Computing the VCO sweep rate limit for a second-order PLL

Phase-locked loops serve important roles in receivers, coherent transponders and similar radio-frequency-based applications. For many of these uses, the bandwidth of the loop must be kept small to limit the detrimental influence of noise, and this requirement makes the natural PLL pull-in process too slow and/or unreliable. To aid the acquisition process in these cases, an external sweep voltage can be applied to the VCO when the loop is unlocked. Hopefully, the sweep voltage will effect a rapid decrease in closed-loop frequency error to a point where phase lock is achieved quickly. For a second-order loop containing a perfect integrator loop filter, there is a maximum VCO sweep rate magnitude, denoted as R_m rad/sec², for which phase lock is guaranteed. If the actual VCO sweep rate magnitude is less than R_m, the loop cannot sweep past a stable phase-lock state without locking correctly. For an applied sweep rate greater than R_m, the loop may sweep past a lock point and fail to achieve phase lock. In the PLL literature, only a trial-and-error approach has been described for approximating R_m given values of loop damping factor p and natural frequency omega_n. Furthermore, no plot exists of R_m/omega_n ² versus rho. This dearth of results is remedied here. A new numerical algorithm is given that converges to the maximum sweep rate magnitude R_m. It is used to generate a plot of R_m/omega_n ² versus rho, a never-before-explored relationship in the PLL literature.