Quartz oscillators: deriving oscillation condition by symbolic calculus

This paper presents the method used to derive the oscillation condition by using symbolic calculus. The program is based on the full nonlinear Barkhausen criterion method. The behaviour of an oscillator is described by a complex polynomial called the characteristic polynomial. This polynomial enables us to calculate the steady state features of the oscillation as well as the differential equation for transient analysis in the time domain. The literal determination of this characteristic polynomial involves lengthy algebraic calculations and cannot be done by hand as soon as the electronic oscillator circuit involves too many components. We recently developed a formal calculus program allowing to automatically obtain all necessary equations for oscillation analysis. We propose new methods to calculate them in an optimal form.