Structural characterization of classical and memristive circuits with purely imaginary eigenvalues

Abstract

The hyperbolicity problem in circuit theory concerns the existence of purely imaginary eigenvalues (PIEs) in the linearization of the time-domain description of the circuit dynamics. In this paper we characterize the circuit configurations which, in a strictly passive setting, yield purely imaginary eigenvalues for all values of the capacitances and inductances. Our framework is based on
branch-oriented, semistate (differential-algebraic) circuit models which capture explicitly the circuit topology, and uses several notions and results from digraph theory.
So-called P-structures arising in the analysis turn out
to be the key element supporting our results. The analysis is shown to hold not only for classical (RLC) circuits but also for nonlinear circuits including memristors and other mem-devices.