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Mathematical Problems in Engineering
Volume 2009, Article ID 256765, 13 pages
Research Article

Fully Connected PLL Networks: How Filter Determines the Number of Nodes

Departamento de Engenharia de Telecomunições e Controle, Escola Politécnica da Universidade de São Paulo, Avenida Prof. Luciano Gualberto, travessa 3, n. 158, 05508-900 São Paulo, SP, Brazil

Received 5 October 2008; Revised 30 January 2009; Accepted 2 March 2009

Academic Editor: Jerzy Warminski

Copyright © 2009 Átila Madureira Bueno 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 plays an important role in telecommunication systems, integrated circuits, and automation systems. Formerly, the masterslave synchronization strategy was used in the great majority of cases due to its reliability and simplicity. Recently, with the wireless networks development, and with the increase of the operation frequency of integrated circuits, the decentralized clock distribution strategies are gaining importance. Consequently, fully connected clock distribution systems with nodes composed of phase-locked loops (PLLs) appear as a convenient engineering solution. In this work, the stability of the synchronous state of these networks is studied in two relevant situations: when the node filters are first-order lag-lead low-pass or when the node filters are second-order low-pass. For first- order filters, the synchronous state of the network shows to be stable for any number of nodes. For second-order filter, there is a superior limit for the number of nodes, depending on the PLL parameters.