Exact lock-in range for classical phase-locked loops

Abstract

Phase-locked loops (PLLs) are are widely used in various applications: Wireless communications, GPS navigation, gyroscope systems, computer architectures, electrical grids, and others. PLLs are inherently non-linear, but in engineering practice, they are actually designed and analyzed mostly using linear methods. Recent development in the manufacturing of electronics has led to ever higher operating frequencies and, hence, to more stringent requirements for the design of PLLs. This dissertation is devoted to the study of phase-locked loops, in particular to their synchronization properties. The ability of the phase-locked loop to synchronize fast without cycle slipping is characterized by the loop’s lock-in range. The problem of determining the lock-in range is called the Gardner problem for lock-in range, named after IEEE Fellow F. M. Gardner, who formulated the problem in the 1960s. Mathematically rigorous formulation of the lock-in range by N. Kuznetsov, however, is only a few years old, and the first approaches to improve the estimates of the lock-in range using non-linear methods were proposed by K.D. Aleksandrov in his doctoral dissertation. This work furthers the mathematically rigorous study of the lock-in range and is devoted to the exact calculation of the lock-in in range for a classical phase-locked loop with active proportionally integrating and lead-lag filters. For phase space, phase-locked loop models with piecewise-linear phase detectors characteristized by an exact lock-in range is obtained for both the considered filter types. For the phase-space phase-locked loop model with an active proportionally integrating filter and tangential that is characteristic of a phase detector, the lock-in range is proven to be infinite. All theorems have strict mathematical proof and have been confirmed by numeric simulation. Keywords: PLL, phase-locked loops, lock-in range, Gardner problem, exact lock-in range