Further Results on the Contour Dependence of the Frequency-Temperature Characteristic of SC-Cut Resonators

Abstract

Vig, Filler, and Washington have reported the change in slope of the frequencytemperature characteristic of plano-convex SC-cut resonators, measured at the inflection temperature, as a function of blank contour. This change in slope can be thought of in terms of an "apparent angle shift." The change, however, is only apparent as the uncontoured side of the blank remains unchanged and retains its original orientation relative to the crystal axes. Stevens and Tiersten, on the other hand, have calculated the real changes in the angles of cut necessary to produce zero-temperature coefficient resonators at 25OC for designs of different radii of curvature. In this paper, the best available approximation for the angle dependence of the SC-cut frequency-temperature characteristic is applied to compare Vig's empirical results and Stevens and Tiersten's theoretical results. In the process, the empirical studies which were begun by Vig et al. were continued and expanded. K e y w o r d s : Q u a r t z c r y s t a l , q u a r t z resonator,SC-cut, frequency-temperature characteristic, blank contour, radius of In the course of establishing this correlation, some related topics of interest will also be discussed. First, the need to reference the recontouring coefficients to some chosen reference temperature will be established. Second, the changes in the inflection temperature during recontouring will be presented. Third, the recontouring coefficients will be shown to be a function of blank contour. Finally, the change in the slope of the SC-cut frequency-temperature characteristic as a result of simple thickness changes will be examined. METHOD OF COMPARISON Denoting the slope of the frequency temperature characteristic as S, we see that Vig has reported (dS/dCITi, where C is the contour in diopters. Stevens and Tiersten presented graphs of vs. R and 0 vs. R, the a n g l e s o f c u t n e c e s s a r y t o p r o d u c e zero-temperature coefficient resonators at 25OC when different radii of curvature are used. To analyze the two sets of data we use4 curvature, frequency-temperature coefficients, angle gradients. and INTRODUCTION Vig et a1.l have reported the change in the slope of the frequency-temperature characteristic of plano-convex SC-cut resonators, measured at the inflection temperature, as a function of blank contour. This change in slope can be thought of in terms of an "apparent angle shift." This change, however, is only apparent as the uncontoured side of the blank remains unchanged, and retains it original orientation relative to the crystal axes. Stevens and T i e r ~ t e n , ~ , ~ o n t h e o t h e r h a n d , h a v e calculated the real changes in the angles of cut necessary to produce zero-temperature coefficient resonators at 25OC for designs of different radii of curvature. In this work, the correlation between these two sets of data will be established. where (d /dR)z5 and (dO/dR)25 are the slopes of the curves published by Stevens and Tiersten, and ( S/ 1 and ( S / 0 ) are the sensitivities of 2ihe slope o?? the frequency-temperature characteristic to changes in the angles of cut. The slopes of Stevens and Tiersten's curves may be approximated by a second order polynomial obtained through least-squares analysis over the range of interest. The angle sensitivities of the frequency-temperature characteristic must be obtained from the approximation for the f requency-temperature characteristic5 and the angle gradients of the coefficients. "US GOVERNMENT WORK IS NOT PROTECTED BY US COPYRIGHT" 400 Using Stevens and Tiersten’s data, we obtain dS/dR for resonators at 25OC, while using Vig’s data, we obtain dS/dR for resonators at the inflection temperature, approximately 95OC. Since the angle gradients of the second and third order temperature coefficients are given as zeroI6 these values should be directly comparable.