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.