Automatic formal derivation of the oscillation condition

The behavior of a quartz crystal oscillator can be described by a nonlinear characteristic polynomial whose coefficients are function of the circuit parameters. Solving the polynomial in the frequency domain leads to the steady state oscillation amplitude and frequency. In the time domain, it gives the oscillator signal transient. Deriving the characteristic polynomial from the circuit description involves lengthy and tedious algebraic calculations if they are performed by hand. They may be now performed by using the symbolic manipulation capabilities of commercially available softwares. However, symbolic analysis using brute force method inevitably leads to an explosion of terms in equations. The paper will present a fully automatic method for generating the coding of an oscillator characteristic polynomial directly from the SPICE description netlist. The code thus generated is eventually compiled and takes place in an oscillator library. Then it is linked with the numerical main program that solves the polynomials. Solutions to overcome problems related to automatic symbolic calculations are presented and discussed. It is shown that the method used leads to concise and efficient code.