The drive level dependence of the resonant frequency of high frequency quartz resonators

The drive level dependence of frequency, which comes from the nonlinear effects of the quartz material, is related to the amplitude of strain in the quartz resonator. High stability, and high frequency resonators are designed in such a way the vibration is trapped in the center of the vibrating plate. Therefore, the amplitude of the effect is also strongly dependent of the trapping of the vibration and the construction parameters of the resonator play a critical role. In this paper we present the basic equations when the nonlinearities are restricted to the mechanical behavior, although the nonlinearity of electrical polarization or other effects can also be introduced. As application, we restrict our attention to the most important case of contoured (plano-convex) resonator