Explicit angular dependence of electroelastic effect of singly rotated extensional mode quartz resonators

Abstract

The theory of the field-induced change of the resonance frequency of piezoelectric extensional mode resonators based on the rotationally invariant nonlinear equations of electroelasticity as stated so far in the literature is formulated in a way that involves sums of a rather large number of terms. This fact is usually disguised by the compact notation allowed by the use of the Einstein summation convention. This form is well suited for numerical evaluation but is not easy to interpret in a general way for its obvious lack of transparence. Using the explicit expressions for the transformed material constants and purely algebraic treatment of these summations it will be shown in a completely rigorous way that the theory may be written in a considerably simplified form for the important case of quartz resonators singly rotated about the X axis. The very concise result is fairly transparent and allows direct comparison with the older phenomenological approach to this problem.