The Dependence of the Frequency of Quartz Piezo-Electric Oscillators upon Circuit Constants

The mathematical theory for the quartz piezo-electric stabilized, vacuum-tube-driven oscillator is given for the following cases: tuned plate circuit, inductance-loaded and resistance-loaded triode with the crystal between grid and plate, and also between grid and filament for each case. The condition for oscillations and the exact expression for the frequencies, damping factors, coupling coefficient, tube constants, etc., is given. In the analysis of the oscillator the equivalent network for the crystal given by Van Dyke has been used. The theory has been checked by measuring the variation infrequency of a quartz stabilized oscillator for variations in impedance of the plate circuit for the tuned circuit and resistance-loaded tube respectively. To satisfy the condition for oscillation it is necessary to use values for the equivalent resistance of the crystal somewhat smaller than those given by Van Dyke's formula. A discussion of the general method by which conditions for oscillation and expressions for the driven frequency of an oscillator may be obtained from the coefficients of differential equations up to the fourth order is included.