SADDLE NODE BIFURCATION AT A NONHYPERBOLIC LIMIT-CYCLE IN A PHASE-LOCKED LOOP

New results are given on the phenomenon of false lock in second-order, Type I phase locked loops (PLLs) with a constant frequency reference of omega(i) and a voltage controlled oscillator (VCO) quiescent frequency of omega0. This loop has only one false lock state whose frequency error approaches omega(i)-omega0 as loop gain delta approaches zero, and this false lock state is stable. This false lock state corresponds to a stable, hyperbolic limit cycle X(s)(t; delta) of the nonlinear equation describing the loop. As gain delta is increased from a value of zero, it is shown that a value delta1 can be reached where X(s) becomes semi-stable and nonhyperbolic. Furthermore, saddle node bifurcation occurs at delta1, and a second limit cycle X(u)(t; delta) branches from this bifurcation point. Limit cycle X(u) is unstable, and it corresponds to an unstable false lock state of the PLL. Furthermore, X(u) can be continued as a function of gain on an interval delta2 < delta less-than-or-equal-to delta1, for some delta2 > 0. Finally, X(s) and X(u) do not exist for delta > delta1. Two numerical algorithms are given to analyse the false locked PLL under consideration. The first is useful for computing the above-mentioned limit cycles and the bifurcation point delta1. The second algorithm can calculate a Poincare map and its derivative which are useful in studying the saddle node bifurcation which occurs at delta1. Also given is a detailed decription of a laboratory experiment which was used to substantiate the theory and numerical techniques. The qualitative theory, numerical method and laboratory procedure are applied to a simple example. The numerical and empirical results are shown to be in close agreement.