Analysis of parametric noise in quartz crystal oscillators

From the designer´s point of view, noise sources may act on oscillator frequency through two main mechanisms: additive noise and parametric noise. The first part of this paper describes how to obtain the analytical expression of the characteristic polynomial through a successive encapsulation process starting from the SPICE netlist description of the circuit. In the time domain, the perturbed characteristic polynomial becomes a non-autonomous differential equation. In contrast to additive noise, in which the excitation appears in the differential equation as inhomogeneity, parametric noise leads to a differential equation with time-varying coefficients. By using an extension of the slowly varying functions method, this differential equation is transformed into a nonlinear differential system in amplitude and phase variables with perturbation terms at the right hand side. Solving this system allows to obtain both amplitude and phase noise spectra of the oscillator