Stability of parametric networks

It is shown using state-variable theory that the s plane poles of the forced response of a linear periodically time-varying system are related simply to the characteristic exponents {¿1} of the A matrix. The poles are given by {±jn¿p ¿1±jn¿p}, where ¿p is the pump frequency and n = 0, 1, 2, .., ¿. It follows that a necessary and sufficient condition for asymptotic stability, given a uniformly bounded input, is that all the poles other than the ±jn¿p have negative real parts. That instability can occur in a physically realisable system is demonstrated using computed examples with known error magnitudes. These results conflict with an assertion by Leon based on his frequency-domain analysis of a class of parametric amplifiers. A further assertion, by Leon and Webber, that a certain class of parametric circuits is unconditionally stable, is demonstrated to be incorrect. The interesting result is obtained that, for a class of simple parametric circuits, there exist two separate regions of stability.