An analysis of nonlinear resonance in contoured-quartz crystal 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 nonlinear resonance in doubly rotated contoured-quartz resonators and all terms present in the general anisotropic case are included. Since the modes of motion in contoured-quartz resonators are essentially thickness modes varying slowly along the plate, only the thickness dependence of the elastic nonlinearities are retained in the equations, as in earlier work. The linear portions of the equations are the same as those that have recently been derived and employed in the analysis of doubly rotated contoured-quartz resonators. The steady-state solutions are obtained by means of an asymptotic iterative procedure and an expansion in the linear eigensolutions while retaining the nonlinear correction to the eigensolution that has a resonant frequency in the vicinity of the driving frequency. The slow variations in the mode along the plate are included in the nonlinear correction by averaging over the plate. Lumped parameter representations of the solutions, which are valid in the vicinity of a resonance and relate the amplitude of the mode nonlinearly to the voltage across the electrodes, are obtained. In each instance the expression for the current through the crystal is determined, the external circuitry is incorporated in the description and an equation relating the mode amplitude nonlinearly to the driving voltage and other circuit parameters is obtained. The analysis holds for the fundamental and odd harmonic overtones. Nonlinear resonance curves are calculated for AT-cut quartz using the known nonlinear coefficients. In particular, it is shown that the order of the harmonic has a more significant influence on the shift in resonant frequency from the linear value than the current through the crystal. An equation relating the change in resonant frequency resulting from the nonlinearity to the current through the crystal, independent of the external circuitry, is derived. This latter equation is employed in the evaluation of the coefficient of nonlinear resonance for SC-cut quartz from measurements of the shift in resonant frequency with current level in contoured resonators.