Analytical modeling of resistance for AT-cut quartz strips

Motional resistance R_m is a key customer specification for quartz plate resonators. Published experimental studies have shown that resistance of z-length quartz strips is strip width dependent. Another resistance characteristic of strip resonators is the steady increase of strip resistance as the frequency of strips gets lower. It is, therefore, very common that low frequency strips are thickness contoured. To explain the observed dependency of resonator resistance on strip width, we propose in this paper that the resistance and, in turn, the Q value, of strip resonators be vibration mode dependent. When the width of strips changes, the vibrations of strips vary from TSH dominant to flexure dominant and, in turn, the resistance fluctuates. To model the relation between resistance and strip width, Mindlin plate equations of coupled TSH and flexure motions are employed. The resistance of the resonators comes from structural damping. The resulting forced vibration problem is solved using the Galerkin method. A numerical example is given using one natural mode in the Galerkin method. The predicted resistance agrees in trend with measured strip discussed observed from experiments. Applications of the proposed method are discussed