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Mathematical Problems in Engineering
Volume 2006 (2006), Article ID 52356, 12 pages

Synchronous state in a fully connected phase-locked loop network

1Departamento de Engenharia de Telecomunicações e Controle, Escola Politécnica, Universidade de São Paulo, São Paulo 05508-900, Brazil
2Pós Graduação em Engenharia Elétrica, Universidade Presbiteriana Mackenzie, São Paulo 01302-907, Brazil

Received 19 April 2006; Revised 17 August 2006; Accepted 13 September 2006

Copyright © 2006 J. 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.


Fully connected phase-locked networks are built with all nodes exchanging phase and frequency signals. The nodes are phase-locked loops (PLLs) with slightly different free-running frequencies. The synchronous state emerges from a dynamic process with the phase interactions generating a common frequency steady state. In this work, an estimation is analytically obtained for the synchronous state in a generic N-node network. Numerical experiments complete the analysis of the fully connected network relating free-running frequencies, node gains, and propagation delays.