Cypress Semiconductor Internal Correspondence

A quartz crystal oscillator can be thought ofas a resonator connected across an amplifier considered asa nonlinear dipole the impedance of which depends on theamplitude of the current that flows through it. The non-linear amplifier resistance and reactance are obtained byusing a time domain electrical simulator like SPICE (Simu-lation Program with Integrated Circuit Emphasis): the res-onator is replaced with a sinusoidal current source of samefrequency and a set of transient analyses is performed bygiving the current source a larger and larger amplitude. AFourier analysis of the steady-state voltage across the dipo-lar amplifier is performed to calculate both real and imag-inary parts of the dipolar impedance as a function of thecurrent amplitude. From these curves, it is then possibleto accurately calculate the oscillation amplitude and fre-quency without having to perform unacceptably long tran-sient analysis needed by a direct oscillator closed loop sim-ulation. This method implemented in the Analyse Dipo-laire des Oscillateurs à Quartz or Quartz Crystal Oscil-lators Dipolar Analysis (ADOQ) program calculates theoscillation start-up condition, the oscillation steady-statefeatures (oscillation amplitude and frequency), and the os-cillator sensitivity to various parameters. The oscillationnonlinear differential equation is solved by using the slowlyvarying function method so that the program quickly andaccurately calculates the current amplitude and frequencytransients. Measurements performed on an actual amplifiershow a very good agreement with the results obtained bythe simulation program.