A piezoelectric crystal has an effective mass of 10−2 kg, stiffness of 1010 N m−1 and damping constant 200 Ns m−1 . The electrical capacitance of the crystal is 1000 pF and the charge sensitivity is 2 × 10−10 C N−1 .

(a) Calculate the series and parallel resonant frequencies of the crystal.

(b) Calculate the magnitude and phase of the overall electrical impedance of the crystal at the above frequencies.

(c) The crystal is incorporated into a closed-loop oscillator system which is to oscillate at the crystal series resonant frequency. Calculate the required gain and phase of the maintaining amplifier at this frequency.