The calculation of electrical parameters of overtone thickness-shear quartz crystal resonators with the higher-order Mindlin plate equations

Abstract

Quartz crystal resonator design achieves frequency targets through the choice of thickness of quartz crystal blank and vibration modes of overtone. The overtone types of resonators have been widely available to meet the ever increasing needs of high frequency resonators. Earlier studies on quartz crystal resonator analysis have been concentrated on analysis of vibrations of the fundamental thickness-shear mode for solutions of frequency and displacements which are essential in the initial design and subsequent calculation of electrical parameters. We are now in the stage to analyze overtone vibrations and provide frequency, electrodes, displacements, and consequently some electrical parameters for the overtone resonators in a manner analogous to the fundamental thickness-shear vibrations. We have derived the higher-order Mindlin plate equations with the consideration of electrical potential and overtone displacements including the thickness-shear family. These equations have been corrected based on the three-dimensional dispersion relations of anisotropic materials like the AT-cut quartz crystal and selected vibration modes are retained as part of the simplification procedure. The elimination and truncation are validated by making comparison with the exact dispersion relations, 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. The formulation of capacitance ratio is done with solutions of displacements and electrical potential, and the results are used for the comparison with measurements. The coupling of vibration modes are examined with frequency spectra from the coupled equations of displacements and electrical potentials at the overtone thickness-shear frequencies. We believe theoretical studies can provide more reliable design guidelines in overtone resonators.