Spur is the key specification that affects the performance of fractional-N frequency synthesizers. The spur for traditional compensation less and ΣΔ modulated fractional-N frequency synthesizers was analyzed theoretically to make the spur components more explicit. An extra phase shift was introduced into traditional compensation-less fractional-N frequency synthesizers to satisfy the locked condition. ForΣΔ modulated fractional-N frequency synthesizers, a relationship was developed for the loop bandwidth as a function of the residual phase error and the high frequency noise rejection using linear analysis. Analysis using the time field model shows that loop nonlinearities, such as the dead-zone in phase-frequency detectors and the mismatch between the charge and discharge currents in the charge pump, can reduce the output noise by 30dB.