A perturbation method for predicting the temperature and stress sensitivities of quartz vibrating structures simulated by finite-element analysis

Thermal and mechanical sensitivities of vibrating structures and wave guides are key parameters for the optimization of high stability resonant devices operating in the ultrasonic frequency range (from a few tenth of kilohertz to a few gigahertz). In this paper, the possibility to simulate and predict temperature coefficients of frequency (TCF) of quartz transducers of any shape as well as their stress sensitivity coefficients is addressed. The theoretical developments based on harmonic finite-element analysis coupled with a variational perturbation method are detailed, showing how to derive the regarded parameters. The proposed approach is validated using a two-dimensional (2-D) model of a plane face-bulk acoustic resonator for which an analytical model can give access to both TCF and stress sensitivity coefficients. It is then applied to a 2-D model of convex plane bulk acoustic resonator of singly rotated quartz and used to compute the first order TCF of a 3-D model of a tuning fork structure. In the latter case, the importance of considering the actual excitation of the device is demonstrated, allowing for the accurate definition of angular loci for which thermal compensation can be expected, in agreement with literature. Possible extensions and improvements of the proposed method is discussed in conclusion.