Amplitude-frequency effects in VHF quartz 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 not to show such effect. These results raise a question as to how these amplitude-frequency effects scale to lower frequencies, what criterion should be used in selecting these characteristics for optimum oscillator phase noise performance and whether it is proper to determine nonlinear elastic constants from AT-cut resonator measurements that assume there is no thermally related effect. LD-cut questions are also raised.