Analysis of noise in quartz crystal oscillators by using slowly varying functions method

By using formal manipulation capability of commercially available symbolic calculation code, it is possible to automatically derive the characteristic polynomial describing the conditions for oscillation of a circuit. The analytical expression of the characteristic polynomial is obtained through an encapsulation process starting from the SPICE netlist description of the circuit: by using a limited number of simple transformations, the initial circuit is progressively transformed in a simplified standard form. In this method, the nonlinear component is described by its large signal admittance parameters obtained from a set of SPICE transient simulations of larger and larger amplitude. The encapsulation process involving linear and nonlinear components as well as noise sources leads to a perturbed characteristic polynomial. In the time domain, the perturbed characteristic polynomial becomes a nonlinear nonautonomous differential equation. By using an extension of the slowly varying functions method, this differential equation is transformed into a nonlinear differential system with perturbation terms as the right-hand side. Eventually, solving this system with classical algorithms allows one to obtain both amplitude and phase noise spectra of the oscillator.