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 ana lytical expression of the characteristic polynomial is obtained through an encapsulation process starting from the SPICE netlist description of the ci rcuit: by using a limited number of simple transformations, the initial cir cuit is progressively transformed in a simplified standard form. In this me thod, the nonlinear component is described by its large signal admittance p arameters 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 pol ynomial. In the time domain, the perturbed characteristic polynomial become s a nonlinear nonautonomous differential equation. By using an extension of the slowly varying functions method, this differential equation is transfo rmed into a nonlinear differential system with perturbation terms as the ri ght-hand side. Eventually solving this system with classical algorithms all ows one to obtain both amplitude and phase noise spectra of the oscillator.