Design and fabrication of a length-extensional mode rectangular X-cut quartz resonator with zero temperature...

Rectangular X-cut quartz crystal resonators with cut angles theta > 5.0 degrees and aspect ratios Rzy (= width 2z0/length 2y0) from 0.3 to 0.5 are investigated. The resonators oscillate mode is a length-extensional mode. A semiempirical frequency equation was derived from the stress expressed in terms of the trigonometric and the hyperbolic transcendental functions with constants estimated by the regression curve fit to the stress simulated by the finite-element method (FEM). Contours on which a point satisfies a zero first order temperature coefficient condition are shown in a cut angle theta and Rzy diagram. We proved that a fabricated resonator with Rzy = 0.400 and theta = 16.0 degrees, whose design parameter is located in the area of the contour, had a zero temperature coefficient.