Time‐domain properties of reactive dual circuits

Abstract

SUMMARYThe present work explores some e ects of the replacement of capacitors by inductors and vice versa in state and semistate models of lumped circuits. Such a replacement, when performed together with an inversion of the capacitance and inductance matrices, yields a transformation of the form → −1 in the system spectra. In the semistate context, this covers in particular extremal cases in which null eigenvalues or inÿnite ones with higher index appear in the matrix pencil associated with the model; these cases describe certain pathological circuit conÿgurations. This approach leads to a discussion of new properties of strictly passive circuits; speciÿcally, from the known fact that the index of strictly passive circuits does not exceed two, we derive that the index of null eigenvalues in this setting cannot exceed one. This precludes in particular Takens-Bogdanov degeneracies, deÿned by an index-two doublezero eigenvalue, in strictly passive circuits. Although the results are addressed in a linear context, they can be extended via linearization to non-linear problems, as it is the case in the transformation of singularity-induced bifurcation phenomena into steady bifurcations discussed at the end of the paper.

Qualitative features of matrix pencils and DAEs arising in circuit dynamics

“…A semistate approach [10,11,18,19,[29][30][31][32][33] based on differential-algebraic equations seems therefore to be of interest in this regard, since the formulation of differential-algebraic models such as Tableau Analysis or Modified Nodal Analysis (MNA) [14,[17][18][19] (the latter being used in SPICE or TITAN) do actually need not much more than the lumped hypothesis. A framework based on matrix pencil theory has been recently proposed for the local stability analysis of semistate-modelled nonlinear circuits [34], supported on previous results concerning the stability of DAEs [7,[35][36][37][38].…”

Zero dynamics and funnel control of general linear differential-algebraic systems

“…The findings of the present paper, in particular the application of the funnel controller, can also be applied to a class of passive electrical networks. A common way of modeling electrical networks is the modified nodal analysis (MNA), see [17,21,40,45]. This modeling procedure results in a description of the circuit by a system of the form (1.1), where the inputs and outputs are appropriately chosen and the matrices E, A, B, C have specific properties, see also [38].…”

Circuit synthesis of passive descriptor systems—a modified nodal approach

“…The requirement on the capacitance, resistance and inductance matrices means that all elements are coupling-symmetric and do not generate energy. For a more detailed mathematical statement of the above conditions and their connection to topological assertions, we refer to [14,17,27].…”

“…A very common modelling technique is the so-called modified nodal analysis (MNA) [14][15][16][17], a method that is based on a graph theoretical consideration of the circuit. The advantages of the MNA are that the modelling can be done automatically, the circuit topology can be read off from the equations and a network list can be directly generated from the involved matrices.…”