Thickness vibrations of flat piezoelectric plates with massy electrodes of unequal thickness

We develop a set of six coupled equations governing the modal amplitudes and phase angles (mode-center offsets) for the flat, piezoelectric plate resonator with massy electrodes of unequal thickness. The case of the AT-cut resonator is used to obtain a set of transcendental equations describing the resonance frequencies and phase angles of an isolated single mode. These show that the phase angle (and corresponding even mode amplitude) is proportional to the electrode mass-loading imbalance, and for a given total electrode mass-loading, the resonance frequencies depend also on the electrode mass imbalance.