Study of the oscillation condition of quartz oscillators by gyrator transformation

The calculation of the oscillation condition is one of the main points of oscillator analysis. Its determination in finite terms allows one to calculate the steady state amplitude and frequency of the oscillator. Symbolic solutions provide an additional insight into the behavior of the circuit. As an example the sensitivity of the oscillator to parameter change can be expressed in an exact form. Numerical solutions are not as helpful as symbolic solutions in the design stage. We present a technique, based on the gyrator transformation, to set up the nonlinear equation network in a form suitable to be solved with analytical methods. We develop a symbolic program based on this technique. As an example, the symbolic program is applied to compute the exact expression of the steady state frequency and amplitude of the Van der Pol oscillator and the Colpitts oscillator