CORRESPONDENCE 253

work obtained with suitable transformers will yield a realization of 8. If A is stable, then from Lemma 1 it follows that matrices P and Q

exist, which satisfy (3). Setting T= Pl’z and d = T-‘AT then yields

A + Al = -& = -T-l&T-‘.

In light of our first observation, therefore, it can be assumed without loss of generality that if A is stable then

and C = B’ yields a system representation whose transfer function matrix Z(s) is diagonal and positive real. Hence, the desired results follows as in Case 1.

A+A'= -Q (6) where Q is a positive semidefinite matrix.

It should be noted from the proof of the theorem that if the restriction that the components of the original state vector x in (1) be voltages or currents in some of the network branches is relaxed, to allow the components of z to be linear combinations of branch voltages and/or currents, then every n X n stable A matrix can be realized without gyrators or transformers, that is, as a network composed of positive resistors and n positive, reactive elements.

The remainder of the proof is divided into two cases.

Case 1 Suppose that the minimal polynomial of A is equal to its charac-

teristic polynomial. Then it is always possible [S] to pick real n vectors bl and bp such that (A, bl) and (S’, bz) are controllable pairs. Moreover, it is always possible to find one n vector bt such that both (A, b3) and (A’, b3) are controllable. To show this, we first establish the following lemma.

LEONARD M. SILVERMAN Dept. of Elec. Engrg.

University of Southern California Los Angeles, Calif. 90007,

REEERENCES

Lemma S Let Ml, M,, N1, and Nz be real n X n matrices, and let Ml and N1

be nonsingular. Then, there exists a real number 01 such that MI + aMz and (YN~ + Nz are both nonsingular.

Proof: By elementary determinant relationships, it is clear that det (Ml f CXM~) = (~ndet Ml det [M;‘M, + (I/&)] anddet (olN1 + Nz) = det N1 det (N;’ N, + ,I). Hence, the only values for which either Ml + cuMz or aNl + Nz is singular are OL = 0, OL an eigenvalue of N;’ Nz and l/01 an eigenvalue of M;’ M2. Hence, save for a finite number of points, any real number o( will suffice to make MI + orMz and olN1 + N, nonsingular.

[l] D. C. Youla and P. Tissi, “n-port synthesis via reactance extraction--Part 1,” 1966 IEEE Internat’l Corm. Rec., vol. 14, pt. 7, pp. 183-205.

PI D. M. .Layton. “State representations. pessi+ty,, reciprocity, and N-port ;~%~y, Proc. 1966 4th Allerton Conf. on Carcud and Systems Theory. pp.

[3] B. D. 0: Anderson and R. W. Brockett, “A multiport state-space Darlington aynthesis,“IEEE Trans. Circuit Theory (Correspondence). vol. CT-14, pp. 336-337, September 1967.

[4] L. M. Silverman,, “Synthesis of impulse response matrices by internally stable and passive reabmtions.” IEEE Trans. Circuit Theory, vol. CT-15. pp. 23% 245, September 1968.

[5] E. S. Kuh, D. M. Layton, and J. Tow, “Network analysis and synthesis via state variables,” Electronic Research Lab., University of California, Berkeley, Tech. Memo. M/69, July 1966.

[S] J. L&a& and S. Lefschetz, Stability by Liapunou’s Direct Method. New York: Academic Press, 1961.

171 B. D. 0. Anderson, “A system theory criterion for positive real mat&s,” J. SIAM on Control, vol. 5, pp. 171-182. 1967.

IS] ~$.ZZ&eh and C. A. Desoer, Lznear System Theory. New York: McGraw-

91 R. iX. Kalman, “Methematical description of linear dynamical systems.” J. SIAM on Control, vol. 1. pp. 152-192, 1963.

To apply this lemma to obtain the desired controllability result, let Ml = [blAbI. .A*+bl], MZ = [bnAbz.. .Anwlbz], NI = [bzA’b,. . . (A’)“-lb21 and Nz = [bl A’bl . . (A’)“-lb,]. If bl and b, are chosen so that (A, bl) and (A’, bz) are controllable, then Ml and N1 are non- singular. Hence, setting b3 = bl + abs it follows from Lemma 3 that there exist values of LY for which (A, ba) and (A’, ba) are controllable.

If we now let b = b3 and c = b& it follows trivially from Lemma 2 that the representation (A, b, c) represents a passive system so that z(s) = c(sZ - A)-1 b is positive real and, since it is a scalar, it is also symmetric. Hence, z(s) has a passive, reciprocal network realization with n reactive elements whose A matrix is necessarily similar to the specified one since we constrained (A, b, c) to be controllable and observable and all minimal realizations of the same transfer function are related by a similarity transformation [9].

Concavity of Resistance Functions

Shannon and Hagelbarger [l] proved that a one-port network of resistors RI, R2, . . . , R, has a port resistance R = R(RI, Rz, - * . , Rn) that is a concave downward function of RI, Rz, . . . , a. A corollary was the fact that any network of linearly wound potentiometers on a common shaft and fixed resistors will give a resistance that is a concave downward function of shaft position, the latter being true also if potentiometers are used that are wound to be concave down- ward functions of shaft position.

Here still another derivation [2], [3] of these results of Shannon and Hagelbarger is given, starting from simple network topology.

Case 2 We first derive the following two theorems about the second

derivative of a driving-point impedance function. If the minimal polynomial of A does not equal its characteristic

polynomial, then it is always possible [S] to find a coordinate trans- formation such that in the new bases A has the form

Theorem 1

A, 0 ... 0

Consider a one-port network that contains n branches whose voltages and currents are the components of the column matrices V = co1 (V,, Vz, . . . , V,) and I = co1 (II,& . . . , L), respectively. These voltages and currents are related by the impedance matrix Z; V = 21. Let Zi, be the port impedance and I be the port (source) current. If each element of Z is a function of a parameter z (in addition to being a function of the complex frequency s = c + j,), and if Z is symmetric, then

where the miniial polynomial of Ai is equal to its characteristic polynomial for i = 1, . . . , k. The Ai may also be chosen to satisfy relationships of the form Ai + A{ = -Qi, where Q< is positive semidefinite, and as in Case 1 we may choose vectors b; of appropriate order such that (Ai, bi) and (A{, bi) are controllable. Setting

B=

i, 0 . . . 0

0 b2 0 . . . . . -1 . . . .

. .

Q 0 hi

Proof: Consider a connected network having b branches, n nodes, and I = b - (n - 1) independent loops. Following the notation of Kuh and Rohrer [4], let V = co1 (VI, VZ, . . . , V,,) be the (b x 1) column matrix of branch voltages, I = co1 (II, &., . . . , 1,) the (b X 1) column matrix of branch currents, 11 the (2 X 1) column matrix of link currents, e the (1 X b) fundamental loop (circuit) matrix, and J the (n - 1 X 1) column matrix whose kth component

Manuscript received October 24. 1968.

254

is the algebraic sum of source currents that appears in the kth fundamental cutset. These quantities Bre related in the following manner:

@V = 0 c-4

IEEE TRAhTSACTIONS ON CIRCUIT THEORY, MAY 1969

REFERENCES

Define the scalar h in the following way:

(4)

Using (3) to evaluate the derivatives of I in (4), we have

By the use of (2), this reduces to

h = -++] _ 2% [-+I + 5’ [-;-I. (5)

Without loss of generality, let the port be located across the nth branch. If the only source current is the port current I, then J = co1 (0, 0, * * * ) I). Then, defining the port impedance as Zi,,, where V, = Zi, I, we can write (5) as

h = a% 2 -T&5- I - 2Zi, g . ( >

2

On the other hand, writing V = ZI, and recalling that h is a scalar, we can evaluate h from (4) as

h = $$(z - z”)I - 2g(g - %)I

Recall that Z is symmetric (Z = ZL ). Therefore,

h=I’eI 8X2

-&& ax ax’

Finally, (1) is ob$ained by equating the values of h in (6) and (8) and setting aZ/ax = 0 in the former because Z is a source current.

Theorem 2 If Z is a symmetric positive semidefinite matrix and PZ/ax” is a

negative semidefinite matrix, then for s = 6, we have a2Zi,/8x2 < 0. Proof: Since Z is symmetric, Theorem 1 is applicable. By hypoth-

esis, the right-hand side of (1) is nonnegative since by virtue of s = 6, Z is real.

Theorem 3 (Theorem of Shannon and Hagelbarger) Consider a network of positive resistances RI, Rg, * . * , R,, each of

which satisfies d2R~/8xZ 5 0. Then a2Ri,,/ax2 5 0. Proof: Z = R is a positive definite diagonal matrix and @Z/ax2 =

azR/as is a negative semidefinite diagonal matrix. By Theorem 2, a=Zi,/ax2 = a2Ri,/ax2 I 0.

ARTHUR J. SCHNEIDER’

Dept. of Elec. Engrg. University of Wisconsin

Madison, Wis.

1 Now with Univac Division, Sperry Rand Corporation, Roseville, Minn. 55113.

[l] C. E. Shannon and D. W. Hsgelbarger, “Concavity of resistance functions,” J. AP& Phys., vol. 27, pp. 42-43, Janusry 1956.

[2] B. Noble, Applications of Undergraduate Mathematics in Engineering. New York: Macmillan, 1967, pp. 213-225.

[3] H. M. Melvin, “On concavity ol resistance functions,” J. Appl. Phys. (Letters), vol. 27. pp. 658-659, June 1956.

[4] E. S. Kuh and R. A. Rohrer, “The state-variable approach to network analysis.” Proc. IEEE, vol. 53, pp. 672+X36, July 1965.

On Scattering Matrices Normalized to Active n-Ports at Real Frequencies

I. INTRODUCTION

Youla [I] developed a means for defining scattering matrices to describe passive networks that are characterized by the following two properties: 1) the matrices are bounded-real in general and are unitary when the network is lossless, and 2) when the ports of the network are terminated in their normalizing impedances the “incident port waves” vaniljh, whereas when terminated in the negative of the conjugate of the normalizing impedances the “re- fleeted port waves” vanish. The procedure accomplishes this even though the normalizing impedances are complex valued as long as their real parts are positive. Rohrer [2] extended the technique to include the possibility of normalization to n X n matrices in such a way that the aforementioned two properties are preserved given only that the n X n matrix has a positive definite Hermitian real part. The technique that we will describe removes the positive definite requirement for the matrix normalization case while pre- serving the two basic properties noted above. At the moment, the new technique is restricted to constant, although complex valued, normalizing matrices and thus the primary applications would be in the study of the real-frequency behavior of active networks, specifically in the design of amplifiers when the given active element is an n-port network.

II. THE NORMALIZATION TECHNIQUE We will assume that a given n-port network is defined in terms

of n X 1 voltage and current vectors V and I, and that we wish to define n X 1 scattering variables (incident and reflected waves) a and b to define the network in a bilinear fashion, i.e., in the form

2H,a = V + AI

2H,b = V+BI

so that the resulting scattering matrix S relating b to a (b = Sa) is a function of the n X n matrices A, B, HI, and Hz. If we want the scattering matrix S to reflect the passivity of the network in the usual bounded fashion, i.e., 1, - S*rS being nonnegative definite, then we must impose constraints on these matrices. Spe- cifically, one may easily establish the following.

Lemma Given the definition of a and b in (l), and the requirement that

V*TI + I*TV = 2(a*Ta - b*Tb) for arbitrary a and b, then it is necessary that B = -A*T, and

A + A*T = 2

H H*T = H,HzT. 1 1

Thus, in particular, the Hermitian real part of A must be non- negative definite.

In the usual normalization, A is identified as the impedance matrix of some given n-port. The lemma then establishes the fact that we cannot proceed in the same way if we are to preserve the bounded property of the scattering matrix. However, we can introduce a related matrix to which the scattering matrix can be normalized, and in doing so preserve the basic features of the

Manuscript received July 24, 1968; revised October 4. 1968.