The Determination of Electrical Parameters of Partially Plated Quartz Crystal Plates with the Consideration of Dissipation

Recently, as the theory for bulk acoustic wave resonator analysis and design moves toward considering the dissipation of quartz crystal through the inclusion of viscosity, we finally have the opportunity to derive electrical parameters through vibration solutions of a crystal plate representing an ideal resonator. In this study, we use the first-order Mindlin plate equations of a piezoelectric plate for the thickness-shear vibration analysis of a simple resonator model which is a partially plated quartz crystal plate with free sides, and expressions of electrical parameters are derived with particular emphasis on the resistance that is related to the imaginary part of complex elastic constants, or the viscosity. All these electrical parameters are frequency dependent, providing the chance to study the frequency behavior of crystal resonators with a direct formulation. We calculated the electrical parameters from our simple resonator of partially plated AT-cut quartz crystal with the first-order Mindlin plate theory