Numerical Solution of Differential-Algebraic Systems Arising in Circuit Simulation

Abstract

In this thesis, quasi-linear Differential-Algebraic Equations (DAE) as they arise in circuit simulation are examined. The circuit is described with help of a netlist, e.g. a list which contains the information about all devices in the circuit and the way these devices are connected to each other. With the help of this netlist a DAE is created by using either the classical or the charge oriented Modified Nodal Analysis. One focus of this thesis is the graph theoretical structure of the circuit, which is reflected in the properties of the circuit DAE. For example, it is known that it is possible under certain conditions to determine the differentiation index of such a DAE based on the graph of the circuit. Moreover, it is possible to determine the derivatives of equations which yield hidden constraints. In the course of this work, two methods have been developed that use this structural information to reduce the differentiation index of a circuit DAE if the DAE has differentiation index 2. To this end the graph theoretical foundations are laid out and applied to the existing results. It is shown that because of the graph theoretical results both index reduction methods work without time consuming rank decisions. Therefore these methods are applicable to large circuits. Moreover, the second method which was developed for DAEs arising from the charge oriented Modified Nodal Analysis allows for a physical interpretation. Hence this method not only can be realized as a numerical method that alters the DAE in order to reduce the differentiation index, but also as a method that alters the circuit itself. The second approach has the advantage that existing software for circuit simulation only has to be adapted slightly to use the index reduction method. Since the charge oriented Modified Nodal Analysis is preferred in industrial circuit simulation, the academical circuit simulator Psim by Robert Melville has been adapted to use this variant to produce the circuit DAE. Following this step the differentiation index of the resulting DAE is determined by graph theoretical algorithms and the index reduction method for DAEs from charge oriented Modified Nodal Analysis as proposed in this thesis is applied. The method has been tested on various examples. A comparison with the numerical results obtained without reducing the differentiation index shows the robustness and efficiency of the method.