Algebraic Derivation of the Oscillation Condition of High–Q Quartz Crystal Oscillators

Abstract: A symbolic–numeric method to simulate a high–Q quartz crystal oscillator is derived in this paper. The proposed method can take into account the isochronism defect of the resonator when high precision simulations are needed. The oscillator circuit is expressed under a canonical 5–terminal network which can be easily handled. The oscillator behavior is then described under the form of a nonlinear characteristic polynomial whose coefficients are functions of the circuit components and of the oscillation amplitude. The steady state oscillation amplitude and frequency are determined by solving the polynomial in the frequency domain. Moreover, if tolerance is assigned to each circuit component, efficient worst case analysis can be performed