For a refined network analysis, we are interested in circuit simulation
including distributed models of semiconductors. We construct a mathematical
model for nonlinear electric networks containing semiconductors
described by the drift-diffusion equations. The focus lies on the coupling
of the network DAEs and the semiconductor PDEs.
Furthermore, we study the behavior of the coupled systems with respect to
time dependent perturbations using an index concept for abstract DAEs.
We present a network topological criterion that guarantees index-1 systems.
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.
In this work we are interested in the numerical solution of a coupled model of
differential algebraic equations (DAEs) and partial differential equations (PDEs).
The DAEs describe the behavior of an electrical circuit that contains semiconductor
devices and the partial differential equations constitute drift-diffusion equations
modelling the semiconductor devices in the circuit.
After space discretization using a finite element method, the coupled system results
in a differential-algebraic system with a properly stated leading term. We
investigate the structure and the properties of this DAE system. In particular, we
develop structural criteria for the DAE index. This is of basic interest since DAE
properties like stability, existence and uniqueness of solutions depend strongly on
its index.