Abstract
This brief studies how mutual synchronization of oscillators can be achieved for cross-coupling time delays much larger than the period of the oscillations. Using the closed loop transfer function for a system of two mutually delay-coupled phase-locked loops (PLLs) and applying the Nyquist stability criterion, the critical time delay for which stable in- and anti-phase synchronized states become unstable is calculated. The analysis reveals the range of feed-forward loop gains for a given time delay value so that stable in- or anti-phase synchronized states can exist. These theoretical predictions are then verified by measurements with PLLs operating at 24GHz and for cross-coupling time delays ranging from the nano- to the microseconds domain. Such delays are equivalent to coupling at distances of up to 500m. The experimental results show a good agreement with the theoretical predictions. Hence, this brief shows how to setup a network of mutually delay-coupled PLLs and achieve stable synchronized states for a given time delay.
Published Version
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