Electrical behaviour of AT-cut quartz crystal resonators as a function of overtone number

An investigation on the electrical behaviour of quartz resonators with fundamental frequencies up to 10 MHz was carried out by impedance analysis on several overtones. The values of the equivalent electrical components of these acoustic sensors were determined and revealed to correctly match those obtained for theoretical one-dimensional crystals. The study on different harmonics allowed to evaluate the variation of the effective vibrating area of each crystal and the effect of energy trapping intrinsic to a crystal of finite surface was then emphasized. It was also shown that the impedance behaviour is affected by the piezoelectric stiffening and the boundary conditions of real quartz resonators. These perturbing effects have a significant magnitude on the fundamental harmonic mode, and to a lesser extent on the third and fifth overtones. Besides, the theoretical models established for viscoelastic analysis of coated layers are based on the assumption that the quartz crystals are ideal with an infinite surface. For this purpose, a model was proposed for different crystals to take into account the various effects influencing both resonance frequencies and bandwidths of quartz resonators.