariant.

iv
Contents

Page

j Preface iii

I Abstract iv

I 1. Background 1

j 2. Introduction 2

j 3. Microwave power measurement 2

' 3.1. Summary of prior art 4

3.2. A measurement procedure for Mgi and Mgm 5

3.3. Examination for dependence upon uniformity 6

4. Microwave "attenuation" measurement 9

4.1. A method for nic3.suring Tjai - 10

4.2. Measurement of Tja 11

4.3. Measurement of A^a; 12

4.4. The measurement of smaU values of rja 14

4.5. Measurement of 7? and /?c 15

4.6. Cascade connected attenuators 16

5. Relationship to existing art 17

6. Summary 17

7. Appendixes 18

7.1. Appendix 1 18

7.2. Appendix 2 18

7.3. Appendix 3 19

7.4. Appendix 4 20

7.5. Appendix 5 20

7.6. Appendix 6 22

7.7. Appendix 7 23

7.8. Appendix 8 23

8. Bibliography 23

List of Figures

Page

'• Figure 1. Basic circuit involved in the application of a terminating type power meter... 3

j Figure 2. A microwave system which includes nonuniform waveguide 3

j Figure 3. Basic form of reflectometer 4

I Figure 4. Reflectometer with tuning transformers 4

|i Figure 5. Reflectometer with nonuniform output arm 5

j Figure 6. Illustrating measurement procedure for Mgi, Mip,, and P,,, 7 Figures 7 & 8. Uniform and perturbed waveguide with termination 8

' Figure 9. Illustrating the extension of measurement capability to systems with non- uniform waveguide 9

1^ Figure 10. Basic circuit for discussion of attenuation 10

j Figure 11. Circuit for measuring t7„; and Xfa 11 Figure 12. Circuit for measuring small values of rja 15

: Fii^ure 13. Illustrating the relationship between R, Rc and 63/64 16

li Figure 14. Basic circuit for discussing cascade connected attenuators.. 16

I Figure 15. Electromagnetic system used in deriving basic reflectometer equation... 19

' Figure 16. System used to analyze wave propagation in perturbed waveguide 20

V

An Introduction to the Description and Evaluation of Microwave Systems

Using Terminal Invariant Parameters*

Glenn F. Engen**

1. Background

The microwave art is characterized by the fact

that a typical dimension of the associated apparatus,

I if measured in wavelengths of the attendant elec-

' tromagnetic radiation, is of the order of unity. Be- cause of this, retardation can no longer be neglected,

^ and the usual circuit concepts, which are extensively

employed at lower frequencies, are no longer

completely valid. There is, however, a type of circuit or network

II theory* which retains its usefulness, and which

I finds widespread application in microwave probi lems. For example, it would be difficult, if not

impossible, in the immediate context, to assign

a meaningful "equivalent circuit" to a slab of

|, resistive material. However, if this slab is mounted

i in a uniform section of waveguide, it now becomes

! possible to assign an equivalent circuit in such

j a way that if this two-port device is incorporated

!iin a given system, the net result on the operation

is predictable.

A sufficient condition for the validity of such a

jj circuit representation is that the different comIjponents be interconnected by ideal (lossless and

ji uniform) waveguide, and of such length that the

interaction is only via single mode."^ Notwithstanding the formal similarity between

microwave and low frequency circuit representaj'tions, the field of microwave measurements remains

I distinct for at least two reasons. First, the lower

Ifrequency concepts of voltage and current lose much

jof their practical significance in the microwave

region and the measurement of power assumes a fundamental role. Second, the other measurement

objectives are usually limited to a determination

| (many times only in part) of the equivalent circuit jparameters. This includes measurements of both

'impedance and attenuation.

i Because the uniform waveguide requirement is |implicit in the usual microwave circuit representation, it is not surprising that a review of the existing

>art finds this requirement deeply enmeshed in icurrent measurement practice. The "standing wave

machine," for example, includes a moving probe

^ *Based on the thesis "A New Concept in Microwave Measurement Techniques"

iin partial fulfillment of the requirements for the degree of Ph. D., University of Colorado. Boulder, Colorado, 1%9.

**Radio Standards Engineering Division, Microwave Circuit Standards, NBS

Laboratories. Boulder, Colorado 80302.

' See for example reference [1]. 'The extension to multimode interaction is straightforward, however this type of operation is avoided in much of the current practice.

which samples the field at different longitudinal

positions in a uniform line. The reflectometer, on

the other hand, measures reflection coefficient and,

in particular, gives a null output when terminated

by a matched (reflectionless) load. The matched load

plays an important role in the adjustment or design

of such devices, and the existing methods of recognizing matched loads include motion within a uniform piece of waveguide. Finally, the usual

definition of attenuation is tied to the reduction in power which results when the two-port device is inserted in a matched (reflectionless) system. The

realization of a "matched" system, in turn, calls for the already cited impedance measurement techniques.

In brief, the field of microwave measurements

represents a highly developed art, and the waveguide uniformity requirement, upon which micro- wave circuit theory is based, is strongly reflected in the existing measurement practice. Indeed, it is difficult to imagine how it could be otherwise . . . at least while attention continues to focus on the

usual microwave circuit theory.

Another feature of the existing art which warrants

consideration is the usual description of a microwave

system which often starts (sometimes implicitly)

with a model in which a match (reflectionless) is assumed for the different components. Then, in order to extend its validity, mismatch corrections

are introduced, and here the circuit concepts come

into full play.

Unfortunately, however, the mathematical

descriptions which usually emerge from such an approach are often complicated in appearance, and

provide little physical insight into the phenomena

they describe. As a result, the subject of mismatch

errors and corrections is but little appreciated, even by many practicing engineers. In addition, there is a considerable amount of permissible arbitrariness in interpreting the mismatch correction. This is reflected by the proliferation of terminology which

exists. In a mismatched system, for example, what

is the most meaningful way to characterize the

generator power? Should it be (a) the power which

would be delivered to a reflectionless load, (b) the

power actually delivered to the given load, (c) the

power associated with the emergent wave amplitude

when the generator is terminated by the given load, or (d) the available power (which is obtained under

conditions of conjugate impedance match)? AU of

these concepts (and perhaps more) are in current
use. The term "matched load" has at least three

meanings: (a) reflectionless, (b) equal to another

load, and (c) complex conjugate of another load (such

that maximum power transfer is effected). (In this dissertation the term "matched" is limited to the

first of these.) This is only the beginning, however;

a recent survey (which was by no means complete)

indicates the existence of at least seventeen different terms for expressing the loss characteristics of a two-port device.

It must be recognized that this proliferation has

been generated, in part, by the need for precision on the one hand, and on the other the desire to limit the number of terms required to describe a particular system. This has been achieved, but at the

virtual expense of inventing a different language for each system.

In summary, the existing measurements art is characterized by a circuit theory which imposes (or

at least does not relax) certain uniformity requirements upon the associated waveguide. The corollary

precision connector requirement results from the recurrent need to separate this uniform waveguide

into two (or more) parts. Perhaps the greatest application for this circuit theory is in connection with

mismatch corrections or other attempts to sharpen

the existing descriptions, but unfortunately there is no consensus of opinion on how best to do this, and

a wide variety of competing descriptions are in current use. Finally, the inevitable deviations from

the assumed uniformity, in any physically realizable

apparatus, represents a limitation on the accuracy

with which the performance or other characteristics

may be evaluated.

2. Introduction

The "new" concept, which is to be developed in this dissertation, begins by observing that many of

the microwave measurement objectives are, of

themselves, quite independent of the assumed waveguide uniformity. Perhaps the best example is power which carries the same physical meaning in a system confined within irregular boundaries as in regular waveguide. In other cases, as will be developed, it is possible to substitute new measurement

objectives, for old ones, such that the uniformity

concept is avoided and yet retain a meaningful

system evaluation. Traditionally, attenuation and

impedance measurements have played a major

support role in extending the dynamic range and

evaluating mismatch corrections. However, since

these quantities are tied to the uniformity concept,

they will be replaced by others for which uniformity

is not required. In particular this implies that the

matched load, which plays a central role in much of the existing practice, is no longer required.

As a consequence, a more meaningful set of

criteria, as to which are the important parameters

in a mismatched system, will emerge, together with

a new a model, for describing the system, in which

the emphasis is on power transfer rather than travel- ing waves, and which provides improved insight into the mismatch corrections. Finally, the uniform waveguide and precision connector requirement will be

greatly relaxed for the class of problems under

consideration.

Throughout the development of this concept, it will prove convenient to retain the existing circuit descriptions as a working tool, and then examine the

final techniques for their independence from uni- formity considerations. The development will pro- ceed with the consideration of specific measurement

areas.

3. Microwave Power Measurement

Because power represents a basic parameter in the microwave art, it is appropriate to apply the new

concept first to power measurement problems.

A large percentage of the microwave power meters

in current use, especially at low power levels, are of the "terminating type." This means that they

(ideally) terminate the waveguide or transmission

line in its characteristic impedance and indicate the

power which they absorb. The practical application

of such devices calls for their connection to the

signal source, in place of the load, to which the

power input is required. Provided that the power

meter and load are of identical impedance, the

power meter reading wiU correctly indicate the

power delivered to the load. In the more general case, where the load and

power meter are of different impedances, it is necessary to multiply the power meter reading by an appropriate "mismatch" factor [2]^^:

P(ii — Pii |i-r,r„ |2(i-|r,p)

i-r,F,|2(i-|r,„|2) (1)

where Pyi and Pym are the powers delivered to the

load and power meter and Fy, F/, F,„ are the reflec- tion coefficients of the generator, load and power

meter. The coefficient of Pgm in eq (1) is a mismatch

factor, whose determination calls for the measure- ment of three complex reflection coefficients and the

indicated computation. Fortunately, in many meas- urement applications (such as the transfer of cahbration between power meters) the parameters of the

generator are at one's disposal, and a substantial

simplification in the mismatch factor is effected by

adjusting the generator impedance such that r,; = 0. In the more general application, this is not possible,

and in many cases the mismatch factor represents an error because of the practical problems in its evaluation.

The application of the new concept to this measurement problem begins with the system

shown in figure 1 where that portion to the left of the arbitrarily chosen (and not necessarily plane)

terminal surface is, by definition, the generator;

that to the right is the load. The entire system,

' FiKures in brackets indicate the literature references at the end of this paper.

2
Generator

-Terminal Surface

Load

to write:

Lossless Region

Power Meter

Figure L Basic circuit involved in the application of a termi- nating type power meter.

including both energy sources and sinks, is enclosed

in a metal envelope,^ a substantial portion of which,

in practice, is in the form of waveguide. The object

of the experiment is to replace the load by the

power meter and from the power meter indication,

predict the power delivered to the load. It is assumed

that the input cross section of the power meter,

while not necessarily the same as that for the load

or generator, is such that the continuity of the

metallic envelope is preserved, (i.e., the power

meter mates with the generator in such a way that there are no "holes" or "gaps" in the system which

would permit the electromagnetic energy to escape).

Of the several ways of characterizing the generator power, which were noted in a previous para- graph, only one satisfies the conditions of being (1) characteristic of the generator alone, (2) independent of the uniformity concept, and (3) independent of the choice of terminal surface (within the

indicated lossless region). This is the available

power, Pfj, which is, by definition, the physically

determined, maximum power that can be delivered

to a passive load by the given generator.

In general, the conditions for maximum power

transfer will not be satisfied, thus it is appropriate

' This is largely a matter of convenience for the discussion to folhiw. If the termination is an antenna it may in principle he replaced by an equivalent load whose characteristics are such that the relationship between the electric and magnetic fields. E, H,

at the terminal surface is unchanged.

^ The validity of this may be recognized in several ways. First, if the conditions for maximum power transfer are satisfied for one choice of terminal surface within the

issless region it must be satisfied for all choices, since if by any means it were possible

1 increase the power flow across one surface, energy ccmservation w()uid require a

imilar increase at all surfaces within the lossless region. In a similar manner if .Myi

'1.9 in a given combination, this suggests that by a suitable deformation of the load

•r generator) boundary le.g.. a waveguide tuning transformer), an additional 10 percent

I power could be realized. Thi' argument is evidently true for any choice of terminal

jrface within the lossh -- rt-^M.in. Alternatively, the pow. r / driven by eq (2) cannot depend upon an arbitrarv divilon of the system into ^'etn-rdlor and load. Thus, if Pa is invariant to the choice of

-rminal surface, the same must be true of M,,,.

^ It is instructive to demonstrate thl^ formjlly for a uniform system using conven-

'>nal circuit theory. This is done in appendix 1.

" If. for example, the bolometric technique is used, the bolometer mount should be

dlibrated in terms of "effective efficiency" rather than "calibration factor."

(2) where Pyi is the power delivered by the generator

to the load and Mgt is a mismatch factor, whose value

ranges between zero and unity, and whose physical

interpretation is that of expressing to what extent

the conditions for maximum power transfer have

been satisfied. (Note that whereas Pi, is a function

only of the generator parameters, yV/y? is a function of

both generator and load. This is reflected in the

choice of subscripts.) In terms of the usual micro- wave circuit theory, Myi is given by: ^

(i-|r»|^)(i-|r;p)

' |i-r^r,|2 (3)

where the reflection coefficients Fy, T; have been

previously defined.

It is to be emphasized that while the uniformity

concept and choice of terminal surface is implicit

in the definition of the reflection coefficients, Myi

is independent of both, ^' and, in fact, retains its physical significance in applications where the usual

definitions of reflection coefficient break down completely. In order to make these points more explicit,

it is useful to consider the system shown in figure 2. As indicated, it is possible to consider the irregular

region as part of the load, part of the generator, or divided between the two in an arbitrary way.

Within the uniform portions of guide, existing theory

permits the specification of P,/ and Myi in terms of

circuit parameters. On the other hand, P,, and Myi

have been selected to characterize the system be- cause they have a precise physical meaning which

is independent of the choice of terminal surface.

The replacement of the load by the power meter

is indicated by substitution of the subscript "m"

for "f

Ptpn PgMgtn- (4)

Provided that the power meter indicates the power

which it actually absorbs,'^ its reading will equal Pym,

and if M„m is known, Pa is determined by eq (4).

Possible Choices of Terminal Surface

Uniform Waveguide

Figure 2. A microwave system which includes nonuniform

waveguide.

352-570 0-69—

3
Finally:

Pal =

My

Ma,

(5)

so that the mismatch factor introduced in eq (1) is the ratio of Mgi to Mgm- The evaluation of the mismatch correction in the use of a terminating type

power meter has thus been reduced to the measure- ment of the two terms, Mgi and Mgm, which are independent of the uniformity concept. Thus, if a method can be devised for measuring Mgi directly,

instead of relying upon the usual reflection coeffi- cient measurements, it may be anticipated that such

a technique might be independent of the waveguide

uniformity requirement.

Before proceeding with a description of the

technique for measuring these terms, it is desirable

to briefly review the prior art. 3.1. Summary of Prior Art

For measurement purposes, one of the most

useful of waveguide devices is the directional

coupler. A pair connected as shown in figure 3,

Detectors

b4t_

02 b2

GeneratorFigure 3. Basic form of reflectometer.

Load

forms a reflectometer. The original use of the

reflectometer was in conjunction with reflection

coefficient measurement, and tuning procedures were developed [3] to compensate for imperfections

in the coupler characteristics. More recently, the

device has been used as a feed through [4] and as a

variable impedance, terminating type [5] power

meter.

In terms of the analysis employed in references

[4] and [5], the electromagnetic fields, which obtain

at the output port (2), are described by the complex

amplitudes a2, 62 of the incident and emergent

traveling waves respectively. These in turn are given by:

Ab4-Cb3

62 = 7 (6)

02 Bbi-Dbz

(7)

where, in a similar manner, 63 and 64 represent the

amplitudes of the emergent waves at arms 3 and 4,

and A, B, C, D, ^ are parameters of the four-arm

junction and the detectors on arms 3 and 4. (Explicit

equations for ^ ... A in terms of the scattering

coefficients are given in the cited references but

will not be needed here.)

If the two couplers have perfect directivity, and

are free of internal reflections, both B and C

vanish and 63 and 64 become proportional, respectively, to 02 and 62- This is the required condition

for a reflectometer. In general, the couplers do not

satisfy these conditions, but the introduction of 1 tuning transformers, Tj, Ty, shown in figure 4

2

-Tuning Transformers

Figure 4. Reflectometer with tuning transformers. makes it possible to adjust for the B — C = 0 condition as described in reference [3]. For power measurement applications, the four- arm junction of figure 4 was fitted with bolometric

power meters (rather than modulation detectors

for example) on arms 3 and 4. With a suitable

normalization, and assuming that B = C = 0,

the net (or actual) power P2 emerging from arm

2 is given by

P2-\b2['- 02

P2 — K1P4 — K2P3 (10)

where P3 and P4 are the powers measured at arms

3 and 4, and Ki, K2 are constants of the system

whose value may be determined by a suitable

calibration or measurement procedure. Equations

(8-10) may be given a simple interpretation: the

(net) power emerging from arm 2 is given by the

difference of the powers carried by the emergent

and incident traveling waves. Although the uniform waveguide concept was

implicit in the derivation of this result, it was also recognized [4] that if the output port is extended

by perfectly conducting boundaries of arbitrary

geometry, and if the enclosed dielectric is lossless,

the (net) power flow across any arbitrary terminal

surface within this region is also given by eq (10). This conclusion follows, quite simply, from energy

conservation. This result, in turn, implies that it should be possible to obtain eq (10) under a set of conditions which is less restrictive thanfi — C = 0. In particular, let the boundaries be extended by means of the (lossless) two-port device of figure 2. The resulting structure is shown in figure 5. If the terminal surface is shifted from the unprimed to

(8)

(9)

4
Power Detectors

Figure 5. Reflectometer icith nonuniform output arm.

primed position, the parameters A ... A will take on new values {A' . . . A') and in general

B' and C will no longer vanish. It has been shown

[4] however that if the structure satisfies the

l(uniformity) conditions under which A ... A

jraay be defined, and if these parameters satisfy

' 'the equation

BD*-AC' 0 (11)

where (*) represents the complex conjugate, this is iboth necessary and sufficient for the output power

!to be expressible in the form of eq (10). In addition,

this equation is also satisfied by the primed variables A' . . . D'.

I The usual method ol achieving eq (11) is by means

of tuner Ty, a moving short at port 2, and observaition of the effect of the motion upon the ratio Ps/PaProvided that Ty has been adjusted such that this ratio remains constant, it can be shown ^ that eq (11)

is satisfied. Although the moving short is usually

provided with a uniform waveguide section, the

only necessary conditions are that the device be

free of loss and that the phase angle of the reflection

be adjustable. Waveguide uniformity is obviously

not required to achieve these conditions.

Having shown that the relationship of interest

{eq 10), once established, continues to give the (net)

'power, irrespective of changes in terminal surface, even in a nonuniform region, it is thus significant

j that a method, which is also independent of the

'uniformity concept, exists for recognizing the proper

|feidjustment of Ty which leads to eq (10).

' In the usual reflectometer application, the only

place, where a uniform section of waveguide is required, is at the output port (2).** Nominal deviaiCions from this ideal, in other parts of the structure,

imay cause some change in the elements of the

equivalent circuit, but these are accounted for n the tuning and calibration procedure. If, however,

he device is to be used as a feed through power

neter, the above arguments indicate, in addition,

hat the uniform section is not required in the

leighborhood of the output port.'" This provides

" See appendix 2, • See appendix 3, Note however that single mode operation was implicit in the derivation of eq (10)

'fid that whereas the output cross section at arm 2 may be such as to permit multimode

peralion, it is, in general, necessary to postulate the existence of a "mode filter" at ome point in arm 2 in order to insure the validity of eq flO). In practice this may be chieved by including a length of guide whose cross section permits only single mode

peration. " To the extent that may be considered lossless, this will not affect the adjustment

f Ty which provides the 60* — /^C'* = 0 condition. In general, it is desirable to use an

iilemative procedure to be described in a following paragraph.

the basis for transferring calibrations between

terminating type power meters without regard for possible connector discontinuity [4]. The prior art thus includes examples which are in the full spirit

of the new concept.

3.2. A Measurement Procedure for

Mgi and Mgm

In common with the foregoing arguments, the

measurement procedure wiU be developed in terms

of the usual microwave circuit theory, and then

examined for its dependence upon the uniformity

concept.

If, as already noted, the junction parameters

satisfy eq (11), the power output at port 2 may be

expressed in the form of eq (10), where [4]:

\A\-^ \B\

|Z)|--|C|

(12)

(13)

In general, this condition may be realized by adjustment of the tuner Ty. This assumes that the couplers

meet certain minimum conditions, which are well

satisfied by available components, as to minimum

directivity, etc.

It is next instructive to consider the equivalent

generator which obtains at port 2 of the "reflectom- eter" in figure 5, if the power, P4, measured by the

detector on arm 4 is assumed to be constant or independent of the load on port 2, the actual generator parameters, etc. (In practice this may be

achieved by means of a servo controlled attenuator between arm 1 and the actual generator.) This

problem has been treated by the author in a prior

paper [6]; the important result in the present context is that there exists an equivalent generator,

at port 2, whose impedance is independent of the

actual source of microwave energy. Following the results obtained in the cited refer- ence, it will prove useful temporarily to replace the

actual generator on port 1 by an adjustable passive

load, and excite the junction via arm 2. The passive

load on arm 1 is first adjusted such that a null obtains at arm 4, and tuner Tx is next adjusted such

that the input impedance at port 2 is equal to the

impedance '' of the generator of figure 1. (For the

present, the various components are assumed to have uniform waveguide.) The condition imposed

upon the "reflectometer," by this adjustment, may

be obtained from eqs (6) and (7) by letting 64 = 0,

bila^^Yg, whence:

c+£)r„=o. (14)

Let F; represent the ratio 02/62 which now obtains

at port 2 with the generator reconnected at port 1. In general, the relationship between 63/64, and F; is

5
given by,

bi~CT, + D

(15)

Substitution of eqs (11) and (14) and taking the

absolute values gives:

(16) bs A 2 P4 D i-r,r.

The factor \AID\'^ may be measured by substituting a totally reflecting termination for Yi, i.e., let 02/62 = 6'^ where 6 is arbitrary. Then,

P4

at

Now:

r *|

2

|i-r,r,

(i-|r,p)(i-|r,n

(17)

-M„ (18)

Equations (16), (17), and (18), thus provide an experimental method of measuring Myi. The measurement

of Mgm obviously requires the substitution of the

power meter in place of the load. A review of these equations shows that if Fg — O, the procedure leads to the usual reflectometer

adjustment B — C = 0 and thus to the value of

|r;|-; in addition, if Ty — O, Myi is given by 1 — |r;|-.

The measurement procedure outlined above may

thus be regarded as a generalization of the reflec- tometer technique, where instead of |r/|'-, the meas- urement yields \Ti—Ty*\-l\l — ryTi\-. In order to clarify the terminology it appears

desirable to reserve the term "reflectometer" for a junction which has been so adjusted as to measure

reflection coefficients. The same general combination of couplers and tuners, but with a different adjustment (one example of which has just been

described) will be called a "generalized reflec- tometer" or "g-reflectometer" for brevity.

This measurement procedure will next be

examined for its dependence upon the uniformity

concept.

3.3. Examination for Dependence Upon

Uniformity

The measurement procedure outlined in the

preceding paragraphs calls first for the adjustment

of Ty and then Tx. However in order to avoid

interaction between the two adjustments an alter- nate form of the g^-reflectometer, in which the order

In practice there is often substantial isolation between the actual source of micro- wave energy and the generator output terminal such that the generator "impedance"

is essentially independent of the energy source. In any case, however, it is not within

the scope of this dissertation to consider all of the practical questions which may

be suggested by the prescribed measurement procedures.

'3 Although Tuner Tjj was also used in obtaining the previous adjustment, it can be shown (see appendix 4) that the C -¥ DVy^O condition is invariant to the adjustment ot n.

" In actual practice it is usually desirable to observe P^lPi with a high quality fixed short, rather than a sliding one.

of the couplers is reversed, is required. Before

evaluating the dependence upon uniformity it is desirable to carefully outline the exact measure- ment procedure.

Referring to figure 6, the items to be considered

include the original generator (1) and load (2) (for which Py and Myi are required), the ^-reflectometer

(3), an auxiliary directional coupler, tuner {Ta),

and signal source (4), a sliding short (5), and power

meter (6). These items are identified by the cor- responding numbers in figure 6. The first step in the procedure is the adjustment

of Tx such that the generalized reflectometer will simulate the behavior of the actual generator (1). The desired adjustment may be recognized with

the help of the auxiliary directional coupler, tuner

(Ta), and signal source (4). In particular, the

actual source of microwave energy in the generator

(1) is turned ofif,'^ the generator (1) is then con- nected to the auxiliary directional coupler and

signal source (4), and tuner Ta adjusted for a sidearm

null. The generator (1) is next replaced by the

g^-reflectometer (3), the input port (arm 1) of which

is terminated in an arbitrary passive load. The

tuner Ty is next adjusted for a null in arm 4 (^4 = 0)

and tuner Tx is adjusted for a null in the sidearm

of the auxiliary directional coupler. The g-re- flectometer parameters now satisfy eq (14), andi

this completes the adjustment of tuner Tx- A signal source is next connected to the input

(arm 1) of the g-reflectometer and the output portj

(arm 2) is connected to the sliding short (5). Tuner

Ty is then adjusted such that the ratio P^P^t re mains constant for all positions of the short. The

g-reflectometer parameters now satisfy eq (11).'^ This completes the adjustment procedure.

The measurement consists of observing the ratio P3/P4 with the shding short," load (2), and power

meter (6) connected in turn. This provides a measure

of Myi and My,,, as already described using eqs (16), (17), (18). Finally, the power meter (6) is connected

to the generator (1), Py,,, is measured, and Pgi cal- culated using eq (5). Note that the available power,

Py, may now be obtained from eq (3), while Mgi

indicates what fraction of the available power is actually delivered to the load. [ Returning to figure 6, it will now be assumed

that the generator (1) and g-reflectometer (3) out puts only are of uniform rectangular waveguide]

When connected to the auxiliary coupler (4) there; may be a discontinuity (in guide cross section) a1, the mating surface; if so, the discontinuity is asy sumed to be identical for the generator (1) and,

g-reflectometer (3). Within the immediate vicinity,

of the mating terminal surface a complete descrip

tion of the waveguide fields will usually require £, number of modes, while at a greater distance thej higher modes will have decayed (exponentially:

to negligible amplitudes and the field may be com

pletely described by the wave amplitudes of the lowest order forward and reverse traveling waves,, Corresponding to the auxiliary coupler (4) sidearrr,

1 lip

'! lis

i

ilmel

pial

'Sill

iS(

ml

Cot

!a(|

Pri

1. i

eter

6
Load (2)

( I ) Generator 0

^ 1 TTT 1 J TTT \1

Generolized Ref lectometer (3)

^ 1 TTT 1 J TTT 0

1 2\|

( I ) Generator Q)

Null Detector

JIL

0

Ta

>- Y

J

Auxiliary Directional Coupler,

Signal Source and Tuner (4)

^=0 Sliding Short

(5)

Load (2)

Power Meter

(6)

Figure 6. Illustrating measurement procedure for Mgi, Mgm and P^].

* 'uU. there is a unique phase and amplitude relation-

'nip between these (lowest order) waves, and if lis is the same for both generator (1) and g^-reflec-

- 'imeter (3), their impedances (by definition) are

|:[ual. Thus, although uniform guide has been

Postulated for the generator and g-reflectometer,

is evidently not required in the auxiliary direconal coupler (4). Continuing with this line of argument, it has

ready been noted that uniformity is not required

1 principle at least) for the phasable short operain. Next, with regard to the load (2) and power

' eter (6), the fields in the mating uniform guide

lain allow a modal expansion in which energy

propagation is only via the lowest order mode.

Associated with this mode is an impedance, and

since the procedure is such as to explicitly account

for this impedance, the operation is independent

of deviations from the assumed uniformity and/or

cross section of the load and power meter.

It is therefore a sufficient condition, for the

prescribed measurement procedure, that the generator and g'-reflectometer only be provided with the

uniform waveguide section. In order to further relax

this requirement, it is necessary to examine the role of the uniform section in greater detail. Referring to figure 7, it will prove instructive to consider the operation of a uniform section of wave-
guide and passive termination. At the input port

(terminal surface 1) it is possible to arbitrarily

prescribe a transverse electric field (subject

only to the appropriate boundary conditions) and

which, in principle, determines the electromagnetic

fields throughout the entire structure. On the other

hand, given an adequate separation between ter- minal surfaces 1 and 2, and assuming all modes

except one are beyond cutoff, the field at terminal

2 will (except for a constant multiplier) be substantially independent of the prescribed excitation

at terminal 1 and dependent only upon the terminating load. Moreover, if the load possesses an adequate length of uniform waveguide at its input,

the field at terminal 2 will consist essentially of the

lowest mode travehng waves.

Uniform Waveguide

i / « Terminal Surfaces ^

—Perturbations —

1 2

Figures 7 & 8. Uniform and perturbed waveguide with

termination. The validity of these statements may be easily

recognized; if the arbitrary field at terminal 1 is expressed in terms of the appropriate modal expansion, it is found that the field at terminal 2 is due

entirely to the single (lowest) propagating mode.^®

The extension of these arguments to a nonuniform

waveguide begins with a statement as to the types of structures which will be considered. Except as may be otherwise noted, it is assumed that the

perturbed guide is at least a "reasonable" approximation to a uniform, single mode structure. In

particular, this excludes arbitrary increases in cross

section or the insertion of dielectric materials which

would permit multimode propagation. In addition

it precludes, for example, the use of a vertical

septum in rectangular waveguide in such a way as

to suppress the TE\o mode while leaving the TEzo

unaffected. As a practical matter, this restriction

is well satisfied by the components in common use. The standard tolerances on rectangular waveguide,

for example, permit dimensional changes in the

range 0.1 percent to 1.0 percent.

Hereafter called terminals 1 and 2 for brevity. Loosely speaking, it is necessary to restrict the "arbitrary" field at terminal 1 tc one in which the coefficients, of the higher order terms in the modal expansion, are ol the same order or less than that of the dominant term. In the projected applicatior

this restriction is usually well satisfied.

Now let perturbations be added to the uniform

section of figure 7 as indicated in figure 8. Since

the usual waveguide modes form a complete set, for any given cross section, one may visuahze the

arbitrary transverse electric field, which occurs at terminal 1 of the perturbed guide, in terms of its modal expansion using for basis functions those of

the unperturbed waveguide. It is then instructive

to consider the resultant field at terminal 2 on a | mode-by-mode basis. To a first approximation the perturbed guide

will continue to propagate the dominant mode while

suppressing the others. Thus, the field at terminal

2 will include traveling wave components associated

with the dominant mode, and higher order, nonpropagating, components which are excited by the

incident dominant wave, as required to satisfy the

j perturbed boundary conditions in the neighborhood | of terminal 2. Higher order modal components which ' may be included in the prescribed excitation or generated in the vicinity of terminal 1 cannot make

a direct contribution of significance to the field at terminal 2 because of the attenuation associated

with their propagation. It may be possible to have

mode conversion from the higher order modes

to the dominant mode, if so this will combine with

the already existing dominant mode to yield a change in the total field amphtude at terminal

2 . . . but not in its functional form. The physical

description of the electromagnetic fields which

emerges from these considerations, is that of the

unperturbed, lowest order propagating mode, to- gether with a superposition of higher order, non- propagating modes, whose amplitudes at any

particular cross section are determined primarily

by the boundary conditions (perturbations) in the

immediate vicinity. In a similar manner, it is useful to consider the

"impedance" which obtains at port 1 of the two

guides. For the unperturbed case (fig. 7) the impedance is usually defined on a mode-by-mode

basis in terms of the ratio of the amplitudes (reflec- tion coefficients) for the ingoing and outcoming

waves. For the lowest mode, this impedance will be a function of the load on terminal 2; for the higher modes, to the extent that the associated

fields are attenuated before reaching terminal 2, the associated impedances are independent of the

load. This same formal definition of modal impedance

can also be applied to the perturbed guide. Again

it will be recognized that, for the lowest mode, the impedance is a function of the terminating load, In contrast with the uniform guide however, if the possibility of mode conversion is allowed, the' impedances for the higher order modes will also b« dependent upon the terminating load. Thus, if one termination is removed from the perturbed guide ol figure 8, and another substituted in its place, thisi may (in principle) be observed as a change in imped

ance for several of the modes at terminal 1. Yi.oy(\

FlCi:

U

"pit is

'As

'bfacl

Itn

loicfn \
ever if the characteristics of the second load are such that the lowest mode impedances observed

at terminal 1 are identical, this is sufficient to en- sure that the impedances (at terminal 1) are the same for each of the modes. The validity of this may be easily recognized from the given description where higher order mode interaction with the

load is possible only through conversion to the

lowest mode where, by hypothesis, the impedances

are identical. As already noted, the ratio of the ingoing to out- coming waves at terminal 1 is determined, on a modal basis, by the associated impedances. Thus,

if the transverse electric field is given, the magnetic

field is thereby determined.

For the purpose of this dissertation, two terminations are equivalent provided that for an arbitrarily

prescribed transverse electric field at the input, the

resultant transverse magnetic fields are the same. The foregoing arguments show that this is equivalent

to requiring an equality of impedances on a mode-bymode basis. Finally, for the type of structure under consideration (perturbed waveguide) it is possible to substitute one termination for another

at port 2 and provided that the impedance observed

at terminal 1 is unchanged for the lowest mode,

the same will be true for all modes.

The application of these results to the power

measurement problem is straightforward. Referring

again to figure 6, it has already been shown that a

sufficient condition for the vahdity of the described

measurement is that the generator (1) and ^-reflec- tometer (3) outputs be provided with the uniform

section. Consider next the arrangement of figure 9, Null

Generator Detector T

TTT 3

Auxiliary Directional Coupler, Tuner and Source

Identical Output Sections

(or the sanne section used in both places)

Figure 9. Illustrating the extension of measurement capability'

to systems with nonuniform waveguide.

" The foregoing arguments have been largely of a descriptive nature with the objective of providing physir:al insight into the problem. An analytical investigation of this topic is given in appendix 5.

'** In practice it may be desirable to insure this condition by actually removing a

I section of the generator output line and connecting it to the ^i-refleclometer. " As thus defined, there is no place for "attenuation" in the new concept. In addition, however, the term "attenuation" is also associated in a general way with the loss

( characteristics of a two port device. Whenever this association is intended the term

. will be inclosed within quotation marks. Finally, in recognition of the prior terminology,

lithe term "attenuator" will continue to be used to represent a two port device whose

l>asic function is that of reducing the power delivered t<i the load.

** The maximum efficiency concept is also implicit in the terms intrinsic insertion

, loss [7], intrinsic attenuation [8], and possibly others.

where the generator and g-reflectometer are pro- vided with identical output sections. Corresponding

to the auxiliary coupler null a unique impedance will exist for the dominant mode and thus for all modes.

Finally, if the same null obtains for the gr-reflectometer, its impedance properties must be identical to the generator as required.

The requirement for a uniform waveguide section

at the generator and g-reflectometer outputs is thus

relaxed to the condition that they be of the same

nominal length and cross section as the uniform

guide and identical to each other.'^

4. Microwave ''Attenuation"

Measurement

The application of these concepts to the area of

attenuation measurement proceeds along the lines already established. As previously noted, at least 17 (and undoubtedly more) different terms have

been introduced into the literature in order to describe the change (usually a reduction) in power

which takes place when a two port device is in- serted between a generator and load. In the most

general case, the determination of this change in power level is by no means a trivial problem since,

in addition to the three complex numbers (four

if reciprocity is not satisfied) required to describe

the two port (attenuator), the complex generator

and load impedances are involved. A complete

description of the problem thus involves 10 (or 12)

parameters. Aside from a certain amount of duplication, the different terms, describing the power

change which follows the insertion of the attenuator, have been generated by imposing different sets of boundary conditions on the system in which

the attenuator is to be installed. "Attenuation,"

for example, is usually defined as the reduction

in power which results if the associated system

is perfectly matched.'^ This invokes the uniformity

concept. "Insertion loss," on the other hand, is typically defined as the change in power (sometimes an increase) which results when the two

port is inserted in an arbitrary system. This avoids

the uniformity requirement but includes the para- meters of the generator and load. Among the descriptions which have been intro- duced, one satisfies the conditions of being characteristic of the two port alone, independent of the

uniformity concept and choice of terminal surface.

This is the maximum efficiency,^" which will be

denoted by r}„. (The letter 17 has been extensively

employed to represent efficiency, the use of a single

subscript reflects the fact that the term is a function

of the parameters of the attenuator only and finally

the choice of the letter "a" is a concession to the

attenuation concept.)

In complete analogy to eq (2), and referring to figure 10, it is now convenient to write:

Pyl=PyMga'naNal (19)

9
GENERATOR ATTENUATOR LOAD

'ga <''al Figure 10. Basic circuit for discussion of attenuation. where the terms are defined as follows:

Mga=

Nal

Power delivered by generator-through

attenuator-to load, as previously defined.

Generator available power as previously

defined.

Mismatch factor between generator and

attenuator-load combination (equivalent

to M,,i except it is important to recognize

that the impedance at the attenuator

input is a function of both attenuator and

load parameters).

Maximum efficiency of attenuator. (Except when ria—l, it is obtained only

for a uniquely determined value of load

impedance.)

Ratio of the attenuator efficiency for the

given load to its maximum efficiency.

This is also a mismatch factor although

of a different nature than Mga.

In particular, (for a two-port) N„i is a. function only

of the attenuator and load parameters, whereas

Mga is a function also of the generator. This fact is reflected by the use of A'^ rather than M to represent

the mismatch factor. As in the previous case, each

of these quantities is invariant to the choice of

terminal surface within an arbitrary lossless region. In terms of rjn and Nai, it is also convenient to write:

r)ai = v<iN«i (20)

where 7}(ii is the efficiency of the attenuator of figure

10 when terminated by the given load. The application of the new concept to "attenuation" measure- ments will begin with a method for the measurement

of 7)(,lSee, for example, reference (9). p. 49. .

'^'^ The term Vi introduced here is not to be confused with the V i of the preceeding

section. ^^The a, b, introduced here are not to be confused with the«], h,. a^, 62, which

are used to represent the incident and emergent wave amplitudes at ports 1 and 2 etc. See a text on complex variable theory.

4.1. A Method for Measuring rjai Referring again to figure 10 it is possible to express r}„i in terms of the attenuator scattering

coefficients and the load reflection coefficient ^

riai- \S2i\Hi-\r,

|l-522r,|2-|(5i2S21-Sn522)r,+ Sn|-

(21)

Although the scattering coefficients are extensively employed in microwave circuit analysis, an

alternative description is more convenient in much

of the discussion which follows. At terminal surface

1 the reflection coefficient, Tin, looking into the attenuator can be written

where:

and

_ aYi + b ^'"~cr,+i

a — S12S21 S11S22,

6=S„,

(22).

(23)

S22.

The parameters a, b, c, thus provide a complete

description of the attenuator if reciprocity {812=821)

is satisfied.

It will also prove convenient to introduce the

auxiliary quantity rjri given by

T7W = |Q-6c|(i-|r,|^)

|i + cr,|2-|ar,+6|2 (24)

Comparison of eqs (21), (23), and (24) indicates

that r)ri and iqai are equal when reciprocity is satisfied. The functional relationship between Fjn and

Fi, eq (22), is in the form of a hnear fractional

transformation, the geometry of which plays an important role in the discussion to follow. In

general, let w and z represent two complex variables

which are related by

az + 13 yz+d (25)

It can be shown that a circle of unit radius and

center at the origin in the z plane is mapped into a

circle in the w plane whose radius R is given by

li all

R =

|8|2-|y|2 (26)

and whose center is displaced from the origin by

the (absolute) amount Rc,

Rc

138*- ay"

|8|2- |y|2 (27| m

' Nil 10
Now let port 2 of the attenuator be connected

to the output port of a g-reflectometer as indicated

|n figure 11 and let the reflection coefficients look- ing to the right at terminal planes 1 and 2 of the Attenuator be designated, by Fi and r2 respectively.

(The nature of the load or other termination on

jport 1 which, produces Fi is, as yet, unspecified.)

In terms of these definitions, and noting that, as jcompared with figure 10, the direction of the at- tenuator has been reversed,

F., aFr

(28)

Substitution of this result into eq (15), and noting

|lhat the F/ of eq (15) now becomes F2, yields:

b3_ {Aa-Bb)r, + B-Ac

64 {Ca-Db)r, + D~Cc (29)

fThe relationship between b3/64 and Fi is thus also

fin the form of a linear fractional transformation.--^

X |hi| X

G -REFLECTOMETER-A.B.CD

(Terminal Surfaces*!

1 I 2 1

Figure 11. Circuit for measuring r)^^ and t)^.

while if the sliding short is connected to arm 2 of

the g-reflectometer (in place of the attenuator) the

corresponding radius R-z is given by:

i?2 =

\AD-BC\

\D\-'-\C[' (31)

Taking the ratio of eqs (30), and (31) yields, after some simplification,

bclila--b

(32)

Comparison of this result with eq (24) indicates

that R1IR2 — iqai provided that CID — — T1 and

reciprocity holds. This type of boundary condition on C and D is identical to that introduced by eq

(14) and may be effected by the tuning procedures

for Tx already described and where the "load" of

figure 10 takes the place of the (inactive) generator of

figure 9. If the losses in the attenuator are small,

the attenuator itself can take the place of the "identical output sections" of figure 9. Otherwise it is necessary to either remove part of the input lead

(to the load of figure 10) and connect it to port 2 of the ^-reflectometer or else assume the sections are

identical. This aspect of the problem is completely

analogous to that previously described in conjunction with figure 9. Given the proper adjustment of Tx {C = — DYi)

. . . note that an adjustment of Ty is not required,

the efficiency 170; is thus given by the ratio of the

radii of the two circles Ri and Rz.-^ The measure- ment of these radii will be the subject of a later paragraph.

It is now instructive to consider the response of the g'-reflectometer (of fig. 11) with arm 1 of the attenuator terminated by a sliding short. For a

sliding short, r^ — e'^ where 0 is a function of the short position. The locus of Fi is thus a circle of unit radius. Comparing eqs (25), (26), and (29), (the transformed circle in the 63/64 plane has a radius Ri given by

^ _ \(B-Ac)iCa-Db) - {Aa-Bb){D-Cc)\

D-Cc\^-\Ca-Db\'

(30)

I " It is a general properly that any succession of linear fractional transformations

; is expressible as a single linear fractional transformation. ^ In the rase of uniform waveguide, a linear function of the short position. " The use of the impedance transformation properties, as a tool for the measurement

of efficiency, has been described by a number of authors. (See references [10, 11, 12,

'13].» However, the present method is useful with terminations of arbitrary impedance,

avoids the measurement of impedance (in the usual sense), and, except for the reel- .procity requirement, is applicable to two port devices of arbitrary characteristics.

4.2. Measurement of rjn The extension of these techniques to the meas- urement of maximum efficiency begins with a determination of that value of load impedance for which the efficiency is a maximum. This could be

obtained from eq (21) by first choosing the argument

of F; (by inspection) such that the denominator is a minimum and then differentiating the resulting

equation with respect to |F;|. The same resuh,

however, may be obtained by an alternative and

somewhat shorter route. Returning to figure 10,

if the parameters of the generator, attenuator, and

load are taken as arbitrary, it would, in general, be

possible to achieve an increase in the power

deUvered to the load by the introduction and ap- propriate adjustment of lossless tuners at terminals

1 and 2. The only exception would occur if the generator and load were such that a conjugate impedance

match already existed at these terminals. Under

these conditions, maximum power is delivered to the load, and the attenuator is evidently operating

352-570 0-69—

11
at maximum efficiency. Thus at terminal 1,

aT,-\-b

1r</ *= r-1 in = (33)

while at terminal 2,

Elimination of F,, between these equations yields

(34)

{c-ab*)r,+ (l-|fe|2-|al2+|c|2)n

+ {c-ab*)* = 0. (35)

This represents the (implicit) condition on F; for which the maximum efficiency is obtained. Equation

35 may be solved for F; and the result substituted

in eq (21) to yield an explicit expression for rja. This

result is not required in the present context, but

is given in a following section.

Next, returning to the system of hgure 11,

it will be assumed that the parameters A . . . D

satisfy eq (11) (as provided by the adjustment

of Ty). In addition, let the ratio CjD (= — F,) be chosen in such a way that the circular locus,

obtained with the sHding short at terminal 1, is centered at the origin of the 63/^4 plane. (Note

that the circle obtained with the short on terminal

2 is centered by virtue of eq (11). The parameters

of the ^-reflectometer and attenuator combination

may be obtained by inspection of eq (29). Substitution of these in eq (27), and assuming Rc = Q

yields:

\{B-Ac){D-Cc)*-{Aa-Bb){Ca-Db)*\=Q

(36)

and imposing the condition of eq (11), and letting

Yi=-CID yields:

\AD\ \{c-ab*)T\^{\-\b\'-\a\'^\c\')T,

+ (c-afe*)*l = 0 (37)

Comparison of eqs (37) and (35) shows that the condition on the ratio CjD which provides /?c = 0 is identical to that for obtaining the maximum

efficiency.

The procedure for measuring maximum effi- ciency, when reciprocity is satisfied, is thus as

follows. Referring again to figure 11, port 2 of the

^ This may be recognized intuitively. If with the sliding short connected to arm 2, Tuner Tx is changed, this is equivalent (in terms of the system response) to motion

of the short. However, if Ty is properly adjusted, the response is invariant to this motion. ^ Alternative procedures for measuring T)a will be found in references (7) and (14). ^Indeed, this would probably be true, for many applications, even if the error were a first order effect.

^' If for any reason a more accurate realization of the assumed tuning conditions (Equations (11) and (37)) is required, this could be achieved, despite the losses inTj,

bv repeating the procedure several times. However, the need for this is not envisioned.

g'-reflectometer is terminated by the shding short

and tuner Ty adjusted such that the ratio 163/64^ = P^IP^ remains constant while the short position

is varied. As previously noted, this estabUshes the

BD* —AC* = 0 condition. The attenuator is next connected to the g-reflectometer (as shown in

fig. 11) and the sliding short to the attenuator terminal 1. Tuner is then adjusted such that F3/P4 again remains constant as the short position

is varied. Provided that tuner Tx is lossless (as

will be assumed for the present) the adjustment

of Ty, as described, is invariant to the adjustment

of Tx- The ^-reflectometer parameters are now

such that eqs (11) and (37) are satisfied and the

maximum efficiency is obtained in terms of the

ratio of the radii of the two circular loci as already

described for r)ai.^^ In practice, tuner Tx is not lossless and some

interaction between the two adjustments is possible.

It is noted, however, that the adjustment of Ty is required only as part of the procedure for obtaining the correct adjustment of Tx which, in turn,

established the ratio of C/D corresponding to that value of F; for which the maximum efficiency obtains. The efficiency measurement itself is independent of the adjustment of Ty. Thus, the only

effect of losses in Tuner Tx is in obtaining an

efficiency measurement for a load which differs from the one which yields the maximum efficiency.

However, since the dependence of the efficiency

upon the load impedance vanishes, as the maximum is approached, a first order error in the ad- justment of Tx wiU yield only a second order error

in the determination of tjq. This is expected to be

completely negligible in practice.^"' In addition to reciprocity considerations, these

techniques for measuring 17 « and 17a/ are further

restricted, by practical considerations, to rather

small values of "attenuation" ~ 10 dB or less. This may be easily recognized from figure 11 where

for large "attenuations" F2, and thus the ratio 63/64 (in terms of which the measurement is made),

becomes independent of Fi. On the other hand, in their range of practical apphcation, these techniques

are easier to implement than the more general

procedures which will follow. First, however, it will prove desirable to consider the measurement

of N„,.

4.3. Measurement of iVal As already noted, Nai is a mismatch factor, and

is, by definition, equal to the ratio rjailyia. Thus the

measurement techniques for tjq and 7)ai, already

described, provide an imphcit determination of

Nai- However because of the practical hmitations,

already described, an alternative procedure is called for. This will now be developed.

As previously noted, Nai is a function of the attenuator parameters and of the terminating

12
load (F/). It is convenient, however to write^^

(i-|r,|^)(i-|r,„p)

Nal =

|i-r,r*|2—r,^;|r,-r.r^

|r,-r,j2

(38)

the displacement, from the origin, of the circular

locus obtained v^ith the sliding short on terminal 1. In terms of Tjr, F/, and Fm, this ratio is given by: ^'^

= 1 |1-F/F^^

|F,-F;„|^

|i-r,F*|2

(39)

^_(i Vr) |i_r,F*

i—n'i

\Ti-T, (42)

|i-r,F*|2

where Fm is that value of load reflection coefficient

for which the maximum efficiency is obtained,

and Tjr is given by eq (24) with F,„ substituted

for F/. (If reciprocity is satisfied, iqr—r),,.)

Referring again to figure 11 it will be assumed

that the parameters of the g--reflectometer satisfy

the two conditions C + Z)F, = 0 and BD*-AC* = Q. (The adjustment of Tx and Ty, by which this is obtained, is as previously described.) In section 4.1 it was shown that 7]ri is given by

As a matter of convenience, in discussing eqs

(41), (42), it will be convenient to make the following

substitutions

Ri

Rci R:

- = v

R2 (40)

where Ri and /?•> are the radii of the two circular

loci obtained with the sliding short at terminal

surfaces 1 and 2. It is also possible to express

Tjri as follows: ^

in-r„,p

"^A' |1-F,F*|

|F,-F,„| _

|1-F,F^| ^•

Equations (41), (42) now become:

1

Rz

1

(43)

(44)

(45)

(46)

(47) = Vrl = -

1 F,„p (41)

|i-r,F:|2

In addition to this expression, the determination

of Nai also involves the ratio Rc\IR2 where Rci is

Equations (46) and (47) may be solved for g and

T/r in terms of the measurable quantities u and v. This is conveniently done by first squaring both

sides of eq (47) and then eliminating^ using eq (46). This leads to the following expression for Tjr:

1

= u

while g is given by:

7Jr = 2^{l+u2-j;2- [(l-a2_j,2)2_4„2j;2]l/2l

g =

Y^

[(l_u2_j^2)2_4„2j,2]l/2}

In terms of v and g, Nai is given by

so that finally:

Nat=l-Vg

A^a/ = 1-2{1 +V'-U^- [(l-a2-j;2)2-4ii2j;2]l/2}

= 1 1 + 1 +

The priMjf of ihese equations will be found in appendix 6. ^This resuli may be obtained from eq (39). and using the relation T)r//i)r=/V„,. " See appendix 7.

1-U2

13

-+

(48)

(49)

(50)

(51)

(52)

(53)

(54)
For a wide range of practical application the

series converges rapidly and eq (54) is the more

convenient of the two expressions.

Equation (49) suggests an alternative method

of measuring rja when reciprocity is satisfied. However as previously noted, the accurate measure- ment of u becomes difficult when its value is small

and this results in a first order error in rja- On the

other hand, as inspection of eq (54) shows, the value

oiNai has a very small dependence upon u and the

errors in its determination.

Comparison of eqs (38) and (39) with eq (18) in- dicates that for large values of "attenuation" (small

values of -qn) the expressions for Na* and Myi be- come identical if Ym is replaced by Tg. This is consistent with the relative independence of F,,, upon

F/ under the same conditions.

4.4. The Measurement of Small Values of 't]a As noted in a preceding section, the previously

described techniques become impractical for small

values of rja. This will become more evident in the following section where the actual measurement

procedures are explained in greater detail. This

section will develop a technique which is appropriate for small values of r)n (large "attenuations")

and which is also applicable when reciprocity is not satisfied. Returning to figure 10, it will prove desirable to employ a description alternative to that associated

with eq (19). In order to make this distinction explicit, however, the first description will be briefly

reviewed. In eq (19) the product iqaNai is equal to the efficiency rjai which the attenuator provides

when terminated by the given load. By definition,

this efficiency is equal to the ratio of the power

delivered to the load, to that delivered to the at- tenuator. The power delivered to the attenuator

is, in turn, given by the product of the available

power P,i and the mismatch factor Mya between the

generator and the attenuator-load combination. The

analysis thus first associates the attenuator with the

load (through the dependence of iq„i upon F;). The

attenuator-load combination is then interpreted as a new "load" and the mismatch between it and the

generator analyzed as was done for figure 1. Notwithstanding the utility of this interpretation

in many applications, an alternative analysis is more useful in the present problem. Instead of

associating the attenuator with the load, its effect upon the system in figure 10 may be interpreted as modifying the generator parameters in such a way

as to reduce the available power and, in general, of

changing the source impedance. The mismatch

Mai, between this new source (the generator-atten- uator combination) and the load, may then be evaluated as was done for figure 1. Let q,jn equal the ratio of the available power at terminal 2 (fig. 10) to that at terminal 1. It is then

possible to write:

Pgl = Pgq.jaMal. (55)

In terms of the generator and attenuator parameters, qga is given by:^^

\S.n\H\-\Ty\^)

^

|1 — SiiFyl ' — I (S12S21 ~SiiS22)F(, -f S22I"

(56)

Comparison of this expression with that for r)ai (eq

(21)) indicates that the two are identical if Fj, is substituted for F/ and the terms Sn, S22 are inter- changed. Provided that reciprocity is satisfied, qga

is thus equal to the efficiency which obtains for an assumed energy propagation in the reverse direction,'" and with the generator impedance taking the

role of a terminating load. The quantity qga, in its functional form, is thus closely related to efficiency,

and this aspect will be exploited in the discussion

to follow. By definition, F,„ is that value of F/ which satisfies eq (35). Solving this equation and substitution in eq (21) yields:

2 1521 p

1 + |5l2.S21 -Su522|^ - |.Sn|^ - |522|^

\ (1 + |S:2S2, -SnS22|2 - \Sn\' - \S,2m

(57)

In common with the foregoing treatment of

efficiency, it will prove useful to write = QijaQa (58)

where qa is the maximum value of qga (obtained

by varying Yg) and Qga is defined by eq (58). An

explicit expression for qa may be obtained, as was

done for tjo, by first finding the value of Tg for which

qa is a maximum and then substituting in eq (56). If this is done it is found that,

''^For ird and |r/„| both less than 0.2 (which corresponds to VSWR of 1.5) and of arbitrary phase, and with "attenuations" as small as 3 dB, the maximum error ir N„i which results from truncating the series after the second term is less than 0.001

percejTt.

3^ At terminal 2, in figure 10, the equivalent generator reflection coefficient PJ, is given by V',, = {aVg — c) I {— br,,+ I) and the available power from the generator- attenuator combination is achieved when F* = V',,. Under these conditions M,i„'f)<,i. Combining these expressions and using eq (21) yields eq (56).

'^ In general the efficiency of a two-port is a function both of the terminating load and the direction of energy propagation. Except as otherwise noted, however, the assumed direction will be from left to right.

Qa^Vn- . (59)

In a completely similar way it is found that the

expression for Qga and Nai are identical, if F^ is substituted for Ti and F„, is understood to represent

that value of generator reflection coefficient for which qa is a maximum.

14
It is thus possible to drop the distinction between

Qa and y)n or between Q and N and write eq (55) in the form:

Py, = PgNya'n„Mal. (60)

The procedure for measuring N,ja is also identical

to that for measuring Nai with the exception that it is now port 1 of the attenuator which is connected

to the g-reflectometer. The technique for measuring small values of 7],,, now to be described, is based upon eqs (2) and (60). Referring to figure 12, the attenuator for which

Tja is to be measured is connected between a

g'-reflectometer and a tunable power detector of

wide dynamic range. It is assumed that tuner Ty

of the g-reflectometer has been first adjusted such

that eq (11) is satisfied, while the adjustment of Tjis abitrary. The tuning transformer which is part

of the power detector is then adjusted for a maximum power indication,^* so that the term Mai (eq

(60)) is equal to unity.'"'

G-REFLEC- TOMETER

ATTENUATOR %

III

WIDE RANGE

POWER

DETECTOR Pd

Figure 12. Circuit far measuring small values of T)a.

By hypothesis, the response of the power detector

is proportional to the power which it absorbs. If

this power indication is denoted by Pit, then

hP, = P, (61)

I where /i is a proportionality factor. In terms of the notation employed in the section on power measurement, the available power,

Py. at the output of the (§'-reflectometer is equal to KiPji where K\ is, as yet, undetermined. After

making these substitutions, eq (60) becomes:

hPda = KiP,„Nyaina (62)

where the subscript a has been added to Pa and

P4 to signify quantities measured with the atten- uator inserted between the g^-reflectometer and

power detector as indicated in figure 12. In a

similar manner, the subscript b will denote the

observed values with the attenuator removed.

^ More correctly, the adjustment is such that the power detector indication is a maximum relative to of theg-reflectometer. ^ It is not necessary to assume that the transformer which provides this adjustment

is lossless since this is taken care of in the procedure to follow.

Then,

hPdb^KxPibMyd (63)

and taking the ratio of eqs (62) and (63) yields. _ PdnPibMyd

PdbP4aNya (64)

The efficiency Tja is thus determined in terms of the

observed quantities Pia-, Pib-, Pda, Pdb, and in terms

of Myd and Nga. These last two quantities are measured as previously described. In brief, the

measurement of Nya calls for the replacement of

the power detector by a sliding short and certain

measurements of P3 and P4, while the measurement

of Mgd requires the values of P3 and P4 with the

power detector and the short connected directly

to the ^-reflectometer.

It is of interest to compare this technique with

the existing art. In the usual attenuation measurement, the ^-reflectometer of figure 12 is replaced

by a matched generator and the power detector

also adjusted for an impedance match. The attenuation is then obtained from the ratio of the power

detector readings, with and without the attenuator

inserted. The wide dynamic range for the detector

is usually achieved by means of another attenuator which may operate either at the signal frequency

or, if hetrodyne detection is employed, at an inter- mediate frequency.

If one considers the g-reflectometer of figure 12 as the counterpart of the necessary instrumentation for establishing the impedance match in the attenuation measurement, the instrumentation

requirements of the two measurements are nominally equivalent. The maximum efficiency measurement, however, requires a number of additional

observations, most of which are associated with

the determination of Myd and A^^a.

4.5. Measurement of R and Rc

As outlined in the preceding sections, the meas- urement of 7]ai, i7rt, N„i has been reduced to the

determination of the parameters R and Rc (eqs (26)

and (27)) for the linear fractional transformation,

which relates the ratio 63/64 to the reflection coefficients which obtain at the different terminal

surfaces. It is the purpose of this section to describe

the measurement procedure in greater detail.

It has been previously noted that the response of

the ratio 63/64 to the motion of the sliding short is in the form of a circular locus whose radius is R

and whose center is displaced from the origin, in the 63/64 plane, by the amount Rc- The relationship

between these quantities is as indicated in figure 13. In actual practice, arms 3 and 4 of the g-reflectometer are provided with power detectors such

that the measured quantities P3 and P4 are pro- portional to I63P and 164!^ respectively. Since only

15
Figure 13. Illustrating the relationship between R, Rc, and

b3/b4.

power ratios are involved, these factors may be

taken as equal to unity (as was also done in eq (16)). As the sliding short is moved, the ratio P3IP4

thus varies and inspection of figure 13 indicates that

P4

1/2 = R + Rc (65)

63 fPs

64 =

min {P4 min / {R-Rc) (66)

where the upper sign is used if, as shown in figure

13, R ^ Rc', otherwise the lower sign is used.

These equations may be solved to yield:

Rc =

P.

P4

P3

P4

1/2 P4

max / \ ' 4

1/2

(67)

(68)

Measurement of the maximum and minimum

values of the ratio P3IP4 thus determines R and Rc

but with an ambiguity as to which is which. As a

practical matter, this ambiguity can often be re- solved on the basis of an approximate knowledge

of the magnitudes of the parameters involved. If not it can be easily resolved by introducing a vari- able attenuator ahead of the sliding short, and using

the upper or lower signs depending respectively

^"The basis for this test is as follows. In the usual microwave circuit description, a shding short represents a reflection coefficient of unit tnagnitude and variable phase.

In appendix 8 it is shown that reflection coefficients of magnitude less than unity are mapped into the mferior of the circle of figure 13. Thus if the circle encloses the origin

it will be possible to make vanish for a suitable reflection as provided by the attenuator-short combination. Note that it is not necessary to carry this procedure to the point where the null is actually achieved. Starting from the point where the ratio Ib^lb^l is a minimum, the introduction of dissipative loss (via the attenuator) will increase the value of [63/64!

if the origin is outside the circle, decrease it. if the origin is inside.

*" In the measurement of tj„;, as prescribed in section 4.1, the adjustment of Ty was

arbitrary, and this could in principle be used to make/?c vanish, such that the -4- sign would apply in eq (67). In the measurement of 17^ the specified adjustment is, ideally, such that ft- vanishes. There is, however, a practical limitation as to the precision and

stability with which the required adjustment can be made. This limits the techniques

of sections 4.1 and 4.2 to small values of attenuation.

upon whether or not, by a suitable adjustment of

the attenuator and short position, it is possible to make 63 vanish.^** The earUer comments about the practical hmi

tations in certain of the described techniques foi measuring small values of r)a and 7}ai may now be

better appreciated. Referring to eq (67), it is the

lower sign which will usually apply if rfai is small,

thus the quantity R, to which the efficiencies are proportional is the difference between two measurements, which individually may be rather large as compared with their difference.'" A similar observation is true for the determination of Rc when it is small with respect to R. The inability to accurately measure small values of Rc is, however, of no con- cern since, as eq (54) indicates, Nai — I — R%JR%

and thus the dependence upon Rc vanishes as Rc

approaches zero.

4.6. Cascade Connected Attenuators

For completeness, it should be noted that there

is a third type of mismatch factor which is of interest

in conjunction with cascade connected attenuators. Referring to figure 14 the power delivered to the

load can be written:

Pyl PgMya^Tf (ijNa ^ail"^ a^l (69)

where except for the addition of the subscripts

1 and 2, to identify the two attenuators, the terms

have been previously defined. Note however that

in the fourth term the additional subscript / is required to indicate that it is the mismatch between

attenuator 1 and the attenuator 2 and load combination, which follows, which is meant. (In an alternative formulation, one might be interested in the

mismatch between attenuator 2 and the generator- attenuator 1 combination which precedes it.) Alternatively, the power delivered to the load in figure 14 may be written (cf eq (19))

P„,^P,jM„aVaNa (70)

where the two attenuators are now considered to comprise a single unit as reflected in the subscript A.

GENERATOR

P9

ATTENUATOR

\,

AHENUATOR LOAD

M9.01 "02,1

Figure 14. Basic circuit for discussing cascade connected

attenuators.

16
In terms of the individual efficiencies 17.4 may be

i written,

T)A = Tqa,Ra,a2Va2 (71)

where Rn,at is a mismatch factor which, from

physical considerations, must lie between zero and

unity. In eq (70) the term Mga, of eq (69). has been

written as MgA since this is more in keeping with

the new interpretation and notation. However, this reinterpretation does not change the load impedance

which is presented to the generator, thus the two

terms are identical. Comparison of eqs (69), (70), and (71) thus indicates that:

An experimental determination of Ra,a2 may thus

be eff^ected by a measurement of the N's, as already

described, and use of eq (72).

5. Relationship to Existing Art

The relationship between the new concept and

the existing art can perhaps be best illustrated by an analogy. The usual circuit description of

a microwave system (by means of the impedance or scattering parameters) represents a substantial

simplification as compared with a description in terms of the field quantities. To a large degree

this simpUfication is the result of suppressing

a great deal of the information which is implicit

in the field description, and focusing attention on those properties of the individual components

which are observable at the respective terminal

surfaces. The new concept carries this simplification

a step further by suppressing much of the detail which is involved in the circuit description, and

instead substitutes the parameters P, 7)a, M, N. R. These permit a system evaluation which is adequate

for many practical purposes and because of their

invariant properties the precision waveguide and

connector requirement is avoided. It may be noted,

in this context, that the usual specification for precision waveguide and connectors results not from ihe fundamental power transmission requirements, but rather to permit a system evaluation.

This elimination of the precision waveguide

and connector requirement is significant both as a consequence of the elimination of this source

of error and also in that it anticipates an appHcation of existing technology to more general types

of systems — e.g., those in which a well defined

**Nol to be confused with the radius of the circular locus introduced in eq (26). As previously noted, it is the (postulated) existence of a lossless region, where the interconnection takes place, that permits the terminal surface to remain arl>itrar>'. Alternatively, the need for this lossless region may be avoided by specifying the

location of the terminal surface.

terminal plane between the diff"erent components

may not exist.*^ In addition, while the described

techniques for the measurement of P, tjq, M, N, R,

are hmited to the case where the perturbed cross

section does not approach the point where multimode propagation becomes possible, the basic

description (in terms of P . . . R) is applicable,

and of potential value in multimode systems. In

fact, the definitions of Pg.Pyi, Mgi are immediately

applicable on a multimode basis. However because

the efficiency, in a multimode problem, depends

upon the generator as well as load parameters,

the given definitions of 17, A^, R would need to be

generalized.

In order to illustrate some of the advantages

which are provided by the new concept, it is useful

to consider the coaxial connector problem. For

many years, the lack of a precision coaxial con- nector was a major hmitation to accurate measure- ments in coaxial systems. The widely employed

Type N connector suffered from the following problems: (1) impedance discontinuity, (2) lack of a well defined terminal surface, (3) its unipolar nature,

(4) nominal diff"erences between "identical" units

due to manufacturing tolerances, and (5) lack of

repeatability when the unit is disconnected and

reconnected. Among this list of shortcomings, only

the last is of concern in the new concept, where it is still necessary to postulate that a device can be

removed and then replaced without changing the

overall system.

Although the new concept calls for the substitution of mismatch factor measurement for impedance

measurement, this does not imply that impedance

measurements are obsolete. In particular the impedance concept makes it convenient to write sys- tem specifications in such a way that the mismatch

factors will be nominally equal to unity and to identify the faulty unit if this design objective is not achieved. The mismatch factor measurement,

on the other hand, permits a direct evaluation of

how well the design objectives have been reahzed

without regard for possible connector discontinuities.

Finally, the described techniques for measuring

M, N, etc., have been developed on the assumption

that the detectors on arms 3 and 4 of the g-refiectometer do not respond to phase information. In

recent years, more sophisticated types of detection

systems have been introduced for use in stepped

frequency measurements, and which include phase

sensitive detectors, memory, and a computer. Given

this type of capabihty, a substantial simplification

in the described procedures may be effected, but

the development of this topic is outside the scope

of this dissertation.

6. Summary

Historically, the quantities power, attenuation

and impedance have been accepted as fundamental

in the description and evaluation of microwave sys17
terns. As noted in the preceding section, the usual

precision waveguide and connector requirement is often imposed to permit the specification and evaluation in terms of these parameters, rather than as a

result of more fundamental considerations.

The new concept, which has been outlined in this dissertation, proposes an alternative description in which power, maximum efficiency, and mismatch

factor (M, A'^, R) are the fundamental parameters.

The advantages of this concept are basically twofold. First, the inevitable departures from waveguide

uniformity in any physically reahzable system no

longer impose a limitation on its experimental

evaluation, and precision measurements are now

possible in coaxial systems which are fitted with

the Type A'^ connector, for example. The second

advantage of this concept is in the more easily

understood physical model which it provides, where

instead of taking account of the traveling wave

amplitudes at the several ports, the attention is directed to the power transfer characteristics. (The

terms M,,,, and Nai in eq (19), for example, take the

place of a rather complicated algebraic expression

involving the six complex reflection and scattering

coefficients.) As a consequence of this simplification,

it should now be practical to instruct field personnel

in the use and application of mismatch corrections.

It should perhaps be noted that many of the

basic parameters, upon which the new concept

is built, are not new. The quantities which are rep- resented by Py, rja, Mfji, for example, are well

known in the prior art. The contribution of this dissertation, is in the recognition of the invariant

features of these parameters and the introduction

of others such that independence of connector

discontinuity is achieved. In addition, practical

techniques, for the measurement of these terms,

have been developed and described.

For completeness, it should be observed that

while the analysis of nonuniform transmission

lines and waveguides is a subject of current interest, the emphasis in this dissertation has been on

the development of a measurement theory and

practice rather than a detailed description of the

fields associated with nonuniform waveguide.

For convenience, the measurement procedures

have been developed in terms of the existing

circuit representation and then examined for their

dependence upon uniformity requirements. In

particular these procedures are characterized by

their dependence only upon a phasable short, and

methods for recognizing the equality of two "impedances." It is possible that these results could

be obtained more directly from an alternative

formulation.

Finally, the discussion has been limited primarily

to the power measurement problem, with attenu-

"^ This may be demonstrated in several ways. From physical considerations it is evident that IFi,,! = 1 when IHI = 1. In terms of eqs (26) and (27) this calls for R=l,

/?r = 0. and which, with the appropriate changes in notation, leads to eqs (A—3) and

(A-4).

ation and impedance taking a supporting role. The potential application of this concept to phase

measurement and multimode systems remains to be investigated.

7. Appendixes

7.1. Appendix 1 A formal demonstration of the invariance of

Myi, to a shift in terminal plane, may be effected

with the help of figure 10. In particular it will be

shown that Mya (obtained at terminal 1) is equal

to Mai (obtained at terminal 2) provided that the "attenuator" is lossless. At terminal 1 in figure 10 the reflection coeffi- ^ cient, Tin, looking to the right is given by:

aTi + b cr,+ 1 (A-1)

while at terminal 2, the reflection coefficient,

r'y , is given by:

F' aFy — c -bVy+l

(A-2)

The terms Fj,, and Fy replace F/ and Tg respectively in eq (18) to yield My,, and Mai.

I'or a lossless and reciprocal twoport, the parameters a, b, c, satisfy the conditions*'*

\a\ = 1 (A-3)

b-ac*^0. (A-4)

Combining eqs (A-4, A-1), and (18) yields:

|aF,+ ac*-F*-cF;F||^

Hi

e(|

M, 1 _ I— — " "

'— (A-'=,\

\l + cri-ar,ry-ac*Yg\^

Since |a| = l, the numerator of the second term

may, without changing its value, be multiplied

(inside the absolute value signs) by a*. Doing

this, rearranging, and again using eq (A-4) yields,

Mga^laVg — c -bFy+l Ff

1-F, bVg+l

(A-6)

which by comparison with eqs (A-2) and (18)

is equal to Mai7.2. Appendix 2

Referring to figure 4, it will be shown that eq (11)

is implied if the ratio P;s/f*4 remains constant for

all positions of a sliding short on arm 2.

18
With arm 2 terminated by a sliding short, 62 ||and 02 are related by

a-i — bie' (A-7)

Substitution in eq (15), and taking the absolute

value yields.

Pi

Ce'^+D (A-8)

By hypothesis the ratio P3IP4 is a constant which

will be denoted by K. Substitution of K and expanjsion of eq (A-8) yields,

'I

I \A\-'+ \B\-' + AB*e'^ + A*B€-'<^

-Ki\C\-^+ IDl-' + CD^ei" + C*De-'»)=0. (A-9)

If this equation is to hold for arbitrary values of

j 9, the coefficients of the several powers of e'^ must

vanish independently. This leads to: AB*-KCD* = 0, (A-10)

\A\-'+\B\-'-K{\C\-'+\D\-^)=0. (A-11)

I (The third equation is the complex conjugate of

' eq (A-10) and contains no additional information.)

Elimination of K between eqs (A-10) and (A-11)

yields:

CD*i\A\'- + \B\-') = AB*{\C['+ \D\-') . (A-12)

Multiplication of this expression by BjB* and using

the result A*BCD* = AB*C*D (which foUows from

eq (A-10) since K is real) leads to:

A-'C*D-AB(\C\^ + |Z)|-) + B'CD* = 0 (A-13)

which may be factored to yield:

{AD-BC){AC*-BD*)=0. (A-14)

This equation admits of two solutions. However,

the AD — BC = 0 solution is trivial in the present

context since it yields a value for P3IP4 which is completely independent of the terminating load. (As a practical matter, this adjustment would

result if one of the couplers were reversed, and is useful in a different application.) This leaves the

condition AC* — BD* = 0, which was to be proved.

In figure 15 let there be an enforced current aiJ in one portion of the structure where J is a three dimensional vector function giving the spatial

distribution and «) is a scalar multiplier. The

current aiJ is the source of an electromagnetic

field throughout the structure and is, by hypothesis, independent of the termination on the uni- form section.

Source

Uniform

Waveguide

Figure 15. Electromagnetic system used in deriving basic

reflectometer equation.

Let the uniform section (arm 2) be first terminated by a matched load. Then the field in arm 2

is completely specified by the term 62 which is a measure of the electric field amplitude associated

with the outgoing traveling wave. In another part

of the structure, let the amplitude of the electric

field component in an arbitrarily chosen direction

be denoted by 63. If the system is linear, it is possible to write,

62= hai

(A-15)

63= kai (A-16)

where h and k are constants. Next let an incoming wave, the amplitude of which will be denoted a-z, be launched in arm 2.*^ By superposition, 62 and 63 now become:

62 = ha\ + ma-z

63 = ka\-\- na-z

(A-17)

(A-18)

where m and n are two additional constants. Elimination of Oi between eqs (A-17) and (A-18) yields:

where

63 = ^02 + 562

hn — km

A = ;

,

(A-19)

(A-20)

7.3. Appendix 3

It is the purpose of this appendix to demonstrate

that for the usual reflectometer, the uniform

waveguide requirement appHes only to the output

arm.

(A-21)

If, in the same way, the component of the field amphtude in a given direction, at another position, is denoted by 64, it is possible to write.

** it makes no difference whether this is achieved by means of a source on arm 2. or by a partial reflection of the wave bz. 64 = Ca-z+Dbo (A-22)

19
where C and D are two additional constants of the

system. Taking the ratio of eqs (A-19) and (A-22)

yields eq (15), which is the starting point in the

usual reflectometer development [3].

7.4. Appendix 4

It will be shown that the adjustment of Tx, to achieve the C + DTg= 0 condition, is invariant to the adjustment of Ty.

This could be done in a completely formal manner by finding the appropriate expressions

for C and D in terms of the parameters of the

two couplers and tuning transformers; this ap- proach, however, is tedious. Instead, in figure 6,

let an auxiliary terminal plane be temporarily in- serted between the tuner Ty and the second coupler

(the one on the right). Within this second coupler

(including Tx) the electromagnetic fields are completely determined by and 62 and, in particular, the field in arm 4 can be written

64 = C"a2 + Z)'62 (A-23)

where C and D' are functions of the parameters

of the second coupler and Tx (but not of Ty). If

64 is eliminated between eqs (15) and (A-23) it becomes apparent that a nontrivial linear rela- tionship exists between 63, 62, and 02 if, and only

if, CjD = CjD' . Since the ratio CID' is invariant

to the adjustment of Ty, the same is true of CjD.

Thus, if this ratio is such that eq (14) is satisfied for one particular adjustment of Ty, the same is true for all possible adjustments.

7.5. Appendix 5

It is the purpose of this appendix to investigate,

analytically, the wave propagation in a perturbed

waveguide.

Beginning with figure 16 it will prove useful

to consider the perturbed waveguide and a circumscribed uniform waveguide. Within the volume V,

which is bounded by the uniform waveguide and the

Uniform Waveguide

Perturbed Waveguide

Figure 16. System used to analyze wave propagation in per- turbed waveguide.

transverse surfaces Si and S2, the electromagnetic

field vectors, E, H, of course, satisfy Maxwell's J equations which, for harmonic time dependence,

may be written:

V X E = -j(o/jiH

V xH=y&>eE + J,

(A-24)

(A-25)

where € and fj. are constant. In particular, the effect of the perturbed boundary is accounted for by the * current density, J. Taking the curl of eq (A-24) and

substituting from eq (A-25) leads to V X V X E = A:2E -ico/LtJ (A-26)

where k'^ = (i)'^tx€. In general the electric field E which satisfies eqs (A-24), (A-25) may be expanded in the form: *^ E = 2 U«(e« + Ea^)e-ya^ + Ba (e« - E«)

]

(A-27)

where a is a mode index, e° and E,,^ are vector

functions of the transverse (but not axial) coordinates, with in the transverse plane and Eaz in the

axial direction, here assumed to be the 2 axis. In eq (A-27) the vectors E^= (e« + E„,)e->a- and E;= (e2-Ea^)e^«-

are the modal solutions of Maxwell's equations for the unperturbed waveguide. Finally, the complex

scalar factors Aa and Ba are functions of z, but not

of the transverse coordinates. Inspection of eq

(A-27) shows that Aa and Ba represent the amplitudes of waves in the positive and negative z directions respectively.

In complete analogy with eq (A-27), the asso- ciated magnetic field H is given by:

H 2 (h;; + H„,) e-V

BaiK - H„.) eya^]

.

(A-28)'

Although a complete definition of the basis fields

e)J, requires two normalization conditions, the

discussion to follow requires only the power normalization which is assigned as follows:

I

e„o X • n c?5 =

1 (A-29)

«cf reference 16, pages 170-223. See for example, reference 15, p. 250.

where the integration is over the transverse plane

and n is in the positive z direction.

It is next necessary to introduce the vector analog

of Green's second identity which states that given

a closed region of space, bounded by a regular sur20
face S, and two vector functions of position P, and

Q, with continuous first and second derivatives,

the following relationship holds:

I

(P XV X Q -Q xVxP) -nds

(A-30)

(Q -VxVx P-P -VxVx Q)dv

where n is now the outward normal from the sur- face which bounds the volume V

In eq (A-30), let the volume of interest be that which is bounded by Si, S2, and the unperturbed

guide in figure 16. Then let: P = E

Q=E;=(e2-E„.)e>a^

which, with the help of eq (A-26), leads to

(ExVxEa"-E,7 XV XE) • n ds

(A-31)

I

^ (A-32) = 1 [E„ • (FE -7w/ttJ) - E V X V X E-] dv.

As already postulated, E„ is a solution of Maxwell's equations, and eq (A-32) may be simplified

with the help of the following:

V X E„ = -jiofxHa (A-33)

VxVxE; = A:2E; (A-34)

ll~=i-K + Haz)eya'. (A-35)

where

After making these substitutions, eq (A-32)

becomes:

1^ (ExH„-E„ XH) -n (/5 =

|

E„ - J dv.

(A-36)

Since the boundary conditions on a conducting

surface require that the tangential electric fields vanish, the evaluation of the left side of this equation requires only the tangential components of

E and H on the surfaces Si and S2. If the tangential

components on Si are represented by Eu and Hk

respectively eqs (A-27) and (A-28) yield:

(A-37)

* See reference 9.

Hh=2 [(^i6e->6-i-fii6e^ft-i)h«]

(A-38)

where Zi is the value of 2 at terminal surface 1, and

Alb, Bib are the coefficients of the expansion which

give the required field at this terminal surface.

If in eqs (A-37), (A-38) the subscript 1 is replaced

by 2, the corresponding expression for the fields at terminal 2 is obtained.

Equation (A-36) may be now written

X ( -h«eVi)

-e^a^ie^X 2 (^166-^6^1 - 5,66^6^1 ) 1 • n dsi

b -II + 2 (^26e->6^2 + 5266^6^2) eg X (-h«eV2)

L 6

I

2be-yb'2-B2ben'-2)hl n ds >

= [e»a-Ea,]eya^ J dv (A-39)

This expression may be simphfied by cancellation

and use of the orthogonality relations:

Sse^a><H-nds = 0. (A-40)

a¥^b

So that eq (A—39) becomes:

2^ la (- e« X hS )

• n <f5 + /,2 2/< 2a (- e« X h«) • n c?5 ^ Sv{e^-Eaz)eya^-Jdv. (A-41)

Finally, observing that the outward normal from the

volume V in figure 16 coincides with the positive z direction at S2, but is opposite to it at Si, and using

eq (A-29) gives:

^2« = ^ia-/.,(e°-E„^) -Jeya^dv. (A-42)

Equation (A-42) is interpreted as follows. At

terminal 2 the coefficients Aia, Aib, ^2c, • • • rep- resent a collection of waves whose direction of

propagation is in the positive 2 direction. From

eq (A-42) each of these coefficients is comprised

of two parts . . . the complex amplitude which

would obtain in the absence of the perturbation,

plus a contribution due to the perturbation. Note

that whereas the coefficients of the field expansion

at terminals 1 and 2 are equal in the absence of the

perturbation, the expression for the transverse field

(cf. eq (A-37)) also includes the propagation constant e^a^ such that the usual attenuation is obtained for those components of the field which

represent evanescent waves.

21
With regard to the second term on the right of

eq (A-42) it is convenient to choose the origin of the

coordinate system such that z—0 on S2. Then z

is negative throughout the volume of integration.

If in eq (A-42), A-za represents the amplitude of one

of the nonpropagating modes, the exponential factor

indicates that the contribution to A2n from the in- duced current density. J. is small for those current elements which are remote from 82- Indeed, it is both possible and convenient to visualize the in- duced current J as a secondary source of waves whose propagation is then governed by the dimensions of the unperturbed guide. In the foregoing treatment, the only type of

perturbation explicitly considered has been that of

a deformed boundary, and this is certainly the problem of greatest practical interest. It is possible to obtain the same general conclusion if an arbitrary

distribution of dielectric and magnetic media is also included in the perturbed guide.

This discussion provides the basis for the description given in section 3c.

7.6. Appendix 6

The derivation of eq (38) begins with the basic

definition of Nai and help of eqs (21) and (23):

Id. IS21IMI

7)a p-q\Yi\- + zYi + z*r*

p - q\r,n\'^ + zrm + Z*T*

\S2i\Hi-\r,„\-n

p

zT„

P

(i-|r,p)

1 -

where

g|r,|^

^ zF, P P P

P=l

q = \a\~ — \ c

z = c — ab*

(i-|r„,p)

(A-43)

(A-44)

(A-45)

(A-46)

(A-47)

Before continuing with the development of the

desired expression for N„i, it is necessary to obtain

expressions for qlp and zip as functions of r,„ andrj,-. In terms of eqs (A-45), (A-46), (A-47), eq (35)

becomes

zr,„|r,„|2+ (p-g)|r,„|2+z*r*=o. (a-48)

If this equation is subtracted from its complex

conjugate, and noting that p and q are real, the

resulting relation can be put in the form:

(zr,„-z*r*)(i-|r,„|2) = o (A-49)

•^This same result may be obtained from eq (A-43) where the seeond factor on the

right is the reciprocal of r}„. By inspection Tj,, is a maximum with respect to the argu- ment of r,„ when zP,/, is real and negative.

and since | F,,, | ^ 1 for a passive termination, this requires, in general,"*^

Equation (A-48) may thus be written:

(A-50)

r,„(i + |r„,|2) + (^i_£j |r,„|2=o. (a-51)

Starting with eq (24), expanding, and using eqs

(A-45), (A-46), (A-47) yields:

\a-bc\Hi-\rm\-'y

(A-52)

(|i + cr„,|2-|ar,„ + 6|2)2 ^ {\a\^+\bc\-'~\c\-'-\ab\''+\c-ab*l'){l-\ T,,, 1 2) (p-g|r,„|2+zr,„+z*r*)2

ipq+\z\pii-\r,„ir- _

{p-q\T,n[' + zY,„ + z*T*Y *

Equation (A-48) may be rearranged in two different

ways to yield:

and

q I r,„ 1 2 - z*r* = [p + zY,n) I r,„ 1 2 (a-54)

\z\-= (9— p)zr,„— zT„ (A- 55)

Substitution of these expressions in the denominator

and numerator respectively of eq (A-53) leads to:

q z Y1 )j >_ P P

p

(A-56)

Equations (A-51) and (A-56) may now be solved

simultaneously to yield the following expressions

for zip and qlp.

z_ (i-r?2,)r*

p l-r/?.|r,„|2

q _ T?r - |r„,|2

p l-r/2.|r,„|2'

(A-57)

(A-58)

Finally, eqs (A-57) and (A-58) may be substituted

in eq (A-44) which may then be factored to obtain

eq (38). Equation (39) may be obtained from eq (38) with

the help of the relationship:

(1- |r,|2)(i - |r„,|2) = |i -r,r*|2- in-r^r^

(A-59)

which may be confirmed by the expansion of both

sides. 22
7.7. Appendix 7

Equation (42) may be obtained as follows. Starting^

with eqs (27) and (29). Rci is given by:

Rc

\{B-Ac)[D-Cc)*- {Aa-Bb){Ca-Db)*\

\D-Cc\^-\Ca-Dbm

(A-60)

Combining this with eq (31), and imposing the condition BD*-AC* = Q, yields:

Ri

I gini- ^ zYi

^ z*rf

(A-61)

where p, q, and z are defined by eqs (A-45), (A—46),

(A-47).

Substituting the values of zip and qlp from eqs

(A-57) and (A-58) into this expression gives eq (42).

7.8. Appendix 8

It is the purpose of this appendix to determine

the conditions under which a reflection coefficient of magnitude less than unity will be mapped into the interior of the circle of figure 13. Returning to eq (27), it will prove convenient for

this appendix to interpret Rc as a complex number,

which locates the center of the transformed circle

in the w plane. Then:

Rc- |8|2- |y|'

(A-62)

By inspection of eq (25), the origin of the z plane

transforms to the point /3/8 in the w plane. Pro- vided that this point is within the transformed circle,

continuity considerations insure that all values of

z, for which |z| <1 will be mapped to its interior. Evidently, the point /3/8 will be found within the

transformed circle provided that:

Rc- < R. (A-63)

Substituting from eqs (26) and (A-62) gives

\Py — a8\

\8\''-\y\- 8 <

|5|2-|y|

which after simplification becomes

(A-64)

(A-65)

The application of this result to the g-reflectometer

yields

\C\<\D\. (A-66)

As a practical matter, this relationship is well

satisfied by the type of system described, and in any case must hold if the tuning condition called out in eq (14) is realized.

The Author expresses a sincere "thank you" to the members of his Advisory Committee, Professors

C. T. Johnk, S. W. Maley, W. A. Reuse, and N. S. Nahman for their encouragement and constructive

criticisms during the preparation of this dissertation. A special measure of appreciation is extended

to Professor D. M. Kerns, under whose direction it was prepared. In addition, a number of constructive

discussions with Dr. P. F. Wacker, of the National

Bureau of Standards, are acknowledged. Finally,

the Author extends his appreciation to Mrs. Margaret

Woolley and to Miss Sharon Foote for typing the

manuscript.

This dissertation was prepared under the auspices

of the Radio Metrology Program.

8. Bibliography

[1

[2

[3

[4

[5

[6

[7

[8

[9

[10

[11

[12

[13

[14

[15

[16

Kerns, D. M., Basis of the application of network equations

to waveguide problems, J. Res. NBS 42 , 515- 40 (May

1949) RP1990.

Beatty, R. W., and Macpherson, A. C, Mismatch errors in microwave power measurements, Proc. IRE 41, 1112-19

(Sept. 1953). Engen, G. F., and Beatty, R. W., Microwave Reflectometer Tech. IRE Trans, on Microwave Theory and Tech. MTT7, 351-55 (July 1959). Engen, G. F., A Transfer Instrument for the Intercomparison of Microwave Power Meters, IRE Trans. Instr. 1-9 , 202-08 (Sept. 1960). Engen, G. F., A variable impedance power meter and ad- justable reflection coefficient standard, J. Res. NBS, 68C

(Eng. and Instr.) No. 1, 7-24 (Jan.-Mar. 1964). Engen, G. F., Amplitude stabilization of a microwave signal source, IRE Trans, on Microwave Theory and Tech. MTT-6, 202-06 (April 1958). Tomiyasu, K., Intrinsic insertion loss of a mismatched micro- wave network, IRE Trans, on Microwave Theory and

Tech. MTT-3, 40-44 (Jan. 1955). Beatty, R. W., Intrinsic Attenuation, IEEE Trans, on Micro- wave Theory and Tech. MTT-11, 179-82 (May 1963). Kerns, D. M., and Beatty, R. W., Basic Theory of Waveguide Junctions, and Introductory Microwave Network

Analysis (Pergamon Press, New York, N.Y., 1967). Montgomery, C. G., Technique of Microwave Measurements, pp 816-21 (McGraw-Hill Book Co., New York, N.Y.,

1947). Alter, W., (^-circles — a means of analysis of resonant micro- wave systems, Proc. IRE 35, 355-61 and 478-85 (April & May 1947). CuUen, A. L., Measurement of microwave transmission

efficiency. Wireless Engineer 26, 255-58.

Beatty, R. W., Determination of attenuation from impedance measurements, Proc. IRE 38, 895-97 (Aug. 1950). Altschuler, H. M., Maximum efficiency of Four-Terminal Networks, Proc. IRE 43, 1016, (Aug. 1955). Stratton, J. A., Electromagnetic Theory (McGraw-Hill Book

Co., New York. N.Y., p. 250, 1941). Collin, R. E., Field Theory of Guided Waves (McGraw-Hill

Book Co., New York, N.Y., 1960).

23

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