A calculation of the resonant frequencies and motional capacitances for the fundamental family of modes of AT-cut quartz strip resonators

Mindlin's equations for the vibrations of elastic crystal plates are employed in the description of AT-cut quartz strip resonators. The electrically driven three-dimensional piezoelectric pure thickness solution is incorporated in the treatment, The driving voltage appearing in this thickness solution is included in the variational principle from which the plate equations are obtained. In this way the resulting Mindlin equations contain the driving voltage and hold for plates with small piezoelectric coupling. The equations are applied in the analysis of strip resonators. The eigensolutions are obtained by solving a sequence of one-dimensional problems that are defined by utilizing the results from the previous problem variationally. The driven solution is obtained by means of an expansion in the eigensolutions. Calculated results are presented for a range of geometries and the influence of the couplings is exhibited.