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Index and solvability of Uncoupled Circuits: a Characterization without Restrictions on their passivity, Topology or Controlling Structure

In this paper, we extend the topological formulae of Maxwell and Kirchhoff, characterizing the non-singularity of node-admittance and loop-impedance matrices, to mixed problems, that is, to circuits combining admittance and impedance descriptions or, in nonlinear cases, involving both voltage- and current-controlled resistors. By means of this mixed formula we analyze the index of differential-algebraic models of nonlinear uncoupled circuits in a very broad setting, namely, without assumptions on their topology, their passivity or the controlling variables for nonlinear resistors. In particular, our approach allows for a characterization of index two circuits in topologically degenerate settings, which had been so far elusive in the non-passive context. As a byproduct we address the unique solvability of mixed resistive circuits, a problem which also arises in connection to the so-called DC-solvability condition of dynamic circuits. For the sake of brevity, we discuss in less detail how to extend the analysis to problems with mixed descriptions in reactive devices.