13. Phase Noise in Autonomous Oscillators

Due to its importance in circuit design, the problem of modeling phase noise in autonomous oscillators has attracted considerable attention over the last 50 years. Noteworthy phase noise studies include a spectrum model based on a feedback circuit oscillator parametrization proposed by Leeson and the impulse sensitivity function model proposed by Hajimiri and Lee. Unfortunately, even though these models offer valuable insights for oscillator design, they exhibit significant defects, such as their failure to predict a Lorentzian power spectral density for the oscillator tones affected by noise. The primary reason for this failure is that most early phase noise theories were based on linear perturbations. The first satisfactory nonlinear phase noise theory was proposed by Kaertner, with significant later refinements by Demir, Mehrothra, and Roychowdury. The key insight of this approach is that for a stable oscillator, random fluctuations in directions away from the oscillator trajectory remain contained due to the inherent trajectory stability, but fluctuations along the trajectory direction accumulate since they are not subjected to a restoring force. The model constructed by this approach relies on rather simplistic approximations, such as the neglect of amplitude fluctuations about the noiseless oscillator trajectory, but it results in a single scalar diffusion equation for the phase noise. This model can be simplified further by noticing that the phase noise evolves on time scale much slower than the rate of oscillation of the oscillator, so that an averaging method can be employed to describe the slow time-scale evolution of the phase noise. This averaging operation reveals that on a slow time-scale, the phase noise behaves like a scaled Wiener process, where the scaling parameter represents therefore the only and most important parameter of the model. This slow time-scale model can then be used to compute the oscillator autocorrelation and its power spectral density, which shows that the effect of phase noise is to smear the pure impulsive spectral lines of the noiseless oscillator into narrow Lorentzian shaped spectra, which conform with experimental observations.