Linear Periodically Time-Varying Circuits and Systems Demystified

An analog/mixed-signal designer encounters time varying circuits everywhere - sample-and- holds, chopper stabilised amplifiers, mixers, switched-capacitor amplifiers and filters, discrete and continuous-time delta sigma modulators, N-path filters. The analysis of signals and noise in these circuits is often associated with messy mathematics and algebra. This talk aims to demystify linear (periodically) time varying circuits. Starting from first principles, intuition behind various aspects of time-varying circuits and systems will be given. This intuition is illustrated with case studies of practical circuits and systems, like chopper-stabilised amplifiers and continuous-time delta-sigma modulators. We also discuss the important class of LPTV networks with sampled outputs, and the use of reciprocity in the analysis of such systems. As an application of the principles learned in this seminar, we will analyze the N-path filter, and show how the analysis of signals and noise can be greatly simplified. Tutorial table of contents : • Time-invariant and time-varying systems basics • LPTV systems and the Zadeh expansion • N-Path systems part : Time and frequency-domain intuition. • Sampled-LPTV systems and equivalence with a time-invariant sampled LTI system • Reciprocity and inter-reciprocity in time invariant and time varying networks • Application of interreciprocity to a cont. time delta sigma modulator • Analysis of chopped amplifiers as LPTV systems, noise folding. • N-path filter analysis based on sampled LPTV approach • Noise in N-path filter