  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.
