The Egan problem on the pull-in range of type 2 PLLs
Kuznetsov, Nikolay V.; Lobachev, Mikhail Y.; Yuldashev, Marat V.; Yuldashev, Renat V. (2020). The Egan problem on the pull-in range of type 2 PLLs. IEEE Transactions on Circuits and Systems II: Express Briefs, Early Access. DOI: 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