The fifth-order overtone vibrations of quartz crystal plates with higher-order Mindlin plate equations

As demands for high frequency quartz crystal resonators rise, we are prompted to design and make overtone devices with the same material, process, and technology. Indeed, high-order overtone resonators have been widely utilized in many applications and further demands in precision products are also growing. The design of overtone resonators and further improvement of the existing ones to meet precision requirements are largely based on empirical approaches, but we found that the technique can be polished with theoretical and analytical efforts as we examine the applications of the Mindlin plate theory in the design and analysis of the fundamental type. Through the extensive improvements of the Mindlin plate theory, we can now analyze the vibration mode couplings, electrode effect, optimal sizes, and thermal behavior, among others. These essential analytical procedures have been implemented in finite element analysis tools with more advanced features such as the nonlinear behavior prediction and eventually circuit parameter extraction. Since it has been proven that the Mindlin plate equations can be used for the vibration analysis of plates at the higher-order overtone modes with accurate prediction of frequency and dispersion relations in the vicinity of cut-off frequencies, we extended the equations to the third-order for the modal behavior and frequency spectra. The results show that earlier knowledge on the proper selection of the sizes of electrode can be proven from our analysis. In addition, the spatial variation and end effects of displacements, particularly of the working mode, can be used in the optimal selection of resonator configuration. The design changes can be used as a way to improve the resonator performance, which has been increasingly degenerating for higher-order overtone types, to meet more stringent requirements. We now extend the plate equations to the fifth-order so the design principle and guidelines can be summarized from more analytical results of overtone vibrations. These predictions on frequency, deformation, and electrode effects from studies with successive orders of equations can be used for resonator design at higher overtone frequencies.