Analysis of capacitance ratio of a rectangular X-cut length-extensional mode quartz crystal resonator by stresses

The electrical properties of an X-cut, length-extensional mode quartz crystal resonator of a cut angle theta around the X-axis were calculated by a variational method using stresses as trial functions. Analytical expressions of stresses were estimated by a linear regression on a cut angle best-fit to the results of finite-element method. The calculated dependence of the capacitance ratio on the cut angle was consistent with the measured results.