Oscillation Theory of Parametric Amplifiers

Abstract

In recent years several papers on parametric amplifiers have appeared in reference to micro-wave amplification. Circuit analogues of these amplifiers were proposed and the analogue circuits have been analyzed in various ways. Those methods of analysis, however, are based upon some assumptions concerning the form of currents in the circuit.
It is the purpose of this paper to prove the validity of the above mentioned conventional methods from the standpoint of the theory of oscillation.
Two types of analogue circuits, linear and nonlinear, are considered and correspondingly a system of linear differential equations with time-varying coefficients and a system of nonlinear differential equations with constant coefficients, which govern the behaviours of the circuits are derived.
Following the so called ‘asymptotic method’ of N. Bogoliuboff et al., it is shown that these systems of differential equations have almost periodic solutions which are asymptotically stable, with the consequent result that the steady state oscillations of the circuits are almost periodic. Average power is calculated and expressions for amplification factor is obtained. These results are in perfect agreement with that of the conventional methods.