Relaxation-based electrical simulation

Circuit simulation programs have proven to be most important computer-aided design tools for the analysis of the electrical performance of integrated circuits. One of the most common analyses performed by circuit simulators and the most expensive in terms of computer time is nonlinear time-domain transient analysis. Conventional circuit simulators were designed initially for the cost-effective analysis of circuits containing a few hundred transistors or less. Because of the need to verify the performance of larger circuits, many users have successfully simulated circuits containing thousands of transistors despite the cost. Recently, a new class of algorithms has been applied to the electrical IC simulation problem. New simulators using these methods provide accurate waveform information with up to two orders of magnitude speed improvement for large circuits. These programs use relaxation methods for the solution of the set of ordinary differential equations, which describe the circuit under analysis, rather than the direct sparse-matrix methods on which standard circuit simulators are based. In this paper, the techniques used in relaxation-based electrical simulation are presented in a rigorous and unified framework, and the numerical properties of the various methods are explored. Both the advantages and the limitations of these techniques for the analysis of large IC's are described.