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 /spl Gamma//sub 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 /spl omega//sub d/ at which the separatrix cycle occurs.<
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