Solution of the Gardner problem on the lock-in range of phase-locked loop

TL;DR

This paper provides a rigorous mathematical solution to Gardner's long-standing problem of precisely defining the lock-in frequency and range in phase-locked loops, with practical computation methods demonstrated on classical PLL examples.

Contribution

It offers the first effective, mathematically rigorous method to define and compute the lock-in range and frequency in PLLs, addressing a problem posed by Gardner decades ago.

Findings

Derived explicit formulas for lock-in range in classical PLLs

Validated the approach with examples on second-order PLLs

Results applicable to various PLL-based communication systems

Abstract

The lock-in frequency and lock-in range concepts were introduced in 1966 by Floyd Gardner to describe the frequency differences of phase-locked loop based circuit for which the loop can acquire lock within one beat, i.e. without cycle slipping. These concepts became popular among engineering community and were given in various engineering publications. However rigorous mathematical explanations these concepts turned out to be a challenging task, thus, in the 2nd edition of Gardner's well-known work, Phaselock Techniques, he wrote that "despite its vague reality, lock-in range is a useful concept" and posed the problem "to define exactly any unique lock-in frequency". In this paper an effective solution for Gardner's problem on the unique definition of the lock-in frequency and lock-in range is discussed. The lock-in range and lock-in frequency computation is explained on the example of classical second-order PLL with lead-lag and active proportional-integral filters. The obtained results can also be used for the lock-in range computation of such PLL-based circuits as two-phase PLL, two-phase Costas loop, BPSK Costas loop, and optical Costas loop, used in intersatellite communication.