IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-20, NO. 1, FEBRUARY 1971 49

Power Equations: A New Concept in the Descriptionand Evaluation of Microwave Systems

GLENN F. ENGEN

Abstract-In the existing art, the description and evaluation By contrast there is a substantial loss or suppression ofof microwave systems is usually in terms of the complex wave detail in going from the field to the circuit description.amplitudes in the (assumed) uniform waveguide by which thecomponents are interconnected. The interrelationships among This loss of detail greatly restricts the domain of appli-these amplitudes are given by the scattering equations or indirectly cability of the circuit equations as compared with the fieldby impedance parameters. equations; but, on the other hand, it is also this loss ofThis paper introduces an altemative description, based on power detail that leads to a greatly simplified description and

equations, where the basic parameter is the net power in the puts the remaining parameters in better focus.waveguide of interest. Compared with the circuit description, In addition to the field and circuit descriptions is athere is a substantial suppression of detail and the remainingparameters are placed in better perspective. In particular, the third that, despite its extensive usage, has perhaps nottechnique provides a simplified method of describing and evaluating been explicitly recognized as such. Provided that themismatch corrections, in which the precision connector and uniform different components that comprise the system arewaveguide requirements have been eliminated. "matched," a particularly simple description is possible.

In this description all terminations are equivalentI. INTRODUCTION (matched or reflectionless) while the description of a

F ONE reviews the history of communications, it is generator requires a single parameter-the power outputapparent there was a time when the field engineers (to a matched load of course). The characterization of aexpected to have a "field" day with the emerging two-port is also achieved by a single parameter-its

microwave art. Because the equipment dimensions, attenuation if the device is passive or gain if active. Inmeasured in wavelengths of the associated radiation, were addition, amplifier noise properties are completelyof the order of unity and retardation effects could no longer specified in terms of noise figure or noise temperature.be ignored, it was easily argued that field theory was the Provided that the hypothesis of matched componentsappropriate tool. is indeed satisfied, the utility of this scheme is self-evident.

Fortunately, it did not take the circuit engineers long Unfortunately, although this condition is included in theto recognize the similarities in the wave propagation that design objectives of most systems, the accuracy demandsoccur in uniconductor waveguide as compared with the in system evaluation are often such that the deviationsalready familiar TEM/I propagation in coaxial line. This from this idealization can not be ignored.was followed by a generalization of the lower frequency One possible response is to return to the descriptionvoltage and current concepts to describe the electro- found in the circuit equations, and no doubt this is done inmagnetic phenomena in uniconductor waveguide, and many instances. In other cases, however, a substantialwith this generalization a large body of circuit theory effort has been made to salvage this simple description bybecame immediately applicable to waveguide problems. patching it up with mismatch corrections and related(For a rigorous treatment of this topic see [1].) Because concepts. Unfortunately, there has been no consensus onthe ultimate justification for this use of circuit theory how best to do this, and the existing art is characterizedrests upon the wave motion in the (assumed) uniform by a number of competing ideas and terminologies.waveguide by which the components are interconnected, (A recent survey of the literature reveals the existence ofit became convenient to adopt the wave amplitudes as the four "kinds" of generator power, a dozen or more termsfundamental parameters, and the "scattering" equations (many redundant) for describing the "loss" characteristicsemerged. of a two-port, etc.) Much of this appears to be rooted, atMicrowave circuit theory also assumes a variety of least implicitly, in the unresolved question of how best to

other forms (e.g., "cascading" parameters); these descrip- characterize the components in a mismatched systemtions are equivalent in that the information content is without reverting to the circuit description.the same though in different form and in that it is possible It is the purpose of this paper to introduce a systemto transform one description to another except for certain description, based on power equations, which explicitlysingularities, accounts for mismatch effects and yet avoids much of the

complexity of the circuit equations. In addition, itManuscript received August 31, 1970. This work was supported includes as a special case the simple description of a

by the National Bureau of Standards, Boulder, Colo, matched system outlined above. In the power equationThe author is with the. Microwa.ve Circuit Standards Section, description, the net power in the wavreguide of interest is

Radio Standards Engineering Division, National Bureau of Stand-ards, Boulder, Colo. the fundamental and real parameter. By contrast the

50 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, FEBRUARY 1971

1 2 3

p pV9g

L L LL AAr

PFig. 1. Elementary model used as starting point in power equation r' r

description. g 9r

IPg

circuit approach uses two complex parameters (e.g., the Fig. 2. A more general model where the generator and load im-

forward and reverse wave amplitudes) to describe the pedances are incorporated by means of lossless two-ports.behavior at a given terminal surface. It will be immediatelyrecognized that the power equation description includes terms of the power that it would deliver to a matched loada further suppression of detail; in consequence the remain- connected at terminal 1. Although this is not possibleing parameters are placed in better focus. In particular (except conceptually), this same power is "available" atthe power equation description provides the following. terminal 2 and is obtained if the load reflection coefficient

1) A systematic and logical basis for the choice of rm is chosen so that the reflection coefficient rP, lookingparameters in mismatched systems (to replace the to the right at terminal 1, vanishes. This quantity is calledmany competing terminologies now in use) . the available power and is the maximum power the

2) A simple system description that provides additional generator will deliver as a function of load impedance.insights into mismatch and related measurement It is worth noting that the available power is dependentproblems. only on that portion of the system to the left of terminal 1,

3) Simplified measurement procedures for evaluating while the generator impedance is determined by thatmismatch corrections. portion between terminaLs 1 and 2.

4) An unanticipated dividend-the precision connector By use of conventional microwave circuit theory, it canrequirement is substantially reduced or eliminated be shown that the power Pgj delivered to the load inin many measurements of practical interest. Fig. 2 is given by the equation

II. GENERAL DESCRIPTION P - (1pg jJi- 2)(r, r12) (1)

In order to describe the power equation approach, it is where P. is the available power from the generator and Pg,useful to consider some specific systems, beginning with rP are, respectively, the generator and load reflectionan assumed impedance match for the different components coefficients.and then generalizing to the mismatched case. Fig. 1 if r, vanishes, P.1 is given byshows an elementary system that consists of a matched(reflectionless) generator and a matched load. As already Pg|jrp = Pg( -Ir, 12) (2)noted, the only additional parameter needed to charac- while if rg vanishesterize the components is the power output from thegenerator. Pglrg=r = Pg(1- IrI2). (3)

This system may be generalized to account for mismatch Finally it is possible to write (1) in the alternative formeffects by the introduction of lossless elements (tuners), asshown in Fig. 2. In particular, the generator now includes Pg1 P=,(1- (r, - r*)/(1 - r,rg)12). (4)two elements, a matched source (as in Fig. 1) and a losslesstwo-port or tuer th elmet of whc hav bencoe In (1)-(4) the different quantities are referred to terminal 2.

two-poyield tuner,theru edevae of g ratr imednchsen.A Next it is instructive to analyze the same system withto yiel th reuie vau of geeao impedance. reference to terminal 1. At this point the generator issimilar observation applies to the load.' In addition to the rfece to t is pointh geto isphysically determined (or actual) interface between matched and it is possible, by comparison with (3)generator and load (which has been designated terminal to writesurface 2 in Fig. 2), it is also desirable to introduce P, = P'(1- Ir, 12) (5)terminal surfaces between the lossless elements and thegenerator (or load) that comprise the "equivalent circuit." where the primes have been added to indicate that theThese have been designated 1 and 3 and are useful con- quantities are referred to terminal 1. However, it has beenceptually even though they have no real existence. postulated that the two-port included between terminal 1Finally, it is convenient to characterize the generator in and 2 is lossless. Therefore

P = P01 (6)1 That such a realization is po.ssible is presumably self-evident

since a los.sless transformer in conjunction with a matched load (or Po Po (7)any load that is not completely reactive) can, in principle, be ad-justed for any desired impedance. It is noted, however, that the and comparison of (4) and (5) indicates thatparameters of these lossless elements are not uniquely determinedsince this would require three quantities, and the complex generator (It - -I r*I)/(t1 -r rj) ( 8)or load impedance determines only two. In particullar, it would be 11 -1 t- 9}\ lpossible to insert an arbitrary length of line (phase shift) between thetuner and generator (or load) without affecting the model. That this is indeed the case can be confirmed mathe-

ENGEN: POWER EQUATION CONCEPT FOR MICROWAVE SYSTEMS 51

matically by evaluating the cascaded properties of thetwo two-ports in question. If the system is analyzed withrespect to terminal 3, a similar argument holds.

It is now convenient to rewrite (1) as

Pg = PMg (9)Mgwhere <ss ( o Jwhere~ ~ 2)1 ENFORCED LOSSLESS LOSSY(/1 -ImrgI|\2)(II-p 12) |I-*l| CURRENT REGION DIELECTRICS

Mgl = 12= 1 - I - rr12 (10) Fig. 3. Illustrating the generality of the power equation description.

From (6)-(9) it is evident that the terms Pgj, Pg, and Mglare terminal invariant. That is, the values are the same By use of (10), it can be easily shown that these ex-irrespective of which of the terminal surfaces (1, 2, or 3) pressions are equivalent, as indeed they must be. Theis chosen as the interface between generator and load. In point to be made here is that whereas the circuit descrip-particular, although Pl # P" and rP # rP, primes may tion points to the three complex parameters PO, I', r,, asbe added in the term the route to the mismatch correction, the power equation

method substitutes the two real quantities Mg, and Mgm.(P2 - I"*)/(l -rPrg)l If r ,rP,m are known, M1lg, and MIgm may, of course, be

without changing its value. (Note that r = 0.) obtained from (10). However the determination of Mg,Referring again to Fig. 2, it shouldg be noted that and Mg. is by no means contingent upon first determining

rg, rP provide a (partial) characterization of the respective Pg, ri, Pm. Indeed it is possible to measure accurately Mg1two-ports. The real termMl on theoand Mgm in situations where the r are ill-defined, if at all.

wo-ports.ofthe rembinalterm MgThnthi thrhadis a specificexIt must also be emphasized that the physical significancepoerlofsindetailthehcombinatiaon hiseista spowei exuam iof of (9) goes far bevond the mere relabeling or regrouping ofthelssction. dtlhtcmaeteo rqtn terms that have taken place. In particular, whereas thedescription. single-mode and uniform-waveguide hypothesis has beenAt this point it may be observed that if one is interested . . .implicit in the discussion to this point, (9) iS valid in ain predicting the generator behavior with arbitrary, or as much more eneral class of systems includin multimodeyet unspecified, loads, one needs both Pg and rg. Alterna- oguyvtively, if one is given a load and wishes to predict its orth us fo iegular swaveguide.operation with arbitrary generators, the important into three regions. In the left-hand portion or "generator"parameter is rP. However, once the generator and load is an enforced current density J that is the source of anhave been chosen, the only parameters that need be electromagnetic field throughout the structure. Themeasured for the type of system evaluation here con- r r c l d m

rsght-hand regaon contaMns lossy dielectric materials thatabsorb energy from the field and thus constitutes a "load."As an example, it is useful to consider a power measure- y

c

ment problem involving a terminating-type power meter. T n regin is lossfee.In applying (9) to the system of Fig. 3, it should firstIn general the use of this device includes its substitution b nin lac oftheloa, fr wichthepowr i reuird; nd be noted that the available power P, iS the same for all

inplacefthelotimateinad,cforawhich thespweraisma rred;a possible choices of terminal surface in the lossless region.(This is a generalization of the behavior that was pre-tion including the terms rg, rP and the meter reflection viously noted in connection with Fig. 2.) The same is

coeffcient Pm iS required. obviously true of P . The term Ilgl expresses to whatThe power equation method also requires the substitu- y gi pbutprvie a difrn approachto extent the conditions for maximum power transfer havetion of meter for load but provides a different approach to been satisfied by the given "load." Because of the terminalthe mismatch problem. In particular, let Fig. 2 and (9) invariant character of Pg and Pg1, M,1 must also share

represent a system in which Pg2 is required. If the power t 2meter iS substitutedl for the load, (9) becomes thisproprtv

Unfortunately it is not possible, at present, to utilizePgm = P M (11) fully this increased generality because the existing

wherePg. is tepoerndcaedyheetrmethods for measuring Mg1, etc. (described below)weePm istepwridctdb h ee ujc o require that the system be a "nominal approximation"3 to

possible efficiency corrections, etc., and Mgm the mismatch aesine-mode stute. Tese arm to,mhowever,m ~~~asingle-mode structure. These arguments do, however,term between it and the generator.Equations (9) and (11) can be combined to yield 2 It is perhaps worth noting, that there are two types of terminal

pnz as / a,> X , \ ~~~invariance. In the case of P¢7 it is possible to extend the output portPl = 1Pgm(lV/lII9/11gm) (12) by perfectly conducting walls of arbitrary geometry without chang-ing this parameter. The terms Pgl and Mg z, on the other hand, are

while the corresponding expression from circuit theory is parameters of the entire sy.stem. Here the terminal invariant prop-erty permits one to be arbitrary in the specification of the interface

I l rr12 {1 r 12\ between generator and load, but there is no occasion for adding top = p0, - PgPXl I(1 - IP12 (_ 13) the system as could be done with Pg.vl-g 1 - PgPij1 (1 - Pm II) 3For a more complete discussion of this point see [2].

52 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, FEBRUARY 1971

a b 2 1 a b 2

Fig. 4. Model used for arbitrary two-port. Fig. 5. Two-port inserted in arbitrary system.

provide the basis for relaxing the tolerances on waveguide the adjacent lossless elements as was done with thecross section, connector properties, etc. generator and load of Fig. 2. However, it is also necessaryIn the discussion that follows, it is convenient to use to recognize the possibility of interaction between these

the existing circuit theory as a tool for deriving these components and the topic must be approached withmeasurement methods. First, however, it is desirable to caution.extend the discussion to include two-port devices or Starting with the two-port, its efficiency with the given"attenuators." load will be designated flal. Then it is convenient to defineThe appropriate model for a two-port, in the power another mismatch factor Nai by the equation

equation description, is a bilaterally matched attenuator(possibly nonreciprocal) that is preceded and followed by t1al = iaNai, (14)lossless tuners4 as shown in Fig. 4. It is also convenient i.e., Nal is the ratio of the efficiency for a given load to theto introduce the terminal surfaces a, b, as shown. Having maximum efficiency. The terminal invariance of Oa hasintroduced this equivalent circuit, the next question is already been noted while f7al and Nal possess terminalthat of characterizing the individual elements. invariance of the "second type."

Beginning with the central element, a convenient It can be shown (see [2]) that Nai is given by thespecification is in terms of its attenuation or efficiency with equationrespect to a matched load. It must be emphasized that 2 2 2 2

these terms refer only to the central element and not the Nat = 1 - [(1 - naDbI]/(1-blaIrbI) (15)two-port as a whole. In general the efficiency X is defined whereas the ratio of power output to power input and is a I1bJ = (1P1- Tm2)/(l- rlrm2)I (16)function of the load impedance. It can be shown that fora bilaterally matched two-port (S,l = S22 = 0), the The similarity between this expression and (10) should beefficiency is a maximum for a matched load. Returning noted. In particular, if Oau goes to zero, Na, approaches Mai.to the actual two-port, which has been represented by the Conversely, if 77a approaches unity, Nag also goes to unity,equivalent circuit of Fig. 4, it is apparent that its efficiency irrespective of the value of rbI.will be a maximum when terminated by a load r, such With these definitions it is now possible to express thethat rb = 0. The maximum efficiency % or intrinsic power Pgi delivered to the loadattenuation of a two-port thus characterizes in part this PgI = PgMga'7aNai (17)central element. (To complete the description, one requiresa phase shift and, if nonreciprocal, a second value of 77a where Mga is the mismatch factor between the generatorand phase shift.) and the attenuator-load combination (regarded as a

The specification of the lossless tuners that precede termination); the other terms have been defined pre-and follow the central element may be effected in terms of viously. The validity of this may be confirmed by inspec-the load reflection coefficients rm1,im2, which, when used tion; the first two factors, on the right-hand side, give theto terminate ports 1 and 2, respectively, yield the maxi- power at terminal 1, and the second two factors give themum efficiency. As before, this specification is not unique ratio of the power at terminal 2 to that at terminal 1.in that a phase term in the tuners is undetermined. It is It should be noted that whereas Nat is readily expressibleconvenient, however, to combine the phase terms and in terms of fla and irbl, no such representation for M,a isassign them to the central element. As is well known, the introduced because of its complexity. Instead, once thespecification of an arbitrary two-port requires four complex attenuator-load combination has been formed, this isparameters. In this representation the left-hand element interpreted as a new "termination" for which the prioris characterized by rm,, the central one by two terms of analysis is applicable. In particular this means that thethe form 71aei5 where O is a "phase factor", and the right- parameters of the left-hand tuner in the attenuator modelhand element by Pm2. From the foregoing discussion it is undergo a transition in forming the "model" for the newevident that fla is terminal invariant while Pm1, rm2 are not. termination. An exception to this occurs if either Pb1 or

If, now, this two-port is inserted between the generator 11a is very small. In this case the changes in tuner param-and load of Fig. 2, the system of Fig. 5 results. At first eters are also small.glance it might appear possible and desirable to combine As already noted, the analysis that leads to (17) first

identifies the attenuator with the load so that a new "load"is formed. Alternatively, it is possible to identify first the

4The basic elements of this representation have previously been ateutrwt h eeao ota e gnrtrused by a number of authors, e.g., [3] and [4]. atnao ihtegnrtrs htanwgnrtr

ENGEN: POWER EQUATION CONCEPT FOR MICROWAVE SYSTEMS 53

is formed. This leads to the equation

PgJ = PgNga71aMal (18)4where the first three factors on the right-hand side repre-sent the available power at terminal 2, and Mal is themismatch between this "generator" and the load. With a 2suitable change in notation, Nga is in the same form asNai, i.e., Fig. 6. Use of a directional coupler with (optional) feedback to

simulate a stable generator.Nga = 1 - [(1 _7-2) |IraP2y/(l _ 2 Ira12) (19)

where significant improvement in accuracy resulted. Unfortun-

ral =(irg - rmiI)/(l1 - rgr*,1). (20) ately, these tuning methods proved both frequencyv|- |D-ml - gml *sensitive and time consuming; and with the furtherThe previous distinction that was made between the development and improvement of broad-band components,

circuit and power equation methods is also applicable swept frequency measurements became popular (even ifhere. In particular, if one is interested in predicting the less accurate). More recently, many of the better featuresoperation of the two-port in arbitrary systems, the of these two methods have been combined in automated"complete" description provided by the circuit parameters stepped-frequency systems where, in particular, the needis required. If, on the other hand, one is only interested in for tuning is avoided by including a phase detectionmaking the mismatch corrections associated with its use to capability.reduce the power in connection with a specific generator- The measurement methods, outlined in the followingload combination, this can be effected by the measurement paragraphs, are closely related to the tuned-reflectometerof Mgl, Mga, and Nat. This assumes the two-port has procedures and thus share the same frequency sensitivitybeen calibrated in terms of its maximum efficiency rather problems. On the other hand, these methods have thethan the more usual attenuation. advantage of requiring a minimum of instrumentationThese examples of the application of the power equation and provide a useful starting point for additional develop-

method are intended to be illustrative rather than ex- ment. It is possible to anticipate some application ofhaustive. In particular an additional application is that swept frequency techniques in this area, although thisof providing a more complete description of amplifier expectation is seriously qualified by recognition that thenoise performance. This, however, is the subject of residual errors in these systems are typically of the sameanother paper [5]. order as those to be evaluated. It is possible to eliminate

the tuning requirement by the introduction of phaseIII. IMPLEMENTATION detection capability, such as is found in automated

Thus far the presentation of the power equation method systems, and indeed the automated system can in principlehas been confined primarily to definitions and illustrations if not in current practice be programmed to provide aof its application in the description and analysis of description in these terms.microwave systems. These ideas have already proved Before continuing with specific measurement methods,their merit in a number of theoretical studies. The it should be noted that a substantial part of the existingpractical aspects and associated measurement techniques art alreadv fits this scheme. In the case of a bolometricwill be considered next. power meter, for example, the appropriate parameter isIn order to make a field application of this scheme, the effective efficiency rather than "calibration factor."

one requires measurement methods to determine the It is of interest that it is this parameter that is determinedparameters Pg, 7ia) M, N, etc. In addition, if the full by both the microcalorimetric and impedance methodspotential is to be realized, these methods should be such of bolometer mount evaluation, and that, in turn, provideas to avoid the waveguide uniformity and precision the basis for the existing National Bureau of Standardsconnector requirements associated with a description in (NBS) power calibration service. The dissemination ofterms of the circuit parameters. From one point of view, this measurand (bolometer mount efficiency) may bethese procedures may be regarded an extension of imped- effected in a way that avoids precision connector require-ance measurements and this provides a convenient ments [6] and the measurement of impedance in the usualstarting point. sense. Note, however, that if the dissemination is in terms

In the early microwave art, the slotted line enjoyed a of "calibration factor," both of these are required. Theprominent role as an instrument for impedance measure- following section includes a "reconstruction" of thisment. With increasing accuracy demands, there was a procedure from a different point of view.shift from the slotted line to the refiectometer; initiallythis was a matter of substituting imperfect directivity A. Measurement of P,, and M,,errors for probe coupling and other slotted-line irregular- In the existing art, it is common practice to provide aities. This was followed by the development of tuning microwave signal generator with an output monitor in theprocedures to eliminate the directivity error, and a form of a directional coupler and sidearm detector, as

54 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, FEBRUARY 1971

P3 zP4 unity (M0, = 0). Observation of P3/P4 thus provides)2

K2/K1. With a power meter connected to port 2, P,1 is__________________________________/\/V_ indicated and K, can be obtained from (21) by substituting

TY the associated values of P3, P4 and K2/KI.rg ' re In the second measurement method, T1 is adjusted soMOe that P3 vanishes when port 2 is terminated by the load

Fig. 7. Addition of second coupler and tuner (T7) to provide rP for which Mg0 is required. If, now, this load is replacedmeasurement of M,1. by a moving short, the ratio P3/P4 will undergo cyclic

variations as a function of the short position.

shown in Fig. 6. It is convenient to postulate a feedback Letloop so that the sidearm power P4 is constant. The Q = V/(P3/P4) max (23)impedance of the "equivalent" generator that obtains at

q _= _24port 2 of the coupler is determined entirely by the coupler = V\(P3/P4) Imi. (24)parameters [7] and can be adjusted, if desired, by a tuning Then it can be shown5 thattransformer at this point.The available power at port 2 is proportional to P4, and Mllgl =1-[(Q - q)/(Q + q)]2. (25)

the proportionality factor is again a function of thecoupler parameters and independent of the actual signal It is of interest to note that the squared expression is justsource. In the event that the feedback loop is not em- the one used to convert voltage standing-wave ratioployed, the same description holds except that the (VSWR) into reflection coefficient magnitude.available power is no longer constant but still proportional If the load happens to be a terminating power meter,to P4. Because the parameters of the equivalent generator P01 is known, and the proportionality factor between P0P9, rg are determined entirely by the coupler characteris- and P4 may again be determined.tics (and the value of P4), it is convenient to think of the The required adjustment6 of T0 is much more easilycoupler and sidearm combination as the "equivalent" effected in this second case but, unfortunately, a separategenerator. adjustment is required for each value of rI. Conversely,A measurement of the mismatch factor Mgl, between the adjustment of T0 is more difficult in the first case but,

this "generator" and an arbitrary load, can be effected once obtained, is the same for all values of rP.by the use of an additional directional coupler with sidearm B Measurement of Ng.detector P3 and tuning transformer T5, as shown in Fig. 7.Although the order of the couplers has been reversed, the The measurement of Nga may also be efected by use ofbasic configuration will be recognized as a reflectometer. theq refiectometer as shown in Fig. 8. The procedureThe prescribed tuning conditions, however, are different, requires first the adjustment of T0 such that the ratioand the device is called a generalized reflectometer P3/P4 iS constant with the sliding short either at terminal 2(g reflectometer). or alternatively at terminal 1 (the "output" of theThere are two different measurement procedures that two-port). It will prove convenient to continue the use

lead to M0l. In the first, T, is adjusted so that the ratio of Q and q to represent maximum and minimum valuesP3/P4 is constant in response to a sliding short at the of P3/P4 in response to the short motion and to use theoutput port. Subject to this adjustment, it can be shown subscripts 2 and 1 to indicate whether the short is at[6], [8] that the net power P51 delivered to an arbitrary terminal 2 or terminal 1. The first of the above tuningload this given by conditions is thus Q2 = q2 and the second, Q1 = q,. Under

the first of these [2]

where ~Pg1 = K1P4[1 - (K2P3/KIP4)] (21)Ng 2\± (±2)1/2(- ()2)1/2)

P,,4 are the observed powers at the respective Ng= 1-!(21 Q2 (1 Q2 Q2detectors and K1, K2 are system parameters. Note that Q2 = (26)K1 depends only on the coupler and detector adjacentto port 2. while under the second

For a given value of P4, Pgl is a maximum when P3 Ng= 1vanishes; therefore

((Q2 I + (q2 _ 2)- 2((Q_ -Q)(q2 -pg = K1P4 ('22) -\((Q(Q- Q2) ± + qY J

and comparison with (9) shows that Ql= q1. (27)

MqIg= 1 - (K2P3/K1P4). 5 A more detailed description of this procedure, and proof of thisrelationship will be included in a paper to follow.

A determination of K2/K1 and K1 is made as follows. 6 It is to be emphasized that in either case the prescribed adjust-' ~~~~~~~~~mentcomnpletely compensates for imperfect directivity, load im-

With a short connected to port 2, K2P13/K1P4 must equal pedance. etc.

ENGEN: POWER EQUATION CONCEPT FOR MICROWAVE SYSTEMS a5

P3 P4 P3 P4

253 1 stJ 2 1

Nga ,,,Fig. 9. System for measuring ,¢.

Fig. 8. Use of 9i reflectometer to measure Ng..It is emphasized that these procedures are not to be

regarded as the ultimate implementation of the conceptThe upper sign is used in (26) if t7a < Pa| otherwise the but rather as a demonstration that it is indeed possiblelower sign is correct. This choice can often be made on to avoid the precision waveguide and connector require-the basis of an approximate knowledge of the parameters ments in measurement practice as well as theory.involved; otherwise it can be resolved as explained in [2].This ambiguity does not arise in (27). IV. APPLICATION

If, instead of Ngg, Nai is required, this may be obtained This presentation of the power equation concept wouldby an additional tuning adjustment of the impedance of not be complete without calling specific attention tothe "equivalent generator" associated with theg reflectom- several areas of potential application. In particular iteter to equal that of the load in question as described now appears desirable to reexamine the existing practice,in [7]. much of which has been formulated on the basis of the

assumed waveguide uniformity and precision connectorC. Measurement of tRa requirement.The measurement of 71a requires, in addition to the First it will be recognized that the power equations

g reflectometer, a detector of wide dynamic range. It is provide a simple, yet significant, generalization of theconvenient to envision the procedure as an insertion loss "everything matched" description. As such, it shouldmeasurement with supplemental measurements of the prove a valuable tool in technician training courses byM and N mismatch factors. In Fig. 9, for example, it is putting the subject of mismatch errors in a more readilyconvenient to adjust the detector impedance for maximum understandable form.detector indication. The system is then described by (18) In spite of its simplicity, the description is adequatewith Mai equal to unity. The other mismatch factor for many practical problems. If more detail is required,Nga is determined by one of the already cited techniques; this can be added without eliminating or obscuring thefinally the detector indication, which is proportional original features. As already noted, the description of ato Pgl, and P4, which is proportional to Pg, is observed. generator begins with Pg. If more detail is required,If the two-port is removed, the system may be described rP is added. The two-port description begins with 27a, toby (9), and if Mg, is now measured, this leads to a pair which rm1, rm2, and the phase term may be added. [Al-of equations that may be solved for 77. It should be noted though the emphasis here is on passive components, thethat it is not necessary to assume the tuner associated with definitions are also applicable to active devices (amplifiers).the detector is "lossless." In the latter case an alternative form of the maximum

In order to keep this section to a reasonable length, efficiency concept is useful: the ratio of the availableit has been necessary to confine it to a brief sketch of some power output to the net power input.] In this way anof the more useful techniques. (The validity of these alternative form of the circuit description is obtained, onemethods has been confirmed experimentally by L. L. which is of considerable interest in its own right. In thisErard in a private communication.) A number of addi- context, it is of interest to note that if 71a is small, rm1 -

tional ones have been developed; some of these are S *, Pin - S *, and, alternatively, if the magnitudes ofdescribed in [2], and others will be reported in a later these terms are small, 'q1a S2112. A more detailed dis-publication. cussion is, unfortunately, outside the scope of this paper.

In an earlier paragraph it was noted that the measure- As previously noted, today's microwave engineer mustment procedures should be such as to avoid the precision cope not only with a proliferation of terminology but alsoconnector requirement. That this criterion is satisfied is with a variety of meanings to these terms. Unfortunately,intuitively evident in that these techniques involve only this continues to exist even in the presence of well-the use of a sliding short, tuning adjustments, and methods considered definitions [9]; it appears just too much tofor recognizing the "equality" of two "impedances." It expect that every engineer is going to become conversantmay also be observed that, to the extent they are lossless, with a dozen or more terms to describe the loss charac-the connector properties appear in the "lossless" elements teristics of a two-port, for example. What is needed is athat were introduced to account for the impedance common denominator for tying this terminology together;properties (cf. :Figs. 2, 4, 5). These elements are then this can be found in the power equation description.combined in a way that avoids the specification of the With regard to the terms Pg, la, it is to be emphasizedinterface between them. A more complete discussion of that these are properties of the respective components,this topic is given in [2]. generator and two-port, in the most basic and elementary

56 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, FEBRUARY 1971

sense. (Note that these have meanings even when the A substantial simplification in the role of the user whoscattering or impedance descriptions do not exist. By wishes to evaluate the mismatch correction results fromcontrast, the "attenuation" tells how the two-port would choosing the terminal invariant quantities, availablebehave in a hypothetical, idealized system.) Beginning power, maximum efficiency, and, for bolometer mounts,with these terms, if one wants a more detailed description effective efficiency as the parameters to be supported byof the components one can add rg, rPin, r,P2, etc., as the calibration hierarchy rather than attenuation anddescribed above. If, on the other hand, one is interested calibration factors. With this change, no longer must thein how the component interacts with a given system, this user determine how his system differs from an ideal oneis provided by the mismatch factors Mll, Nal, etc. but only how it interacts with the component in question.

In this context it is convenient to introduce the sub- This point is well illustrated by comparing the equationsscript 0 to represent a matched (reflectionless) component associated with the use of an "attenuator" to reduce theso that MI, represents the mismatch between an arbitrary power delivered to a termination. If P,,1 and P,,, representgenerator and matched load, No, the mismatch between a the initial and final powers, respectively, the powermatched two-port and arbitrary load, etc. Subject to this equation method givesconvention, the following interrelationships emerge. (P/121P, 1) = (Mg,aNat/Mgi)?7,, (28)For a generator: while the corresponding expression from circuit theory is

available power = Pg; P912 = 1 -rgrl2 S21 2net power (PgI) = P,Ml; PgS12 S2- 12821 1

S22)rrgenerator power = P,1M10; (29)incident power = Pg(Mg1/M0).(

In (29) the term I81,, is determined by the attenuation;For a two-port: thus the remainder of the expression is a mismatch correc-

insertion loss = -10 log (Mga?7aNa1/AIg1); tion involving five complex parameters (six, if reciprocityattenuation = 10 logMoa-q Nao; is not satisfied). By contrast, the mismatch correctiontransducer loss = -10 log Mgan7aNal; in (28) involves only three scaler quantities. In addition,efficiency = n7aNal; uniform waveguide and precision connector requirementsabsorption coefficient = 1 - Ngaa; are implicit in the circuit theory method but absent inintrinsic attenuation = -10 log ?7a. the mismatch factor measurement. This is not to imply

that the measurement of Mga, Nal, Mg1 is a trivial matterThis list is intended to be illustrative rather than but only an easier route to the desired correction.

exhaustive. In particular this scheme has the advantage There is also a potential advantage to the user whothat the interrelationships among the different terms are ignores the mismatch in that the error limits are "one-evident upon inspection. sided." If in a receiver sensitivity measurement, for

It is next desirable to consider the application of example, the signal generator is calibrated in terms ofpower equation methods in the calibration support P, and the attenuator by fla, this leads to an upper limitproblem. The prevailing philosophy among calibration for the delivered power and a lower limit to the receiverlaboratories appears to be that of telling the customer sensitivity even though the mismatch corrections arehow his component (attenuator, bolometer mount, etc.) ignored.will behave in an ideal system (meaning that of the This proposed use of the terminal invariant parameterscalibration laboratory). In the event that the user's system as the basis of calibration support also puts the respectivedoes not satisfy this criterion, he must make the adjust- roles of the calibration laboratory and user in better focus.ment. In general, the user who wishes to make the most In general the impossibility of calibrating a component souse of a calibration result must first determine to what as to predict its behavior in arbitrary systems is wellextent his system differs from the ideal one in which the recognized by the calibration laboratory personnel, butcomponent was calibrated. That is, he must make a much less so by the user groups. The power equationmismatch correction. formulation provides a well-defined interface between

Unfortunately, the practical problems in doing this have those parameters that are properties of the individualbeen substantial, and the correction is often ignored. As a components (P,, U) as contrasted with those that areconsequence, the accuracy expectancy of field measure- dependent upon the choice of components that form thements may be limited more by the VSWR specifications system (Mga, Nat).on the individual components than by the calibration In general, the adaption of this philosophy will makethehierarchy. The unfortunate feature in all this is that the role of the calibration laboratory ea.sier in the area ofcomponent tolerances are usually less stringent at the power calibration, while the substitution of maximumfield level, and thus the most important mismatch efficiency for attenuation will entail some extra effort, atcorrection falls to the lot of the ultimate user, who is least initially. Even this, however, appears a small priceusually the least prepared to cope wvith it. to pay for the projected improvements. (Indeed, unless

ENGEN: POWER EQUATION CONCEPT FOR MICRONVAVE SYSTEMS 57

Circuit Equations replace the field equations. In particular the role of thepower equations appears to be in system evaluation or

Field Equations testing as contrasted with design and specification. Thecircuit description, for example, makes it possible to writecomponent specifications in such a way that the mismatchfactors Mg,, Nal, etc., are close to unity. The powerequation methods provide a "direct" indication of howwell this design objective is realized but fail to identifythe faulty unit if it is not.

It should perhaps be noted that most of the basicparameters upon which this description is built are not

Power Equations new. The quantities represented, by Pg, f7a, and M1, for10. Illustrating the relationships among the power, circuit, example, are well known in the prior art. The contribution

Fig.fildeuai n of this reformulation is in the recognition of the invariantand field equations.

features of these parameters and the introduction of otherssuch that independence of connector discontinuity is

the calibration laboratory can show the user an improved achieved.route through the mismatch problem, it appears only a On the other side, the acceptance of this philosophy doesmatter of time until serious questions are raised about call for some reorientation in the existing practice.its relevance.) Although the questions involved are primarily matters ofAn initial application of these ideas has already been arbitrary convention, some type of general evaluation

made in conjunction with the NBS power calibration and ultimate agreement is required if these potentialservice, and the anticipated benefits realized. In particular, benefits are to be realized. (Between the calibrationan experiment has demonstrated that it is now possible laboratory and user, for example, it is obviouslv desirableto propagate a power calibration between laboratories at to have some understanding as to what property or10 GHz and, using type-N connectors, with an unprece- properties of a device are to be measured.)dented accuracy. This, however, is the subject of a paper Finally it should be recognized that the system descrip-to follow. tion that evolves from the power equations coincides

Finally it should be noted that the parameters, available with the existing practice in the case of a "matched"power, intrinsic attenuation (= -10 log fla), and effective system. The only difference is in the methods of accountingefficiency become identical with generator power, attenua- for the effects of mismatch.tion, and calibration factor in the limit of matchedcomponents. The difference in the two descriptions is thus ACKNOWLEDGMENTsmall in many cases of practical interest and may beignored. The author expresses his appreciation to Dr. D. M.

Kerns, Dr. P. F. Wacker, and Dr. S. W. Maley for theirV. SUMMARY contribution in the development of the power equation

The power equation method, which is based on terminal concept and preparation of this manuscript.invariant parameters, represents a basic tool for thedescription and evaluation of microwave systems. Its REFERENCESrelationship to the field and circuit descriptions is as [1] D. M. Kerns, "Basis of the applications of network equations toshown in Fig. 10 where the rectangle represents the domain waveguide problems," J. Res. Nat. Bur. Stand., vol. 42, May

1949, pp. 515-540.of applicabilitv of field theory. This domain includes [2] G. F. Engen, "An introduction to the description and evaluationthat of both the circuit and power equations. The justifi- of microwave systems using terminal invariant parameters,"

Nat. Bur. Stand., Mono. 112.cation for the existence of circuit theory lies in its simplicity [3] H. A. Wheeler, "The transmission efficiency of linear networksand ease of application as compared with the field equa- and frequency changers," Wheeler, Mono. 10, May 1949.

[4] H. M. Altschuler, "A method of measuring dissipative four-polestions. The domain of the power equations overlaps that of based on a modified Wheeler network," IRE Trans. Microwavecircuit theory but also includes some problems for which Theory Tech., vol. MTT-3, Jan. 1955, pp. 30-36.

[5] G. F. Engen, "A new method of characterizing amplifier noisecircuit theory is not valid. The region of overlap includes performance," IEEE Trans. Instrum. Meas., vol. IM-19, Nov.certain problems involving mismatch correction and 1970, pp. 346-351.

[6] , "A transfer instrument for the intercomparison of micro-evaluation; here the power equations provide a simplified wave power meters," IRE Trans. Instrum., vol. I-9, Sept. 1960,approach. In addition, the uniform waveguide and pre- []pp. 202-208... ."~~.'Amplitude stabilization of a microwave signal source,"cision connectbor postulate iS not required in the power IRE Trans. Microwave Theory Tech., vol. MTT-6, Apr. 1958,equation description; this is reflected in the nonoverlapping []pp. 202-206.[] ."A methlod of determining the mismatch correction inregion. microwave power measurements," IEEE Trans. Instrum. Meas.,

It is emphasized that the power equations do not replace []vol. JM-17, Dec. 1968, pp. 392-395.[]R. W. Beatty, "Insertion loss concepts," Proc. IEEE, vol. 52,

the circuit equations any more than the circuit equations June 1964, pp. 663-671.