Analysis of intermodulation in thickness−shear and trapped energy resonators

Abstract

Electroelastic equations containing terms up to cubic in the small mechanical displacement field, but no higher than linear in the electric variables, are applied in the analysis of intermodulation in rotated Y−cut quartz oscillators. Both pure thickness−shear vibrators and essentially thickness−shear trapped energy resonators are treated. In the linear part of the analysis of the trapped energy resonator, a closed−form asymptotic expression for the frequency wavenumber dispersion relation for the fundamental and odd overtone thickness−shear branches near cutoff is obtained from the three−dimensional linear equations. Lumped parameter representations of the solutions, which are valid in the vicinity of a resonance, are presented for the linear and nonlinear portions of both the pure thickness−shear and trapped energy thickness−shear problems. The influence of the driving and detecting circuitry is included and, in particular, in each case the relation between the intermodulation and driving voltage is obtained. Application of this relation to a number of AT−cut quartz fundamental and third overtone trapped energy resonators yields good agreement with experiment and an estimate of the fourth−order elastic constant cE46666 for the AT−cut. Subject Classification: 40.24, 40.20, 40.65; 85.32.