Multi-Time PDEs for Dynamical System Analysis

Abstract

RF systems typically involve mixing, sampling and frequency — modulation, operations that generate signals with widely separated time scales (WSTS). SPICE-like simulation of such systems is often impossible, and even spe-cialized techniques such as harmonic balance have difficulty when there are sharp nonlinearities, e.g., switches. In this paper, we describe the use of apowerful tool for handling such systems, i.e., using multiple time scales at the differential equation level. We first describe the use of multiple time scalesfor non-oscillatory systems forced by WSTS inputs. We introduce multitime signals and the Multirate Partial Differential Equation (MPDE) andexplain how they form an efficient means of working with WSTS problems. Next, we describe how warped time scales are needed to represent WSTSsignals arising from forced oscillatory systems, like VCOs. We present the Warped MPDE (WaMPDE) for such systems. We also show how multiple time scales are useful for macromodelling RF blocks that perform frequency-translation or sampling (i.e., linear time-varying blocks).