Control theory of mutual synchronization in networks of delay-coupled phase-locked loops

This post is related to a manuscript that we have submitted for publication to IEEE Access and I will extend this summary here step by step.

Introduction

In classical phase-locked loop (PLL) theory concepts like the gain and phase margin  and application of the Nyquist criterion help to analyze the stability of a system. These are accessed from the PLLs transfer function in Laplace space and analysis at the critical point for zero real part of the Laplace variable. Hence, the stability of the system is studied where it changes from stable to unstable, in dynamical systems theory we call this the marginal stable case. At this point, the above mentioned concepts rely on the variation of the complex part of the Laplace variable, the frequency. From this, Bode and Nyquist plots are constructed and gain and phase margins defined.
This is straightforward for the case of the entrainment of a PLL by a periodic reference signal. However, in the literature on classical PLL theory one aspect is often missing — the effect of the synchronized or entrained state itself. Here, I will discuss this topic and show how the transfer functions of PLLs and networks of PLLs change because of it.

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