Linear Approach for Synchronous State Stability in Fully Connected PLL Networks

Synchronization is an essential feature for the use of digital systems in telecommunication networks, integrated circuits, and manufacturing automation. Formerly, master-slave (MS) architectures, with precise master clock generators sending signals to phase-locked loops (PLLs) working as slave oscillators, were considered the best solution. Nowadays, the development of wireless networks with dynamical connectivity and the increase of the size and the operation frequency of integrated circuits suggest that the distribution of clock signals could be more efficient if distributed solutions with fully connected oscillators are used. Here, fully connected networks with second-order PLLs as nodes are considered. In previous work, how the synchronous state frequency for this type of network depends on the node parameters and delays was studied and an expression for the long-term frequency was derived (Piqueira, 2006). Here, by taking the first term of the Taylor series expansion for the dynamical system description, it is shown that for a generic network with ð ‘ nodes, the synchronous state is locally asymptotically stable.