Mathematical models for the behavior of fractional-N phase-locked-loop frequency synthesizers (Frac-N) are presented. The models are intended for calculating rms phase error and determining spurs in the output of Frac-N. The models describe noise contributions due to the charge pump (CP), the phase frequency detector (PFD), the loop filter, the voltage control osicllator, and the delta-sigma modulator. Models are presented for the effects of static CP gain mismatch, CP dynamic mismatch and PFD reset delay mismatch. A simple analytic expression shows the level of 16 sequence noise caused by static CP current mismatch. We further show that un-equal rise time and fall time constants of the CP result in dynamic mismatch noise. Reset delay mismatch in PFD is shown to also contribute significantly to close-in phase noise. The model takes into account the reduction in CP thermal and flicker noise due to the changing duty cycle of Frac-N CP. Our model is therefore useful in characterizing the noise performance of Frac-N at the system-level, simplifying the design of fractional-N synthesizers and transmitters. Analytical and simulated results are compared and show good agreement with prior published data on Frac-N realizations.Index Terms-Charge pump (CP), delta-sigma, dynamic mismatch, dynamic mismatch corner frequency, flicker noise, flicker noise corner frequency, fractional-N frequency synthesizer (Frac-N), frequency synthesizer, gain mismatch, gain mismatch corner frequency, phase frequency detector (PFD), phase noise, reset delay mismatch, rms phase error, spurs, thermal noise, voltage-controlled oscillator (VCO).