Electric equivalent circuit for flexural vibrations in piezoelectric materials

General expressions for the values of the components of an electric equivalent circuit are derived for flexural vibrations. These expressions are applied to some interesting electrode configurations. The obtained values are in good agreement with experimental values and values specified by manufacturers of watch crystals. It is noted that solving Laplace´s equation for the dielectric field is sufficient in order to obtain the values of the vibration amplitude, the piezoelectric current, and the equivalent components. The piezoelectric part of the electric field need only be considered if very accurate values of the resonance frequencies are desired. It is shown how accurate estimates can be obtained without the need of advanced calculation tools.<>