The pull-in range and a counterexample to the Egan conjecture for the fourth-order type 2 analog PLL

Phase-locked loop is classical nonlinear control system for frequency and phase synchronization in electrical circuits. According to the Egan conjecture, the type 2 loops have an infinitely large pull-in range. This article reconsider the pull-in problem for the fourth order type 2 PLLs. The global stability domain is estimated by applying Lyapunov direct method for the cylindrical phase space. An analytical estimate of the pull-in range of the fourth-order PLL is presented for the first time in the literature. Parameters violating the pull-in conditions are determined by computer simulation. The results have revealed that stable oscillations may develop in the higher-order type 2 PLL in steady-state which is a clear counterexample to the Egan’s conjecture.