Phase-plane analysis of phase-locked loops used for clock recovery

The transient analysis of phase-locked loops is not an easy analytical task. Although linearisation of these devices is common practice, the exact nature of the loop is not obtained. Using phase-plane techniques it is possible to determine both the transient and steady-state responses of phase-locked loops. We investigate the application of phase-plane techniques to a second-order phase-locked loop and to a more complex clock-recovery loop. A number of interesting results were obtained and are discussed including the effects on the phase-plane plot for a variety of loop filter bandwidths and loop gain values.