IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 6, JUNE 1988 667

Unified Theory of Compatibility Impedances YI-SHENG ZHU, MEMBER, IEEE, AND WAI-KAI CHEN, FELLOW, IEEE

Abstract -The paper presents a unified summary of recent progress on compatible impedances. It is shown that Chen and Satyanarayana's theory is completely equivalent to Wohlen' solution to the problem of compatible impedances, and that the problem of finding a lossless coupling network to transform one impedance to another specified one is equivalent to that of designing an equalizer to match an arbitrary passive lumped load to a resistive generator to yield a preassigned transducer power-gain character- istic. Thus the two problems can be solved by the same set of formulas. Finally, examples are given to illustrate the procedures.

I. INTRODUCTION WO impedances Z , ( s ) and z2(s) are said to be com- T patible if Z,(s) can be realized as the input imped-

ance of a lossless two-port network terminated in z2(s), as shown in Fig. 1. The problem has many applications in broadband matching, filter design, and cascade synthesis. Many papers have been published in this area.

Schoeffler [l] was first to study the conditions of com- patible impedances. The conditions were derived by using the Darlington theory and cascade synthesis approach. However, his result is restricted in that no common factors can be inserted into the numerator and denominator of

To obtain a general solution, Wohlers [2] gave a scatter- ing matrix existence theorem. The scattering matrix of the two-port network N of Fig. 1, normalized to 1 52 and z2(s), is first expressed in terms of Z l ( s ) and z2(s) and the admittance matrix of the augmented network of Fig. 2, from which a set of necessary and sufficient conditions for two impedances to be compatible were derived. Wohlers' approach however does not concern itself with the actual synthesis of the coupling network but is satisfied with the determination of a realizable scattering matrix.

The common factors inserted into the numerator and denominator of Z l ( s ) are referred to as the augmenting factors. Using different augmenting factors, a set of output reflection coefficients are obtained, all of which have the same jw-axis magnitude but different angles. This is equiv- alent to the insertion of different real regular all-pass functions in the output reflection coefficient. The proce- dure may lead to the realization of two compatible imped- ances or the realization of an equalizer to match two complex impedances to yeld a preassigned transmission characteristic. Ho and Balabanian [3] considered the rela- tions between the augmenting factors and the real regular all-pass functions and proposed a procedure for the real- ization of two compatible impedances.

Zl(S) .

Manuscript received July 20, 1987; revised November 23, 1987. This paper was recommended by Associate Editor Y. V. Genin.

The authors are with the Department of Electrical Engineering and Computer Science, University of Illinois at Chicago, Chicago, IL 60680. IEEE Log Number 8820485.

In the design of a communication system, a basic prob- lem is to design a coupling network between a given source and a given load so that the transfer of power from the source to the load is maximized over a given frequency band of interest. When the source and/or load are frequency dependent, the problem is referred to as broad- band matching. Youla [4] developed a new theory based on the principle of complex normalization. The significance of Youla's theorem is that it gives conditions (see Appendix) under which there exists a match between a passive load and a resistive generator with a preassigned transducer power-gain characteristic.

Using Youla's result, Satyanarayana and Chen [SI, [6] gave another impedance compatibility theorem. The pur- pose of the present paper is to show that Satyanarayana and Chen's theorem is completely equivalent to Wohlers' conditions. Moreover, a unified theory is presented which states that the broadband matching and compatible im- pedances are two facets of the same problem. Therefore, they can be solved by the same set of formulas. Finally, examples are given to illustrate the procedure.

11. PRELIMINARY CONSIDERATIONS Consider two non-Foster positive-real impedances Zl ( s)

and z,(s). Write

where the M(s)'s, m(s))s , and N(s)'s, n(s)'s stand for the even and odd parts of the relatively prime polynomials, respectively. The even parts of Z,(s) and z2(s) can be expressed as

1 2 R l ( s ) = - [ z,( s) + z,( - s)]

1

(3)

(4)

0098-4094/88/0600-0667$01.00 01988 IEEE

668 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 6, JUNE 1988

lossless, reciprocol two - port network 2, ( 5 ) -

N

Fig. 1. The schematics of the impedance compatibility problem.

lI ! ; lossless

two-port network

I

I I L __--__ 2 _ _ _ _ _ _ _ _ J

N

N

Fig. 2. An augumented two-port network for the study of impedance compatibility problem.

In (3) and (4), the factorizations are to be performed so that Wl(s) and w2(s ) include all the zeros in the open left half of the s-plane (LHS), Wl( - s) and w2( - s) include all the zeros in the open right half of the s-plane (RHS), and the zeros on the jw-axis are equally divided between Wl(s) and Wl( - s) or w2(s ) and w2( - s). Write

r 2 ( 4 = h 2 ( 4 h 2 ( - s) ( 5 )

( 6 )

where

w2( - 4 '2'') = m 2 ( s ) + n 2 ( s ) .

We recognize that h 2 ( s ) / h 2 ( - s) is a real regular all-pass function, and can be expressed as

where

(7)

(9)

For a given impedance z 2 ( s ) , a closed RHS zero so of multiplicity k of the function

is called a zero of transmission of order k of z 2 ( s ) . For convenience, the zeros of transmission are divided into the following four mutually exclusive classes. Let so = uo + jw,. Then so belongs to one of the following classes depending

on uo and z2(so) , as follows:

transmission. Class I: uo > 0, which includes all the open RHS zero of

Cluss II: uo = 0 and z2 ( j w o ) = 0. Class 111: uo = 0 and 0 < Iz2(jwo)l < CO.

Class IV: U, = 0 and I z 2 ( j w o ) I = CO.

(W (W (W

Assume that Zl(s) and z 2 ( s ) are compatible. Then there exists a lossless two-port network N , whose input impedance is Z , ( s ) when the output is terminated in z 2 ( s ) , as shown in Fig. 1. Let

be the scattering matrix of N normalized to

If the two-port network N is reciprocal, then using the para-unitary property of S(s) we can write

M 2 ( s ) - N , ( s ) - M l ( s ) + N l ( s ) .- W l ( 4 8 2 ( s )

M,(s)+N,(s)+M,(s)+N,(s) W1(-4 s 2 h ) =

(16) where e(s) is an arbitrary real regular all-pass function. The necessary and sufficient condition for a matrix to be the scattering matrix of an n-port network was first given by Wohlers [ 2 ] .

Theorem 1: The necessary and sufficient conditions for an n X n rational matrix S(s) to be the scattering matrix of a linear, lumped, time-invariant, and passive n-port network, normalizing to the reference impedance matrix z(s), the elements of which are non-Foster positive-real functions, are that:

the Hermitian matrix U - S * ( j w ) S ( j w ) be non- negative-definite for all 0 ;

the matrix

2 Y h ) = h - ' ( s ) [ h ( s ) h - ' ( - s ) - S ( s ) ] h - ' ( s ) (17)

be analytic in the open RHS; and either

det{U-[z(s)-U]Y,(s)} # O (18) for all s in the open RHS, or the matrix

{ U - 144 - U1 Y m ) [ z b ) + U1 (19) have at most simple poles on the real-frequency axis, and the residue matrix evaluated at each of these poles be Hermitian and nonnegative-definite;

ZHU AND CHEN: THEORY OF COMPATIBLE IMPEDANCES 669

where S*( j w ) denotes the transpose and complex conjugate transformation of S(jw), z(s) = diag[z,(s), z 2 ( s ) ; . e , z n ( s ) ] and h ( s ) = diag [ h , ( s ) , h 2 ( s ) , - -, h n (s 11.

111. COMPATIBILITY THEOREMS Using Theorem 1, the compatibility of two positive-real

impedances, as given by Wohlers, is stated as [2]. Theorem 2: The necessary and sufficient conditions that

two non-Foster positive-real impedances Z,( s) and z 2 ( s ) be compatible with a reciprocal realization are:

1) at each zero so of r2(s ) of order m , (i)

(ii)

if z 2 ( s ) is analytic at so,so as a zero of R,(s) must be to the order of at least m, if z 2 ( s ) is singular at so = j w , but Z,(s) is not,

is uniquely specified by z2( s), satisfies the conditions that 1) y22n(s) is analytic in the open RHS, and that 2) at each real-frequency axis zero so = jwo , of r2(s ) of

if z2(s) is analytic at so = j w , then y220(s) can at most have a simple pole at so with rea and positive residue, or

(ii) if t Z ( s ) is singular at so = jw,, then

order k, (i)

fin? Z 2 ( 4 Y 2 2 0 ( ~ ) (25) S -* /WO

must be a real number less than or equal to one; if, and only if, +2(s) is a bounded-real reflection coeffi- cient satisfying the coefficients constraints (Appendix) at each zero of transmission s o of z 2 ( s ) of order k.

From (16), let

e 2 ( s ) = B&)e;(S) (26) so as a zero of R , ( s ) must be to the order of at least m + 2, and the sum of the orders of so as a zero of R,(s)

2) a real regular all-pass function O(s) exists so that the

where do(s) is another real-regulu all-pass function and write (iii)

and r2 must be an even integer for Res, > 0; S22(s) = B,(s)Z(s). (27)

function Substituting (26) and (27) in (16) gives

M A S ) - N2(s)- Mds) + Nds) Y 2 2 0 W

.- wl(s) B2(s)e;(s). (28) W1(- 4

with K,(s) and K;’(s) being analytic in the open RHS, and

(i) if z 2 ( s ) is analytic at so= j w o on the real- frequency axis, then y220(s) can at most have a simple pole at so with real and positive residue, or if z 2 ( s ) is singular at so = ju,, then (ii)

lin? Z z ( s ) Y 2 2 0 ( s ) (22) S + J %

must be a real number less than or equal to one.

In Theorem 2, two important requirements are included. One is about the order of the zeros of R,(s), and the other is about the admittance yZ2Js ) . The latter can be com- pletely characterized by Y oula’s coefficient conditions according to Satyanarayana and Chen [5 ] .

Theorem 3: Let z 2 ( s ) be a prescribed, rational, non- Foster positive-real function and &(s) a real, rational function of the complex variable s. Then the function defined by the relation

Y 2 2 0 ( 4 = L 4 2 ( 4 - + 2 ( 4 / F 2 ( 4 (23)

F2(s) = 2 r 2 ( + 4 2 ( 4 (24)

where A2(s) is given in (8) and

By imposing $2(s) = ((s) in Theorem 3, we have [5 ] , [6] Theorem 4: Let Z,(s) and z 2 ( s ) be prescribed, ra-

tional, non-Foster, positive-real functions. Then Z,(s) is compatible with z 2 ( s ) with a reciprocal two-port network realization if, and only if, there exists a real regular all-pass function eo($) such that the function defined by the rela- tion

M 2 ( s ) - N2b) - N s ) + Nib) +2(s) = M 1 ( s ) + N , ( s ) + M 2 ( s ) + N 2 ( s )

is a bounded-real reflection coefficient satisfying the coef- ficient constraints at each zero of transmission so of z z ( s ) of order k.

IV. UNIFIED THEORY It may first appear that condition 1) of Theorem 2 is

missing in Theorem 4, and Theorem 4 is not sufficient. In this section it is shown that Satyanarayana and Chen’s theory (Theorem 4) is completely equivalent to Wohlers’ solution (Theorem 2). Since Theorem 4 is developed from Theorem 2 [5 ] , we need only to prove that if Theorem 4 is true, then Theorem 2 is also true.

Proof of the Equivalent Condition: Consider Theorems 2, 3, and 4. Assume that the condi-

tions of Theorem 4 are satisfied, we show that conditions of Theorem 2 are also satisfied. From Theorem 4, we see

610 IEEE TRANSAC~ONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 6, JUNE 1988

that t # ~ ~ ( s ) is a bounded-real reflection coefficient which i) satisfies Youla's coefficient constraints, and ii) is not arbi- trary, but defined by (29). Because +2(s) satisfies Youla's coefficient constraints, according to Theorem 3, condition 2) of Theorem 2 is satisfied. The problem is then to show that, with specified q2(s) as given in (29), Theorem 4 implies condition 1) of Theorem 2.

We first study Class I zero of transmission of z2(s) and show that if Theorem 4 is true, the sum of the orders of so as a zero of R, ( s ) and r2(s) must be an even integer for Re so > 0.

Let so = U, + j w , be a k th-order Class I zero of trans- mission. Then we can write

[ ( S + ~ o ) 2 + W ; ] m [ ( S - - o ) 2 + ~ ; ] m f ( ~ ) f ( - S )

m w - +) r 2 ( 4 = 9

u,>O (30)

where f(s) is a Hurwitz polynomial, since the zeros of the even part of a positive-real function cannot occur at a pole of the function except on the real frequency axis [2], we have

k = m (31) for a Class I zero of transmission, and

The Laurent series expansion of (8) about so is

+ Jm-,(s - s o y - , + - * - (33)

a, f 0. (34)

where

Substituting (32) in (29) and expanding the resulting func- tion give

M 2 ( s ) - N2(s)- M h ) + N l ( 4 W s ) +2(s) = M , ( s ) + N 1 ( s ) + M 2 ( s ) + N 2 ( s ) -w,o

= 6, + I$,( s - so) + - . + 6m-l( s - so) -l+ * * . (35)

To satisfy the conditions of Theorem 4, being equivalent to satisfying Youla's coefficient constraints for the k th-order Class I zero of transmission, we require that (Appendix)

Since 2, is not zero, for the coefficient 4, in (35) not to vanish, the factor [(s - uo)2 + U;]" in (35) must be com- pletely cancelled by that in the denominator in (35). Ob- serve that O,'(s) is a real regular all-pass function and M , ( s ) + M 2 ( s ) + N,(s) + N2(s) is strictly Hurwitz by as-

A A

Ai = + i , i = 0,1,.. ., k -1. (36)

sumption. They do not contain factors like [(s - Thus we can write

+ a;].

(37)

x > m . (38)

-- W ) g(s)[(s + 4 w-4 - g(-s)[(s-u,) + , ; I x 2

Substituting (37) and (38) in (3), we can show that all open RHS zeros of r2(s) must be contained in those of Rl(s). From Theorem 4, (29) is a bounded-real reflection coeffi- cient, being devoid of the poles in the close RHS. The factor [(s - uo)2 + w ; ] ~ - ~ must be cancelled by an even- order real regular all-pass function for the reciprocal reali- zation of the equalizer [7]. This means that x - m is an even integer, so is x + m. Thus we conclude that the sum of the orders of so as a zero of R , ( s ) and r2(s ) must be an even integer for Res, > 0.

We next show that if z 2 ( s ) is analytic at so, so as a zero of R, ( s ) must be to the order of at least m, and if z2(s) is singular at so = j w , but Z , ( s ) is not, so as a zero of R , ( s ) must be to the order of at least m +2. We now consider the jo-axis zeros of r2(s). Let so = j w , be a kth-order zero of transmission of z2(s) , and write

r 2 ( 4 = ( 3 - j Q o ) m f ( s ) (39) where f(s) is rational and analytic at jw,. Then

k = m - 1

for a Class I1 zero of transmission, k = m

for a Class I11 zero of transmission, k = m + l

for a Class IV zero of transmission. Consider the admittance function of the augmented

network of Fig. 2:

(43)

where Z2,( s) is the back-end short-circuit driving-point impedance of N in Fig. 2. Appealing to Youla's coefficient constraints as given in the appendix and using the condi- tion of Theorem 4, we obtain

1 - A 2 ( - ~ ) + 2 ( 3 ) = ( ~ 2 + ~ ; ) x j ( ~ ) (46)

where

x a m - 1 (47)

x > m + l (48)

for a Class I1 or I11 zero of transmission,

for a Class IV zero of transmission, and j ( s ) is rational and analytic at jw,. Combining (49, (46) and (47) shows

671 ZHU AND CHEN: THEORY OF COMPATIBLE IMPEDANCES

y 2 2 a ( ~ ) has at most simple poles on the jw-axis for Class 11 or Class 111 zeros of transmission of z 2 ( s ) . From (45), (46) and (48) we can demonstrate that y 2 2 a ( ~ ) is not only analytic on the jw-axis, but also possesses a zero at j w , of order 1 for Class IV zeros of transmission of z2(s ) .

The above statement shows that the positive-real prop- erty of y 2 2 a ( ~ ) determines the orders of the zeros of the function 1 - A2( - S ) + ~ ( S ) , which is closely related to those of the transmission coefficient S12(s) and R,(s). To see this, we obtain from (l), (3), and (15)

Rib) =P(s)s12(s)s12(- 4 (49) where

. (50) [ M , ( s ) + M2(4I2- "s)+ N2(4I2

4 [ M 2 W - N 2 W l 4 s ) = B( - s ) + A ( - s ) z 2 ( - s)

A ( - s)+ C( - s) z20(- s > + z 2 ( - 4- Substituting the solution of ~ ~ ( s ) from (46) and using the (60) equation S2,(s)S2,( - s) = ~ + ~ ( s ) + ~ ( - s) in

Consider (47). The minimum order of x is m - 1. Sub- s22(s>s22(-s>+S12(S)s12(-s)=l (51) stituting x = m - 1 , (45), and (46) in (43) shows

we get z 2 o ( ; ~ o l + .2(@0> = 0. (61)

(52) with the exception that

= ( s 2 + ai)"[ g(s) + g( - s)]

-(?+ w ; ) 2 x g ( s ) g ( - s). (53)

Equation (53) indicates that Sl2(s)Sl2( - s ) possesses at least the xth-order zero at j w , when the compatibility conditions of Theorem 4 are satisfied. If p ( s ) is analytic at jw,, R,(s) also possesses at least the xth-order zero at

We now proceed to investigate p ( s ) . Suppose that the conditions of Theorem 4 are satisfied, and that Z,(s) is compatible with z 2 ( s ) . Let Z2,(s ) be the back-end driv- ing-point impedance of a realized two-port network N of Fig. 2, and write

J"o.

(54)

where A ( s ) , B(s ) , C( s ) , and D ( s ) are the numerator polynomials of the chain parameters of N , A ( $ ) , and D ( s ) are even polynomials, and B ( s ) and C ( s ) are odd poly- nomials, or vice versa. Then we have

Two cases are distinguished: Case 1: When jo, is a Class I1 or Class 111 zero of

transmission of z 2 ( s ) , z 2 ( s ) is analytic at j w , and

A ( j w , ) = B ( j w , ) = O . (63) Notice that (62) contributes a zero at jo, to R,(s). From (47), (49), (53), and (62), we see that the zero of R,(s ) at jog, being a Class I1 or Class I11 zero of transmission of z2 ( s ) , must at least be of order m, when A ( j w , ) and B(jw,) do not vanish simultaneously. The even part of (55) can be written as

A ( s ) D ( s ) - B ( s ) C ( s )

R1(s) = [ c ( s ) z 2 ( s ) + D ( s ) ] [ - C ( s ) z , ( - s ) + D ( s ) ]

. (64) . - 1 . l l ( s>m2(4- 4 + 2 ( 4

m ; w 4(4 Letting s = jo, and substituting (61) and (63) in (64):

(65) m1( j oo ) m 2 ( . b o ) - n l ( juo) n 2 ( ; W O )

mS(&o)- m , o ) Rl( jwo) =

showing that the above conclusion is still true. In (60), if z2( jw,) = Z2,( jo,) = 0 and only B( jo,) = 0, (62) remains valid.

Case 2: When jo, is a Class IV zero of transmission of z 2 ( s ) , z 2 ( s ) is singular at j w , and Z,(s) is analytic at this point by assumption. From ( 5 5 ) we obtain

Ml(jQ0) = A ( j o 0 ) (66)

%( jO, ) = 0 (67)

M2(jQO) = 0 (68)

N2(@0) = W . 0 ) . (69)

612 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 6, JUNE 1988

Substituting (66)-(69) in (50) yields with

- s 2 + 1 r 2 ( s ) = ( s + 2 ) ( - s + 2 )

Since Z , ( s ) is analytic at ju,, so is ~ ( s ) . Combining (48), (49), (53), and (70) shows that so as a zero of R,(s) to the order of at least m +2, because the zeros of the even part of a positive-real function are of even order on the real- frequency axis. This completes the proof of equivalency.

Theorem 5: Given two non-Foster positive-real imped- ances

shows that zeros of R,(s) contain those of r 2 ( s ) to at least the same order.

2) - s 2 + 1

A($) = - - - (79) r 2 ( 4

where M(s)’s , m(s)’s, and N(s)’s , n(s)’s denote the even and odd parts of the relatively prime polynomials, respec- tively. Then Z , ( s ) is compatible with z2(s) by a reciprocal and/or nonreciprocal lossless two port if, and only if, the following conditions are satisfied.

1) The zeros of r2(s ) = $ [ z 2 ( s ) + z2( - s)] are included among those of R,(s ) = + [ Z , ( s ) + Z,( - s)] to at least the same order.

2) There exists a real regular all-pass function q ( s ) such that the function defined by the relation

is a bounded-real reflection coefficient satisfying the Youla’s coefficient constraints (Appendix), where

Reo, > 0. (74) I

Theorem 5 follows directly from the above proof of the equivalency of the two theorems. The significance of Theo- rem 5 is that it not only simplifies (29), but also shows that the broadband matchmg and compatible impedances are two facets of the same problem. Therefore, they can be solved by the same set of formulas.

V. ILLUSTRATIVE EXAMPLE Example I : Let

1 s + -

3 s + 3

1 s + -

2 s + 2 .

(75) z,(s) = -

(76) z 2 ( s ) = - Solution: 1) Comparing

- s 2 + 1

R1(s) = (s + 3)( - s + 3) (77)

when s approaches 1. Choose

Substituting (81) and (75) in (73) gives - s + a 4 s + a 3 s + 5

+2( s) = f ~ * -

.(-s+l)+ a . . . (82)

a = 5 (83)

I From (A4) with k =1, we obtain

and we choose the plus sign in (82).

Z2,(s ) is found to be 3) From (A8) the back-end driving-point impedance

6s +45 z20(4 = (84)

which can be realized as the driving-point impedance of a lossless nonreciprocal two-port network terminated in a 1-Q resistor. The removal of this 1-Q resistor and the termination of the load z2(s) yield the realization of Fig. 3.

Example 2: Let

18s2 + 24s + 13 (85) zl(s) = 27s3 + 36s2 + 30s + 14

26 z 2 ( 4 =E. (86)

Solution: 1) Comparing

R h ) 182

- - ( - 2 7 ~ ~ + 3 6 ~ ~ - 3 0 ~ + 1 4 ) ( 2 7 ~ ~ + 3 6 ~ ~ + 3 0 ~ + 1 4 )

(87)

ZHU AND CHEN: THEORY OF COMPATIBLE IMPEDANCES 673

b d

Fig. 3. A realization of the problem considered in Example 1

with 728

(88) r 2 ( s ) = (21s +28)( -21s +28)

shows that zeros of R,(s) contain those of r 2 ( s ) to at least the same order.

2) 28

- r 2 ( 4 A(s) = - - z 2 ( s ) -21s +28

has a Class I1 zero of transmission of order 1 at the infinity, and

8 32 - - - - I+- - - 9 t

- 3 s + 4

3s + 4 s s 2 A ~ ( s ) = ~ -

208 F2(s) = 2r2(s)A2(s) =

7(3s + 4)2 208 1664 - - 63 189 = 0 + 0 + 7 - - + . * a .

S s3 Substituting (85) and (81) in (73) gives

- s + U

s + U

27s3 - 1 8 ~ ’ + 6 ~ - 1 27s3 +54s2 +54s +27 +2(s) = * ~.

8 16 32 2 a + - 2 u 2 + - u - -

3 3 9 - . (92) I =f -l+-- I S S2

1

From (A5) we choose u = O (93)

and the plus sign in (92).

to be 3) The back-end driving-point impedance Z2,(s ) is found

36s2+24s+14 21s + 14 (94) z 2 0 ( s ) =

which can be realized as a lossless two-port network terminated in a 1 4 resistor. The removal of this 1 4 resistor and the termination of the load z2(s) yield the realization of Fig. 4.

XH 7 p-j+$

18s2+24s+13 26 zl(s)= 27s3+36s2+30s+(4 z P ( s ) ’ ~ 21s+28

Fig. 4. A realization of the problem considered in Example 2.

VI. CONCLUSION

In the paper, we have presented a unified theory of the compatible impedances, and showed that Satyanarayana and Chen’s theorem is completely equivalent to Wohlers’ solution. A simplified formula was given to show that the compatible impedance problem and the broadband match- ing theory are two sides of the same problem. Therefore, they can be treated by the same method and solved by the same formulas.

APPENDIX

BASIC COEFFICIENT CONSTRAINTS The basic coefficient constraints on the bounded-real

reflection coefficient +2(s) are stated in terms of the coefficients of the Laurent series expansions about the zeros of transmission so of z2(s) of the following func- tions:

00

A,(s) = A,(s (AI)

F2(s) = 1 F x ( s (A2)

+ 2 ( 4 = c i i ( S - s o l X (A3)

x = o oc

x = o

cc

x = o

where A 2 ( s ) is defined in (8), F2(s) in (24), and G2(s ) in (29) or (73).

For each zero of transmission so of order k of z2(s), one of the following four sets of coefficient conditions must be satisfied, depending on the classification of so given by (11):

(2) Class [I: 4, = I$~, for x = 0,1,2, . . . , k - 1,

(3) Class<ZI: AxA=&, !or x = 0 , 1 , 2 ; . . , k - 2 ,

(4) Class IT: 2, 4x, A for x = 0,1,2; . ., k -1,

. . A

(1) CIUSSI: A x = + x , f o r x = 0 , 1 , 2 ; . . , k - l . (A4)

and(Ak-+k)/Fk+l> 0. 645)

and (Ak-1- +k-l)/Fk ;z 0. (A6)

(-47) and Fk-l/(Ak - + k ) > C l , the residue of z2(s) evaluated at the pole &.

The importance of these coefficient constraints is sum- marized in the following theorem first gven by Youla [4].

674 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 6, JUNE 1988

Theorem A: Let z2(s) be a prescribed, rational, non- Foster positive-real function and c+~(s) a real, rational function of the complex variable s. Then the function defined by

is positive real if, and only if, $I~(s) is a bounded-real reflection coefficient satisfying the basic constraints

REFERENCES

(A4)-(A7).

[l] J. D. Schoeffler, “Impedance transformation using lossless network,” IRE Trans. Circrff‘t Theory, vol. CT-8, pp. 131-137, 1961.

[2] M. R. Wohlers, Complex normalization of scattering matrices and the problem of compatible impedances,” IEEE Trans. Circuit The-

[3] C. W. Ho and N. Balabanian, Synthesis of active and passive compatible impedances,” IEEE Trans. Circuit Theory, vol. CT-14,

[4] D. C. Youla, “A new theory of broadband matching,” lEEE Trans. Circuit Theorv. vol. CT-11. DD. 30-50. 1964.

ory, vol. CT-12, pp. 528-535, 196:;

pp. 118-128, 1967.

[ 5 ] C. Satyanarayka and W. K.khen, “Theory of broadband matching and the problem of compatible impedances,” J . Franklin Inst., vol.

[6] W. K. Chen, “Unified theory of broadband matchng,” J. Franklin Inst.. vol. 310. DD. 287-301. 1980.

309, pp. 267-279, 1980.

[7] Y . S: Zhu and W. K. Chen,’ “Realizability of lossless reciprocal and nonreciprocal broadband matching network,” J . Franklin Inst., vol. 319, pp. 325-340, 1985.

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Yi-Sheng Zhu (M’87) was born in D a h n , Chna in 1945. He graduated from Qing-Hua University, Beijing, in 1970.

He was a Teaching Assistant with the Department of Radio Engi- neering and Electronics, Qing-Hua University from 1970 to 1978. He then transferred to the Department of Electronic Engineering,

Dalian Marine College, Dalian, China, and be- came a Lecturer in 1982. From 1983 to 1985, he was a Visiting Scholar with the Department of Electrical Engineering and Computer Science, University of Illinois at Chcago. Since Septem- ber 1986, he has been a Research Engineer in the Department of Electrical Engineering and Com- puter Science, University of Illinois at Chcago, on leave from Dalian Marine College. His re- search interests are mainly in network theory.

Mr. Zhu is a member of the Chinese Institute of Electronics.

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Wai-Kai Chen (S’61- M61- SM’71- F’77) re- ceived the B.S. and the M.S. degrees in electrical engineering from Oh0 University in 1960 and 1961, respectively, and the Ph.D. degree from the University of Illinois at Urbana-Champaign in 1964.

From 1964 to 1981 he was with Ohio Univer- sity, where he rose from Assistant Professor to become a Distinguished Professor of Electrical Engineering in 1978. Since September 1981, he has been Professor and Head of the Department

of Electrical Engineering and Computer Science at the University of Illinois at Chicago. He is the author of eight books, and his current interests are in broadband matching, active networks, filters, and applied graph theory, especially its applications to parallel computations.

Dr. Chen is a recipient of the 1967 Lester R. Ford Award of the Mathematical Association of America, and a 1972 Research Institute Fellow Award from Ohio University. He was awarded Honorary Profes- sorships by seven institutions in China, an Alexander von Humboldt Award (Senior U.S. Scientist Award) by Alexander von Humboldt- Stiftung of Germany in 1985, a JSPS Fellowship Award from the Japan Society for the Promotion of Science, a Senior University Scholar Award from the Unitersity of Illinois in 1986, and a Medal of Merit from the Ohio University in 1987.