asymptotic stability hyperbolicity differential-algebraic equation circuit model modified nodal analysis graph topology matrix pencil Topological analysis of qualitative features in electrical circuit theory Ricardo Riaza Riaza Ricardo Caren Tischendorf Tischendorf Caren Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

Topological analysis of qualitative features in electrical circuit theory

Ricardo Riaza , Caren Tischendorf

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

34D20 Lyapunov stability
94C05 Analytic circuit theory

Abstract
Several qualitative properties of equilibria in electrical circuits are analyzed in this paper. Specifically, non-singularity, hyperbolicity, and asymptotic stability are addressed in terms of the circuit topology, which is captured through the use of Modified Nodal Analysis (MNA) models. The differential-algebraic or semistate nature of these models drives the analysis of the spectrum to a matrix pencil setting, and puts the results beyond the ones already known for state-space models, unfeasible in many actual problems. The topological conditions arising in this qualitative study are proved independent of those supporting the index, and therefore they apply to both index-1 and index-2 configurations. The analysis combines results coming from graph theory, matrix analysis, matrix pencil theory, and Lyapunov theory for DAEs. The study is restricted to problems with independent sources; qualitative properties of circuits including controlled sources are the focus of future research.


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