Frequency warping in time-domain circuit simulation

Time-domain simulation of dynamic circuits and, in general, of any physical model characterized by ordinary differential equations or differential algebraic equations, implies the use of some numerical integration method to find an approximate solution in a discrete set of time points. Among these methods, the class known as linear multistep includes many well-known formulas such as the backward Euler method, the trapezoid method, and the implicit backward differentiation formulas used in most circuit simulators. All these methods introduce a very subtle effect that is, here, called the warping error. As shown, it is equivalent to a perturbation of the eigenvalues of the linearized ordinary differential problem. The perturbation introduced depends on the integration time step; it is often very small and in most cases irrelevant or even not noticeable. Nevertheless an exception to this situation is found when simulating high-quality factor circuits where even very small warping errors can lead to qualitatively wrong solutions. In this paper, we demonstrate that higher order linear multistep methods, while characterized by weaker stability properties, introduce less of a warping error and are well suited to the simulation of high-quality factor circuits.