Direct analysis of time-varying continuous and discrete difference equations with application to nonuniformly switched-capacitor circuits

The exact time-varying transfer functions of circuits or systems described by matrix continuous difference equations and discrete difference equations are obtained. The method is a time-domain analysis without nsing any transform technique and the familiar
s, z
and nonuniform fraction of z variables are introduced in the process of analyss. The transfer functions at Input signal frequency and harmonic frequencies are given explicitly In closed forms. Criteria for stability and asymptotic stability In the sense of Lyapunov are given. Well-known examples of switched-capacitor (SC) circuits are discussed, and new insights are given. A general SC N-path circuit structure is proposed and its time-varying transfer function is obtained. An example of 3-path circuit is given as an illustration to show that its frequency characteristics such as center frequency and Q are well defined by its switching frequency and a capacitor ratio.