Forced vibrations of KT-cut width-extensional mode quartz crystal resonators

The author describes an analysis of forced vibrations for KT-cut width-extensional mode quartz crystal resonators with a zero temperature coefficient at room temperature and an absolute value of its second order temperature coefficient β smaller than that of the DT-cut. The objectives are to clarify the influence on frequency temperature behavior of a piezoelectric term and errors of temperature coefficients of elastic stiffness constants and the piezoelectric term used in the calculation. A frequency equation and a transcendental equation are derived including a piezoelectric term, i.e., the frequency equation is given as a one-dimensional-approximate equation. The frequency constant and temperature coefficients of α, β, and γ are calculated from the frequency equation. The calculated values of frequency temperature behavior were compared with the measured ones for a WE resonator with frequencies of fs = 13.6, 16.3, and 30.7 MHz, when the error of temperature coefficients for the elastic stiffness constants and the piezoelectric term used in this calculation was taken onto account