Abstract
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.
Highlights
P HASE-LOCKED LOOPS (PLLs) are classical nonlinear control systems for phase and frequency synchronization in electrical circuits [1]–[3]
The present paper introduces a counterexample to the Egan conjecture: a type 2 PLL with an infinite hold-in range and a persistent oscillation, which indicates the emptiness of the pull-in range
For type 2 PLLs the Egan conjecture states that an infinite hold-in range implies an infinite pull-in range
Summary
P HASE-LOCKED LOOPS (PLLs) are classical nonlinear control systems for phase and frequency synchronization in electrical circuits [1]–[3]. Viterbi applied the phaseplane analysis and stated that the second-order type 2 PLL models have infinite (theoretically) hold-in and pull-in ranges for any loop parameters [8, p.12], [1]. The observed stable oscillation and the Egan conjecture naturally pose the following 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. For such a class of PLLs the global stability conditions can be obtained by straightforward linear methods. Using the direct Lyapunov method for the cylindrical phase space we obtain a sufficient condition of the pull-in range infiniteness, thereby providing a solution to the Egan problem
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