Slowly varying function method applied to quartz crystal oscillator transient calculation

By using an approach based on the full nonlinear Barkhausen criterion, it is possible to describe oscillator behavior under the form of a nonlinear characteristic polynomial whose coefficients are functions of the circuit components and of the oscillation amplitude. Solving the polynomial in the frequency domain leads to the steady state oscillation amplitude and frequency. In the time domain, the characteristic polynomial represents a nonlinear differential equation whose solution gives the oscillator signal transient. It is shown how symbolic manipulation capabilities of commercially available softwares can be used to automatically generate the coding of the oscillator characteristic polynomial from the SPICE description netlist. The numerical processing of such an equation in the time domain leads to unacceptable computer time because of the high quality factor of the oscillator circuits involved. Nevertheless, by using the slowly varying amplitude and phase method, it is possible to transform the initial nonlinear differential equation into a nonlinear first order differential equation system in the amplitude and phase variables. The solution of this system directly gives the designer the most relevant features of the oscillation; that is, the amplitude, phase, or frequency transients which can be accurately obtained within a short computer time by using classical numerical algorithms.