On the Synthesis of R Networks

Abstract

A study is made of a class of real symmetric matrices, and a new set of necessary and sufficient conditions are defined on the entries of these matrices such that they can be synthesized, without using ideal transformers, as an R-network with a minimum number of terminals. The conditions are stated in terms of the terminal graph used in representing the terminal characteristics of multiterminal components. The new class of matrices then represents "terminal equations" corresponding to a path-tree terminal graph. It is also shown that one of the well-known class of matrices, referred to as dominant matrices, represent the terminal equations for a Lagrangian-tree terminal graph. It is further indicated that any such class of real symmetric matrices is distinguishable by a particular terminal graph. For the cases when a real symmetric matrix of order n cannot be synthesized by an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> network with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)</tex> terminal vertices, an "enlarged" matrix is formed, and the necessary and sufficient conditions for realizability of these matrices are given.