Time-varying, frequency-domain modeling and analysis of phase-locked loops with sampling phase-frequency detectors

This paper presents a new, frequency-domain based method for modeling and analysis of phase-locked loop (PLL) small-signal behavior, including time-varying aspects. Focus is given to PLLs with sampling phase-frequency detectors (PFDs) which compute the phase error only once per period of the reference signal. Using the harmonic transfer matrix (HTM) formalism, the well known continuous-time, linear time-invariant (LTI) approximations are extended to take the impact of time-varying behavior arising from the sampling nature of the PFD, into account. Especially for PLLs with a fast feedback loop, this time-varying behavior has severe impact on, for example, loop stability and cannot be neglected. Contrary to LTI analysis, our method is able to predict and quantify these difficulties. The method is verified for a typical loop design.