Stability Limited PLL bandwidth Derivation using Impulse Invariance Method

The continuous-time (CT) model of the charge pump based (CP) phase-locked loop (PLL) is a linear time-invariant (LTI) model independent of the reference frequency $f_{ref}$. If all poles and zeros of the loop are scaled by $\alpha \gt 1$, the loop stability remains unaffected but the unity loop gain frequency $f_{u}$ scales by α. The CT model hence does not have any stability imposed limit on $f_{u}$. In reality, PLL is a sampled system due to the presence of the divider and the phase frequency detector (PFD), which makes it a hybrid loop. These two blocks are functionally nonlinear too. With reasonable assumptions, a linear discrete-time (DT) loop gain $L(z)$ is developed using the impulse invariance method applied to the hybrid model of the PLL. The location of closed-loop poles, which are zeros of $1 -L(z)$, imposes an upper limit of $f_{ref}/\pi$ on $f_{u}$. The analysis is validated by the PLL transient simulation.