Noise conversion in crystal oscillators

This paper discusses the noise performance of crystal oscillators. It is assumed that the crystal resonator motional losses, inductance, and capacity as well as its static capacity are flicker noisy. The Johnson additive thermal noise of the motional losses is also taken into account. To convert the intrinsic noise sources to the amplitude and phase fluctuations, the crystal resonator and the feedback amplifier are performed by noisy impedances. The crystal oscillator is reduced to the closed loop of the resonator impedance and the feedback amplifier impedance. The transformation coefficients are derived to convert the resonator and amplifier amplitude and phase power spectral density functions to those of the oscillator amplitude and phase. Noise performance of a crystal oscillator with different excitation angles of a resonator is studied in detail.