Some Theorems on Properties of DC Equations of Nonlinear Networks

Abstract

Several results are presented concerning the equation F(x) + Ax = B (with F(·) a “diagonal” nonlinear mapping of real Euclidean n-space E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> into itself, and A a real n × n matrix) which plays a central role in the dc analysis of transistor networks. In particular, we give necessary and sufficient conditions on A such that the equation possesses a unique solution x for each real n-vector B and each strictly monotone increasing F(·) that maps E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> onto itself. There are several direct circuit-theoretic implications of the results. For example, we show that if the short-circuit admittance matrix G of the linear portion of the dc model of a transistor network satisfies a certain dominance condition, then the network cannot be bistable. Therefore, a fundamental restriction on the G matrix of an interesting class of switching circuits is that it must violate the dominance condition.