Harmonic balance analysis of pull-in range and oscillatory behavior of third-order type 2 analog PLLs

Abstract

The most important design parameters of each phase-locked loop (PLL) are the local and global stability properties, and the pull-in range. To extend the pull-in range, engineers often use type 2 PLLs. However, the engineering design relies on approximations which prevent a full exploitation of the benefits of type 2 PLLs. Using an exact mathematical model and relying on a rigorous mathematical thinking this problem is revisited here and the stability and pull-in properties of the third-order type 2 analog PLLs are determined. Both the local and global stability conditions are derived. As a new idea, the harmonic balance method is used to derive the global stability conditions. That approach offers an extra advantage, the birth of unwanted oscillations can be also predicted. As a verification it is shown that the sufficient conditions of global stability derived by the harmonic balance method proposed here and the well-known direct Lyapunov approach coincide with each other, moreover, the harmonic balance predicts the birth of oscillations in the gap between the local and global stability conditions. Finally, an example when the conditions for local and global stability coincide, is considered.