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

Abstract

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.

The Analytic Determination of the PPV for Second-Order Oscillators Via Time-Varying Eigenvalues

“…Consider an arbitrary second-order LTV system of the form (6) where and are the state variables of the system, and , , , and are scalar functions of the time variable . According to [9], to find a complete set of linearly independent solutions for system (6), it has to be transformed first into a diagonal system using the following coordinate transformations:…”

“…Some research work has been undertaken to obtain analytical expressions for the PPV associated to some types of oscillators. Many of the ideas presented in [3] and [5] were used successfully in the analytical determination of the PPV for simplified models of ring and inductance-capacitance oscillators in [6] and [7]. In [6], the variational system obtained for the simplified model of the ring oscillator could be solved directly because it was linear time invariant (LTI).…”

“…Many of the ideas presented in [3] and [5] were used successfully in the analytical determination of the PPV for simplified models of ring and inductance-capacitance oscillators in [6] and [7]. In [6], the variational system obtained for the simplified model of the ring oscillator could be solved directly because it was linear time invariant (LTI). In [7], harmonic balance methods were used to obtain a very accurate approximation of the PPV assuming that the dc term and the first positive and the first negative Fourier coefficients of this vector would suffice to represent it.…”

An efficient, fully nonlinear, variability-aware non-monte-carlo yield estimation procedure with applications to SRAM cells and ring oscillators

“…Because a line in m-D space can always be expressed in the form of parametric equations, equivalent to m − 1 linear constraints, we augment the scalar equation with m parametric equations (6) of the line, with one more unknown s introduced. In equation (6), p 0 is the coordinate of a point the constraint line (parameter axis in this case) goes through, u is the unit vector denoting the direction of the line.…”

“…To align all the points evenly on the boundary surface, we again apply the line constraint on the solution using equation (6) such that the line goes through the centroid of the points got from previous iterations, and is perpendicular to the manifold we have already constituted. Then line search with Newton-Raphson method can be applied to solve the augmented system.…”