Mutual synchronization of geographically separated oscillators

A control scheme for synchronizing the frequencies of geographically separated oscillators connected by communication links consists of averaging the phases received at each station from remote oscillators, comparing the result with the local phase, and applying the filtered error signal as a correction to the local oscillator frequency. The system was studied by V. E. Beneš who found a sufficient condition for the stability of the system using advanced mathematical techniques. In this paper, the stability condition is derived (for a slightly more general control scheme) using only the transfer function concept of linear systems and some properties of determinants. A practical difficulty regarding the final frequency of the oscillators is discussed and a modification of the control scheme is shown to alleviate the difficulty. Also examined are the questions of sensitivity to parameter changes, the effect of jitter noise on the performance of the system, and the effect of failure of an oscillator or transmission link.