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For a second-order PLL with a lead-lag (i.e., imperfect integrator) loop filter, the pull-in range Omegap 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 omegaDelta that lie in the range 0les |omegaDelta|<Omegap, 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|omegaDelta|<Omegap, pull-in will occur from all points on the (phi,phidot) phase plane (hence the name pull-in range for Omegap). For values of loop detuning omegaDelta that lie in the range Omegaples|omegaDelta|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.