The determination of electrical parameters of quartz crystal resonators with the consideration of dissipation

Recently, as the dissipation of quartz crystal through material viscosity is being considered in vibrations of piezoelectric plates, we have the opportunity to obtain electrical parameters from vibration solutions of a crystal plate representing an ideal resonator. Since the solutions are readily available with complex elastic constants from Mindlin plate equations for thickness-shear vibrations, the calculation of resistance and other parameters related to both mechanical deformation and electrical potential is straightforward. We start with the first-order Mindlin plate equations of a piezoelectric plate for the thickness-shear vibration analysis of a simple resonator model. The electrical parameters are derived with emphasis on the resistance that is related to the imaginary part of complex elastic constants, or the viscosity. All the electrical parameters are frequency dependent, enabling the study of the frequency behavior of crystal resonators with a direct formulation. Through the full consideration of complications like partial electrodes and supporting structures, we should be able obtain electrical parameters for practical applications in resonator design.