The separatrix cycle for a class of phase-locked loops

The vector field model for a class of second-order, PLLs is shown to have rotational properties which imply the existence of a separatrix cycle. The external stability of this separatrix cycle is shown to be an indicator of the type of bifurcation which terminates the limit cycle Γ_s associated with the PLL´s stable false lock state. A formula is given for determining the separatrix cycle´s stability, and this leads to the conclusion that the cycle is externally stable for sufficiently small valves of closed loop gain. A novel algorithm is presented for computing the PLL´s separatrix cycle. For a given loop filter and valve of closed loop gain, the algorithm uses an iterative approach to find the valve of the PLL´s detuning parameter ω_d at which the separatrix cycle occurs