General theory of broadband matchingProf. Wai-Kai Chen, M.Sc, Ph.D., Fel. I.E.E.E., Fel. A.A.A.S., and

Chandra Satyanarayana, M.Sc, Ph.D., Mem. I.E.E.E.

Abstract: Given the source and load impedances and a preassigned transducer power-gain characteristic, thenecessary and sufficient conditions are given for the existence of a lossless reciprocal equaliser which, whenoperating between the two given impedances, yields the desired transducer power-gain characteristic. Thesignificance of the present approach is that the realisation of the equaliser involves only driving-point syn-thesis by the Darlington theory.

1 Introduction

The problem of designing an optimum lossless equaliser tomatch a load composed of the parallel combination of acapacitor and a resistor to a resistive generator was firststudied by Bode [1]. Fano [2] extended Bode's work andsolved the problem of impedance matching between an arbi-trary passive load and a resistive generator, in its full gener-ality. Fano's approach, however, suffers two main draw-backs in that the replacement of the load impedance by itsDarlington equivalent leads to complications in translatingcertain of its properties into structural properties of theassociated Darlington, and that the technique is not readilyamendable to an active load. Based on the principle of com-plex normalisation [3 ,4 ] , Youla [5] developed a new theorywhich circumvents these difficulties. Chan and Kuh [6]generalised this theory by considering the active load. Analternative approach to the matching problem was suggested byRohrer [7].

In many practical situations, the internal impedances of theavailable electronic sources are not purely resistive, especiallyat high frequencies. In the design of interstage couplingnetworks, for example, the output and the input impedancesof the stages involved may not be approximated by pureresistances. In these cases, the design of a lossless equaliserto match out two arbitrary passive impedances is necessary,as indicated by Ku and Peterson [8]. Fielder [9] was anearlier worker along this line, and considered the problemof broadband matching any load whose Darlington equivalentis restricted to be a lossless ladder terminated in a resistor,to a resistive generator embedded in a prescribed losslessladder. A general solution to this problem was recently givenby Chien [10], who presented the necessary and sufficientconditions for the existence of a scattering matrix with pre-scribed transducer power-gain characteristic. The desiredmatching network is obtained by realising this scatteringmatrix, using any of the known techniques. This, by itself,is not a simple matter.

The purpose of the present paper is to present a generalmatching theory between two arbitrary passive impedances.The theory circumvents the need for obtaining the Darlingtonequivalents of the source and load impedances. The realisation

Fig. 1 Schematic for study of broadband-matching problem betweenarbitrary source and load systems

Paper 1883G, received 9th December 1981Prof. Chen is with the Department of Information Engineering,University of Illinois at Chicago, Chicago, IL 60680 USA, and Dr.Satyanarayana is with the Faculty of Engineering, Department ofElectrical Engineering, University of Alfaatih, Tripoli, Libya

of the equaliser involves only driving-point synthesis by theDarlington theory.

2 Preliminary considerations

In the 2-port network N of Fig. 1, let zx (s) and z2 (s) be twopassive non-Foster terminating impedances. Write

n(s) = z l ( - s ) ] /2 = ht(s)ht(rs) i = 1,2 (1)

where the factorisation is to be performed, so that h{(s)and h'ix (— s) are analytic in the open right half of the s-plane*(RHS). We recognise that /i,-(s)//i,-(— s) is a real regular all-passfunction, whose poles include all the open left half of thes-plane (LHS) poles of zt(s). Thus, it can be written as theproduct of the real regular all-pass function

A,(S) = n; = 1 S + dj

> 0, i = 1,2 (2)

defined by the open RHS poles aj(j = 1, 2, . . . ,vt) of z,(—s)and another real regular all-pass function Bt(s) defined by theopen RHS zeros of r,(s), namely

ht(s)lht(-s) = At(s)Bt(s) (3)

Observe that, for a positive-real z,(s), ft,(s) and /*,-(— s) areanalytic on the real-frequency axis and that, from eqns. 1and 3,

2h](s) = 7rt(s)At(s)Bt(s) = Ft(s)B,(s)

where

F,(s)= 2ri(s)Ai(s) i = 1,2

(4)

(5)

As is well known, the reflection coefficients Sn (s) and S22 (s)at the input and output ports normalising to the terminatingimpedances zx (s) and z2 (s), respectively, can be expressed asin Reference 3:

where

i = 1,2

= At(s)Zu^-Zii-s)

(6)

(7)

96 0143-7089/82/030096 + 07 $01.50/0

Z22 (s) and Zn (s) being the driving-point impedances lookinginto the output and input ports (Fig. 1), when the input andoutput ports are terminated in zt(s) and z2(s), respectively.Since the 2-port network N is lossless,t the transducer powergain G(CJ2) can be expressed in terms of p1{jcS) by therelation

G(co2) = l - | 5 2 2 ( / w ) | 2 = l - | 5 i i ( / w ) | 2

= l - | p , ( / w ) | a / = 1,2 (8)

Thus, to study the class of transducer power-gain character-

*This requires that all the zeros of h(s) be restricted to the closedRHS and all the poles to the open LHS

tldeal transformers are includedIEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982

istics compatible with prescribed load impedances z,(s), itsuffices to consider the bounded-real reflected coefficientsPi(s). At times we shall find it necessary throughout thepaper to use Ss(s) = Bi(s)pi(s), in order to take advantageof the additional degrees of freedom introduced by the factorBt(s).

For a given impedance z,(s), a closed RHS zero soi ofmultiplicity kt of the function ri(s)/Zi(s) is called a zero oftransmission of order kt of z,(s). The zeros of transmissionare conveniently divided into the following four mutuallyexclusive classes: Let sQi = o0i + jcjOi. Then sOi belongs toone of the following classes depending on o0i and zi(sOi)',as follows:

Class I

oQi>0

which includes all the open RHS zeros of transmission

Class II

oQi = 0 and z ^

Class III

aOi = 0 and 0 < |zf(/co0,-)l < °°

Class IV

aOi = 0 and |zf(/co0i)l = °°

For each zero of transmission soi of z,(s), consider the Laurent-series expansions* of the following functions about the points0i:

x=0Axi(s-soif

= Ix=0

x=0pxi(s-sOiy

(9)

(10)

(11)

2.1 Basic constraints on pt (s)For each zero of transmission sOi of order k( of z,(s) 0 = 1 , 2),one of the following four sets of coefficient conditions mustbe satisfied, depending on the classification of sOi:

Class I

Axi = pxi for x = 0 ,1 ,2 , . . . , kt - 1

Class II

Axi = pXi for x = 0, 1,2,. . . , kt - 1

and

(12a)

(12*)

Class III

Axi = pxi for x = 0 , 1 , 2 , . . . , k t - 2

and

U<*,- - I)« - P(fef -1 >/] lFkii > 0 02c)

tThese are actually Taylor-series expansions. We use the more generalLaurent-series expansions for the situation where the zero of trans-mission soi is located at infinity

Class IV

Axi = pxi for x = 0, 1, 2, .. . , kt - 1

and

• -I)«/(4fe,i - (\2d)

a-u being the residue of z,(s) evaluated at the pole/co0;.The importance of these coefficient constraints is sum-

marised in the following theorem first given by Youla [5].

Theorem ILet z{(s) be a prescribed, rational, non-Foster positive-realfunction and p,(s) a real, rational function of the complexvariable s. Then, the function defined by the relation

i = 1,2 (13)

where A{(s) and Ft(s), as given by eqns. 2 and 5, are uniquelyspecified by z((s), is positive-real if, and only if pt(s) is abounded-real reflection coefficient satisfying the basic con-straints (eqn. 12).

3 General theory

Given two arbitrary non-Foster positive-real impedances zx(s).and z2 (s) as the internal impedance of the generator and loadas shown in Fig. 1 and given a real rational function G(to2),bounded by unity for all real co, as the transducer power-gaincharacteristic, our main objective is to determine conditionsunder which there exists a lossless reciprocal 2-port networkwhich, when inserted between zt(s) and z2(s), will realisethe given G(co2), and furthermore, if one exists, to realisethe matching 2-port network.

Refer again to the network of Fig. 1. Let Z n (s) and z2 (s)be two positive-real impedances. Then Z u (s) and z2 (s) aresaid to be compatible if Z n (s) can be realised as the driving-point impedance of a lossless 2-port network terminated inz2 (z). The problem of compatible impedances was consideredby Schoeffler [11], Wohlers [12] and Ho and Balabanian[13]. Schoeffler and Wohlers studied the problem for positive-real impedances, and Ho and Balabanian extended their resultsto include the situation where the impedances may be eitherpassive or active. We note that, in a compatible impedanceproblem, the impedances Zn (J) and z2(s) are specified,whereas, in a matching problem, the transducer power-gaincharacteristic G(co2) and the terminating impedances zx (s)and z2(s) are prescribed. Thus, in the former case Zn(s),and hence px (s), normalising to a chosen impedance, arefixed, whereas in the latter case we have freedom in changingZ\i (s) by introducing open RHS zeros in px (s). Since G(u2)and py (/CO) are related by eqn. 8, the compatible impedanceproblem can be considered as a special case of the broadband-matching problem.

In the following, we state the necessary and sufficientconditions of the compatability of two passive impedancesin terms of the coefficient constraints [14].

Theorem 2Let Zn (s) and z2 (s) be prescribed, rational, non-Foster,positive-real functions. Then Z n (s) is compatible with z2 (s)if, and only if, there exists a real regular all-pass function60(s), such that the function defined by the relation

1 ~ ^ ' — x — l ± ^ - B2 (s)dl (s) (14a)

x=0

IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982 97

is a bounded-real reflection coefficient satisfying the coef-ficient constraints (eqn. 12), with / = 2 and <px2 replacing px2

at each zero of transmission S02 of Z2 (S) of order k2 , where

Rn(s) = ( - s ) (15)

with Mn (s) and AfjV (s) being analytic in the open RHS. § Thefunctions A2 (s), B2(s) and F2 (s), as given in eqns. 2, 3 and 5,are uniquely determined by z2 (s), and their series expansionsabout So2 are given in eqns. 9 and 10.

Furthermore, if Zn(s) and z2(s) are compatible, theimpedance function defined by

Z20(s) =A2(s)-(p2(s)

(16)

can be augmented by multiplying its numerator and denomin-ator polynomials by the same factor, so that the resultingimpedance function can be realised as the driving-point impe-dance of a lossless reciprocal 2-port network terminated ina 1Q, resistor, the input impedance Z'[\ is) facing the 1£2resistor being the desired Zn (s), i.e. Z"n (s) = Zn (s) (seeFig. 2).

Z"n(s) = Zn(s)

Fig. 2 Lossless 2-port network terminated with resistive generatorand passive load impedance

A proof of this theorem can be found in Reference 14.The basic idea of our approach to the broadband matching

of two passive impedances over a band of frequencies with apreassigned transducer power/gain characteristic can be simplyoutlined as follows.

From a prescribed transducer power-gain characteristicG(co2), obtain G(— s2) by appealing to the theory of analyticcontinuation and form the bounded-real reflection coefficient

Pi = ± 17 (s)pm(s)

from

(17)

Pt^p^-s) = l-G(-s2) = pm(s)pm(-s) (18)

where rj(s) is an arbitrary real regular all-pass function andpm(s) is the minimum-phase factorisation of 1— G(—s2).Determine 77(5), if one exists, so as to satisfy the coefficientconstraints (eqn. 12) for / = 1. From theorem 1, the functionZn(s), as defined by eqn. 13, is positive real. For the 2-portnetwork N of Fig. 1 to exist, Zn (s) must be compatiblewith z2(s). According to theorem 2, this is equivalent toconstructing a bounded-real reflection coefficient 02 ofeqn 14a by choosing an appropriate real regular all-passfunction 60(s), so that the related coefficient constraintsare satisfied. If no such do(s) exists, the given G(co2) is notphysically realisable. We present the main result of the paperas a theorem.

Theorem 3Given two non-Foster positive-real rational functions zY (s) andz2(s) and an even rational function G(co2), 0 < G ( c o 2 ) < lfor all co, of the real-frequency variable co, rational functions

§This requires that all the zeros of M,, (s) be restricted to the closedLHS, and all the poles to the open LHS

98

Ai(s), Bt(s) and Ft(s), i= 1,2, and pm(s) are uniquely deter-mined as in eqns. 2, 3 and 5, where pm(s) is the minimum-phase solution of 1 —G(—s2). Then the necessary and suf-ficient conditions for the existence of a lossless reciprocal2-port network, which when operating between a generator

.of internal impedance zl(s) and a load impedance z2(s)yields the transducer power gain G(co2), are that:

(i) there exists a real regular all-pass function 17 (s), suchthat the function defined by

Pi(s) = ±r}(s)pm(s) (19)

be bounded-real, satisfying the coefficient constraints (eqn.12) with / = 1 at each zero of transmission s01 of z j (s) oforder kx

(ii) there exists a real regular all-pass function 60(s), suchthat the function defined by

l-Zn(-s) Mn(s)x

1+Zn(s) Mn(-s)

<t>x2 (? -

(20a)

(20Z>)

be bounded-real, satisfying the coefficient constraints (eqn. 12)with / = 2 and 0x2 replacing px2 at each zero of transmissions02 of z2(s) of order k2, where {Zn(s) +Zn(—s)}/2 =Mn(s)Mn{—s), with M n ( s ) and M~x\ (s) being analyticin the open RHS, and

(21)Al(s)-pl(s)

Proof:Necessity: Let the 2-port network N of Fig. 3- realise thetransducer power-gain characteristic G(to2). Let Sn(s)be the input reflection coefficient of N normalising to zx (s).

Su(s)Fig. 3 Lossless 2-port network with preassigned tranducer power-gaincharacteristic

Then we have

|P l(/co)|2 = |pm(/co)|2 = 1-G(co2)

= lSn(/co)|2 1 (22)

for all co, showing that S n (s) and pm(s) differ only by a realregular all-pass function 77 j (s) or

As in eqns. 6 and 7, the input impedance Z n{s) with theoutput port terminating in z2 (s) can be expressed as

.(*) (24)11 w ii^o-^e^nto i w

Since Z n (?) is positive-real, from theorem 1 the function

Bi(-s)Su(s) = ±Bl(-s)n1(s)Pm(s) (25)

must be bounded-real, implying that 771 (s) must containBi (s) as a factor or

T?! (0 = 77(5)5, (s) (26)

IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982

for some real regular all-pass function 77(5), yielding

Bi(TS)Sn(s) = ±Tl(s)pm(s) = Pl(s) (27)

and condition (i) follows directly from theorem 1. Fromeqns. 21, 24 and 27jve have Z n (s) = Zn (s). This, togetherwith the fact that Zn(s) and z2(s) are compatible, showsthat, from theorem 2, condition (ii) is necessary.

Sufficiency: Assume that conditions (i) and (ii) are satisfied.We show that a desired 2-port network exists. From condition(i) and theorem l , Z u ( s ) of eqn. 21 is positive-real and non-Foster. With Zn (s) and z2(s) being non-Foster and positive-real, from condition (ii) and theorem 2, Zn (s) is compatiblewith z2 (s), and furthermore the impedance Z2o (s) defined byeqn. 16 can be augmented by multiplying its numerator anddenominator polynomials by the same factor, so that theresulting function can be realised as the driving-point impe-dance of a lossless reciprocal 2-port network N terminatedin a 112 resistor, the input impedance facing the 112 resistorwith the output terminating in z2 (s) is Zn (s), as depictedin Fig. 2. We demonstrate that this 2-port networks, whenoperating between a generator of internal impedance zx (s)and a load impedance z2(s), as shown in Fig. 1, yields thetransducer power-gain characteristic G(co2).

To see this, let Sn(s) be the input reflection coefficientof N, normalising to the impedance zx (s). Then we have

n (s) = (28)

Substituting eqn. 21 in eqn. 28 in conjunction with eqns. 3, 5and 19 gives

Su(s) = = ±v(s)B1(s)pm(s) (29)

The transducer power gain of the network of Fig. 1 can beexpressed as

2 =l - | p m ( / c o ) | 2 = (30)

This completes the proof of the theorem.In the special situation where the source impedance zx (s) is

purely resistive, theorem 3 degenerates to the known resultsofYoula [5].

Corollary!Given a non-Foster positive-real rational function z2 (s) and aneven rational function G(co2 ), 0 < G(co2 ) < 1 for all w, of thereal-frequency variable co, rational function A2(s), B2(s),F2 (s) and pm(s) are uniquely determined as in eqns. 2, 3 and5, where pm (s) is the minimum-phase solution of 1 — G(— s2).Then there exists a lossless reciprocal 2-port network, which,when operating between a generator of internal resistance Rxand a load impedance z2 (s), yields the transducer power gainG(CJ 2 ) , if, and only if, there exists a real regular all-passfunction £(s), such that the function defined by

(31)

is bounded-real, satisfying the coefficient constraints (eqn. 12),with i = 2 and 0x2 replacing px2 at each zero of transmissions02 of z2 (s) of order k2 .

Proof.In theorem 3, let zl(s) = Rl. Then condition (i) is alwayssatisfied a n d Z u ( s ) of eqn. 21 becomes

Zn (*) =

IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982

(32)

whose even part is obtained as:

- P i Pi

G(-s2)

giving

Mn(s)=

(33)

(34)

where Pi2m(s) is the minimum-phase solution of G(— s2) =Pi2m(s)Pi2m (~s). Substituting eqn. 32 and 34 in eqn. 20a,in conjunction with eqn. 19, yields

02(5) = ±V(s)pm(-syB2{s)OUs)Pl2m(s)lp12m(-s)

Now we can write

(35)

( \l t \ y («\y f o\ (iau\

S)/p i2m\—S) — s3^/s4(.— $) \JOu)

where ^x (s), %2 (s), £3(5) and £4(5) are real regular all-passfunctions defined, respectively, by the open RHS zeros ofpm(—s), open RHS poles of pm(— s), open RHS poles ofPi2m(— s), and open RHS zeros of p 1 2 m (— s). If there existsa 60(s) satisfying condition (ii) of theorem 3, then do(s)must contain the open RHS zeros of T?(S), p^,1 (— 5) and

(37)d

for some real regular all-pass function 6i(s). Substitutingeqns. 36 and 37 in eqn. 35 yields eqn. 31 with

On the other hand, if the conditions of the corollary aresatisfied, we can choose the real regular all-pass functionsr}(s) and Q0(s) with

(39a)

(39b)

Substituting eqns. 36 and 39 in eqn. 35 gives

02 (s) = ±S(s)p«(s) (40)

showing that condition (ii) of theorem 3 is implied by thecondition of the corollary. This completes the proof of thecorollary.

4 Method of synthesis

In theorem 3, we indicated that the function Z20 (s) of eqn. 16can be augmented, if necessary, so that the resulting functioncan be realised by Darlington theory as the driving-pointimpedance of a lossless reciprocal 2-port network N termin-ated in a 112 resistor; the input impedance facing the 112resistor with the output port terminating in the given loadz2 (s) is Z n (s) of eqn. 21. This is always possible since, fromtheorem 1, the impedance

A2(s)-<p2(s)-z2(s) (41)

is positive-real. An augmenting procedure was given by Ho andBalabanian [13]. The 2-port network N is a desired matchingnetwork which, when operating between a generator ofinternal impedance z t(s) and a load z2 (s), yields the pre-

99

assigned transducer power-gain characteristic G(co2). In thespecial situation where the source impedance Zl(s) is purelyresistive, no augmentation of Z20(s) is necessary. To see this,we solve <p2 (

s) m ecln- 41 and obtain

2 0) = A 2 (s)Z20(s)-z2(-s) = S22 (s)Z20(s) + z2(s) B2(s)

(42)

where S22 (s) is the output reflection coefficient normalisingto z2 (s) of the 2-port network N realised directly from Z20 (s)without augmenting factors. The transducer power gain of theterminated network becomes

= l - | p m ( / c o ) i 2 =2 = (43)

as required.For computational purposes, it is convenient to express

Mn (s) in eqn. 20a explicitly. For this we compute the evenpart of Zn(s), giving

rl(s)G(-s2)

which can be factored to obtain

hl(s)pl2m(s)Mn (s) =Ai(s)~Pi(s)

(44)

(45)

where p12m(s), as before, denotes the minimum-phase factor-isation of G(— s2). Substituting eqns. 21 and 45 in eqn. 20ayields

0 ( s ) = x

-Zl(S))-Pl(rS)

If zx (s) =£ 1, eqn. 46 can be written as:

A /- \ Pl2m(s) 1 Pi ( •02 (s) = ——

where

B2(s)d20(s)

(46)

(47)

(48)

We illustrate the above results by the following example.

Example 1We wish to design a lossless equaliser to match the sourceand load systems of Fig. 4 with

Rx = 1 Cl,R2 = 1 fi,Ci = 2F, C2 = 0.25 F

The equaliser is required to attain the 3rd-order lowpassButterworth transducer power-gain characteristic

0 < (49)

c2TFig. 4 Lossless 2-port network required to match two parallel RCimpedances

with maximum DC gain K3 for a 3dB radian bandwidthcoc = 1 rad/s.

The source and load impedances are obtained as:

1

giving

2 5 + 1 '

- 1 - 1 6

s2 - 1 6

(50)

(51)

Thus, zx (s) and z2 (s) each possess a class II zero of trans-mission at the infinity of order 1 or ki — k2 = 1 and s01 = S02= °°. The other desired functions, together with their Laurent-series expansions about SQI =$02 =oo> a r e computed as follows:

2s - 1

(0 =

F2(s) =

5 —

5 +

(25

—

+ 1

4

4

-2

+32

I)2

1 0.5

— + —8 32

+ — +S 5

_ 3 2

52

(52a)

(52b)

(53a)

(53b)

and Bx (s) = B2 (s) = — 1. Appealing to analytic continuation,eqns. 49 and 18 can be written as:

G(- : j = P 12m (OP 12m (~S)

(~s) = Pm(s)pm(-s) = a1 - v 6

(54)

(55)

where a6 = 1 — K3 andjy = s/a. The minimum-phase solutionsof eqns. 54 and 55 are given by

Pl2m(0 =

Pm(s) = «3

1 + 2s2 + 25 + 1

y3 +2y2 +2y+ 1s3 + 252 + 25 + 1

(56)

(57)

Write the arbitrary real regular all-pass function r}(s) in eqn. 17explicitly as:

b s—p:

= n -TTRep,- > 0 (58)

for some positive integer b. Substituting eqns. 57 and 58in eqn. 17 gives

. b s — n,= n

s3 + 2a52+ 2a2s + o

s3 + 2s2 + 25 + 1

i=l S+pt

2 [ l -.

(59)

For &i = 1, the class II constraints eqn. 12Z> with / = 1 im-posed by zx (s) become

-<4oi =

> 0

(60)

(61)'21

100 IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982

To satisfy constraint 60, we must choose the plus sign ineqn. 59 or eqn. 19, since A01 = 1. From eqns. 52a, 53aand 59, constraint 61 becomes:

2(1 -Kz)1/6 > 1 + 2 £ pt

(62)

Thus, to maximise K3, we let p{ = 0 for all /, yielding

K3,max = 63/64 (63)

and from eqns. 56 and 57, we obtain the minimum-phasesolutions of eqns. 54 and 55 as:

Pl2mO) = s3 + 2s2 + 2s + 1

_ s3 +s2+s/2+ 1/8Pm(s) ~ S3 +2s

2+2s+\

and Pi (s) = Pm (s)- Substituting these in eqn. 47 gives

4s2 - 3s - 1 „

(64)

(65)

(66)

which is recognised as a lossless ladder terminated in a resistor.The desired matching network is shown in Fig. 5. For ourpurposes, we recompute the input impedance Z[\ (s) facing thesource impedance Zi(s):

2 " ( S ) =12s

= Z l l W3(2,+ 3)

where Zn (s) is defined by eqn. 21.

(75)

in 12/7H

5/12FTJT

Fig. 5 Lossless equaliser obtained in example 1

where

f(s) = 1 — 1/5 (67)

As in eqn. 58, express the real regular all-pass function 60(s)explicitly as:

0o(s)= Reqt >0 (68)

for some positive integer c. Using 60 (?) in eqn. 66, <p2 (?) canbe expanded in the Laurent series about the point S02 = °°as:

01202 (S) = 002 + + . . . = 1

s s. . . (69)

For A:2 = 1, the class II constraints eqn. 12b with i = 2imposed by z2 (s) become

(70)02 = 002

12 ~012> 0 (71)

'22

Constraint 70 is always satisfied. From eqns. 52b, 53b and 69,constraint 71 becomes

5 > 4 I q, (72)

For simplicity, choose qx = 0 for all /, yielding do(s) = 1 and

4s2 - 3s - 102 (s) = 4s2 + 9s + 8

(73)

This shows that conditions (i) and (ii) of theorem 3 are satis-fied, and a desired equaliser can be synthesised.

To realise this equaliser, we compute the impedance func-tion Z2o (?) from ecln- 41 and obtain

_ 4(12i2 + 57s + 36) 48s+ 36~ (s + 4) (20s2 + 15s + 28)~ 20s2 + 15s + 28

1

5s/121

(74)

12s/7 + 9/7IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982

5 Conclusions

Necessary and sufficient conditions are presented for theexistence of a lossless reciprocal equaliser, which, whenoperating between a generator of given internal impedanceand a given load, yields a preassigned transducer power-gaincharacteristic. The results are significant in that it circumventsthe need for obtaining the Darlington equivalents of thesource and load impedances, and also avoids the necessity ofrealising a complex-normalised scattering matrix, which byitself is not a simple matter. In the present approach, therealisation of the equaliser involves only driving-point syn-thesis by the Darlington theory.

Design procedure and illustrative examples for using thebroadband matching theory will be reported in another paper.Specifically, we will derive gain-bandwidth limitations forseveral practical source and load systems that achieve thelowpass Butterworth or Chebyshev transducer power-gaincharacteristics of arbitrary order.

6 References

1 BODE, H.W.: 'Network analysis and feedback amplifier design'(Van Nostrand, Princeton, 1945)

2 FANO, R.M.: "Theoretical limitations on the broadband matchingof arbitrary impedances', J. Franklin Inst., 1950, 249, pp. 57—83and 139-154

3 CHEN, W.K.: "Theory and design of broadband matching networks'(Pergamon Press, 1976)

4 YOULA, D.C.: 'An extension of the concept of scattering matrix',IEEE Trans., 1964,CT-11, pp. 310-312

5 YOULA, D.C.: 'A new theory of broad-band matching ', ibid.,1964, CT-11, pp. 30-50

6 CHAN, Y.T., and KUH, E.S.: 'A general matching theory and itsapplications to tunnel diode amplifiers', ibid., 1966, CT-13,pp. 6-18

7 ROHRER, R.A.: 'Optimal matching: A new approach to thematching problem for real linear time-invariant one-port networks',ibid., 1968, CT-15, pp. 118-124

8 KU, W.H., and PETERSEN, W.C.: 'Optimum gain-bandwidthlimitations of transistors amplifiers as reactively constrained activetwo-port networks', ibid., 1975, CAS-22, pp. 523-533

9 FIELDER, D.C.: 'Broadband matching between load and sourcesystems', IRE Trans., 1961, CT-8, pp. 138-153

10 CHIEN, T.M.: 'A theory of broadband matching of a frequency-dependent generator and load - Part I: Theory', /. Franklin Inst.,1974, 298, pp. 181-199

101

11 SCHOEFFLER, J.D.: 'Impedance transformation using losslessnetworks', IRE Trans., 1961, CT-8, pp. 131-137

12 WOHLERS, M.R.. 'Complex normalisation of scattering matricesand the problem of compatible impedances', IEEE Trans., 1965,CT-12, pp. 528-535

13 HO, C.W., and BALABANIAN, N.: 'Synthesis of active and passivecompatible impedances', ibid., 1967, CT-14, pp. 118-128

14 SATYANARAYANA, C, and CHEN, W.K.: 'Theory of broad-Chandra Satyanarayana received the B.E. degree in electricalcommunication engineering from the Indian Institute ofScience in 1963, the M.Tech degree in electrical engineeringfrom the Indian Institute of Technology, Kanpur, India, in1966, and the Ph.D degree from Ohio University in 1975.

He was an Associate Lecturer and then a Lecturer atOsmania University, Hyderabad, India from 1963 to 1970.He was appointed a Research Assistant at the University ofIowa in 1970. From 1972 to 1975, he was a Graduate Associ-ate at Ohio University. Since 1976, he has been an AssistantProfessor at the University of Alfaatih, Tripoli, Libya.

His current research interests are communication theoryand network theory.

band matching and the problem of compatible impedances', /.Franklin Inst., 1980, 309, pp. 267-279

15 SATYANARAYANA, C: 'A general theory of broadband matching'.Ph.D. dissertation, Ohio University, Athens, Ohio, March 1975

16 YOULA, D.C.: 'A new theory of cascade synthesis', IRE Trans.,1961, CT-9, pp. 244-266

17 YOULA, D.C.: "A new theory of cascade synthesis'. Correction',ibid., 1966, CT-13, pp. 90-91

Wai-Kai Chen received the B.S. and the fjM.S. degrees in electrical engineeringfrom Ohio University in 1960 and 1961,and the Ph.D degree from the Universityof Illinois at Urbana-Champaign inFebruary 1964.

From 1964 to 1981 he was withOhio University, where he became aDistinguished Professor of ElectricalEngineering in 1978. Since September1981, he has been Professor and Head ofthe Department of Information Engineering at the Universityof Illinois at Chicago.

Dr. Chen is a recipient of the 1967 Lester R. Ford Awardof the Mathematical Association of America. He received aResearch Institute Fellow Award from Ohio University in1972, an Outstanding Educator of America Award in 1973,and a Baker Fund award in 1974 and also in 1978 from OhioUniversity. He is a Fellow of the IEEE and of the AmericanAssociation for the Advancement of Science.

102 IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982