A calculation and validation of electrical resistance of quartz crystal resonators with structural viscosity

We study the calculation of electrical resistance of AT-cut quartz crystal resonators with the consideration of structural viscosity. A theoretical analysis of electrically forced vibrations of the coupled fundamental thickness-shear and spurious modes of a rectangular resonator model which is a partially electroded quartz crystal plate with free edges is performed. The equations are derived for calculating the electrical parameters, which can be used for the characterization of electronic devices, are obtained from the first-order Mindlin plate theory with the consideration of equivalent viscous dissipation of piezoelectric plates. Numerical results of the electrical resistance of resonators are obtained. It is found that through adding proper equivalent viscosity coefficient in electroded portion of the crystal plate, the calculated results of resistance is in good agreement with measurements from actual product samples.