Analysis of the pull‐in range of phase‐locked loops by the galerkin procedure

Numerous ways to analyze the pull-in range of the phase-locked system have been proposed, These include harmonic balance, the phase plane method and numerical analysis. In most of these methods, however, it was difficult to determine the actual pull-in range with high accuracy. This is because of the limitations in the applicable phase detector characteristics and loop filters, or the need to perform a complex numerical analysis. This paper proposes Galerkin's procedure to determine the pull-in range of the phase-locked system, demonstrating that the pull-in range can be determined with a high accuracy in a relatively simple way. Galerkin's procedure is a method of numerical analysis which can determine the periodic function for the nonlinear ordinary differential equation. By applying the procedure, the periodic solution of the second kind, corresponding to the out-of-lock condition of the system, is determined. From the critical input-output detuning for the solution to exist, the pull-in range is determined. This method can be applied to any phase-detector characteristics and loop filters, and by taking higher harmonics into consideration, a highly accurate solution can be determined. In this paper, as examples, the pull-in ranges are determined for a wide range of natural angular frequency and damping factor, for the phase detector characteristics which are sinusoidal, triangular, tanlock and sawtooth with fast jitter.