Structural analysis of electric circuits and consequences for MNA

The development of integrated circuits requires powerful numerical simulation programs. Of course, there is no method that treats all the diierent kinds of circuits successfully. The numerical simulation tools provide reliable results only if the circuit model meets the assumptions that guarantee the successful application of the integration software. Because of the large dimension of many circuits (about 10 7 circuit elements) it is often diicult to nd the circuit conngurations that lead to numerical diiculties. In this paper, we analyze electric circuits with respect to their structural properties in order to give circuit designers some help for xing modelling problems if the numerical simulation fails. We consider one of the most frequently used modelling technique, the modiied nodal analysis (MNA), and discuss the index of the diierential algebraic equations (DAEs) obtained by this kind of modelling. 1 Structural analysis In the following we discuss lumped electric circuits containing nonlinear and possibly time-variant resistances, capacitances, inductances, voltage sources and current sources. Usually circuit simulation tools are based on these kinds of network elements. For two-terminal (one-port) lumped elements, the current through the element and the voltage across it are well-deened quantities. For lumped elements with more than two terminals, the current entering any terminal and the voltage across any pair of terminals are well deened at all times (cf. 3]). Hence, general n-terminal elements are completely described by (n?1) currents entering the (n ? 1) terminals and the (n ? 1) branch voltages across each of these (n ? 1) terminals and the reference terminal n.