Lee plate equations for electroded quartz crystal plates with the consideration of electrode density and stiffness

Lee plate equations for high-frequency vibrations of piezoelectric plates have been established and improved over the last decades with the sole objective of obtaining the accurate prediction of frequency and mode shapes to aid crystal resonator design. The latest improvement includes extra terms related to derivatives of the flexural displacement to adjust the accuracy and consider electrodes for practical applications. As part of the efforts to make the equations applicable for resonator design with the improved frequency accuracy and consideration of electrodes, we derived Lee plate equations for electroded plates by changing the integration limits in the dimension reduction procedure to signify the dominant role of the crystal plate. By modifying the density terms in plate equations to include the contribution of both electrode stiffness and density, the accuracy of the thickness-shear vibration frequency of electroded plates is improved for commonly used electrode materials.