ANALYSIS OF DOUBLY ROTATED LAME-MODE PIEZOELECTRIC RESONATORS BY THE ENERGY METHOD

The frequency constant and frequency temperature characteristics of do ubly rotated (x and y' axes) Lame-mode piezoelectric resonators made w ith quartz and (alpha-AlPO4 of trigonal crystals are discussed. The el ectrical characteristics of quartz crystal resonators are also discuss ed. A frequency equation is derived first from the (S, E)-type e-form piezoelectric equations via the energy method. Then, the frequency con stant and the first- and second- order temperature coefficients alpha and beta are derived as a function of cut angle phi (theta = 45 degree s). For both quartz and alpha-AlPO4 resonators there exist two cut ang les phi where alpha = 0. The two quartz cuts are designated LQ(1)T (ph i = 36 degrees) cut and LQ(2)T (phi = 130 degrees) cut. Furthermore, t wo cut angles phi exist for quartz crystal resonators where beta = 0. The calculated values of alpha and beta for the quartz crystal resonat ors show good agreement with measured values. However, no cut angle ph i exists for alpha-AlPO4 resonators where beta = 0. Next, assuming par tial electrodes and higher mode harmonics, motional inductance L-1 and motional capacitance C-1 are calculated via the (S, E) type e-form eq uations. Finally, the capacitance ratio is calculated and the results are compared with those for fundamental quartz crystal resonators. The calculated capacitance ratios are small and show good agreement with measured values.