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 no nlinear characteristic polynomial whose coefficients are functions of the circuit components and of the oscillation amplitude. Solving the p olynomial in the frequency domain leads to the steady state oscillatio n amplitude and frequency, In the time domain, the characteristic poly nomial represents a nonlinear differential equation whose solution giv es the oscillator signal transient, It is shown how symbolic manipulat ion capabilities of commercially available softwares can be used to au tomatically generate the coding of the oscillator characteristic polyn omial from the SPICE description netlist. The numerical processing of such an equation in the time domain leads to unacceptable computer tim e because of the high quality factor of the oscillator circuits involv ed, Nevertheless, by using the slowly varying amplitude and phase meth od, it is possible to transform the initial nonlinear differential equ ation into a nonlinear first order differential equation system in the amplitude and phase variables, The solution of this system directly g ives the designer the most relevant features of the oscillation; that is, the amplitude, phase, or frequency transients which can be accurat ely obtained within a short computer time by using classical numerical algorithms.