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The hyperbolicity problem in electrical circuit theory
Author(s) -
Riaza Ricardo,
Tischendorf Caren
Publication year - 2010
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1312
Subject(s) - mathematics , matrix pencil , network analysis , eigenvalues and eigenvectors , digraph , electrical network , linear algebra , electronic circuit , topology (electrical circuits) , algebraic number , algebra over a field , jacobian matrix and determinant , pure mathematics , mathematical analysis , discrete mathematics , geometry , combinatorics , physics , quantum mechanics , electrical engineering , engineering
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 © 2010 John Wiley & Sons, Ltd.