TY - JOUR

T1 - Exact Pull-In Range and the Hidden Boundary of Global Stability for PLL With Lead-Lag Filter

AU - Кузнецов, Николай Владимирович

AU - Лобачев, Михаил Юрьевич

PY - 2025

Y1 - 2025

N2 - 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.

AB - 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.

KW - Cycle of the second kind

KW - Kapranov conjecture

KW - Lyapunov methods

KW - global stability analysis

KW - heteroclinic trajectory

KW - lead-lag filter

KW - nonlinear analysis

KW - nonlinear control systems

KW - phase-locked loop (PLL)

KW - pull-in range

KW - cycle of the second kind

KW - Phase-locked loop (PLL)

UR - https://ieeexplore.ieee.org/document/11015452

UR - https://www.mendeley.com/catalogue/2a59c33e-397c-3e41-bd8e-f3e04fe1853a/

U2 - 10.1109/access.2025.3573693

DO - 10.1109/access.2025.3573693

M3 - Article

VL - 13

SP - 94785

EP - 94821

JO - IEEE Access

JF - IEEE Access

SN - 2169-3536

ER -