A Stability Criterion for Linear Networks Containing Time-Varying Capacitors

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 capacitance
c(t)
. We say that the overall network is stable if (and only if) for an arbitrary set of initial conditions at
t = 0
, the voltage across the time-varying element both approaches zero as
t\rightarrow \infty
and is square-integrable on
(0,\infty)
. Let
Z(s)
denote the driving-point impedance faced by the time-varying capacitor, and let
m
and
M
denote positive constants such that
m < M
and
m \leq [c(t)]^{-1}\leq M
for
t \geq 0
. Then, assuming that
Z(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]
.