Piecewise-linear analysis of the pull-in range for second-order PLLs

To the memory of Dr. Charles R. Cahn (1929-2021).In the present work, we consider a second-order phase-locked loop with a lead-lag loop filter and a piecewise-linear phase detector characteristic. Developing ideas of Charles R. Cahn, we offer an effective analytical-numerical method for determination of the global stability boundary and the pull-in range. Due to piecewise-linearity of the corresponding nonlinear system, we can analytically integrate the trajectories on the linear segments and provide necessary and sufficient conditions of global stability. The results are compared with the Cahn pull-in range estimate, which relies on approximations.