Analysis of charge pump phase locked loop in the presence of loop delay and deterministic noise

Rigorous math is provided in this paper to obtain the transfer function of a charge pump phase locked loop (CPLL). The math clearly explains when and how the impulse invariant transform based method works and corrects its errors in some cases. The loop transfer function is then used for loop stability evaluation and jitter calculation. It is shown that the loop stability can be improved by increasing loop gain and loop delay, revealing a counterintuitive application of loop delay, i.e., a loop delay can be used to speed up phase locking process. As for the jitter calculation, a closed form expression is derived to predict the output jitter due to deterministic input noise.