Self-Coupling and Mutual Pulling in Phase-Locked Loops

The influence of coupling of oscillators in phase-locked loops (PLLs) was studied here. Particularly, numerical analyses based on nonlinear differential equations of the self- and mutual coupling systems were performed and compared with the results of the linear approximation model under synchronous conditions. The analysis revealed that coupling using the reference clock caused greater PLL instability than that caused by self-coupling for a given coupling strength. This shows that the phase error of mutual coupling exhibits the same behavior as that of reference-clock coupling in the single PLL under synchronous conditions, and the noise sensitivity of mutual coupling is twice that of the reference-clock coupling. In addition, the phase error of mutual coupling in the quasi-synchronous condition strongly depends on the relationship between the frequency difference of the two PLLs and the PLL bandwidth. We designed and fabricated the test circuits for theoretical verification using the 0.18-
$\mu \text{m}$
standard CMOS process. From the measurements, adjustment of the timing difference between the two PLLs is effective in mitigating mutual coupling under synchronous conditions. Moreover, we implemented mutual injections between two oscillators in the test circuit and confirmed their effectiveness. The proposed methods will be useful for achieving the compact and effective cancellation of the mutual coupling between PLLs.