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Electric circuits are present in a number of applications, e.g. in home computers, television, credit cards, electric power networks, etc. The development of integrated circuit requires numerical simulation. Modern modeling techniques like the Modified Nodal Analysis (MNA) lead to differential algebraic equations (DAEs). Properties like the stability of solutions of such systems depend strongly on the DAE index. The paper deals with lumped circuits containing voltage sources, current sources as well as general nonlinear but time-invariant capacitances, inductances and resistances. We present network-topological criteria for the index of the DAEs obtained by the classical and the charge oriented MNA. Furthermore, the index is shown to be limited to 2 for our model-class.

The development of integrated circuits requires powerful numerical simulation programs. Of course, there is no method that treats all the different 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 difficult to find the circuit configurations that lead to numerical difficulties. In this paper, we analyze electric circuits with respect to their structural properties in order to give circuit designers some help for fixing modelling problems if the numerical simulation fails. We consider one of the most frequently used modelling technique, the modified nodal analysis (MNA), and discuss the index of the differential algebraic equations (DAEs) obtained by this kind of modelling.

Circuit simulation is a standard task for the computer-aided design of electronic circuits. The transient analysis is well understood and realized in powerful simulation packages for conventional circuits. But further developments in the production engineering lead to new classes of circuits which may cause difficulties for the numerical integration. The dimension of circuit models can be quite large (105 equations). The complexity of the models demands a higher abstraction level. In this paper, we analyze electric circuits with respect to their structural properties. We discuss the relevant subspaces of the resulting differential algebraic equations (DAEs) and present algorithms for calculating the index as well as consistent initial values.

For index-1 DAEs with properly stated leading term, we characterize dissipative and contractive flows and study how the qualitative properties of the DAE solutions are reflected by numerical approximations. The best situation occurs when the discretization and the decoupling procedure commute. It turns out out that this is the case if the relevant part of the inherent regular ODE has a constant state space. Different kinds of reformulations are studied to obtain numerically well formulated systems. Those reformulations might be expensive, hence, in order to avoid them, criteria ensuring the given DAE to be numerically equivalent to a numerically well formulated representation are proved.

differential algebraic systems (ADASs), i.e., differential algebraic systems with operators acting in real Hilbert spaces are introduced for a systematical treatment of coupled systems of PDEs, DAEs and integral equations. Using the finite-dimensional decoupling theory for DAEs as motivation, this paper will examine what one appropriate analogue is for infinite-dimensional systems. This leads to an index definition for ADASs. Thereby, instead of the inherent regular ODE one obtains an explicit (abstract) differential equation. In particular, when discussing PDAEs, the inherent regular differential equation is actually a parabolic PDE. The decoupling procedure provides, additionally, appropriate initial and boundary conditions for unique solvability of the coupled systems. The concept to handle ADASs is explained in different case studies.

For linear index-2 DAEs with properly stated leading term we characterize contractive and dissipative flows. We study under which conditions the qualitative properties of the DAE solutions are reflected by the numerical approximations. This is the case if the discretisation and the decoupling processes commute. Commutativity is achieved if two subspaces associated with the index-2 DAE are constant; in this case we say that the index-2 DAE is numerically well formulated. If both subspaces are time dependent, the problem should be reformulated; in order to avoid numerically equivalent reformulations, a criterion is given. If only one of these subspaces is constant, the problem is in some sense close to a numerically well formulated one and thus depending on the problem context, no reformulations are needed. Contractive and dissipative flows induced by the DAE are characterized and results on qualitative properties of the numerical solution are given.

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.

A model for a linear electric circuit containing semiconductors is presented. The modified nodal analysis leads to a differential algebraic equation (DAE) describing the electric network. The non-linear behaviour of the semiconductors is modelled by the drift diffusion equations. Coupling relations are defined and a sensitivity analysis concept that generalises the DAE index for finite systems to infinite ones is presented and applied to the resulting partial differential algebraic equation (PDAE). It is shown that the coupled system is of index 1 if the voltages applied to the semiconductors are low and the network without semiconductors is of index 1.

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.

In electric circuit simulation the charge oriented modified nodal analysis may lead to highly nonlinear DAEs with low smoothness properties. They may have index 2 but they do not belong to the class of Hessenberg form systems that are well understood. In the present paper, on the background of a detailed analysis of the resulting structure, it is shown that charge oriented modified nodal analysis yields the same index as the classical modified nodal analysis. Moreover, for index 2 DAEs in the charge oriented case, a further careful analysis with respect to solvability, linearization and numerical integration is given.

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 modeling 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.