A PLL is a feedback control system that aligns the phase of a voltage-controlled oscillator (VCO) with a reference signal. PLLs are ubiquitous in electronics: FM demodulation, clock recovery, frequency synthesis, and GPS.
The phase detector measures the phase error (here via sin(φ)), the loop filter (first-order, time constant τ) smooths noise, and the VCO shifts its frequency by K·v(t).
Lock condition: The VCO can lock if |ω₀ − ω_ref| < K (lock range). With a large loop gain K, locking is fast but may overshoot. Large τ filters noise but slows acquisition.
Watch the reference (purple) and VCO (orange) signals align as the loop locks. The lower panel shows the phase error decaying to zero.