Analyzing circuits with widely separated time scales using numerical PDE methods

Widely separated time scales arise in many kinds of circuits, e,g,, switched-capacitor filters, mixers, switching power converters, etc. Numerical solution of such circuits is often difficult, especially when strong nonlinearities are present. In this paper, we present a mathematical formulation and numerical methods for analyzing a broad class of such circuits or systems. The key idea is to use multiple time variables, which enable signals with widely separated rates of variation to be represented efficiently This results in the transformation of differential equation descriptions of a system to partial differential ones, in effect decoupling different rates of variation from each other. Numerical methods can then be used to solve the partial differential equations (PDEs), In particular, time-domain methods can be used to handle the hitherto difficult ease of strong nonlinearities together with widely separated rates of signal variation. We examine methods for obtaining quasiperiodic and envelope solutions, and describe how the PDE formulation unifies existing techniques for separated-time-constant problems. Several applications are described. Significant computation and memory savings result from using the new numerical techniques, which also scale gracefully with problem size.