Revisiting stability analysis of Phase-Locked Loops with the Popov Stability Criterion

This paper revisits the problem of stability analysis of Phase-Locked Loops (PLLs), focusing specifically on stability conditions derived using the Popov Stability Criterion. A new form of the Popov based frequency domain criterion is derived in which the size of the stability region, the PLL locking range, appears independently of the loop transfer-function. This enables one to maximize the stability region graphically and directly on the Popov plot, rather than iteratively. Various numerical and analytic results available in the literature are shown to be particular cases of the proposed new stability test. It is also shown that for PLLs of type $r$, in which $r$ denotes the number of integrators in the loop, it is not possible to achieve full locking range if $r$ is larger or equal than three.