Abstract
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.
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Notes
Singular diffusion simply means that the matrix valued function \(\sigma : {\mathbb R}^k \rightarrow {\mathbb R}^{k \times k}\) in (8) is singular, i.e., for all \(X \in {\mathbb R}^k\), \(\det {(\sigma (X))} = 0\).
\(\Vert \cdot \Vert\) can be any vector norm but is taken to be the Euclidean norm in this paper.
The circuit was simulated in MATALB® using the SDEtools library [15].
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Acknowledgments
We thank Professor Andrew Heunis for bringing Theorem 1 to our attention.
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Baker, R.J.A., Leung, B. & Nielsen, C. Phase-locked loop stability based on stochastic bounds. Analog Integr Circ Sig Process 85, 323–333 (2015). https://doi.org/10.1007/s10470-015-0606-z
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DOI: https://doi.org/10.1007/s10470-015-0606-z