A new theory for electroded piezoelectric plates and its finite element application for the forced vibrations of quartz crystal resonators

Abstract

For crystal resonators, it is always desirable to calculate the electric properties accurately for application purposes. Such calculations have been done with analytical solutions from approximate equations and simplified models with good results, but for better consideration of the actual resonators, finite element method has been used for the free vibration analysis with excellent results to aid the analysis and design. The finite element analysis based on the higher order Mindlin plate theory is particularly effective and easy to implement and expand. As an extension of the Mindlin plate theory based finite element analysis of crystal resonators, a new theory for the electroded plates is derived and the piezoelectrically forced vibrations are formulated and implemented in this paper in a manner similar to our previous work. The effect of the electrodes and the electric boundary conditions are taken into consideration through the modification of the higher order plate equations by changing the expansion function of the electric potential for this particular problem. Through the conventional discretization of the new plate theory, the linear equations for the piezoelectric plate under thickness excitation are constructed and solved with efficient numerical computation techniques such as the sparse matrix handling. The solutions of mechanical displacement and electric potential are then used for the computation of the capacitance ratio of the electroded plate with emphasis on its derivation with the two-dimensional plate theory. The applications of these results in crystal resonator modeling are discussed and demonstrated in detail. Numerical examples showing good predictions of the resonance frequency and capacitance ratio of electroded crystal plates of AT-cut quartz are presented with experimental data.