An exact formula for the half-plane pull-in range of a PLL

A second-order phase-lock loop (PLL) that is based on a triangular-characteristic phase detector and imperfect-integrator loop filter is found in many applications where simplicity and economics are major considerations. For many of these applications, digital-logic-compatible reference and VCO signals are used, an exclusive-OR gate implements the phase detector, and the loop filter is constructed from passive components. When designing these loops, the half-plane pull-in range Ω2 is of interest. Until now, this important loop parameter could only be calculated by using a computer-based technique that numerically integrated the nonlinear differential equation that describes the PLL model. This requirement/limitation is removed here by the development of an exact closed-form formula for Ω2, the main contribution of this paper. More generally, the value of Ω2 is dependent on the PLL phase detector characteristic that is used, be it triangular, sinusoidal, or something else. With regard to the value of Ω2 produced, a comparison is given of two PLLs, both described by the same linear model so that the comparison is meaningful. The first PLL is based on a triangular-characteristic phase detector; the second loop is based on a sinusoidal phase detector.