An efficient numerical method in calculating the electrical impedance different modes of AT-cut quartz crystal resonator

In this paper, we first solve for the eigenmodes of the AT-cut quartz crystal resonator without piezoelectric effect using the Lee-Brebbia FEA method, based on Mindlin´s 2D elastic equations. By assuming weak piezoelectric coupling, we rearrange the electrical potential to the RHS of the elastic-piezoelectric equations of motion. These terms become the forcing terms of the pure elastic equations of motion. We also consider the constant damping terms in the equations of motion. Then, we use the mode superposition method to calculate the weight of each eigenmode. The elastic displacement field with electrical potential effect can then be calculated as the weighting sum of each eigenmode. The electrical charges on the quartz crystal resonator can be obtained from the elastic displacement and electrical potential field at each node point. Finally, we can calculate the electrical impedance from the electrical charges and the potential. This method is considered much more efficient than the full-blown 3D elastic-piezoelectric analysis or the simplified Mindlin´s 2D elastic-piezoelectric analysis.