A Fractional-Order Phase-Locked Loop with Time-Delay and Its Hopf Bifurcation

A fractional-order phase-locked loop(PLL) with a time-delay is firstly proposed on the basis of the fact that a capacitor has memory.The existence of Hopf bifurcation of the fractional-order PLL with a time-delay is investigated by studying the root location of the characteristic equation,and the bifurcated periodic solution and its stability are studied simply by using "pseudo-oscillator analysis".The results are checked by numerical simulation.It is found that the fractional-order PLL with a time-delay reduces the locking time,and it minimizes the amplitude of the bifurcated periodic solution when the order is properly chosen.