Relationship between amplitude and resonant frequency in quartz crystal resonators

In this paper we report results of measurements on the resonant frequency of crystal resonators as a function of drive current (amplitude). Previous studies indicate that the resonant frequency increases as the square of the driving current. If we assume that the limiting factor in the flicker frequency noise of crystal resonators is the noise in the drive current, then the quadratic dependence suggests that crystal resonators should be driven at low current for better frequency stability. Frequency versus amplitude measurements were made on SC-cut, 5th overtone, 100 MHz crystal resonators using a network analyzer. As expected, the measurements show a general quadratic dependence of frequency versus drive current. Nevertheless, some crystals exhibit phase (frequency) jumps at certain drive currents and certain temperatures. Phase modulation (PM) noise measurements were made in test oscillators at several currents to see if there is a correlation between the amplitude-frequency effect and flicker of frequency noise. Our results indicate that the flicker of frequency noise varies with current, but the current at which the flicker of frequency noise is the lowest is not necessarily the lowest current (as the quadratic relation of v₀ versus drive current suggests)