1/F Noise in crystal resonators

It has been found that factional frequency 1/f noise in crystal oscillators is generated by the energy dissipated in the resonator. Earlier investigations found the same origin of 1/f noise also in quartz resonators. In the present contribution we shall show that the losses connected with the current flow through the resonator generate frequency fluctuations which are changed into 1/f phase fluctuations due to the transportation delay. Further we shall prove that the magnitude of the corresponding Power Spectral Density (PSD) is independent of the resonator resonance frequency, f_o, since it is function of the material constant Qf_o. In addition, we will present a theoretical PSD model of the phase noise at the output of the resonator with a discussion on the 1/f² and 1/f³ slopes. Finally, we shall investigate contribution of the white noise on the measurement and shall discuss the Allan variance for the resonator and an eventual crystal oscillator.