A simple equation predicting the amplitude of motion of quartz crystal resonators

The amplitude of motion of quartz crystal resonators, u(0), has been calculated on the basis of the transmission line model by Mason [Piezoelectric Crystals and Their Applications to Ultrasonics (Van Nostrand, Princeton, 1948)]. It is predicted to be u(0)=4/(pi n)(2) Qd(26)U(el,0), where n is the overtone order, Q is the quality factor, d(26) is the piezoelectric strain coefficient, and U(el,0) is the amplitude of the driving voltage. This simple result is in good agreement with previous numerical calculations, with an experimental value from the literature, and with our own experimental checks. As a side result, an equation is provided which allows to estimate the active area of the crystal from the product of the motional resistance R(1) and the Q factor. (c) 2006 American Institute of Physics.