Exact Results for a Parametrically Phase-Locked Oscillator

An exact expression for determining the form and stability of solutions of the second-order linear differential equation governing a simple tuned circuit with square-wave variable conductance is derived. This expression is numerically evaluated to provide mode and stability diagrams particularly relevant to applications where it is desired to generate oscillations which are phase-locked to an external signal and experimental verification of some of the data is given. Relative to nonlinear element approaches to the synthesis of phase-locked oscillators, the principal advantages of the present method would appear to be that the locking-range accuracy and the condition for oscillation do not rely upon a particular nonlinearity but, instead, on the extent to which a square wave can be generated. Depending on the application, one disadvantage may be that, in contrast to the results which have been obtained for previous special variation cases of Hill's equation, it is difficult to lock the oscillator frequency to an even integral multiple of one half the pump frequency.