The Egan Problem on the Pull-in Range of Type 2 PLLs

• Published in: IEEE transactions on circuits and systems. II, Express briefs Vol. 68; no. 4; pp. 1467 - 1471
• Main Authors: Kuznetsov, Nikolay V., Lobachev, Mikhail Y., Yuldashev, Marat V., Yuldashev, Renat V.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: describing function; Detectors; Egan conjecture; Egan problem on the pull-in range; Frequency control; Gardner problem on the lock-in range; global stability; harmonic balance method; hold-in range; Lyapunov functions; Lyapunov methods; non-linear analysis; Phase locked loops; Phase-locked loop; PLL; Stable oscillations; Stationary state; Transfer functions; type 2; type II; Voltage-controlled oscillators
• ISSN: 1549-7747; 1558-3791
• DOI: 10.1109/TCSII.2020.3038075

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, this brief shows that the Egan conjecture is 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.