On exponential fitting for circuit simulation

Some of the theoretical and practical aspects of using exponential fitting for MOS digital circuit timing analysis are described. The multivariate test problem x˙=-Ax where A∈R^n×n and is assumed to be irreducibly diagonally dominant with positive diagonals is considered, as this models the equations resulting from the way MOS circuits are treated in timing simulation programs. It is shown that for these problems, the Cinnamon exponentially fit algorithm is A-stable, and an example is given where the algorithm is XPSim is unstable. A semi-implicit version of the XPSim algorithm is described, and it is shown that this semi-implicit algorithm is A-stable. Examination of examples demonstrates that neither the stabilized XPSim algorithm nor the Cinnamon algorithm produces satisfactory results for very large timesteps. The effect of ordering on the accuracy and stability of the integration methods is also examined, and it is shown that ordering always enhances accuracy, though not significantly for large timesteps, and that the XPSim algorithm can be made more stable with a carefully chosen ordering