TY - JOUR

T1 - The Gardner Problem on the Lock-In Range of Second-Order Type 2 Phase-Locked Loops

AU - Kuznetsov, Nikolay V.

AU - Lobachev, Mikhail Y.

AU - Yuldashev, Marat V.

AU - Yuldashev, Renat V.

AU - Tavazoei, Mohammad S.

PY - 2023/12/1

Y1 - 2023/12/1

N2 - Phase-locked loops (PLLs) are nonlinear automatic control circuits widely used in telecommunications, computer architecture, gyroscopes, and other applications. One of the key problems of nonlinear analysis of PLL systems has been stated by Floyd M. Gardner as being “to define exactly any unique lock-in frequency.” The lock-in range concept describes the ability of PLLs to reacquire a locked state without cycle slipping and its calculation requires nonlinear analysis. The present work analyzes a second-order type 2 phase-locked loop with a sinusoidal phase detector characteristic. Using the qualitative theory of dynamical systems and classical methods of control theory, we provide stability analysis and suggest analytical lower and upper estimates of the lock-in range based on the exact lock-in range formula for a second-order type 2 PLL with a triangular phase detector characteristic, obtained earlier. Applying phase plane analysis, an asymptotic formula for the lock-in range which refines the existing formula is obtained. The analytical formulas are compared with computer simulation and engineering estimates of the lock-in range. The comparison shows that engineering estimates can lead to cycle slipping in the corresponding PLL model and cannot provide a reliable solution for the Gardner problem, whereas the lower estimate presented in this article guarantees frequency reacquisition without cycle slipping for all parameters, which provides a solution to the Gardner problem.

AB - Phase-locked loops (PLLs) are nonlinear automatic control circuits widely used in telecommunications, computer architecture, gyroscopes, and other applications. One of the key problems of nonlinear analysis of PLL systems has been stated by Floyd M. Gardner as being “to define exactly any unique lock-in frequency.” The lock-in range concept describes the ability of PLLs to reacquire a locked state without cycle slipping and its calculation requires nonlinear analysis. The present work analyzes a second-order type 2 phase-locked loop with a sinusoidal phase detector characteristic. Using the qualitative theory of dynamical systems and classical methods of control theory, we provide stability analysis and suggest analytical lower and upper estimates of the lock-in range based on the exact lock-in range formula for a second-order type 2 PLL with a triangular phase detector characteristic, obtained earlier. Applying phase plane analysis, an asymptotic formula for the lock-in range which refines the existing formula is obtained. The analytical formulas are compared with computer simulation and engineering estimates of the lock-in range. The comparison shows that engineering estimates can lead to cycle slipping in the corresponding PLL model and cannot provide a reliable solution for the Gardner problem, whereas the lower estimate presented in this article guarantees frequency reacquisition without cycle slipping for all parameters, which provides a solution to the Gardner problem.

KW - Aerospace electronics

KW - Cycle slipping

KW - Detectors

KW - Frequency control

KW - Mathematical models

KW - PLL

KW - Phase locked loops

KW - Stability analysis

KW - Voltage-controlled oscillators

KW - gardner problem

KW - lock-in range

KW - lyapunov methods

KW - nonlinear control systems

KW - phase-locked loops

KW - stability analysis

UR - https://www.mendeley.com/catalogue/316a3169-a46f-3430-8891-e9819f6ddb3b/

U2 - 10.1109/tac.2023.3277896

DO - 10.1109/tac.2023.3277896

M3 - Article

VL - 68

SP - 7436

EP - 7450

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 12

M1 - 12

ER -