A Stability Criterion for Linear Networks Containing Time-Varying Capacitors

Abstract
A frequency-domain stability criterion is presented for networks containing linear, but not necessarily lumped, timeinvariant elements and an arbitrary finite number of linear timevarying capacitors. The criterion is applicable whether or not the capacitors vary periodically. In order to indicate the character of the result, consider the special but important case in which a passive lumped constant RLC network is terminated with a single time-varying capacitor with capacitancec(t). We say that the overall network is stable if (and only if) for an arbitrary set of initial conditions att = 0, the voltage across the time-varying element both approaches zero ast\rightarrow \inftyand is square-integrable on(0,\infty). LetZ(s)denote the driving-point impedance faced by the time-varying capacitor, and letmandMdenote positive constants such thatm < Mandm \leq [c(t)]^{-1}\leq Mfort \geq 0. Then, assuming thatZ(0)is finite and that Z(s) is not a reactance function, it is proved that the overall network is stable if for all real\omega,j\omega Z(j\omega)takes on values outside the circle of radius(M - m)centered in the complexplane at[-(m + M), 0].