Verifying start-up failures in coupled ring oscillators in presence of variability using predictive global optimization

This paper describes a simulation-based approach to establish whether a ring-oscillator always converges to the correct mode of operation regardless of its initial conditions and variability conditions. The verification is performed using a predictive global optimization algorithm that looks for a problematic initial state from a discretized state space. The algorithm explores the initial states that can maximize the settling time for the oscillator to reach its final steady state. If any of these initial states visited during the search is found exhibiting false oscillation behaviors for certain variability conditions, the initial state is reported as problematic. On the other hand, if the initial state with the globally maximum settling time is found without discovering such problematic states, the oscillator is reported free of start-up failures. It can be shown that despite the finite number of initial state candidates considered and finite number of Monte-Carlo samples to model variability, the proposed algorithm can verify the oscillator to a prescribed confidence level. Demonstrated on various even-stage differential ring oscillators, the algorithm was able to validate the circuit for 99% yield with 99.9% confidence level by evaluating 7~60 initial states each with 1,000 Monte-Carlo samples. To our knowledge, this is the first algorithm ever reported to address start-up failures with variability.