The half-plane pull-in range of a second-order phase-locked loop

The pull-in range Ωp of a second-order, Type I phase-locked loop (PLL) is defined as the maximum value of loop detuning ωos for which pull-in occurs from anywhere on the PLLʹs phase plane. That is, pull-in is guaranteed from anywhere on the phase plane if |ωos| < Ωp. Simple approximations are available for computing Ωp. The concept is expanded here, and a definition is given for the PLLʹs half-plane pull-in range Ω2. Simply stated, pull-in is guaranteed from anywhere on the phase planeʹs lower-half if 0 < ωos < Ω2. Unlike the parameter Ωp, a simple approximation for Ω2 is not available. However, a Galerkin based algorithm is presented for computing the PLLʹs half-plane pull-in range Ω2, and it is applied to a simple example.