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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. [ABSTRACT FROM AUTHOR]

The memory-resistor or memristor is a new electrical element characterized by a nonlinear charge-flux relation. This device poses many challenging problems, in particular from the circuit modeling point of view. In this paper, we address the index analysis of certain differential-algebraic models of memristive circuits; specifically, our attention is focused on so-called branch-oriented models, which include in particular tree-based formulations of the circuit equations. Our approach combines results coming from differential-algebraic equation theory, matrix analysis and theory of digraphs. This framework should be useful in future studies of dynamical aspects of memristive circuits. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: Augmented nodal matrices play an important role in the analysis of different features of electrical circuit models. Their study can be addressed in an abstract setting involving two- and three-colour weighted digraphs. By means of a detailed characterization of the structure of proper and normal trees, we provide a unifying framework for the rank analysis of augmented matrices. This covers in particular Maxwell’s tree-based determinantal expansions of (non-augmented) nodal matrices, which can be considered as a one-colour version of our results. Via different colour assignments to circuit devices, we tackle the DC-solvability problem and the index characterization of certain differential–algebraic models which arise in the nodal analysis of electrical circuits, extending several known results of passive circuits to the non-passive context. [Copyright &y& Elsevier]

We examine in this article so-called B-critical points of linear, time-varying differential-algebraic equations (DAEs) of the form A(t)(D(t)x(t))' + B(t)x(t) = q(t). These critical or singular points, which cannot be handled by classical projector methods, require adapting a recently introduced framework based on Π-projectors. Via a continuation of certain invariant spaces through the singularity, we arrive at a scenario which accommodates both A- and B-critical DAEs. The working hypotheses apply in particular to standard-form analytic systems although, in contrast to other approaches to critical problems, the scope of our approach extends beyond the analytic setting. Some examples illustrate the results. [ABSTRACT FROM AUTHOR]

Time-domain models of dynamical systems are formulated in many applications in terms of differential-algebraic equations (DAEs). In the linear time-varying context, certain limitations of models of the form E(t)x'(t) + B(t)x(t) = q(t) have recently led to the properly stated formulation A(t)(D(t)x(t))' + B(t)x(t) = q(t), which allows for explicit descriptions of problem solutions in regular DAEs with arbitrary index, and provides precise functional input-output characterizations of the system. In this context, the present paper addresses critical points of linear DAEs with properly stated leading term; such critical points describe different types of singularities in the system. Critical points are classified according to a taxonomy which reflects the phenomenon from which the singularity stems; this taxonomy is proved independent of projectors and also invariant under linear time-varying coordinate changes and refactorizations. Under certain working assumptions, the analysis of such critical problems can be carried out through a scalarly implicit decoupling, yielding a singular inherent ODE. Certain harmless problems for which this decoupling can be rewritten in explicit form are characterized. Some electrical circuit applications, including a linear time-varying analogue of Chua's circuit, are discussed for illustrative purposes. [ABSTRACT FROM AUTHOR]

Abstract: We consider in this work linear, time-varying differential-algebraic equations (DAEs) of the form through a projector approach. Our analysis applies in particular to linear DAEs in standard form . Under mild smoothness assumptions, we introduce local regularity and index notions, showing that they hold uniformly in intervals and are independent of projectors. Several algebraic and geometric properties supporting these notions are addressed. This framework is aimed at supporting a complementary analysis of so-called critical points, where the assumptions for regularity fail. Our results are applied here to the analysis of a linear time-varying analogue of Chua''s circuit with current-controlled resistors, displaying a rich variety of indices depending on the characteristics of resistive and reactive devices. [Copyright &y& Elsevier]