The frequency-temperature analysis equations of piezo electric plates with Lee plate theory

Frequency-temperature analysis theory of crystal plates based on incremental field theory and Mindlin plate theory has been widely used in the vibration behavior analysis of crystal resonators for both analytical solutions and numerical approximations. As one of the two major plate theories for resonator analysis, Lee plate theory based on trigonometric series expansion of displacements has also been extensively used for both analytical and numerical analyses of quartz resonators, and efforts have also been made in correcting and perfecting the theory for much broader applications. In this paper, the earlier frequency-temperature relations based on the incremental field theory is further extended to the trigonometric series expansion in a systematic manner. By incorporating the frequency-temperature relations into Lee plate theory, the complete analysis of crystal resonators can be made in a more consistent way. The thickness-shear and flexural vibration equations have been compared with the previous power series theory to demonstrate the similarity and consistency