A probabilistic analysis of the crystal oscillator behavior at low drive levels Yuriy S. Shmaliy a and ⋆ R´ emi Brendel b a Guanajuato University, FIMEE, 36730, Salamanca, Gto, Mexico, b ⋆ LPMO du CNRS, l’Universit´ e de Franche-Comt´ e-Besan¸con, 25044, Besan¸ con, Cedex, France ABSTRACT The paper discusses a probabilistic model of a crystal oscillator at low drive levels where the noise intensity is comparable with the oscillation amplitude. The stationary probability density of the oscillations envelope is derived and investigated for the nonlinear resonator loses. A stochastic explanation is given for the well-known phenomenon termed sleeping sickness associated with losing a facility of self-excitation by a crystal oscillator after a long storage without a power supply. It is shown that, with low drive levels leading to an insufficient feedback, a crystal oscillator generates the noise-induced oscillations rather than it absolutely ”falls in sleep”. Keywords: Oscillator, nonlinear crystal resonator, noise, drive level dependence 1. INTRODUCTION Crystal oscillators are now used widely in different applications whenever a low cost precision and accurate fre- quency source is needed. When stored for a few days-decades without connection to a power supply, however, the crystals often develop a condition known as sleeping sickness. 1–7 Their facility for self-oscillation then becomes strongly attenuated and may even disappear. The origin of this phenomenon is not well understood, although it is known to be associated with increased resonator losses caused, in particular, by a surface contamination. An overview of possible courses and observable effects associated with this phenomenon was recently given in. 8 To circumvent, provision often has to be made for an increased driving amplitude at switch-on. In this paper * we present a probabilistic model of a crystal oscillator with low drive levels. Based upon, we give a stochastic explanation of sleeping sickness, viewing it as a nonlinear phenomenon self-induced by the Johnson noise in the circuit with an insufficient feedback. As we show, the picture we propose seems to fit principle features of sleeping sickness reported in the literature. We consider, in particular, what happens at very low drives where the natural frequency of the crystal will be least “pulled” by the external circuit, focussing our attention on physics close to the noise floor where the amplitude of the oscillatory drive is comparable with the internal noise intensity. 2. A STOCHASTIC MODEL OF A CRYSTAL OSCILLATOR In, 6 it was shown experimentally that the following approximation fits reasonably well the drive level dependence (DLD) of the crystal resonator losses R 1 , measuring from the noise floor and gradually increasing the piezoelectric current up to normal drive levels, for which the losses are R 10 . In the fractional term δ R (I r )= ΔR 1 (I r ) R 10 , where ΔR 1 is a DLD increment of the losses, such a DLD model is δ R (I r )= γ R I n r , (1) * The material was discussed at the 2005 Int. Conf. on Precision Oscillations in Electronics and Optics (POEO): Theory and Applications, 15-17 Sept., Yalta, Crimea, Ukraine. 9 Second International Conference on Advanced Optoelectronics and Lasers, edited by Igor A. Sukhoivanov, Vasily A. Svich, Yuriy S. Shmaliy, Proc. of SPIE Vol. 7009, 70091H, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.795284 Proc. of SPIE Vol. 7009 70091H-1 2008 SPIE Digital Library -- Subscriber Archive Copy