On the Analysis and Synthesis of Single-Element-Kind Net-works

Abstract

This paper presents a method for the realization of a given nth order node conductance (or capacitance or reciprocal inductance) matrix as an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)</tex> -node network with all positive elements and no ideal transformers. The first part establishes simple relations between branch conductances and elements in the given matrix which are assumed to be based upon a linear tree. These relations are analogous to the well-known ones pertinent to a so-called "dominant" matrix based upon the starlike tree. In an analysis problem they enable one to compute a single driving-point or transfer impedance with a minimum of computations. The second part of the paper develops a method whereby one can readily determine whether a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> matrix is based upon a tree (the necessary condition for its realization) and find the pertinent geometrical tree configuration when one exists. Once the latter is established, realization is simple and straightforward. The entire process requires no repeated trials and proceeds toward the desired goal with a minimum of effort.