Phase-Locked Loop — Acquisition & Lock Dynamics

A PLL consists of a Phase Detector (PD), Loop Filter (LF), and Voltage-Controlled Oscillator (VCO). It locks onto an input frequency by feedback. Phase error φ_e obeys a nonlinear ODE: dφ_e/dt = Δω − K_d·K_v·sin(φ_e) — a pendulum equation! Lock range, pull-in range, and transient response are all visible here.