Time-Frequency Formulation for the Nonlinear Analysis of Coupled Phase-Locked Loops

In this work, harmonic balance (HB) and envelope- transient formulations of coupled phase-locked loops (CPLLs) are presented. The CPLL has the added difficulty of its autonomous behavior since no reference oscillator is present. The new formulation takes into account the autonomy of the system, introducing a special set of state variables, which depend on the autonomous frequencies. The hysteresis phenomenon in CPLLs is analyzed in detail, efficiently obtaining the pull-in and hold-in ranges through HB. The pole analysis of the perturbed HB system enables an accurate prediction of instabilities and resonances. Due to the CPLL autonomy, there exists an inherent noise accumulation effect. This effect is taken into account, analyzing the perturbation in terms of accumulation and deviation components. The envelope formulation allows simulating the CPLL behavior in the presence of modulation signals. The influence of the stability of the steady-state solution on the modulated signals is investigated. The simulation results have been successfully compared with the measurements in a manufactured CPLL system at 2 GHz.