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False Lock and Bifurcation in the Phase Locked Loop

John Stensby
SIAM Journal on Applied Mathematics
Vol. 47, No. 6 (Dec., 1987), pp. 1177-1184
Stable URL: http://www.jstor.org/stable/2101669
Page Count: 8
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False Lock and Bifurcation in the Phase Locked Loop
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Abstract

New results are given on the phenomenon of false lock in phase locked loops (PLL). First, it is proved that for small values of closed loop gain δ, there is only one function ωf(δ) which represents the frequency error in a false locked Type I PLL. That is, there is only one false lock equilibrium point for δ in a neighborhood of the origin. Furthermore, this equilibrium point is stable. The proof establishes that bifurcation of periodic solutions does not occur at δ = 0 in the nonlinear differential equation which describes the classical Type I PLL. This differs from the currently accepted theory presented in the literature. The existing theory claims that a multiplicity of such frequency error functions can exist for δ in a neighborhood of the origin and that each function corresponds to a point of false lock equilibrium. Furthermore, it is erroneously claimed that these (one or more) equilibrium points may be unstable. Next, a perturbation-based method for analyzing false lock in Type I PLLs is given. The technique expresses ωf(δ) and the periodic solution of the equation describing the false locked loop as power series in δ. The algorithm is applied to the classical second order loop containing an imperfect integrator.

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