A Method for Calculating Phase-Locked Loop Performance Near Threshold

Abstract
Although several methods have evolved over the past few years for analyzing the near-threshold behavior of phaselocked devices, none so far have been able conveniently to integrate all desirable aspects of an engineering method which requires both accuracy and simplicity of formulation. This paper contains a method conceptually as simple as the linear and quasi-linear techniques, but with the added advantage that it is much more accurate near loop threshold-an accuracy, by the way, which increases as the order of the loop increases and as the loop bandwidth decreases. For the simple first-order loop, which is the one this method predicts with least fidelity, the result follows the known exact carrier tracking result with a maximum error of less than 0.08 radians in the rms phase error at system threshold. Compared to actual measurements of carrier tracking secondorder loop performance, the method provided excellent agreement, nominally within 0.05 radians at a\sigma^{2} = 1.