Search

In the last years, several mathematical models coupling differential algebraic equations and partial differential equations for describing the behavior of electrical circuits have been proposed in the literature. Most of them investigate the properties of coupled systems that include one-dimensional drift–diffusion equations for describing the highly sensitive semi-conducting elements in the circuit. Here, we extend the results to coupled systems with higher dimensional drift–diffusion models that allow a proper modeling of semiconductor devices constituted of regions with different material properties. For stability reasons, we investigate a monolithic simulation approach and consider two common variants of PDE discretizations for semiconductors in the system: besides a finite element method that may be combined with the Scharfetter–Gummel approach, a mixed finite element discretization. The resulting differential algebraic equations share important properties that allow us to show that their index is always less or equal to two and depends only on the circuit’s topology. The information about the index enables us to conclude unique solvability of the associated initial value problems as well as to choose appropriate time-stepping methods for them.

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.

We discuss the analysis of linear constant coefficient differential algebraic equations on infinite dimensional Hilbert spaces. We give solution concepts and discuss solvability criteria which are mainly based on Laplace transform. Furthermore, we investigate the decoupling of these systems motivated by the Kronecker normal form for the finite dimensional case. Applications are given by the analysis of mixed systems of ordinary differential, partial differential and differential algebraic equations.

Conclusion We have seen that the stationary solutions of the regularized equations are asymptotically stable if the stationary solution of the original quasilinear index-2 tractable DAE is so. In other words, the given regularizations supply a solution sharing the stability properties of the solution of the unperturbed equation.

We present a little overview about different regularization approaches for electrical circuits. Most of them treat model equations arising from the modified nodal analysis (MNA). They are of the most practical relevance since the well established circuit simulators used in semiconductor industry apply the MNA procedure to generate the circuit equations. Additionally, we discuss some hybrid modeling approaches that are proved to provide more stable model formulations than the standard MNA approach.

Circuit simulators used in semiconductor industry are based on lumped element models described in form of net lists. In order to be able to incorporate the mutual electromagnetic influence of neighboring elements (e.g. cross talking), one needs refined models based on a sufficiently exact discretization of the full Maxwell equations. Here, we present an interface model for such refined models that can be used for lumped circuit net lists.