Phase-locked loop stability based on stochastic bounds

In this paper we study the stability of a phase-locked loop (PLL) in the presence of noise. We represent the noise as Brownian motion and model the circuit as a nonlinear stochastic differential equation, with the noise lumped at the phase detector input. We show that for the PLL, the theory of asymptotics of singular diffusions can be applied and we use this theory to develop a new figure of merit which we call a stability margin. The stability margin provides easily computable bounds on the acceptable noise levels for which stability is guaranteed. Through simulation, we show that such a sufficient bound provides a realistic prediction for PLL stability.