An asymptotical noise model of a crystal oscillator with a nonlinear resonator

A generalized noise model is discussed of a crystal oscillator, in which the resonator equivalent parameters of the fundamental modes, overtones, and anharmonic branches are assumed to be drive level dependent (DLD) and flicker-noisy. An additive thermal noise of the resonator losses is also taken into account. A feedback amplifier is presented by the generalized nonlinear noise model, in which the flicker fluctuations of the amplitude and phase are involved along with the thermal noise. The model is designed in a frame of the asymptotical methods allowing studying the crystal oscillator nonlinear effects, such as "sleeping sickness", "second level of drive", DLDs with low and high drive levels, transients in the amplitude and phase, and others. An example is given of the model use in the low drive levels sub range.