zbMATH — the first resource for mathematics

Numerical analysis of DAEs from coupled circuit and semiconductor simulation. (English) Zbl 1073.65094
The paper is devoted to the problem of numerical solving a system of differential-algebraic equatins (DAEs) and partial differential equations (PDEs) that present an electrical circuit containing semiconductor devices. The desired functions are node potentials, currents through inductors, currents through voltage sources that are functions of time and are connected by ordinary DAE, as well as semiconductor’s electrostatic potential, densities of electrons and holes that depend on time and one space variable. The last three objects are connected by PDEs consisting of the Poisson equation and continuity equations (drift-diffusion equations). The DAEs and PDEs systems are connected by boundary conditions of the potential. The Poisson equation can be replaced by the equivalent energy conservation equation.
The investigated numerical method is the semi-discretization of the PDEs on the spatial variable. The resulting system is an ordinary DAE with properly stated leading term. The main results of the paper are three theorems that present structural criteria of the circuits which provides to the differentiation index of system of value 1 or 2.

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L80 Numerical methods for differential-algebraic equations
35Q60 PDEs in connection with optics and electromagnetic theory
82D37 Statistical mechanics of semiconductors
34A09 Implicit ordinary differential equations, differential-algebraic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDF BibTeX Cite
Full Text: DOI
[1] Ali, G.; Bartel, A.; Günther, M.; Tischendorf, C., Elliptic partial differential algebraic multiphysics models in electrical network design, Math. models methods appl. sci., 13, 9, 1261-1278, (2003) · Zbl 1046.94519
[2] Brenan, K.E.; Campbell, S.L.; Petzold, L.R., The numerical solution of initial value problems in ordinary differential algebraic equations, (1989), North-Holland Amsterdam · Zbl 0699.65057
[3] Estévez Schwarz, D.; Tischendorf, C., Structural analysis of electrical circuits and consequences for MNA, Internat. J. circuit theory appl., 28, 131-162, (2000) · Zbl 1054.94529
[4] Fosséprez, M., Non-linear circuits: qualitative analysis of non-linear, non-reciprocal circuits, (1992), Wiley Chichester
[5] Gajewski, H., On existence, uniqueness and asymptotic behaviour of solutions of the basic equations for carrier transport in semiconductors, Z. angew. math. mech., 65, 101-108, (1985) · Zbl 0579.35016
[6] Gajewski, H.; Gröger, K., Semiconductor equations for variable mobilities based on Boltzmann statistics or fermi – dirac statistics, Math. nachr., 140, 7-36, (1989) · Zbl 0681.35081
[7] Gajewski, H., Analysis und numerik von ladungstransport in halbleitern, GAMM mitteilungen, 16, 35-57, (1993)
[8] Gajewski, H., On the uniqueness of solutions to the drift-diffusion model of semiconductors devices, Math. models methods appl. sci., 4, 121-133, (1994) · Zbl 0801.35133
[9] Gajewski, H.; Heinemann, B.; Langmach, H.; Nürnberg, R.; Kaiser, H.-Chr.; Bandelow, U., WIAS-tesca. two dimensional semi-conductor analysis package. manual, (1999), Weierstrass Institut für Angewandte Analysis und Stochastik
[10] I. Higueras, R. März, Differential algebraic equations with properly stated leading terms, Preprint 00-20, Institute of Mathematics, Humboldt-Univ. zu Berlin, 2000, Comput. Math. Appl., in press · Zbl 1068.34005
[11] Higueras, I.; März, R.; Tischendorf, C., Stability preserving integration of index-1 daes, Appl. numer. math., 45, 175-200, (2003) · Zbl 1041.65065
[12] R. Lamour, R. März, C. Tischendorf, PDAEs and further mixed systems as abstract differential algebraic systems, Technical Report 01-11, Institute of Mathematics, Humboldt University of Berlin, 11, 2001
[13] Markovich, P.A., The stationary semiconductor device equations, (1986), Springer Berlin
[14] R. März, Nonlinear differential – algebraic equations with properly formulated leading term, Technical Report 01-3, Institute of Mathematics, Humboldt University of Berlin, 2001
[15] März, R., Differential algebraic systems with properly stated leading term and MNA equations, (), 135-151 · Zbl 1052.34006
[16] Mock, M.S., Analysis of mathematical models of semiconductor devices, (1983), Boole Press Dublin · Zbl 0532.65081
[17] Selberherr, S., Analysis and simulation of semiconductor devices, (1984), Springer Berlin
[18] Tischendorf, C., Topological index calculation of DAEs in circuit simulation, Surveys math. indust., 8, 187-199, (1999) · Zbl 1085.94513
[19] Tischendorf, C., Modeling circuit systems coupled with distributed semiconductor equations, (), 229-247 · Zbl 1045.94545
[20] C. Tischendorf, Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation, Habilitation Thesis, Humboldt University of Berlin, 2003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.