In this paper, a novel time-frequency method for the analysis of phase-locked loops (PLLs) has been developed. It enables an efficient and realistic simulation, taking into account the spurious frequency components generated at the phase detectors and intrinsic to their nonlinear performance. The variables of the loop equations are expanded in a Fourier series, at these spurious frequencies, with time-varying phasors. This generalizes the envelope-transient analysis to loops containing three different time scales, provided by the frequency of the voltage-controlled oscillator, external signals, and noise or modulations, respectively. The new technique substantially reduces the computational cost in the simulation of transients and acquisition times in the common case of a narrow-band loop. It also allows considering low-frequency noise perturbations, while taking into account the spurious frequency components. In the absence of modulation, the harmonic-balance (HB) formulation of the loop equations enables an efficient analysis of the variation of the PLL solution versus any parameter of interest. In addition, the conversion matrix approach, based on this HB formulation, can be used to determine the output noise spectrum from frequency-domain descriptions of the input noise sources, while taking the spurious content into account. To show the generality of application of the techniques, two different PLLs have been considered here: a frequency synthesizer with a tri-state comparator and charge pump, and an automatic frequency-control loop. In each case, the results have been successfully compared with time-domain simulations and measurements.
