Multiphysical finite element modeling of a quartz micro-resonator thermal sensitivity

A length-extension quartz micro-resonator was developed for time & frequency applications in order to provide an alternative to silicon MEMS resonators for oscillators. Quartz presents two main advantages over silicon: its high intrinsic quality factor and its low thermal sensitivity especially for appropriate quartz cuts. Reference [2] introduces the new micro-resonator and its peculiar structure, focusing on the structural design of the resonator itself to achieve a high quality factor by reducing mounting losses. Its predicted high quality factor frequency product (Q.F) is very close to the state of the art of macro shear mode quartz resonators used in Ultra Stable Oscillators (USOs). This paper focuses on how to achieve a low thermal sensitivity for this resonator. After describing the resonator itself, it focuses on the calculation method to obtain the thermal sensitivity of a resonator with respect to a certain quartz cut. The thermal sensitivity was calculated using the finite element softwares Samcef® and Oofelie® Multiphysics. Those two combined softwares enable us to calculate both thermo-mechanical and piezoelectrical behaviors of the resonator for rotated quartz cuts. At the end of this paper some results are presented to validate the approach on the VIG (Vibrating Integrated Gyro), comparing measurements with Finite Element Modeling (FEM). Eventually the thermal behavior in the extended temperature range (-55°C to 125°C) of four different quartz cuts (the Z cut, the X cut and Z and X rotated cuts) for the new length extension quartz micro-resonator is presented.