The coefficient of the voltage induced frequency shift measurement on a quartz tuning fork

We have measured the coefficient of the voltage induced frequency shift (VIFS) of a 32.768 KHz quartz tuning fork. Three vibration modes were studied: one prong oscillating, two prongs oscillating in the same direction, and two prongs oscillating in opposite directions. They all showed a parabolic dependence of the eigen-frequency shift on the bias voltage applied across the fork, due to the voltage-induced internal stress, which varies as the fork oscillates. The average coefficient of the VIFS effect is as low as several hundred nano-Hz per millivolt, implying that fast-response voltage-controlled oscillators and phase-locked loops with nano-Hz resolution can be built.