Exact Pull-In Range and the Hidden Boundary of Global Stability for PLL With Lead-Lag Filter

Phase-locked loops (PLLs) are nonlinear automatic control circuits widely used in grid synchronization, gyroscopes, wireless communications, and other applications. One of the main tasks of a PLL is to synchronize an internal oscillator, both in frequency and phase, with a reference signal. The pull-in range concept describes the range of reference frequencies such that synchronization occurs for an arbitrary initial state, making its accurate determination essential for the reliable design and operation of PLLs. This work analyzes a classical PLL model with a lead-lag loop filter and a continuous piecewise-linear phase detector characteristic. An efficient approach for global nonlinear analysis is developed, providing counterexamples to Kapranov’s conjecture and enabling the rigorous closed-form derivation of the exact pull-in range formula, along with its asymptotics. The derived analytical formulas rectify inaccuracies from previous works and significantly refine existing engineering approximations. Within the framework of the theory of hidden oscillations, this approach provides a complete solution to the problem of determining the boundary of global stability and revealing its hidden parts corresponding to the nonlocal birth of hidden oscillations. To support practical implementation, bifurcation diagrams for the pull-in range calculation are presented, accompanied by code for their construction.