Design of quartz crystal resonators with an analytical procedure based on the Mindlin plate theory

By using the first-order Mindlin plate equations with the consideration of electrode and thermal effects, a theoretical model of AT-cut quartz crystal resonator is established. This one-dimensional model and analytical solutions can provide frequency, mode shapes, temperature effect, and electrical parameters like the capacitance ratios. The frequency temperature relations and capacitance are also measured from samples with exact parameters for calculations. The mode coupling, mode conversion, and occurrences of spurious mode are examined through the measurement and compared with analytical results for mode identification. The parameters examined include that of the crystal blank, orientation, electrode, materials for fine tunings. Mindlin plate equations with couplings of thickness-shear, flexural, face-shear, extension, and other modes and electrical field have been considered for the frequency, displacements, and electrical potential solutions. Measurements are made for the frequency, which is also calculated with known equations with the consideration of electrodes to validate the equations and initial design.