Exact analytical equations for predicting nonlinear phase errors and jitter in ring oscillators

In this paper, we present a simple analytical equation for capturing phase errors in 3-stage ring oscillators. The model, based on a simple but useful idealization of the ring oscillator, is provably exact for small noise perturbations. Despite its simplicity and purely analytical form, our model correctly captures the time-dependent sensitivity of oscillator phase to external perturbations. It is thus well suited for estimating both qualitative and quantitative features of ring oscillator phase response to internal noises, as well as to power, ground and substrate interference. The nonlinear nature of the model makes it suitable for predicting injection locking as well. Comparisons of the new model with existing phase models are provided, and its application for correct prediction of supply-noise induced jitter in PLLs, as well as for capturing injection locking, demonstrated. Requiring knowledge only of the amplitude and frequency of the oscillator, the model is ideally suited for early design exploration at the system and circuit levels. An interesting feature of the analytical equation is its strong connection with the number 1.618034 (the Golden Mean), celebrated since ancient times for its significance in a variety of mathematical, aesthetic and scientific disciplines.