PROCEEDINGS OF THE I.R.E.

Resistor-Transmission-Line Circuits*PAUL I. RICHARDSt, ASSOCIATE, I.R.E.

Summary-Necessary and sufficient conditions are derived for afunction to be the driving-point impedance of a physically realizablenetwork consisting (essentially) of lumped resistors and losslesstransmission lines. The circuits so developed are thoroughly prac-tical for pure reactances and in many other special cases, but, ingeneral, ideal transformers are sometimes required. A rigorous cor-respondence between lumped-constant circuits and line-resistor cir-cuits is established. This correspondence immediately extends theusefulness of a wealth of theorems and techniques.

I. INTRODUCTION

T, HROUGHOUT this paper we shall consider net-works which are composed of lumped resistorsand of lossless transmission lines whose electrical

lengths are commensurable. Our goals are (1) to find amethod of recognizing physically realizable driving-point impedance functions for such networks, and (2) toconstruct at least one such circuit having a prescribeddriving-point impedance. We shall thus be concernedwith an extension of the well-known results of Brunelin the field of lumped constants. The word "impedance"will be used throughout only in the sense of "driving-point impedance," unless otherwise stated. Except forthe purely reactive circuits, it has, unfortunately, notyet been possible to eliminate ideal transformers in gen-eral, so that these, too, must be allowed in the class ofcircuits to be studied.As is usual in this type of analysis,",2 we shall find it

convenient to introduce a complex frequency variables-='y+jco. The "real-frequency" axis is then the imagi-nary s axis, and it is on this line that the truly significantvalues of the network functions are assumed. The useof the entire complex plane, however, with the conse-quent extension of the functions involved, enables theuse of the powerful methods of the theory of functionsof a complex variable.

II. NECESSARY CONDITIONS

It has been shown"3 that any physical impedancewhatever must have a positive real part in the righthalf-plane, y> 0. Then our first necessary condition is

* Decimal classification: R117.11 XR143. Original manuscript re-ceived by the Institute, March 12, 1947. This paper contains a por-tion of the author's doctor's thesis, the research for which was carriedon under the direction of Dr. P. LeCorbeiller at Harvard University.

t Brookhaven National Laboratory, Upton, L. I., N. Y.1 0. Brune, "Synthesis of a finite two-terminal network whose

driving-point impedance is a prescribed function of frequency,"Jour. Math. and Phys., vol. 10, pp. 191-236; October, 1931.

2 H. W. Bode, 'Network Analysis and Feedback Amplifier De-sign," D. Van Nostrand Co., New York, N. Y., 1945.

3 P. I. Richards, "General impedance-function theory," QuartAppl. Math., scheduled for publication, January, 1948.

R . 0 in y > 0 (1)

where R =real part of Z.Since only transmission lines and lumped resistors

(and ideal transformers) are involved in the circuit, Z willbe a rational function of factors of the form ebs. More-over, the restriction that the line electrical lengths becommensurable insures that only terms of the formenas= (eas)n will appear, where a is an appropriate funda-mental phase parameter, and n is an integer. Hence, oursecond necessary condition is

Z(s)-rational function of eas. (2)

III. CHANGE OF THE FREQUENCY VARIABLE

Let us define a new frequency variable by the equa-tion

ea8- 1S(s) = tanh (as/2) = _ + =-r ± jQ.

ea+ 1 (3)

We can then solve for el,=(I+S)/(1 -S); if we sub-stitute this value for ea8 into the impedance function, weobtain from (2)

Z = rational function of S. (4)

Moreover, it is easily verified that the complex func-tion S has a positive real part r whenever 7 .0. Inother words, (3) maps the right-half of the s plane intothe right-half of the S plane. Thus, from (1),

R > 0 in r _ o. (5)Of course, this mapping is not one-to-one, but themultiple-valuedness of the inverse corresponds merelyto the periodicity of Z by (2).

IV. SUFFICIENCY OF (1) AND (2)

Consider the new variable S as the independent fre-quency variable. The conditions (4) and (5) have beenshown by Brune' to be both necessary and sufficient forZ to be a physically realizable lumped-constant imped-ance.

Using Brune's method, let us develop this lumped-constant impedance. This development will, in general,yield a circuit containing inductive tees with one nega-tive element (equivalent to a pair of perfectly coupledcoils). These may be replaced, however, with induct-ances and ideal transformers by means of the equiva-lences shown in Fig. 1. The lumped circuit will thencontain only positive L, R, C, and ideal transformers.Now replace each inductance L by the input of a

short-circuited line of electrical length ac/2 (c = velocity

1948 217

PROCEEDINGS OF THE I.R.E.

of light) and of characteristic impedance L, and replaceeach capacitance C by an open-circuited line of electricallength ac/2 and of characteristic impedance 1/C. Theresult will be to replace the impedances Ls and l/Cs bythe impedances L tanh (as/2) =LS and 1/CS. Thus thelumped-constant circuit will go over into a network hav-ing exactly the required impedance, and sufficiency of(1) and (2)-or their equivalents, (4) and (5)-has beenproved.

La

LiLa+ L,L3+ L2L3 0

La>0 i . <O

, L3L1

e 1 |

Fig. 1-Equivalents of perfectly coupled coils.

The circuit produced by the above procedure will,in general, involve the eminently impractical idealtransformer. For pure reactances and in other specialcases these may be eliminated, as will be discussed later.However, no completely successful general method ofelimination has yet been found.

many lumped-constant theorems may be "translated"into theorems about line-resistor networks. For exam-ple, Bode's well-known "phase-area law"2 becomes, forline-resistor circuits,

W= 7r /2

:=0

o t an (r ( 7r RO)Xd glog tan- =-R -)-R(O) ) (7)

Again, if the circuit is purely reactive,

dX X

do, sin aw(8)

This correspondence may also be extended to side-circuit or transfer properties. The appropriate equiva-lence is shown in Fig. 2. (The line length (ac) appears,instead of (ac/2) as before, because transfer propertiesare only singly periodic in line lengths while input prop-erties are doubly periodic.) The only restriction to beimposed is that the line section must be run in the bal-

ZI

0~~~~~ oac-

NV

L = ZGFig. 2-Side-circuit equivalence properties.

V. CORRESPONDENCE WITH LUMPED CIRCUITS

The last section shows that a completely rigorous cor-

respondence exists between the driving-point imped-ances of circuits of the type under consideration andthose of lumped constants. This immediately makesavailable a wealth of material. By use of the corre-

spondences

S-plane(lumped nets)

= 00

S = O

dw

s-plane(line-resistor nets)

s = r/a

s-O

j tan (wa/2)

adco

2 cos2 (wa/2)

anced condition, where the currents in the two conduc-tors are equal and opposite. Thus, for coaxial lines, theremust be no current traveling on the outer surface of theshield. With this restriction, any of our line-resistor net-works can be replaced by an exactly equivalent lumpedcircuit. A conventional circuit analyzer may then beuised, and the results translated back into terms of thetrue frequency merely by means of a table of tangents.

VI. REACTIVE CIRCUITS

From the work of Brune and Foster, it is known thatthe equivalent lumped circuit (and hence the final net-work) will not contain ideal transformers if the circuitis purely reactive. This removes the greatest practicalobjection to the method of Section IV, but the circuit

218 February

Richards: Resistor Transmission Lines

may not yet be in a form enabling completely shieldedconstruction. This objection may also be removed.

Consider the impedance reduction represented byFig. 3(a), where

.z - sz1z-S=z

1 - -Sz1

Z, = (the value of Z at S = 1)1It may be shown4 that Z' will again satisfy (1) and (2) ortheir equivalents, (3) and (4). Moreover, it may also beshown4 that Z' is of lower degree than Z. Thus a repeti-tion of (9) will eventually lead to an impedance whichis merely a multiple of S or 1/S.

Zs Z

ac

C?r. imp. Z

(a)

s) rt orop" circuit

(b)

Fig. 3.-(a) Fundamental step in reactance realization.(b) Canonical form for reactances.

Any rectance of the type considered can be realized inthe canonical form shown in Fig. 3(b).

In actual calculation, Z' will at first appear to be ofhigher degree than Z. However, the factor (S2-1) willalways cancel from numerator and denominator. Thiscancellation can be performed in a single step by thefollowing device. To obtain a given coefficient in theresult, add the corresponding coefficient of the originalto the previous coefficient of the result. For example,

S7 + 3S5-2S3-2S Ss + 4S3 + 2S

SI + 2S4-2S2-1 S4 + 3S2 + 1

The rule may be proved by observing the results of theprocess of synthetic division.Of course, the canonical form of Fig. 3(b) is not the

only shielded circuit which has the prescribed reactance.For instance, a zero of Z may be removed at any stagein the procedure. The resulting circuit will then haveseveral stubs connected in shunt across the main line.This flexibility may be of considerable assistance inmeeting mechanical or other practical requirements.

4 P. I. Richards, "A special class of functions with positive realpart in a half-plane," Duke Math. Jour., September, 1947.

An interesting reciprocity theorem applies to thecanonical form of Fig. 3(b). Namely, it is easily seenfrom (9) that if we are required to realize a new react-ance equal to Ro2/Z, where Z is the old reactance andRo a (real) constant, then the new canonical circuit canbe obtained from that for Z by (a) reciprocating eachcharacteristic impedance with respect to Ro, and (b)changing the final short- (open-) circuited to an open-(short-) circuited termination.

VII. ELIMINATING IDEAL TRANSFORMERS

It is known4 that the procedure of (9) and Fig. 3(a)will always yield a physically realizable Z' of no higherdegree in S than Z. (A factor (S-1) will always cancel.)Moreover, Z'(S) will be of lower degree than Z(S) if,and only if, Z(-1)=--Z(1).

In view of this fact, if for all w

Ro_ Ro(Z() +Z(- 1)) > Of

then we may remove a series resistor Ro and apply (9) toobtain a Z' of lower degree than Z. If this is not pos-sible, it may be that the same method can be appliedto Y= 1/Z (removal of a shunt conductance). In gen-eral, however, both may fail. Occasionally either maysucceed after a few applications of (9), but again thiswill not always happen. Despite the lack of generality,however, these methods are often of considerable assist-ance.

(a)

(b)Fig. 4-(a) A minimum-resistance-minimum-reactance circuit. (b)

The first section of the Brune development of (a) if all elementsthere are given the value of unity.

The above devices will almost never be applicable to"minimum-resistance-minimum-reactance" functions;that is, functions that have no resonances but whichhave zero resistance at some frequency. These functionswill always require mutual coupling in the Brune repre-sentation. This does not mean, however, that such

1948 219

PROCEEDINGS OF THE I.R.E.

coupling is inevitable. For example, the circuit of Fig.4(a) has an input impedance of the minimum-resistance-minimum-reactance type. The first section of its Brunedevelopment is shown in Fig. 4(b). While the latterform would require ideal transformers in the line-resistorequivalent, the former goes into the very practical cir-cuit shown in Fig. 5.

Fig. 5-Line-resistor equivalent of Fig. 4(a).

Thus it appears that there may be hope of eventuallyfinding a general way of eliminating ideal transformers.The principal difficulty appears to be that the presenceof an ideal transformer does not affect the mathemati-cal form of the impedance in any easily detectablemanner.

VIII. DISCUSSION

The reader has undoubtedly noticed that the presenttheory derives its simplicity from two postulates defin-ing the class of circuits studied. Without these postu-lates the simple statements (2) and (4) cannot be made.The assumption that the lines are lossless is no great

bar to practical application. First, these losses are wellknown to be extremely small and, for a host of practicalproblems, may be entirely neglected. Second, the fol-lowing device for including small losses may be carriedover from the theory of lumped circuits: Let r, 1, g, c bethe series resistance, series inductance, shunt conduct-ance, and shunt capacitance, all per unit length, for agiven line. Assume, as a first approximation, that for alllines in the circuit (g/c) = (r/l) = d and that d has the

same value for all the lines. Then the propagation con-stants and characteristic impedances may be written:

/(r + Is)(g + cs) = (s + d)\/lc/r + Is /1

Thus the circuit behavior can be reduced to the losslesscase by the familiar device of changing the frequencyvariable. Note that this change must be made in the(true frequency) s plane and not in the (equivalent) Splane.Our second fundamental postulate requires that the

electrical lengths of the lines be commensurable. Al-though most engineers will feel that this is not a seriouspractical consideration, it ought to be mentioned thatapplication of the theory to a noncommensurable circuitassumes not only convergence but uniform convergence.'The reader will note immediately that the restriction

is an "unnatural" one. In truth, given two actual lines itwould be impossible to determine whether their lengthswere commensurable; ever-present experimental errorwould always limit us to a finite range of ratios, withinwhich there would always lie both rational and irra-tional numbers. The reason why the mathematical the-ory asks a question which experiment cannot answerlies, of course, in the fact that the simple theory ignoresthe "smoothing" effect of discontinuity reactances aris-ing at the junction of dissimilar lines. In general, prob-ably no actual physical impedance will be periodic infrequency.Yet our theory is rooted in an assumption of perio-

dicity. The author feels that this is no serious objectionto its physical meaning. A similar situation arises inlumped-circuit theory, where lead inductance, distrib-uted capacitance, and radiation are neglected. This isunimportant for most purposes, but at the same timeinsures that the impedance function will not be correctfor high frequencies. Yet the mathematical theory treatsZ as having physical meaning beyond the frequency ofvisible light and, indeed, at s-= oo. As usual, a theoryboth tractable and useful has been achieved by idealiz-ing the physical facts in such a way that there are noheuristic difficulties or internal contradictions. It is feltthat the same is true of the present theory.

6 This was pointed out by P. LeCorbeiller.

C~~2A~5D

Fe-bruary220