Conditions on Structure Matrices of n-Port, Immittance-Equivalent Electric Networks

This paper deals with the relations between the structure matrices of electric networks which are n-port equivalent to a given network composed of passive linear time-invariant bilateral two-terminal elements, whose associated linear graph is connected and has no self-loops. The -ports are defined as (i) ports created by cutting designated links of a cotree, or (ii) ports created by connecting to designated branches of a tree. Matrix representations for the electromagnetic, topological, and port structures of networks are described. It is found that the structures of equivalent networks can be formally represented by means of a pair of square matrices, one of which describes the change of electromagnetic structure, and the other the change of topological and port structures. The principal result of the paper is the determination of necessary and sufficient conditions on these matrices for n-port immittance equivalence. This result opens the possibility of directly specifying the electromagnetic, topological, or port structures of an equivalent network, and then formally solving for the remaining information. This is in contrast to existing methods, based on W. Cauer\´s linear transformation of port variables, in which the effect on the three structures is "mixed" in a single matrix of transformation.