An Analysis of a Piezo-Electric Oscillator Circuit

In this paper there is presented a mathematical analysis of a crystal "oscillator" circuit based on the suppositions: (a) that the oscillating crystal plate may be replaced by an equivalent electrical network; (b) that the "oscillation characteristic" of the tube may be represented by a three term nonlinear expression. The principal results are contained in an expression giving the resulting a-c plate voltage as a function of the plate tuning capacity, and the tube and circuit parameters. From this equation certain conclusions are drawn as to the suitable values for these parameters. The principal limiatations of the analysis are those due to the assumptions: (a) that the internal grid current is negligibly small; (b) that the resistance drop terms in the differential equation may be neglected in comparison with the reactance drop terms. It is pointed out that while the former assumption does not materially affect the results obtained, the latter, by introducing a certain discontinuity which has no physical counterpart, leaves much to be desired in the interpretation of the results for a certain important range of values of the independent variable.