An Index Analysis from Coupled Circuit and Device Simulation

Abstract

Nowadays the semiconductor devices in an electrical circuit are modelled by equivalent circuits containing basic network elements described by algebraic and ordinary differential equations. But the correct adjustment of these circuits has become a very difficult task for the network design. In [2] a new model for electrical circuits containing semiconductor devices is proposed and in [1] its well-posedness is studied. In both articles the differential algebraic equations (DAEs) for the basic circuit’s elements are coupled to partial differential equations (PDEs), more specifically to one-dimensional Drift-Diffusion (DD) equations, modelling the semiconductor devices in it. Systems of this type are called Abstract Differential Algebraic Systems (ADAS). In [9] the tractability index [5, 9] of this model is analysed and in [8] it is proved that the DAE obtained after discretization in space of the DD equations in it has the same index as the abstract system. In this work we study the tractability index of an abstract system where higher dimensional PDEs describe the behavior of the semiconductor devices in the circuit. The index of the DAE obtained after discretization in space of the PDEs in the system is also analysed. In the next section the model is briefly described. The Sect. 3 is devoted to the study of the index of the system, as ADAS. Finally, in Sect. 4 it is shown that the DAE that is obtained after discretization in space of the DD equations has the same index as the abstract system. In what follows we consider electrical circuits with only one semiconductor device, the results can easily be generalized to circuits containing more semiconductor devices.