Straight crested wave analysis of quartz MEMS ring electroded mesa resonators

An analytical technique for designing high Q, thickness shear micro electromechanical, ring electroded mesa quartz resonators is proposed. The method is demonstrated using two-dimensional straight crested wave analysis. The design method is based on the two characteristics of a stable resonator: (a) The mode is energy trapped and relatively isolated from its supports, and (b) the motional impedance of the mode is low. The root mean squares of vibration displacements are employed to characterize the modes of vibration, and the thickness shear mode has a large rms u₁ displacement in the x₁ direction (diagonal axis). The rms displacement is used to compare the energy trapping of the thickness shear mode as a function of the electrode and plate geometry. For each mode of vibration, the electric flux density D₂ is calculated at the quartz to electrode interface to yield the electric current at the electrodes. Given a constant driving voltage, the magnitude of the electric current is inversely proportional to the motional impedance. Hence the electric current for a mode as a function of the electrode and plate geometry is employed as a further means for comparing the merits of different resonator designs. Results are shown for a 1 GHz inverted mesa AT-cut resonator.