Computing the Pull-In Range of Second-Order Phase Locked Loops with Tanlock and Sawtooth Phase Detectors

In previous work, we have shown that second-order phase locked loop (PLL) with sinusoidal phase detector characteristics have a separatrix cycle for a certain value of closed loop gain. It was verified that bifurcation from a stable separatrix cycle is the mechanism responsible for breaking the limit cycle associated with the PLL’s out-of lock state and the loop pulls in (phase lock). The value of the pull-in range (detuning frequency) where bifurcation occurs was determined using Galerkin method. In this paper, we analyze the pullin range for a second-order phase locked loop with tanlock and sawtooth phase detector characteristics. An algorithm is proposed for computing the PLL’s separatrix cycle. It is shown that for a given loop filter parameters (poles and zeros) and value of closed loop gain, the algorithm iteratively find the value of PLL’s detuning frequency (pull-in range) where the PLL reached the phase locked state. A comparison is drawn between the values of the pull-in range obtained for different phase detector characteristics (sinusoidal, tanlock and sawtooth). Results show that the PLL with sawtooth phase detector characteristics has the wider pull-in range followed by the tanlock and sinusoidal, respectively. Key-Words: Phase-Locked Loops, Bifurcation, Separatrix Cycle, Phase-Detector, Pull-in, Phase-lock.