Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 364084, 13 pages
Research Article

Linear Approach for Synchronous State Stability in Fully Connected PLL Networks

José R. C. Piqueira,1 Maurízio Q. de Oliveira,1 and Luiz H. A. Monteiro1,2

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.


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.