Electroelastic interactions, biasing states, and precision crystal resonators

Abstract

In the interaction of the quasistatic electric field with deformable insulators, the condition of rotational invariance causes a combination of the electric field and the deformation gradients to occur in the constitutive equations along with the finite strain. Since the resulting system is intrinsically nonlinear, the linear dynamic equations are more general than those of linear piezoelectricity when a bias is present and reduce to them only in the absence of a bias. Even in the simplest case of stress-free thermal deformation, which is just about always present, the more general equations arise when the fixed reference coordinates at the reference temperature are employed. The advantage of the use of reference coordinates, which cannot be employed within the usual linear theory, in the accurate calculation of the temperature sensitivity of high precision contoured quartz resonators is shown. In the treatment, the equation for the perturbation in eigenfrequency of the piezoelectric solution due to a bias, which is obtained from the more general linear equations, is employed. However, both the biasing state and the vibrational solution are obtained by solving systems of unbiased linear equations. The change in frequency resulting from any bias may readily be calculated from the perturbation equation when the linear piezoelectric solution and biasing state are known. The importance of the phenomenon of energy trapping in crystal resonators is discussed and means of controlling it are noted.