Stationary principles and potential functions for nonlinear networks

This paper presents a novel graph theoretic approach for deriving various stationary principles for single-element type nonlinear networks. The concepts of total content, co-content and hybrid content are generalized to that of total parametric content for resistive nonlinear networks containing multivalued elements. The results by Brayton and Moser on potential functions for complete m-ports are generalized to non-complete m-ports in terms of three pseudopotential functions; namely, the pseudo-content, pseudo- co-content and the pseudo-hybrid content. The precise criterion for which each of these pseudo-potential functions reduces to a legitimate potential function is shown to be the unique solvability of an implicit set of equations given explicitly in terms of standard topological matrices. Simple circuit theoretic sufficient conditions are given which require merely the positive-definiteness of the incremental resistance matrix of an auxiliary complete current-controlled p-port and the incremental conductance matrix of an auxiliary complete voltage-controlled q-port. These conditions also guarantee that a non-complete dynamic nonlinear network can be represented by a system of normal form differential equations in terms of a single mixed potential function.