The hyperbolicity problem in electrical circuit theory

The time-domain characterization of qualitative properties of electrical circuits requires the combined use of mathematical concepts and tools coming from digraph theory, applied linear algebra and the theory of differential-algebraic equations. This applies, in particular, to the analysis of the circuit hyperbolicity, a key qualitative feature regarding oscillations. A linear circuit is hyperbolic if all of its eigenvalues are away from the imaginary axis. Characterizing the hyperbolicity of a strictly passive circuit family is a two-fold problem, which involves the description of (so-called topologically non-hyperbolic) configurations yielding purely imaginary eigenvalues (PIEs) for all circuit parameters and, when this is not the case, the description of the parameter values leading to PIEs. A full characterization of the problem is shown here to be feasible for certain circuit topologies. The analysis is performed in terms of differential-algebraic branch-oriented circuit models, which drive the spectral study to a matrix pencil setting, and makes systematic use of a matrix-based formulation of digraph properties. Several examples illustrate the results. Copyright (C) 2010 John Wiley & Sons, Ltd