Efficient analysis of phase-locked loops through a novel time-frequency approach, based on two envelope transient formulations

A novel time-frequency analysis for phase-locked loops (PLL) is presented here, based on two nested envelope transient formulations. The outer envelope is centered, as usual, about the VCO frequency. The inner-envelope is obtained when considering a frequency domain expansion of the loop state variables about the reference frequency and its multiples, with slow-varying phasors. Thus, the influence of these frequency components on the loop behavior is taken into account. The double envelope approach enables an efficient and realistic simulation of the acquisition times in the common case of slow time response of the loop. On the other hand, the phase-noise can be analyzed from very low offsets from the carrier, while taking into account the spurious content due to the reference frequency. In the absence of noise perturbations, the application of harmonic balance to the VCO envelope enables an accurate and efficient analysis of the PLL versus any parameter of interest. The combination with continuation techniques to circumvent the possible turning points is straightforward. The stability is analyzed through the calculation of the eigenvalues of the harmonic-balance characteristic matrix. Very good agreement has been obtained in the comparison with time domain simulation and experimental results.