Nonlinearities for parametric pumping of quartz UHF oscillators

The feasibility of a parametric amplification method for improving the Q of UHF quartz oscillators is considered. A fundamental thickness shear resonator operating at about 500 MHz was studied for estimating the magnitude of fractional change in stiffness, Δk/k that could be obtained at resonance and off-resonance. The Δk/k is employed in a parametric pumping of the thickness shear mode of vibration. A relationship between the Δk/k at resonance and Δk/k at off-resonance was derived. For a 532 MHz fundamental thickness shear resonator, a Δk/k = 3.8×10⁻⁴ /V at resonance, and a Δk/k = 1.6×10⁻⁷ /V at off-resonance was found. The off-resonance data compares well with measured data of Δk/k = 1.7×10⁻⁷ /V. Our off-resonance study of Δk/k established that it is independent of the sign of electric potential drive that is Δk/k is a rectified excitation and therefore appears at twice the excitation frequency. The parametric amplification phenomenon is governed by the Mathieu equation. MATLAB Simulink models of the Mathieu equation were developed to establish the baseline criteria for parametric amplification to improve the resonator Q. The resonator model was excited at a frequency ω<inf>A</inf> over a range that included the natural frequency ω<inf>0</inf> and the third overtone ω<inf>3</inf>. The parametric drive frequency ω<inf>P</inf> was set equal to 2ω<inf>A</inf>/n (n=1, 2, 3, ….). The model results showed that for parametric drive frequencies of ω<inf>P</inf> = ω<inf>A</inf> (for fundamental mode operation), and ω<inf>P</inf> = ω<inf>A</inf>/3 (for third overtone operation), the Δk/k needed for parametric amplification is in the range of 0.001 to 0.003 for a resonator Q of 9,000 to 15,000. It was observed that there was shift of the resonance frequency with the parametric amplification resulting from the change in dc stiffness Δk/k of the thickness shear mode.