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

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5 Author(s)
Brendel, Remi ; CNRS, Univ. de Franche-Comte, Besancon, France ; Ratier, N. ; Couteleau, L. ; Marianneau, Gilles
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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.

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Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on  (Volume:46 ,  Issue: 2 )