* UNITED STATES DEPARTMENT OF COMMERCE

PUBLICATION NBS TECHNICAL NOTE 637

U.S. :PARTMENT

OF

COMMERCE

National

QC

\oo

no. 437

THEORY OF UHF AND

MICROWAVE MEASUREMENTS

USING THE

POWER EQUATION CONCEPT
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and

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« Part of the Center for Building Technology.
toe 2?: ^Theory of

UHF and Microwave Measurements

Using the Power Equation Concept

Glenn F. Engen

Electromagnetics Division

Institute for Basic Standards

US National Bureau of Standards

Boulder, Colorado 80302

>^

NBS Technical notes are designed to supplement the

Bureau's regular publications program. They provide

a means for making available scientific data that are of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

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

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued April 1973
National Bureau of Standards Technical Note 637

Nat. Bur. Stand. (U.S.) , Tech. Note 637, 68 pages (April 1973) CODEN: NBTNAE

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

(Order by SD Catalog No. 03.46:637)
I.

II.

III.

IV.

V.

VI.

VII.

VIII

IX.

X.

CONTENTS

Page

INTRODUCTION 1 MEASUREMENT OF MISMATCH 7 TWO-PORT MEASUREMENTS 20

MEASUREMENT OF P 29

g

IMPLEMENTATION OF THE VARI-Z POWER METER 36

IMPLEMENTATION OF A VARI-Z POWER METER BY A

SINGLE COUPLER 42

MEASUREMENT OF SMALL VALUES OF n 46

a

S UMMARY 4 8 ACKNOWLEDGMENT 50

REFERENCES 51

APPENDIX I 53

APPENDIX II 57

APPENDIX III 59

APPENDIX IV 61

APPENDIX V 63

in
LIST OF FIGURES

Page

Figure 1. Basic Circuit for Defining6 P , P „, M „ g gl' g^

Figure 2. Circuit for Discussion of M „ Measurement in

Conjunction with an Output Monitor

Figure 3. Addition of a Second Coupler to Provide for

Measurement of M „ g£

Figure 4. Illustrating the Relationships Among R, R , b 3 C

^_

Figure 5. An Alternate Application for the Measurement

of M . g*-

Figure 6. Complete Instrumentation for Alternate M « Application

Figure 7. Basic Circuit for Discussion of 2-port Applications

Figure 8. Basic System for Measurement of n , n , q , ga' a.2. Figure 9. Alternative System, with Tuners, for

Measuring N „ cl JO Figure 10. A "Variable- Impedance" Power Meter for

Measurings P

g

Figure 11. Addition of a Lossless (LL) Two-Port to

Provide a General Analysis

Figure 12. Alternative Form of Vari-Z Power Meter

Figure 13. Use of 3-port as a Vari-Z Power Meter

Figure 14. Several Specific 3-port, Vari-Z Power Meter

Configurations

Figure 15. Circuit for Measuring Small Values of n Figure 16. Circuit for Further Discussion of "Alternate"

M „ Application

iv

8

10

12

17

17

21

24

29

30

31

39

43

45

48

53
THEORY OF UHF AND MICROWAVE MEASUREMENTS USING

THE POWER EQUATION CONCEPT

by

Glenn F. Engen

ABSTRACT

Theoretical considerations indicate that, in

addition to other advantages, UHF and microwave

measurements based on the power equation concept

should exhibit reduced sensitivity to certain

connector imperfections. This has now been experimentally confirmed in a number of laboratories,

and interest in these methods is increasing. The

previously described measurement techniques, for

implementing this concept, have relied heavily

upon tuning procedures, while current trends are towards automation where tuning is impractical if

not impossible.

By assuming a phase detection capability

(such as is found in automated systems) it is

possible to eliminate the tuning requirement.

Moreover, this generalization has made it possible

to summarize the earlier results in a compact

form. In addition, this paper includes a number

of new measurement ideas

.

It is expected that this report will serve

as a major reference in the continuing development

of detailed measurement procedures which exploit

the power equation concept.

Key Words: Microwave; Microwave measurements; Power

equation; Power measurement; Terminal invariant.

INTRODUCTION

Although the initial acceptance has been rather slow,

recent laboratory confirmations [l] 1

, [2], of the theoretically

x The figures in brackets [ ] represent literature references

at the end of this paper.
predicted benefits of the power equation [3] methods, have

stimulated a growing interest. This paper will present an important extension to the basic theory, namely the theoretical

basis for some additional measurement techniques, in which a key feature is the elimination of the tuning adjustment.

Ordinarily, at radio- frequencies , it is much easier to

measure the amplitudes of the associated parameters (e.g.

v, i, etc.) than the phase . In the microwave region, phase

measurements have usually been achieved by conversion to a lower frequency. If one reviews the past two decades, it

will be recognized that only amplitude detection or measurement capability was assumed in a majority of the microwave

measurement techniques developed and proposed. Presumably

this restriction on the assumed detection process was motivated,

to a substantial degree, by the desire to avoid the complexity

of the phase measurement process. During the latter part of

this era, a variety of tuning procedures were developed

such that, for example, one could start with a pair of

arbitrary directional couplers and tuners, and "construct"

an ideal reflectometer [4]. Tuning procedures were also

developed to satisfy a variety of other criteria. These

tuning methods generally provided a significant improvement in accuracy and the need for phase information was largely suppressed.
Unfortunately, these tuning adjustments were usually

frequency sensitive, often time consuming, and in the development of automated, stepped frequency measurement systems,

these tuning procedures were eliminated, but at the price of

re - introducing a phase detection capability. 2

The techniques, which have been described to date, for

use in effecting measurement via the power equation concept,

have also relied heavily upon tuning procedures. In some

cases, a substantial simplification in these procedures has

been realized. These methods have the advantage of permitting

a substantial improvement in the accuracy potential 3 of many

existing calibration systems which do not possess a phase

detection capability.

On the other hand it must be recognized that there is,

today, a substantial trend towards automation, and phase

detection capabilities in particular. As was the case in

the conventional approach, the availability of phase information permits the elimination of tuning requirements in

measurements via the power equation concept.

It of course is obvious that the relative merit of any

model, for describing the real world, rests to a large degree

2 It must be emphasized that the absence of tuning requirements

in existing automated systems rests solely on this phase

detection feature.

3 For example: relaxation of precision connector requirements,

explicit accounting for mismatch corrections, etc.
upon the ease (or difficulty) with which the parameters

upon which it is based may be evaluated or measured. As

already noted, this paper will provide the theoretical

basis for a collection of measurement techniques which

exploit the power equation concept. This will be done from

a broader base than previously employed and include the

prior results as special cases. (These will be noted as the

theory is developed.) In addition to the elimination of

the tuning requirement in the existing methods, this paper

will also include a number of other techniques which have no counterpart in the existing art.

A key feature of the power equation concept is that it

retains its validity in non-uniform waveguide systems. On

the other hand, the measurement techniques developed to date

(including those to be described in this paper) require that

the waveguide in question be a "reasonable" approximation to

a single mode structure. (In practice this means that the

cross section must not be allowed to approach those dimensions

which represent either cut-off or multimode propagation.) A

measurement technique which satisfies the terminal invariant

(power equation) criteria must also be valid in the limit of

an ideal waveguide. For this reason, it has proven convenient

to utilize the existing microwave circuit description in the

development of measurement procedures. At the same time,
however, it must be emphasized that the concepts of reflectionless termination, directivity, VSWR , and impedance

generally have no meaning in the power equation context. It

is thus necessary to work within the following constraints.

1) The different components, waveguide junctions, etc.,

which comprise the measurement system are of

arbitrary parameters. (This does not mean, however,

that all junctions are equally satisfactory for a given application! In general, one uses the circuit

theory to recognize a junction whose properties are approximately optimum for a given application. For

example, where a directional coupler is specified

in what follows, one should understand this to mean

a three-port whose parameters approximate those of

an ideal device, etc.)

2) The remaining tools at one's disposal include a terminating power meter (of unknown impedance)

which indicates net power and an adjustable waveguide short, of unknown phase angle. (Equivalently

,

one has a variable, or collection of fixed reactances

whose values are unknown.)

3) In the case of a multi-arm waveguide junction (eg

"reflectometer") it is possible to provide the

"sidearm" with detectors which indicate relative

amplitudes (only) and phase difference (optional).
Because the "tools" noted in (2) above satisfy the

terminal invariant criteria, it should be intuitively evident

that the same will be true of measurement techniques which

are derived therefrom. A more complete discussion is given

in [5] .

In the interests of economy it will only be possible to

present a broad theoretical base. It is anticipated that

this paper will become a reference document for the development of detailed measurement procedures. This paper will

build on the general introduction already given in [3], [5]

and include only enough background material to ensure con- tinuity .
II. MEASUREMENT OF MISMATCH

Referring to figure 1, the system is described by the

equation

,

P „ = P M . (1) g£ g g£ < J

where P is the available power of the generator, P . is the

net power delivered to the load, and M . is defined by (1).

The parameter, M „, ranges between zero and unity, and

g^

expresses to what extent the (impedance) conditions for

maximum power transfer are satisfied.

At this point a brief comment on the notation is desirable. Wherever a single subscript is employed (eg. P ), this

indicates that the parameter in question is a function only

of a single component (eg., generator). A double subscript

(eg., P „, M „ ) is used where the parameter is a function of 6 gl' gl

two or more components (eg. , generator and load) . This scheme

will be further developed in the section which follows.

G>

M 9 -WV

9l

Figure 1. Basic Circuit for Defining P , P „ , M

g gJi gJi
It is common practice, in measurement applications at

least, to provide the generator with an output monitor in

the form of a directional coupler and sidearm power detector.

If one agrees to interpret the sidearm power as a measure of

the generator level, it has been shown [6] that the parameters

of the "equivalent" generator, referred to port 2, in figure 2,

are determined by the coupler parameters and independent of

the actual signal source. The measurement of M „ will first

g^

be discussed in this context.

Referring to figure 2, if one assumes the coupler is

terminated by a load of arbitrary reflection coefficient, r„,

one can write,

p„o = |b 9 | 2 (i - |r | 2

) (2)

b

! , al M—h-*- VWiJrP.

7<

X

>0WER METER

a 2| b

2 -*—i—

I To

g£

Figure 2. Circuit for Discussion of M „ Measurement in

Conjunction with an Output Monitor.

The "characteristic" impedance is chosen to equal unity.
where b~, a

? are the emergent and incident wave amplitudes at

port 2, and r. = 2l~I\>2'

The wave amplitude, b., in the sidearm 4 is a linear

function of a~ , b„

b. = Ca

2 + Db2

, and a

2 = b

2 r„, so that:

p

4

- ib 4 r

- b

2

l 2 |cr

£

* D|*

Equations (2), (4) may now be combined to yield,

V1 - r 1 2

) v J

n| 2

I i + - r I 2 u \ I J- D £

! and with further manipulation (5) may be written:

(3)

(4)

(5)

g*

D| 2 d - |r I 2

)

where ,

- J.

1 - r £

" F 8

w 1 - Vg

c

> D

(6)

(7)

and moreover, r is the reflection coefficient of the equiva

lent generator 5 associated with the directional coupler.

* By1 inspection, v P . is a maximum when r„ = r (which of

gi * g course is only to be expected if r indeed represents the

source reflection coefficient). By comparison with eq. (1) ,

5 Ordinarily, the equation for a source is written,

b = b + a-,r where b is characteristic of the generator.

2 g 2 g g

Equation (3) may be written, b~ - b

4 /D - a~ (C/D) . By analogy,

b./D becomes the generator amplitude (b ) so that r = -C/D.
P

g

= P

4 [|D| 2 (1 - iTgl

2 )]" 1

. (8)

The factor [ | D | 2 (1 -

| r | 2

) ] ~ is a property only of the

coupler- sidearm detector, and represents a parameter of some

importance in power calibration since its product with P. determines the available power. It is designated by the

symbol K.. The immediate concern of this section, however,

is in the mismatch factor, M - . Again by comparison with (1),

g ~

M

r r l 1

(9)

The measurement of M . calls for the addition of a

second coupler, as shown in figure 3, such that a four-arm

junction is obtained, and of which the initial directional

coupler and sidearm detector present a sub-assembly. By

1. &

-aw

x X

a 2i b

2

I »

I

I r I

i

g£

Figure 3. Addition of a Second Coupler to Provide for

Measurement of M „

10
analogy with (3), one can write, 6 ' 7

b

3 = Aa2 + Bb

2

, (10)

and the ratio of (10) and (2) yields,

b Ar + B — = — . (11)

b

4 cr

£

+ d

In contrast with most of the prior treatments, it is

assumed that a detection capability exists such that the

complex ratio r-5- may be measured. (In practice this may

D i+ be effected in a variety of ways, the specific details are

outside the scope of this paper.) Let I\ = e , where 9

is arbitrary. Then, with a little manipulation, (11) may be

written

,

^1 = Ae

16

+ B = BD* - AC* + (AD-BC)e l9 (D*+C*e~ l6

) b

4 Ce

19

+ D |D| 2

- |C| 2 (|D| 2 -|C| 2 )(D+Ce 10

)

By inspection, if 9 is permitted to vary, the locus of

r-2- in the complex plane is a circle with center R , and

bi( c radius R where

,

8

6 The terminology has been chosen in such a way as to permit a ready comparison with a number of earlier results.

7 Here it will be recognized that the values of A and B

depend upon both couplers. (Note that the "original" coupler

may be considered an extension of the main arm of the coupler

which has just been added.) By contrast, C, D, continue to

depend only on the "original" coupler parameters. This con- clusion is implicit in the above development, a more detailed

proof is found in [7].

"Note that the ratio (D*+C*e

10

) / (D+Ce lQ

) has a magnitude of

unity

.

11
R = c

R =

BD* - AC*

D| 2

" |C| 2

AD - BC

(13)

d[ 2 - |cr t 14 )

This relationship is indicated in figure 4. In particular

it should be noted that R is a complex parameter.

Figure k. Illustrating the Relationships Among R, R , r-2- C D ij

The measurement of M . requires, in part, a determination of R , R. This may, quite obviously, be done in a variety

of ways. For example, the values of ,-?- may be observed for

three values of 6 (offset shorts, "opens," reactances, etc.),

and the circular locus determined from the standard formulas

of analytic geometry. Alternatively, if a moving short is

used, inspection of figure 4 indicates that if is adjusted

such that |r-2-| is a maximum,

arg R = arg —

,

b

4

(15)

12
R + R =

c

(16) max

while if 9 is adjusted such that |t-^-| is a minimum,

b. R R. (17)

min

where the upper sign is used if, as in figure 4, the argument of t-^- now differs from the previous value by n. If, on

D k the other hand, the origin is outside the circle, the argument of t-^- will have the same value for both maximum and

minimum values of \rr^-\ and the lower sign should be used. For

the systems considered in this paper, R > |R | and this is

assumed in what follows. (See Appendix 5). Equations (15),

(16), (17) may be easily inverted to obtain R and R.

In addition to the measurement of R , R, the determinab

,

tion of M . requires a measurement of *-*- when r o pertains to

the specified load for which M . is required. This will be

designated by w. Evidently:

w = AT

l + B

cr

£

+ d" (18)

Equation (18) may be solved for r £

. This together

with (7) leads to

(|D| 2- |C| 2 r - r ) w - (BD*-AC*)

1 - r £

r

, AD - BC

(19)

13
and this may be combined with (9), (13), (14) to yield,

|w - R

c

| 2 V

= x —^—

^ 2 °)

Equation (20) is the desired relationship.

This result explicitly demonstrates several important

features of the power equation formulation. In principle, either (9) or (20) could serve as the basis for a determination

of M .. The former, however, calls for a measurement of r and

r„; this represents the "conventional" approach, and generally

involves impedance standards, precision slotted lines, or directivity considerations. Equation (20) demonstrates, on the

other hand, that it is possible to avoid the impedance measurement routine; it is only necessary that w, R , and R be measured,

In particular it should be noted that this result is valid

for "imperfect" couplers and loads of arbitrary impedance.

Moreover, the determination of R and R does not require

c

n that the phase of the sliding (or offset) shorts be known. 9

The measurement method thus satisfies the "terminal invariant"

criteria [5]

.

In the prior art, a tuning transformer was included

between the two couplers. This was adjusted either for

b, = with the load I\ connected, or such that |t-2-| was a

constant with respect to a moving short at port 2 . From

9 In terms of v, i description, this means that the terminations should be "completely reactive." The value of reactance

need not be known.

14
the foregoing discussion it is apparent that the first

condition leads to w = 0, the second to R =0. In either

event the phase detection is not required.

If w = 0, the measurement of M . calls for observing

\^-\ and |t-2-| in response to a sliding short. Then, max min.

V =

x R I 2 c

= 1

b 3 b 3

* b k max

b* min

b 3 + b 3 b- max

b„ min

On the other hand, if R = , it suffices to observe

with the load and a short connected. In this case,

(21)

b 3 I BT 1

M 1

-

b^ 2 b*

r

ki 2 bu ^

(22)

A major application of the foregoing treatment is in the area of power calibration. To briefly summarize, the power

output, P , from the coupler in figure 2, may be written,

;£ \l = P

4 KA

Mg£> (23)

where P. is the sidearm power, K. is a property of the couplersidearm power meter combination and M „ is a mismatch factor

g£

whose measurement has been described above. If the termination is a "standard" power meter, its reading determines

P „, which, together with the associated values of P. and

gx, ' ° 4 M „, determines K. . When the termination is another power

gx, A

15
meter, its reading may be compared with P . and thus its

calibration obtained.

It will be recalled that the reflection coefficient, r , g associated with the directional coupler was the negative of

the ratio of C to D, (7). Moreover, these parameters are independent of the properties of the second coupler which

was added to permit the measurement of M . . Indeed, unless

this were true, of C/D at least, r would not be a property

of the original coupler alone. It will next be instructive

to consider a different combination of components, but in

which the foregoing criteria are only partially satisfied.

There are a number of practical situations in which this

alternate configuration is useful. One example is in the

comparison of two noise sources via radiometric techniques

[8]. Another example is in the "receiving end" of an antenna

calibration setup [9]. In these applications, and others,

the use of a directional coupler to monitor the "source"

behavior is obviously impractical, if not impossible. It may be possible, however, to add a (first) directional

coupler to the detector input as shown in figure 5. Ordinarily the coupling ratio, measured in dB , should be 20 or

more. To complete the system, the "second" directional

coupler and other components as shown in figure 6 are added.

16
I

X

J

SIGNAL SOURCE

e.g. , HEATED TERMINATION

ANTENNA, ETC.

-H>-

DETECTOR

e.g., RADIOMETER,

RECEIVER, ETC.

Figure 5. An Alternate Application for the Measurement

of M „

SOURCE

(INACTIVE)

:',

©— : X- Ml

X

DETECTOR

>-©

r r

9 A

Figure 6. Complete Instrumentation for Alternate M n Application

Now it is assumed that the "active" element in the source may be "turned off" without affecting r . In the case of the antenna, for example, the illuminating source

is disconnected, or the antenna pointed in a different

direction. In the noise measurement application, it may

be that the radiometer design is such as to recognize the

17
difference between a cw and noise source. Or, the level

of the cw signal in figure 6 may be such as to completely

mask the noise power input. Alternatively, the radiometer

input may contain isolators or similar components, ahead

of the signal amplifying elements, which determine the

"detector" impedance properties. When this is the case,

it may be desirable to switch in an auxiliary detector,

following the isolator, and replacing the amplifier for the

measurement of M .

.

In a formal sense, the only difference between the con- figurations of figures 6, and 3, is a matter of terminology

and in the resultant values for the parameters A, B, C, D.

With regard to terminology, the roles of r and r - have been

g *> reversed. In particular the terminating "load" in figure 6 now has the reflection coefficient 'T ," while the ratio

-C/D becomes "r„." Moreover, the parameters C, D are now

dependent upon both couplers, If, however, the coupling

ratio for the "first" coupler is "large" (20 dB or greater)

the ratio C/D has only a weak dependence upon the properties

of the "second" coupler.

The validity of this statement may be recognized with

the help of (3). In general (3) is valid under all modes

of operation. It is convenient, therefore, to assume that

excitation has been applied via port 2 in figure 6, and that

port 1 has been terminated such that b. = 0. Then,

18
b—2 _ C = h

=

- -' (24)

a

2 D

which provides an explicit demonstration of an earlier

assertion. Because the coupling between the primary and

secondary line is "small," it may now be recognized that

the input reflection coefficient at port 2 will have

only a weak dependence upon the characteristics of the second

coupler. 10 The addition of these components thus represents

little more than an extension of the input port of the

original detector.

The measurement of M „ may again be obtained in several

gl

ways, as previously described. For example, T may be

adjusted such that the detector output vanishes when excitation is applied via port 1, and the "source" is connected to

port 2 but "turned off." Following this adjustment, a moving

short is connected to port 2 , and the maximum and minimum

values of I t-M are observed and substituted into (21). Alternatively, T may be adjusted such that [t-^-| is a constant in

response to the short motion, in this case (22) becomes the

appropriate equation. Finally, it may be possible to obtain

the argument of t-^-, in this event (20) applies and the tuning

D it operation is not required.

10 For example, if ideal components are assumed, r„ = 0. If

the second coupler is replaced by a "short," the value of

| T„ I becomes . 01.

19
Before concluding this discussion, several other items

are worthy of mention. Referring once more to figure 6, the

contribution of the "second" directional coupler is rather

small, other than providing the phase reference, and can usually

be deleted if desired. 11 In addition to the questions already

considered, it is also recognized that the operation of the

tuner T will have a small but ordinarily negligible effect

upon the power transmission "efficiency" between port 2 and

the detector input. These questions will be considered in

greater detail in Appendix I. Finally, there will undoubtedly be "field" situations

where the permanent introduction of these couplers, etc., is

untenable. In this case it is possible to measure P . by a "variable -impedance" power meter which has been adjusted to

duplicate the required load impedance. Methods for implementing this and measuring P will be described in following

sections. This also permits the determination of M . via (1).

8 «

III. TWO -PORT MEASUREMENTS

The extension of the power equation methods to describe

two-ports (attenuators) begins with the system shown in

figure 7, and the equation,

11 If one postulates that the sidearm auxiliary signal source is

stable and that it is replaced by a passive termination of iden

tical impedance, the "second" coupler contributes nothing other

than the "phase reference" and may be deleted when the tuning

is employed. In real life the instability of the signal source

affects only the difference between M . and unity, while the

failure to satisfy the impedance & condition is "attenua

by the coupling ratio.

20

ted"
V

= P

g

M

ga^a£> (25)

where n „ is the efficiency of the attenuator for the given

load, and the other terms have been previously defined. The

validity of this result is easily established. In particular,

M represents the mismatch factor between the generator and

ga v & the attenuator-load combination. The product P M thus

represents the net power delivered to the attenuator, while

n (by definition) is the ratio of the power output of the

3. A/ attenuator fP n ) to the power input (P M ). In this

description the attenuator is "combined" with the load to

yield a new "load" such that (1) applies at terminal 2 in

figure 7.

GENERATOR

©•

"ATTENUATOR 1 LOAD

Figure 7. Basic Circuit for Discussion of 2-port Applicat i ons

Alternatively, it is possible to write,

P „ = P q M „

,

r 2 61

where P ., P are already defined, P q is the available gV g

' g H ga

power from the generator-attenuator combination (at terminal

21
11 . and M „ is the mismatch factor between this "modified

generator" and the load. Here it is noted that (1) has

been applied at terminal 1 rather than 2. Moreover the

parameter q is the ratio between the available power at

terminal 1 and the available power at terminal 2 . By

contrast, n is a ratio between net powers. In spite of this

distinction, when reciprocity is satisfied, q and n a ^ are

related in a rather simple manner. In figure 7, n a £ is

defined in conjunction with an assumed power flow from

terminal 2 to terminal 1 and is a function of the load, but

not the source, impedance. If one assumes the active element

associated with the generator in figure 7 is "turned off,"

such that it becomes a passive load of identical impedance

properties, and excitation is applied via port 1, the

"efficiency" for power flow from terminal 1 to terminal 2

is equal to q , and is a function of the generator impedance.

§ a

It has also proven useful- to introduce the maximum value

of n „ (as a function of load impedance) by n , and the

a. 36 3, parameter N . such that

3. J6

n a£

=

n aN a£' (27)

where N . is another type of mismatch factor which again

varies between zero and unity. Moreover, if reciprocity is

satisfied, the maximum value of qn will also equal n and

ga n

'a it is possible to write

q ga

= n aN

ga> (28)

where N is defined by (28).

22
The parameters N , N o thus express to what extent

n

aJ^ and q differ from their maximum possible value, n . Loosely speaking, the values of these parameters are determined by the impedance conditions found at terminals 1 and

2 respectively. The same, however, could be said of M

3. As and MfT0 . The distinction lies in that N . is a function ga ail only of the attenuator and load parameters while M „ also

ax, depends, to some extent at least, upon the generator. A

similar statement could be made in reference to N and M

ga ga

In summary, the "M" type of mismatch depends, to some extent at least, upon all of the components found in the

system. The subscripts indicate the two components between

which the mismatch is to be evaluated. The subscripts play

a similar role in the "N" type of mismatch, but here the

functional dependance includes only the two-port on one side

of the terminal surface of interest, and all of the components on the other side. With regard to n vs . n , the

3. a a, former depends only on the two-port while the latter depends

upon the load as well. Perhaps in the interests of consistency

the term P should have been introduced, but this has not gax, been deemed desirable. For a further discussion [3] and [5]

should be consulted.

The measurement of n a£ , n &

, q , N , N &£ may be con- veniently realized by means of the measurement system 12 shown

12 The conventional ordering of the directional couplers would

be equally convenient in many cases.

23
in figure 8, provided that reciprocity holds and the value

of n is not too small. (The measurement problem where these

a criteria are not satisfied will be considered in a following

section.) As compared with figure 7, the "generator" has

been replaced by the measurement system of figure 3. In

harmony with the terminology already introduced, the complex

ratio b,/b. will be denoted by w when the system is connected

as shown in figure 8. In addition the terms R-, , R , , and

R- , R - will represent the parameters of the circular locus

2 c2

obtained with moving or fixed "shorts" connected to ports

1 and 2 respectively.

y.i

&

X

AAAr yX wv

Figure 8. Basic System for Measurement of T\ - , T] , q , N , N . ax a ga

ga ax,

It has been previously shown [5] that,

R

l R

2 Together with (8) and (23) this means that the available

power from terminal 2 is P

4 K., while that from terminal 1

24
is P.KA

R

1 /R2

. This result is of interest in that q is

the one parameter which requires neither phase information,

nor tuning to effect its measurement.

An equation for n a£ , in terms of w, R, , R„-, , R , R,. cl ^c2 may now be easily obtained. Since P . , P represent the

same values in (25) and (26), it immediately follows,

M n n = q M „ . f 3m ga 'a£ n ga a£ < JU J

By means of (20), the terms M , M „ may now be evaluated

in terms of w, R

1

, Rcl , and w, R

2

, R - respectively; q is

given by (29) so that: w - R .

2 cl '

'a£

2I A

R

2

[ 1

- w - R .1 2 ^ c2 * (31)

The value of n may next be obtained from (31) as

a follows. In general there will be a unique relationship

between w and the load impedance. For some value of load

impedance, n „ will be a maximum; this must be reflected in

a corresponding value for w. It is therefore sufficient to

find the maximum value of (31) as a function of w. This

computation, the details of which will be found in Appendix II,

leads to

!

3

1 3 A second solution, the reciprocal of that given, is also obtained but of no interest in this context.

25
= H - /ff 1, (32)

where

,

H = R

l + R

2

- l R

c2 - R

cl'

2

2 Rl R

2

(33)

If the value of r> is small, the following expansion

3. should be used,

n

= F[l + F

2 + 2F" + 5F 6 + •••]

3. (34)

where

F = R

1 R2 2H R

2 + R2 - |Rc2

- R

cl

|*

(35)

An expression for N may now be obtained with the f ga

help of (28) and (29). From (32),

1/n = H + /H 2

- 1, d

[1 - F

2 - F* - 2F £ ],

(36)

(37)

so that,

ga

£* = *2. f H + v^ 2 - l] , ^a R

2

(38)

2 R t R t Ri c2 cl ' 1

2 2 R

2 R2

J

2 .C 1* _ or-G 1-F 2 -F 4 -2F (39)

In a similar manner,

26
iw-Rcl m

'a A a£ w-Rc2 | 2 H + Ai 2 -l\ , (40)

c2 cl1

2 2 R? R? ;

1

- w-R n

I 2 c2 '

1 - w-Rcl l 2 l-F^F 1* -2F'

(41)

As a general rule this collection of measurement techniques is useful only for "large" values of n (i.e.

3.

n > 0.1). The reason for this limitation is easily

a explained with the help of figure 8. Loosely speaking,

the measurement procedure calls for observation of the

change in impedance at port 2 in response to a phasable

short at terminal 1. As n becomes small, these changes

a

& become smaller and are ultimately lost in the noise and

instability of the measurement system. In the case of ri o > 3. 36 (31), not only does R

q become difficult to measure, but the

difference between w and R , will be small lk so that

cl

|w - R , | /R, becomes indeterminant

.

The measurement of N , however, stands in contrast to

ga'

this general rule. Inspection of (39) indicates that if

i

i* Note that as L -* , the impedance at port 2 becomes the same for all terminations and w = R

cl'

27
Rn -*

, R ^ -» w, and,

1 ' cl

lim N

n +0

'a

ga

1

- w - R

c 2

l 2 ga (42)

This result is to be expected since if n =0 the two-port

a. behaves like a termination.

Because of the limitation on the foregoing techniques,

it is necessary to devise alternative procedures when n

is small, or when reciprocity is not satisfied. In a following section, this more general problem will be considered,

In the foregoing equations, phase detection is required

to evaluate the expressions |w-R ,|, |w-R _|, |R_„-R_-il. As c2 cl

before, this requirement can be avoided by the use of tuners.

For example, the insertion of a tuner between the two couplers

in figure 8 permits one to adjust the system such that one of

the parameters w, R, , R

2

> vanishes. This is sufficient for

the measurement of n„ > n > and N . If the phase requirement

is to be avoided in the measurement of N . , however, two of

3. A/ these three parameters must vanish. This may be realized by

the addition of a second tuner at terminal 2 in figure 8.

However to avoid interaction in the adjustments, the position

of the two couplers should also be exchanged, such that the

configuration of figure 9 results. Tuner T (figure 9) may

now be adjusted such that w vanishes, and either R n or R ~ J cl c2

reduced to zero by use of T . Alternatively, T may be adjusted such that R n vanishes and T used to eliminate R ,

.

J c2 x cl

28
In the former case the adjustments are independent provided

that they are performed in the given order; in the latter

case a small amount of interaction is possible due to losses

in T .

AAAr

&

X

J

u± X

-VNA*

_LLL

Figure 9- Alternative System, with Tuners, for Measurinj

IV. MEASUREMENT OF P

g The measurement of P in conjunction with a power monitor

g (directional coupler and sidearm detector) has already been

described (cf. (8), (23)). In particular the parameter "K."

determines the available power from the monitor output port

for a given monitor (sidearm power) indication. It remains

to consider the more general problem where the only access

to the generator is via its output port

.

The measurement of P , in this more general context,

g may be effected by use of a "variable impedance" (vari-Z)

power meter. This device is comprised of a two-port, one

arm of which represents the power meter input, while the other

29
is terminated by a variable load. Provided that the two -port

efficiency is reasonably large, changes in the variable load

impedance will result in impedance variations at the input

port. In addition, instrumentation is provided which indicates

the net power input (P „) . Perhaps the most straightforward use of this device is to

connect it to the generator of interest, and vary the load

impedance until the indicated value of P . is a maximum. Pro

* g£

vided that the conjugate of the generator impedance falls within

the range of impedance values to which the vari-Z power meter

can be adjusted, this procedure will yield P . Alternatively, the need for providing arbitrary but un- known impedance values can be avoided in an alternate scheme

where the "variable load" takes the form of a moving short. This

device is indicated schematically in figure 10. The theory

for this alternative application will now be developed; further

details for implementing the vari-Z power meter will be found

in the next section.

VARI-Z POWER METER

(INDICATES P J

&

1

i b

2

jL

T SLIDING

SHORT

F

g = °

F £ Figure 10. A "Variable-Impedance" Power Meter for Measurinj

P

30
The two-port parameters will be taken as arbitrary; how- ever as an expedient, an impedance match (r = 0) is assumed

for the generator. Although at first this represents a very

special case, it is, in fact, possible to derive a measurement

technique which is valid for arbitrary values of generator impedance provided that certain "rules" are followed. This re- presents a powerful analytical tool and perhaps warrants a brief digression in order that the approach is clearly understood.

Referring to figure 11, a lossless (LL) , but otherwise

arbitrary, two-port has been inserted between the generator

and vari-Z power meter. If this lossless two-port is con- sidered as part of the generator, its parameters may be so chosen as to yield arbitrary values of generator impedance,

but its presence will not affect the value of P .

©"

LOSSLESS

TWO-PORT

LL

A.

T

Figure 11. Addition of a Lossless (LL) Two-Port to

Provide a General Analysis

31
On the other hand, the lossless two port may also be considered an integral part of the vari-Z power meter. Although it

introduces an element of indeterminacy into the parameters

of the associated two-port, (which are, as yet, unspecified),

the addition of this component will not affect the operation

(calibration) of the vari-Z power meter because of its

lossless character.

In order to complete the picture, it is evident that

some further information, relative to the vari-Z power meter

parameters, will be required because a determination of P obviously requires more than the measurement of P . for gl

several unknown values of load impedance. From the foregoing

discussion it should be evident, however, that if a method

for measuring P when r = can be devised, and in which

only terminal invariant properties of the vari-Z power meter

are involved, this procedure will work equally well when

r ? 0. It has already been noted that the specified indication (in terms of net power) satisfies this condition.

Referring to figure 10, it will be assumed that the

lossless two-port has been "absorbed" by the power meter.

The reflection coefficient, I\ , presented by the power meter,

can be written in the form (cf. (12))

_ ae + b , . „.

T l

~ 19 I

» ( 43 ) ce + 1

32
where a, b, c are parameters of the power meter, and 9 is

determined by the position of the moving short. For a matched generator, the emergent wave, b_ , is independent

of I\ , and from (2), |b

2

| 2 = P . Combining this result

with (2) and (43) gives,

f P . = P g£ g

1

- ae + b

2 "I

ce +1

As 9 varies (i.e. the short position is changed), P will also vary, and using arguments similar to those found

in section II

(44)

where

,

V(max) = Ml - Cr-rc)*],

V(min) = V1 " Cr + rc

)*.].

be

(45)

(46)

(47)

ac

(48)

By1 hypothesis, P „ , ., and P , . > may be measured by '* ' g£(max) ' g£(min) J }

the vari-Z power meter so (45) and (46) represent two equations

in the (as yet) unknown quantities P , r, r . To complete

g c the determination, one more condition is required, moreover,

if the procedure is to be independent of r , the condition

must be a terminal invariant one. One candidate is the

maximum efficiency, n > of the two-port associated with the

a vari-Z power meter. It is given by [5],

33
2T

* /r

(49)

4T 2 where

,

be

T =

1 + |a| 2

- |b| 2

With a little algebra, (50) becomes:

r T =

1 + r

:

(50)

(51)

Equation (49) may easily be inverted to obtain T in

terms of n , and n may be measured by the previously

a a described techniques so (51) represents a relationship

between r and r . Equations (45), (46), (49), and (51) may

be combined to yield:

pq

p + q

1 + 2 /pq

p + q

1 (1+ <

)2 h>-qj

2

) + qJ 4n.

<i l

»

(52)

where p, q, represent P nr . and P n r , . . respectively. ' n ' r gJl(max) gil(min) r 7 The measurement o£ P may thus be effected by a variable impedance, terminating type power meter, in which

the impedance variations are ultimately produced by a moving short. In use the device is connected to the

generator of interest and the maximum and minimum values

34
of net power in response to the short motion are observed. 15

The maximum efficiency between the input port of the vari-Z

power meter and the port where the moving short is connected

must also have been determined. These values are then substituted in (52) to yield the desired result.

As previously rioted, this scheme avoids the need for

providing prescribed impedance values and is perhaps better

suited for certain potential automation applications.

Although a detailed study has not been made, it would

appear that the preferred value for n would lie in the

a nominal range 0.1-0.8. In some cases the generator limitations on variations in load impedance will be the determining

factor. If n is large, the VSWR presented by the vari-Z

a power meter will also be large and the extrapolation to

more typical values of load impedance may suffer. On the

other hand, if n is small the power meter impedance will be

a nearly constant and the extrapolation again suffers.

These conclusions may also be inferred from (52) where

it is noted that the last factor in the numerator contains

2 -

the term (1-ri ) , and the second term in the denominator

a contains the factor n . With the help of (45), (46), (49),

cL and (51), this latter term may be written,

15 Although the most immediate use of this result is in the

frequency regions where sliding shorts are practical, it may

be possible to extend its application to lower frequencies by

the use of adjustable reactances (eg. variable condensers, etc)

Some study has also been given to a counterpart scheme using

several fixed reactances; this approach appears promising but

requires further investigation.

35
d+n^)

2 (p-q) 2 r^l + r

2 -^)

2

4n

a (p+q) (l-r 2 -r

2

) (53)

From (49) and (51) it is evident that if n * 0, r -* a

2 and the right-hand-side in (53) approaches r so the term

does remain finite as n "*" 0. As a practical matter, how3- ever, ri is more difficult to accurately measure when small,

a moreover the difference between p and q becomes small and

difficult to determine.

The details for implementation of the vari-Z power

meter will be found in the next section.

V. IMPLEMENTATION OF THE VARI-Z POWER METER

Although a major projected use for the vari-Z power

meter is in conjunction with measurements of P , other

potential applications have been noted (section II). Moreover, the idea is not new [10]. The power equation concept,

however, provides for a substantial expansion of the original

idea.

It is convenient to begin with figure 3 and (20) , (23)

which may be combined to yield:

V

= KA

P

4

|w - R

c

| 2

R' (54)

Although the discussion and choice of terminology leading

to this result have been built around an assumed excitation

at port 1, and a passive termination at port 2, the focal

36
point is port 2. No explicit reference is made to terminal

1 and the given result is also valid for excitation via

port 2. To be more specific, (54) may also be used to

determine the power input ' 6 from a signal source connected

to port 2 and an arbitrary termination (including a moving

short) on port 1.

Equation (54), however, is not in the most convenient

form for this type of application since for certain terminations P. -> 0, w + », and the expression becomes indeterminate

With a little algebra (54) becomes:

" KA

P

3 g* R I 2 c

R v +

1

- R I 2 c

'

R

: R I 2 c

'

(55)

where P, = lb-,1 2 and v = r^- = —. The similarity between $ ' o

b 3 w J

(55) and (54) should be noted. Moreover, the minus sign

may be deleted if it is understood that port 2 is now the

input to a "termination" rather than the output of a "generator.

To extend this analogy it is convenient to define the

parameter K

T

, K

A (56) Krp - D2 R I 2 c

'

16 Note that "P " is now negative

37
and write

,

1 7

P

gJl " KT

P

3 L c£ (57)

where

L o = 1

-

V + R I 2 c

'

R | 2

(58)

Procedures for the measurement of K. , R, and R have A' ' c been given in section II, these together with (56) may be

used to determine K~. It is usually preferable, however,

to evaluate K

T via an alternative procedure. Referring to

(57), the device may be provided with a known power, P .

,

by use of the techniques for calibrating terminating type

power meters described in section II. The evaluation of

L . assumes the knowledge of R, and R ; this permits the

determination of K~.

The use of (55) presupposes a phase detection capability

since the complex ratio t-5- is required. As before, this

o 3 phase detection requirement may be eliminated by the introduction of tuning transformers. For example, as previously

noted, the appropriate adjustment of a tuner inserted between

the two couplers will cause R to vanish. This case is the

subject of reference [10].

'The minus sign has been dropped.

38
In addition to satisfying the requirements for the

measurement of P , there are other potential uses for

the vari-Z power meter. It has already been noted that

there are field situations where the permanent installation of a directional coupler is untenable. Here the best

solution to the measurement of P may be to terminate the

vari-Z power in such a way that its input impedance duplicates that of the load, for which it is then substituted,

and P . measured. Either of the foregoing arrangements (with g^ or without tuning) are potentially applicable in this problem.

There is, in addition, another tuning adjustment which

is of interest. For this application the order of the

couplers should be reversed, and, indeed, for reasons which

will emerge, this is generally desirable in the prior cases too

The system is implemented as shown in figure 12. The

merit of this arrangement lies in that KT

is a property only

of the coupler included between port 2 and T , and is indepedent of the adjustment of T or T . y z

b 3i_

L

wv- J X

J_i

X X

AM

J_LL -WV

Figure 12

y z Alternative Form of Vari-Z Power Meter

39
To demonstrate this point it is sufficient to recall

that K. = [|D| 2 (1 - |r | 2 )]" which may be combined with

(7) to yield,

KA

= (|D| 2

- ICl

2 ]" 1 (59)

Equations (13) , (14) may be combined to obtain,

. , ., |A|2 |B|2

R

2

- R

c

2 - (60)

|D| 2 - |C| 2

If these results are substituted into (56) ,

KT

= (|A| 2- IBI

2 ]" 1

. (61)

Now by arguments similar to those employed in relation to

C and D the parameters A and B, (and thus K„) depend only on

that portion of the system "between" the detector which

measures P~ and port 2.

This arrangement is used as follows

:

1. Excitation is applied via port 2, port 1 is terminated by a load with nominal impedance match, and T

adjusted so b . = .

2. T is adjusted to yield the required input

impedance

.

3. Port 2 of the system is connected to the generator

for which P „ is required and P, noted.

g£ H 3 4. The system is next excited via port 1, a sliding

short is connected to port 2, \j^-\ and |-j-=-|

"* max "* min

are observed.

40
Using (16) and (17), R and |R | may be determined

This leads to

P

g£

K

T

P

3

r b 3 b- max

b 3 min

L

b 3 b,

+ max

b 3 b*

min-*

(62)

(Note that this assumes K~, has also been measured.)

As compared with the previously described technique

[10], the tuning adjustments are more easily realized, but

R and |R | have to be remeasured every time the device is

adjusted for a new value of input impedance. This is avoided

in the previous case but the required adjustment is more difficult to realize.

Comparing figures 12, and 3, also (23), (57) it is noted

that in one case the measurement of "P." plays a major role,

in the other case the measurement of "P . " The choice of

terminology is, of course, arbitrary. This particular

convention is the result of showing how (55) could be

obtained by use of (54) . It will prove instructive to assume

that the coupler which yields "b " is in reality that previously

used to obtain "b. . " The arrangement shown in figure 2 thus provides the "key" element in both applications. Now since

K., K„ are associated with the same coupler it is of interest

to note (see Appendix IV) they are related by the equation,

(63) KA

- K

T

n 12

P

4 =

41
where n 1? is the "efficiency" of the coupler in figure 2 for

an assumed power flow from port 2 to port 1, where the latter

has been terminated in such a way that P. = . This

efficiency may be measured as previously described (29)

,

and provides another method of obtaining K

T

when K. is

known, or conversely.

VI. IMPLEMENTATION OF A VARI-Z POWER METER BY A SINGLE COUPLER

The previously described implementations have required a

pair of directional couplers and associated power detectors.

It is also possible to satisfy the requirements for the

measurement of P by a single coupler arrangement, although

the vari-Z power meter, thus obtained, does not have the

additional applications noted in the previous section. Moreover it is not possible, generally speaking, to avoid the

tuning requirement in the single coupler arrangement.

Referring to figure 13, it is convenient to begin with

an arbitrary three port junction, where port 1 is the (vari-Z)

power meter input, port 2 is terminated by a power meter, and

port 3 is connected to a moving short. In order to satisfy

the requirements of section IV, it is required that the

efficiency between ports 2 and 1 be independent of the short

position. By use of the arguments found in [10], there is

no loss in generality in assuming that the power meter which

terminates port 2 is reflectionless . The efficiency of the

42
two-port, ru-i may then be written 1 8

S 21

+ S S e

11

1 - S 33 e

'21

S ll

+ S 13 S 31

e

S 33 e

(64)

where the S are the scattering coefficients of the 3-port, and

mn 6 r >

is a function of the short position. The next step is to

ask under what condition n~, will be independent of the

choice of 9. Although it is possible to give a condition which

is both necessary and sufficient, the discussion of this

general case is tedious. Instead a number of special cases will be considered and the nature of the general solution

briefly described.

LJ

Figure 13. Use of 3-port as a Vari-Z Power Meter

1

8

In terms of scattering parameters, the efficiency of a two

port terminated in a matched load is given by n

=

] S

?1 | 2 /(l- | S-.-. | 2

) . To generalize this result for a three port

one arm of which is terminated in a moving short, one sets

i A a 3 = b 3 e . The three-port scattering equations then reduce to

two-port form, from which the scattering parameters of the re- sultant "two-port" may be determined by inspection and substituted in the above equation.

43
From (64) a sufficient condition that n ?1 be independent

of 6 is for S^ , S 33 and either S 21 or S-, to vanish. In

figure 14 is shown four possible arrangements, built around

a directional coupler and tuning transformers, by which these

criteria may be realized. To obtain this adjustment in figure

14a or 14b, the junction is first excited via port 3, and

port 1 terminated by a matched (reflect ionless) termination.

Transformer T, is now adjusted for no signal at port 2

(So? = 0) and T., adjusted for a match (reflect ionless) looking into port 3 (S,^ = 0). Excitation is now applied at port

1, port 3 is terminated in a matched load, and T

? adjusted

for a match looking into port 1 (S,, = 0). The above criteria are now satisfied.

The procedure for adjusting the configurations of

figure 14c and 14d is similar. With excitation applied at

port 1, and a matched termination at port 3, T.. is adjusted

for a null at port 2 (S 21= 0) and T, adjusted for a match

at port 1 (S,, = 0). Excitation is next applied at port 3,

the matched termination is connected to port 1, and T ? adjusted

for a match at port 3 (S,, = 0) . The above criteria are again

satis f ied.

Although the foregoing adjustments are sufficient to

yield the desired behavior, they fall outside the power equation philosophy. However it is easily recognized that

tuners T, , T are unnecessary except as a means of recognizing

44
LJ

N^\ — oo © T7T

oj X TTT

LJ

hO

IX

CNI ©

LJ

hO

csi

>o

Ph

ISI

I •H>

-po

Ph to

I El ro o

•H o -p •h cd <« !m •H pi O M

(D >H

ft <m CO flO H O

> -P

<D a;

0)

hO •H

45
the proper adjustment of T~ . The basis for this observation

is the "terminal invariant" character of the subsequent operations at terminals 1 and 3. In particular, terminal 3 is

terminated by a moving short (whose phase is not required)

,

while only the net power is of interest at terminal 1. This

indicates that it should be possible to satisfy the basic

requirements by an adjustment of T- alone.

By means of techniques described in section II, it is

possible to construct a feed-through meter which reads net

power. With port 1 (figure 14) connected to the output of

such a device, it should be possible to adjust T~ for a constant ratio of indicated to delivered power for all short

positions. A more detailed discussion is outside the scope

of this paper.

VII. MEASUREMENT OF SMALL VALUES OF n a As previously noted, the methods for measuring n , described in section III, becomes impractical as the value

of n becomes small (typically 0.1 and less), or if reciprocity

3. is not satisfied. As a consequence, other procedures must

be devised. The previously described solutions [3], [5] to

this problem called for tuning. This is now to be eliminated.

By use of the foregoing methods, a variety of techniques may

be devised.

46
The basic setup is shown in figure 15. The "power meter"

which terminates port 1 may assume a variety of forms, including

(but not limited to) the vari-Z arrangement.

By the use of (25), (27), the power output, P . , from

g^ i the two-port under test, may be written

Vi = P

g

M

ganaNa£ (65) or alternatively, by use of (26), (28),

Vi = P

g

N

gan aMar (66)

With the two-port removed, the power output, P . becomes,

P „ = P M .. (67)

g^2 g g£

Assuming P is held constant, equations (66) and (67)

5

now yield,

P . N n M . g£i ga 'a aZ -&-L = -* » (68)

g£ 2 g£

and this provides a determination of r\ provided that the

mismatch factors and the ratio P . /P . are measured. The

measurement of P and M is perhaps most conveniently

obtained by the arrangement of figure 6. The terms N

and M „ are then obtained as described in section III.

g£

Alternatively, a similar expression results if (67) is

combined with (65). Instead of N„ , M , it is necessary

to measure M , N .. Although the prior discussion has

centered around its use in measurement of the "M" type of

mismatch, the arrangement of figure 6 may also be used to

determine N „

.

3. A,

47
X X

POWER

METER

-Gn

a

Figure 15- Circuit for Measuring Small Values of r|

The foregoing methods have the disadvantage that a signal source is required in conjunction with the terminating

"power meter." The need for this signal source may be eliminated by use of one of the schemes for measuring available

power described in sections IV, V, VI. Referring to (66) and

figure 15 the available power, P , at terminal 1 is given by

P = P N n , f69) gi g ga a' >- uy J

while that at terminal 2, P , is just,

P = P . (701

g2 g L/UJ

Assuming P , P , and N _ are measured, n may be determined

gi g2 ga a from (69) , (70) .

VIII. SUMMARY

This paper has presented the theoretical basis for a wide

collection of measurement techniques which satisfy the con- straints of the power equation concept. In making this collec

tion the emphasis has been on completeness, and little or no

48
attempt has been made to assess the practical merit of the

many alternatives. Although a number of these ideas, particularly those described in Section II have already demonstrated their merit, there are others whose validity or practical value awaits experimental confirmation.

The rationale adopted in this paper differs from prior

descriptions in that, for the most part, a phase detection

capability is assumed. Although this represents a substantial increase in the complexity of the associated equipment,

this feature also permits the elimination of the tuning

transformer and leads to procedures which are compatable with

today's automated systems.

The assumed phase detection capability has also made it

possible to present the theory in a concise form in which the

phase detection requirements are explicitly displayed. This

also permits one to recognize by inspection the many and

varied alternatives for the use of tuning to eliminate the

phase measurement requirement. Such methods come out of the

general theory as special cases, and are of value in those cases where the investment in instrumentation must be kept to

a low level.

With the exception of recent work on a six-port [11] and

a specific procedure for implementing Kerns' impedance method

of evaluating bolometer mount efficiency [12], this paper

49
represents a fairly complete catalogue of the measurement

methods for implementing the power equation concept as they

exist at this writing.

IX. ACKNOWLEDGMENT

As noted earlier, this work is in response to a growing

interest in the power equation concept. In particular the

author has been stimulated by the efforts of Fred R. Hume,

Autonetics Division, North American Rockwell, and R.D. Moyer,

Sandia Laboratories, who are actively reducing these concepts

to practical measurement practice. In addition, some of the

theoretical results, contained herein, have been independently

derived by Mr. Moyer in an alternative form [2]. The author

also expresses his appreciation to Mr. C.A. Hoer, and Mr. Moyer

for their constructive suggestions in the preparation of this

manuscript

.

This effort was sponsored by the DoD Calibration

Coordination Group.

50
X. REFERENCES

[1] Hume, F.R., Koide, F.K., and Dederich, D.J., Practical

and Precise Means of Microwave Power Meter Calibration

Transfer, IEEE Trans, on Instr. and Meas . , IM-21, No. 4,

457-466, (Nov. 1972).

[2] Moyer, R.D., Terminal Invariant Parameter Determinations

from Complex Signal Ratio Measurements, IEEE Trans, on Instr. and Meas., IM-21, No. 4, 532-537, (Nov. 1972).

[3] Engen, G.F., Power Equations: A New Concept in the

Description and Evaluation of Microwave Systems, IEEE

Trans, on Instr. and Meas. IM-20, No. 1, 49-57, (Feb. 1971).

[4] Engen, G.F., and Beatty, R.W. , Microwave Ref lectometer

Techniques, IRE Trans. Microwave Theory and Tech., MTT-7,

351-355, (July 1959)

.

[5] Engen, G.F., An Introduction to the Description and

Evaluation of Microwave Systems Using Terminal Invariant

Parameters, NBS Monograph 112, (Oct. 1969).

[6] Engen, G.F., Amplitude Stabilization of a Microwave Signal

Source, IRE Trans, on Microwave Theory and Tech., MTT-6,

No. 2, 202-206, (April 1958).

[7] Engen, G.F., A Method of Determining the Mismatch Correction

in Microwave Power Measurements, IEEE Trans. Instr. and

Meas., IM-17, No. 4, 392-395, (Dec. 1968).

[8] Miller, C.K.S., Daywitt, W.C., and Arthur, M.G., Noise

Standards, Measurements, and Receiver Noise Definitions,

Proc. IEEE, 55, No. 6, 865-877, (June 1967).

51
[9] Bowman, R.R., Field Strength Above 1 GHz: Measurement

Procedures for Standard Antennas, Proc. IEEE, 55, No. 6,

981-990, (June 1967)

.

[10] Engen, G.F., \ Variable Impedance Power Meter and Adjustable Reflection Coefficient Standard, J. Research, NBS

,

68C, 7-24, (Jan. -Mar. 1964).

[11] Engen, G.F., and Hoer, C.A., Application of an Arbitrary

6-port Junction to Power Measurement Problems, IEEE Trans

on Instr. and Meas . , IM-21, No. 4, 470-474, (Nov. 1972).

[12] Engen, G.F., A Bolometer Mount Efficiency Measurement

Technique, J. Research, NBS, 65C, 113-124, (April-June

1961) .

52
APPENDIX I

In the earlier discussion of figure 6, certain sources

of error were noted which will now be considered in greater

detail. This is perhaps most conveniently done in terms of

a specific example.

By use of the arguments contained in [6] , it is possible

to replace the "second" directional coupler and source by a simple generator, of reflection coefficient r as shown in

figure 16. In addition, the tuner (T ) and load of figure 6 have been replaced by an adjustable termination r . A 20 dB

value will be assumed for both coupling ratio and directivity,

Finally, the magnitudes of the reflection coefficients are assumed to be typically 0.2 (VSWR = 1.5) or less. Following

the measurement of M „, as explained in Section II, the

generator on arm 1 is "turned off" or replaced by a passive

termination of the same impedance.

X H>-

-*—i

** |—*. r r r

g A

[ d

Figure l6 . Circuit for Further Discussion of "Alternate"

M . Application

S *

53
Two questions must now be answered:

1) What change in M „ is engendered by failure of the

passive load impedances to duplicate the active one?

2) How does the power transmission efficiency from

terminal 2 to terminal 3 depend upon r and r ? With regard to the first question it should be noted that

whereas r. has a (small) dependence upon r , this is automatically

taken care of since the cited procedure measures M . for whatever value of T. results when the adjustment of r is completed.

i Let r and r refer to the active and passive terminations

s s r

t respectively, and let r», I\ represent the corresponding

values at terminal 2. Then, if only the first order terms

are retained,

T

l

- h~- S

12

S 21^s " V' CA-D

and for the assumed 20 db coupling factor,

|ri - rj « |r; - r

s

| * io" 2 (A -2)

To continue the discussion, a value of 0.2 is assumed for

i |r - r | . (This means, for example, if the generator on arm 1

is ref lectionless , the passive termination which replaces it

may have a VSWR as large as 1.5 and conversely.) Finally, the

values of |r | and |r | are also assumed to be 0.2.

If, now, these values are substituted in (9), and the

worst phase conditions are assumed, this leads to an error of

0.161 in M „ . Except for the most careful work, this error

54
is ordinarily negligible. Moreover, it can obviously be

reduced by ensuring that r is more nearly equal to r . This

example demonstrates that even a "token" effort in this area

is adequate for most practical purposes.

With regard to the second question, the power transmission

efficiency is determined primarily by | S, ? | 2

. As before a prime

will be used to differentiate between initial and final values,

and only the first order terms will be retained. Then

S 23- S 23

= ^SS^VV + S 12 S31<V r s>

+ S

12

S 51S 35^ r

s

r tr

s

r t)

i i Again values of 0.2 will be assumed for |r - F |, |r - V [,

i i and as an upper limit for the values | r | ,

| r | ,

| T | , | T \

.

For the assumed directivity and coupling values, | S- .S., | and

' S 12 S 31 1 are nominall y

10

" 3 while I S 12 S 51 S 35 ' ~ 10

~ 2, The last

term in (A-3) is thus the important one, and for the assumed

conditions may be as large as .0008. This, in turn, may

change the power transmissions efficiency by as much as 0.16%.

i As before, this error may be reduced by making r more

nearly equal to r . The value of r , however is dictated by

the value of r and less subject to user control, at least

g

without restricting the range of potential application. On

the other hand, the sensitivity to the |r - T \ term may be

t reduced by the proper choice of r . If T - T , (A-3) becomes,

55
S 23" S 23 " ( S

52

S 35

+ S

12

S 51S 35r

S HVV (A-4)

and the right hand side vanishes for the proper choice of T .

In practice the correct adjustment of r may be recognized by

exciting the system via port 2, and adjusting V for a null

in arm 5

.

The foregoing example demonstrates that the assumed 20 dB

coupling and directivity are ordinarily sufficient to insure

negligable errors except in the most exacting applications.

Moreover, this error may be reduced by the cited measures or by

increasing the coupling and directivity values.

56
APPENDIX II The purpose of this appendix is to find the maximum value

of n „, as given by (31), as a function of w. Conceptually,

this is a straightforward exercise in differential calculus,

however the effort involved may be substantial unless certain

"short-cuts" are observed.

As it stands, (31) is not an analytic function of w because of the absolute value operation. However if x and y

represent the real and imaginary parts of w, it is convenient

to consider w and w* as two different functions of the independent variables x,y. Equation (31) can then be written as

an analytic function of w and w*,

R 2 fR 1 2 -(w-R

Ci )(w

ft -R^)|

n a£

= — ,

* * • (A " 5)

Ri R 2

2 -(w-R )(w -R ) c 2 c 2

A necessary and sufficient condition (see Appendix 3) that n «

have an extremum is for

' = 0, (A-6)

3w

(and in which w* is assumed to be a constant)

.

Together with (A-5) this leads to,

2 Ri - I w-R I 2 w-R

Ci c i R 2

- Iw-R I 2

2 w-R ' c 2

' c 2

as the condition on w for which n ofl is a maximum (n„)

.

a jc d

57

(A-7)
It is thus permissible to substitute n for n in (A-5)

,

provided that w satisfies (A-7). The remaining step is to

eliminate w between (A-5) and (A-7) . In theory, (A-7) would

be solved for w and substituted in (A-5) but this approach

is tedious. Comparison of (A-5) and (A-7) shows that,

R 2 ^- R

Cl ) n a

= . (A-8)

Ri (w-R ) C2

After cross multiplication and cancellation, (A-7) may be

written,

w-R Ri + (w-R )(R* -R* ) C i C i C i c 2 w-R R2 c 2 And this may be combined with (A-8) to obtain,

Ri + (w-R. )(R* -R* ) C i C i c 2 n a

= (A-9)

RiR2 At this point (A-8) is solved for w, and substituted in

(A-9) . This leads to (32)

.

58
APPENDIX III

Given an analytic function p, of the complex variables

z, z* such that p is real for all choices of z, it will be

shown that a necessary and sufficient condition for an extremum of p is that,

3£ = 0> (A-10)

9z

It will prove convenient to introduce some additional

notation

z = u = x + jy (A-ll)

z* = v = x - jy (A-12)

where x and y are respectively the real and imaginary components of z.

Then,

p = f(u,v)

and following the usual rules for differentiation,

dp = M du + — dv

9u 9v — (dx+jdy) + — (dx-jdy)

9u 9v —9f 9f), .f9f 3fl

+ — dx + j 9u 9v 9u 9v

dy

'* 9v

(A-14) becomes,

dp = (a+B) dx + j(a-B) dy

(A-13)

(A-14)

3 f 9 f

To further simplify the notation, let ^— = a, -^r = 3. Then

a U

CA-15)

59
By hypothesis, p is real for all choices of x and y.

Therefore dp must be real for arbitrary dx, dy. This is only

possible if (a+3) is a real number, and (oc-3) is imaginary.

This leads to the following equations,

it A a+3 = a + 6 , (A-16)

a-B = -a* + 3*, (A-17)

from which,

a = 3*. (A-18)

From (A-18), if a vanishes, the same is true of 3, and

(A-15) shows this is sufficient to ensure dp = . Conversely,

it is easily recognized from (A-15) that if dp vanishes for

arbitrary dx, dy, this requires a = 3 = 0. This completes

the proof.

60
APPENDIX IV

This appendix will outline derivation of (63) . As noted

in the discussion attending (63) , it is instructive to let

the circuit of figure 2 represent alternatively the "output"

element of figure 3 and the "input" element of figure 12.

In the former case, no change in terminology is required, but

in the latter, the "P " and "port 2" of figure 12, become

the "P/' and "port 1" of figure 2 respectively. As previously

noted, the parameters C, D are determined by the "output"

coupler of figure 3, while A, B are parameters of the "input"

coupler of figure 12.

In figure 2, as previously noted,

b 4 = Ca 2 + Db 2 (A-19)

while the change in terminology outlined above leads to

b 4 = Aai + Bbi

.

(A-20)

It will now prove convenient to consider the scattering

description 18 of figure 2.

bi = S ai + S 12 a 2

b 2 = S 12 a! + S 22 a 2 (A-21)

bi* = S 4i a i + S 42 a2

where reciprocity (S,

7 = S^,) has been assumed. This system

of equations (A-21) can be solved for b 4 as a linear function

of any two of the parameters ai a 2 , bi, b 2 . Equations (A-19),

18 By use of the arguments contained in [10] it is permissible

to assume a 4 = .

61
(A-20) are two such equations. 19 In addition,

S.21

11

(Dbi - Aa 2 ) (A-22)

The efficiency, n,., is given by,

bi ai

'12 (A- 2 3) a 2 subject to the condition, bi» = 0. This condition, in combination with, (A-19), (A-20), (A-22), (A-23) leads to,

|A| 2 -|B| 2

'12 P

4

= D

(A-24)

Finally, this result may be compared with (59) and (61) to

obtain (63) .

1 9 Explicit expressions for A'»«D in terms of the S could

be introduced but these are not required. mn

62
APPENDIX V

Throughout this paper it has been assumed that the described systems are such that R > |R | . Returning to (13) and

(14) , this will be true if,

AD - BC

Dl 2 - C

BD*-AC*

D

2

- C

Squaring both sides and cancellation yields,

AD BC BD AC

(A-25)

(A-26)

This may be simplified to obtain,

(|A| 2 -|B| 2 )(|D| 2 -|C| 2) > o, (A-27)

So that R > |R | provided that

|A| > |B|

,

(A-28)

|D| > |C|

.

(A-29)

In terms of (3) and (10), this means that b k must be

more strongly coupled to b 2 than to a 2 , while the converse

must be true of b 3

. Alternatively, one may have |B| > |A|

and |C| > |D| , but this is of no interest in the immediate

context

.

63
FORM NBS-114A (1-71) 2. Gov t Accession No.

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEETS

1. PUBLICATION OR REPORT NO. NBS TN-6373. Recipient's Accession No.

4. TITLE AND SUBTITLE

Theory of UHF and Microwave Measurements Using the Power

Equation Concept

5. Publication Date April 1973

6. Performing Organization Code

7. AUTHOR(S)

Glenn F. Engen

8. Performing Organization

10. Project/Task/Work Unit No. 2721393

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

11. Contract/Grant No. CCG Project

12. Sponsoring Organization Name and Address Calibration Coordination Group

Department of Defense

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) Theoretical considerations indicate that, in addition to other advantages, UHF

and microwave measurements based on the power equation concept should exhibit reduced

sensitivity to certain connector imperfections. This has now been experimentally

confirmed in a number of laboratories, and interest in these methods is increasing.

The previously described measurement techniques, for implementing this concept, have

relied heavily upon tuning procedures, while current trends are towards automation

where tuning is impractical if not impossible.

By assuming a phase detection capability (such as is found in automated systems)

it is possible to eliminate the tuning requirement. Moreover, this generalization

has made it possible to summarize the earlier results in a compact form. In addition,

this paper includes a number of new measurement ideas

It is expected that this report will serve as a major reference in the continuing

development of detailed measurement procedures which exploit the power equation

concept

17. KEY WORDS (Alphabetical order, separated by semicolons)

Microwave; Microwave measurements; Power equation; Power measurement; Terminal

invariant

18. AVAILABILITY STATEMENT

l~yl UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

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