Effects of electromagnetic radiation on the Q of quartz resonators

The quartz resonator Q with aluminum electrodes was studied with respect to its fundamental thickness shear mode frequency and its viscoelastic, viscopiezoelectric, and viscopiezoelectromagnetic behaviors. The governing equations for viscoelasticity, viscopiezoelectricity, and viscopiezoelectromagnetism were implemented for an AT-cut quartz resonator. To simulate the radiation conditions at infinity for the viscopiezoelectromagnetic model, perfectly matched layers over a surface enclosing the resonator were implemented to absorb all incident electromagnetic radiation. The shape of the radiation spectrum of a 5.6 MHz AT-cut quartz resonator was found to compare relatively well the measured results by Campbell and Weber. The mesa-plate resonator was studied for a frequency range of 1.4 GHz to 3.4 GHz. The resonator Q was determined to be influenced predominantly by the quartz viscoelasticity; however at frequencies greater than 2.3 GHz, the quartz electromagnetic radiation had an increasingly significant effect on the resonator Q. At 3.4 GHz, the electromagnetic radiation accounted for about 14% of the loss in resonator Q. At frequencies less than 2 GHz, the calculated resonator Q compared well with the intrinsic Q_x provided by the formula Q_x = 16 times 10⁶/f where f was in MHz. At frequencies higher than 2.3 GHz, the aluminum electrodes had significant effects on the resonator Q. At 3.4 GHz, the electromagnetic radiation loss in the electrodes was an order of magnitude greater than their viscoelastic loss; hence, the vibrating aluminum electrodes became an efficient emitter of electromagnetic waves. The effects of electrical resistance in both the electrodes and quartz were determined to be negligible.