Dynamic measurement of amplitude-frequency effect of VHF resonators

Dynamic amplitude-frequency effect measurements made of VHF quartz resonators indicate the presence of two separate mechanisms, the relative contribution of which varies for different cuts. For the SC-cut a step-change in drive level leads to a frequency shift with very short time constant (1 ms) that is presumably the classic direct nonlinear elastic effect. Others such as AT-, BT-, and LD-cuts show only a much longer time constant (200 ms), presumably a thermally related indirect nonlinear elastic effect. Near the SC-cut, resonators exhibit a combination of both short- and long-time-constant frequency changes. In some cases the two mechanisms produce frequency shifts in opposite directions. This behavior is problematic for some amplitude-frequency-effect measurement approaches because the measured shift depends on the timing of those measurements. It is not surprising that the SC-cut response shows only the direct nonlinear elastic effect, but it is unexpected that the AT- and BT-cuts appear riot to show such effect. These results raise a question as to what amplitude-frequency effect criterion is appropriate to use, as well as whether it is proper to determine nonlinear elastic constants from AT-cut resonator measurements that assume the effect is not thermally related.