Analysis of the Internal Noise of Quartz-Crystal Oscillators

This paper describes the influence of the parallel capacitance of a quartz-crystal resonator on the amplitude-frequency coupling and particularly on the internal noise spectra of the oscillator working at the series resonance. A theoretical analysis which is a first order perturbation method is used. It is shown that the parallel capacitance of the quartz-crystal resonator increases the amplitude-frequency coupling and drastically modifies both amplitude and phase spectra of the internal noise. The 1/f2 phase spectrum of the internal thermal noise is transformed into a white phase spectrum for noise component frequencies greater than f0 + f² or less than f0 - fΔ, where f0 is the resonator series resonant frequency and fΔ, the difference between antiresonant and resonant frequencies of the quartz crystal. A "noise quieting" phenomenon appears when the noise component frequencies are in the vicinity of the antiresonant frequency fp. A good agreement between theoretical and experimental results for different values of the parallel-capacitance proves the validity of the mathematical model.