Title :
An efficient numerical method in calculating the electrical impedance different modes of AT-cut quartz crystal resonator
Author :
Pao, S.Y. ; Chao, M.K. ; Lam, C.S. ; Chang, P.-Z.
Author_Institution :
Inst. of Appl. Mech., Nat. Taiwan Univ., Taipei, Taiwan
Abstract :
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.
Keywords :
crystal resonators; eigenvalues and eigenfunctions; electric impedance; finite element analysis; piezoelectricity; quartz; AT-cut crystal resonator; Lee-Brebbia FEA method; Mindlin 2D elastic equations; SiO2; eigenmodes; elastic displacement field; elastic-piezoelectric equations of motion; electrical impedance calculation; electrical potential field; mode superposition method; piezoelectric effect; resonator modes; weak piezoelectric coupling; Circuit stability; Damping; Design engineering; Electric potential; Equations; Finite element methods; Impedance; Oscillators; Piezoelectric effect; Temperature;
Conference_Titel :
Frequency Control Symposium and Exposition, 2004. Proceedings of the 2004 IEEE International
Print_ISBN :
0-7803-8414-8
DOI :
10.1109/FREQ.2004.1418488