The Egan problem on the pull-in range of type 2 PLLs
Kuznetsov, N. V., Lobachev, M. Y., Yuldashev, M. V., & Yuldashev, R. V. (2021). The Egan problem on the pull-in range of type 2 PLLs. IEEE Transactions on Circuits and Systems II: Express Briefs, 68(4), 1467-1471. https://doi.org/10.1109/tcsii.2020.3038075
© 2020 the Authors
In 1981, famous engineer William F. Egan conjectured that a higher-order type 2 PLL with an infinite hold-in range also has an infinite pull-in range, and supported his conjecture with some third-order PLL implementations. Although it is known that for the second-order type 2 PLLs the hold-in range and the pull-in range are both infinite, the present paper shows that the Egan conjecture may be not valid in general. We provide an implementation of the third-order type 2 PLL, which has an infinite hold-in range and experiences stable oscillations. This implementation and the Egan conjecture naturally pose a problem, which we will call the Egan problem: to determine a class of type 2 PLLs for which an infinite hold-in range implies an infinite pull-in range. Using the direct Lyapunov method for the cylindrical phase space we suggest a sufficient condition of the pull-in range infiniteness, which provides a solution to the Egan problem.
PublisherInstitute of Electrical and Electronics Engineers (IEEE)
phase-locked loop PLL type II type 2 hold-in range Egan conjecture Egan problem on the pull-in range Gardner problem on the lock-in range Lyapunov functions nonlinear analysis global stability describing function harmonic balance method säätöteoria differentiaaliyhtälöt värähtelyt elektroniset piirit
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Additional information about fundingThe work is supported by the Russian Science Foundation (project 19-41-02002).
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