International Symposium on Nonlinear Theory and its Applications


Session Number:2-1-1



Entrainment of driven oscillators and the dynamic behavior of PLL’s

Judith Schmackers,  Wolfgang Mathis,  


Publication Date:2005/10/18

Online ISSN:2188-5079


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Since the so called entrainment phenomenon was observed as disturbance in early radio receivers, several mathematical models were developed by Moller, van ¨ der Pol and many others, however, detailed understanding was included in a paper of Andronov and Witt in 1930. At the same time, first ideas were published to use the entrainment effect in a constructive manner for synchronization devices. A related concept for the synchronization of signals was developed in electronics and AFC (automatic frequency control) around 1940 which was denoted as phase? locked loop (PLL) in the late 1950s. Although many papers about the analysis of PLLs were published during the last forty years, no clear relationship between entrainment and PLL behavior is available. In this paper it is shown that the describing equations are very different but they are included in the same class of nonlinear differential equations with limit cycles and an excitation. Furthermore it is shown that both behaviors are essentially the same.