On rlc network matrices

Abstract

SUMMARYA generalization of the coefficient conditions which hold for an RLC network as well as of the paramount property obeyed by the matrix of a resistance network is derived. This result is a property of either the nodal or mesh matrices for either the admittances or impedances of a general RLC network. It states that if P is any principal minor or such a matrix and Q any other minor constructed from the same rows (columns) as P then P, P + Q , P-Q are rational functions of the complex frequency variable which, before cancellation of common factors, have only non-negative coefficients.Much has been written concerning RLC (transformerless) multiport networks. Nevertheless the basic realization problem remains unsolved except for special cases such as the RC and RLC voltage transfer function4" and the resistance n-port having exactly n + 1 ~e r t i c e s .~ Even necessary conditions which derive peculiarly from the transformerless character of the network are rather scarce. Two such properties of some generality which are inherent to networks without transformers are (i) the coefficient conditions for RLC 2-ports,' (ii) the paramount character of the node admittance matrix and of the mesh impedance matrix and their inverses for a resistance network.2 The purpose of this paper is to derive a generalization of these properties which includes both of them. PRELIMINARIESWe first summarize some results of graph theory and their application to electric network theory which are needed in the sequel. The details inherent in this summary are given in References 2 and 6. Let be a connected RLC (transformerless) network with b branches and n + 1 nodes. Of course, b > n. Any complete tree T ofThe two nodes of each branch of T define a port, so that an n-port N is thus defined by means of T Similarly, each link may be associated with the loop formed by it and the unique path in T between the vertices of the link. In this way, the system of fundamental loops or meshes M is defined by means of T (formed by deleting one row from the incidence matrix) and B the (6 -n ) x 6 fundamental loop matrix corresponding to the loops of M . Both A and B are unimodular ; that is, the determinant of any square submatrix of either A or B is -1, + 1 or 0.has n branches and b -n links.Let A be the n x 6 reduced incidence matrix ofThe n x n nodal admittance matrix Y, and the nodal impedance matrix Z, of the n-port are given byHere Y is the diagonal matrix of branch admittances, enumerated in the same order as the columns of A, and the prime denotes the transpose of the matrix. Since A, Y are of rank n, 6 respectively, Y, must be nonsingular. For the fundamental loop system M , the (b -n) x ( 6 -n ) mesh impedance matrix Z, and the mesh admittance matrix Y , are given by Z, = BZB', Y, = Z,'