A Simplified Algorithm for Computing the Half-Plane Pull-In Range of a PLL

For a second-order PLL with a lead-lag (i.e., imperfect integrator) loop filter, the pull-in range Omega_p and the half-plane pull-in range Omega₂ are important parameters. Both parameters play key roles in determining if phase lock will (or can) occur once the loop is closed. For values of loop detuning omega_Delta that lie in the range 0les |omega_Delta|<Omega_p, the PLL will pull-in to phase lock from all values of initial phase error phi(0) and frequency error phidot(0). That is, for 0les|omega_Delta|<Omega_p, pull-in will occur from all points on the (phi,phidot) phase plane (hence the name pull-in range for Omega_p). For values of loop detuning omega_Delta that lie in the range Omega_ples|omega_Delta|lesOmega₂, the PLL will pull-in to phase lock from initial points that lie in a half-plane portion of the (phi,phidot) phase plane (hence the name half-plane pull-in range for Omega₂). In the literature, a numerical algorithm is described for computing Omega₂. It involves a differential equation that becomes indeterminate at a saddle point from which a separatrix must be computed by using a complicated numerical procedure. In what follows, this algorithm is simplified by transforming the differential equation to remove the indeterminacy. The modified algorithm is applied, and numerical results are given for a practical and useful range of loop parameters.