PLL bifurcation analysis via smooth periodic extension

To aid the acquisition process in an out-of-lock PLL, an external sweep voltage can be applied to the VCO. The goal is to sweep the system state towards a stable lock point. For a second-order loop containing a perfect integrator loop filter, there is a maximum VCO sweep rate magnitude, denoted as R
_m
rad/sec
²
, 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. This is not true for a sweep rate greater than R
_m
. As the sweep rate increases through R
_m
, an unstable phase plane 2π-periodic limit cycle bifurcates from a 2π-periodic separatrix cycle. This separatrix cycle must be approximated in order to approximate R
_m
. This task is made difficult by the fact that the separatrix cycle has a discontinuous first derivative at each saddle point. To eliminate this problem, the 2π-periodic separatrix cycle is used to construct a smooth (i.e., has a continuous derivative) 4π-periodic solution of an equation derived from the PLL dynamic model. This smooth periodic extension is approximate accurately by a simple finite Fourier series, and an approximation is derived for R
_m
.