A Comsol analysis of vibrations of quartz crystal plates with Mindlin plate equations

In the design and optimization of quartz crystal resonators, the finite element method has been adopted for the analysis of quartz crystal plates with complications for the vibration frequency and mode shapes. Further extension of such analysis can be made by using complex solutions to extract the electrical parameters of quartz crystal resonators because it is a typical electronic component in circuit applications. With Comsol which is widely used in industry and academic research, the Mindlin plate equations for the analysis of quartz crystal plates at thickness-shear mode are implemented for demonstration and validation purposes. To demonstrate the usefulness of Comsol in the vibration analysis of quartz crystal plates with coupled equations, we choose to utilize the first-order Mindlin plate equations and establish the analytical model with the input of differential equations in standard form as specified by Comsol. Then we found the equations can be processed and solved with the built-in solver of Comsol and results are reasonable and accurate. With the procedure and results presented in this study, it is clear that high frequency vibrations of quartz crystal resonators can be analyzed by the Mindlin plate equations and Comsol for accurate solutions needed in the optimization and design. The results are compared with analytical solutions and other FEM tools for accuracy. It is clear that the Mindlin plate can be conveniently used for the analysis, and results can be obtained quickly by taking advantage of Comsol.