Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 364084, 13 pages
doi:10.1155/2008/364084
Research Article
Linear Approach for Synchronous State Stability in Fully Connected PLL Networks
1Escola Politécnica, Universidade de São Paulo, Avenida Professor Luciano Gualberto, Travessa 3, no. 158, 05508-900 São Paulo, SP, Brazil
2Escola de Engenharia, Universidade Presbiteriana Mackenzie, Rua da Consolacao, 896 São Paulo, SP 01302-907, Brazil
Received 10 October 2007; Accepted 7 March 2008
Academic Editor: Jerzy Warminski
Copyright © 2008 José R. C. Piqueira et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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 N nodes, the synchronous state is
locally asymptotically stable.