Analyzing circuits with widely separated time scales using numerical PDE methods

Widely separated time scales arise in many kinds of circuits, e,g,, switche d-capacitor filters, mixers, switching power converters, etc. Numerical sol ution of such circuits is often difficult, especially when strong nonlinear ities 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 resu lts in the transformation of differential equation descriptions of a system to partial differential ones, in effect decoupling different rates of vari ation from each other. Numerical methods can then be used to solve the part ial differential equations (PDEs), In particular, time-domain methods can b e used to handle the hitherto difficult ease of strong nonlinearities toget her with widely separated rates of signal variation. We examine methods for obtaining quasiperiodic and envelope solutions, and describe how the PDE f ormulation unifies existing techniques for separated-time-constant problems . Several applications are described. Significant computation and memory sa vings result from using the new numerical techniques, which also scale grac efully with problem size.