Analysis of First- and Second-Order Digital DS Modulator Used in Fractional-N PLLs

In this paper, we analyze for the first time behavior in the time domain of the accumulated quantization error induced by the first- and second-order digital DeltaSigma modulator (DSM). The DSM is adopted in fractional-N PLLs to dither frequency division factor. From the analysis, difference in behavior of the accumulated quantization error in the two cases is clearly explained. Furthermore, by mean of this, the reason of using second-order DSM is required for the calibration loop of digital/time converter canceling the quantization error is revealed. It also explains why there is variation in convergence time even with the second-order DSM when fractional part of the division factor changes.