The Analysis Of Higher-order Overtone Vibrations Of Quartz Crystal Plates With The Consideration Of Piezoelectric Effect

Quartz crystal resonator design achieves frequency targets through choices of thickness of quartz crystal blank and the functioning mode. The development of electronic circuits and resonator manufacturing technology will push for higher frequency. The overtone quartz crystal resonators have been widely utilized in various electronic products for their higher vibration frequencies, which are increasingly out of reach with the fundamental thickness-shear vibration mode. However, the design of overtone resonators and further improvement of the existing ones to meet precision requirements are largely based on empirical approaches. It is preferred that we can analyze the overtone vibrations and provide exact vibration frequency, mode coupling, and consequently some electrical parameters for the overtone type resonators in a manner analogous to the fundamental thickness-shear vibrations.In this study, we have derived the fifth-order Mindlin plate equations with the consideration of electrical potential and overtone displacements including the thickness-shear family. The elimination and truncation are validated by making comparison with the exact dispersion relations from three-dimensional equations, and the correction is done with the exact solutions of the cut-off frequencies of the thickness-shear mode family and slopes of some other modes. The frequency spectra are computed in the vicinity of overtone and fundamental thickness-shear modes to study the coupling and interactions with spurious modes, and the solutions have been used for the determination of optimal size of the crystal blank as in the fundamental thickness-shear mode. The formulation of capacitance ratio is done with solutions of displacements and electrical potential.The predictions on frequency, deformation, and electrode effects from studies with successive orders of equations can be used for resonator design at higher overtone frequencies. We believe this study can provide more reliable design guidelines in overtone resonators, as the results have been expanded from vibration frequency spectra and mode shapes. The capacitance ratio based on the displacement and electrical potential solutions in the vicinity of the overtone resonant frequency will be an important parameter for validation of theoretical results with actual measurements. The proper determination of these parameters, in conjunction with experimental results, will help the optimal design for better performance and, more importantly, reduced time and cost related to resonator design.