Phase-Locked Loop (2D)

What a phase-locked loop does. A phase detector (PD) compares the phase of an input carrier with the phase of a voltage-controlled oscillator (VCO). The PD output passes through a loop filter and steers the VCO frequency until the VCO follows the input in phase (it is locked).

Model used here (phase domain). Phase error e = θ_in − θ_vco. Sinusoidal PD: u = sin(e) (K_d = 1 V/rad near lock). VCO: dθ_vco/dt = K_vv_c relative to the carrier. With loop gain K = K_dK_v and the active PI filter F(s) = (1 + sτ₂)/(sτ₁):

e′ = Δω − K(a sin e + x),   x′ = sin(e)/τ₁,   a = τ₂/τ₁.

For small e this is a second-order system: ω_n = √(K/τ₁), ζ = ω_nτ₂/2, closed loop H(s) = (2ζω_ns + ω_n²)/(s² + 2ζω_ns + ω_n²). The error after a phase step is e(t) = Δθ e^−ζω_nt(cos ω_dt − (ζω_n/ω_d) sin ω_dt), ω_d = ω_n√(1−ζ²). The 3 dB bandwidth is ω_n√(1 + 2ζ² + √((1 + 2ζ²)² + 1)); the noise bandwidth is B_L = (ω_n/2)(ζ + 1/(4ζ)).

Lock ranges. First-order loop (no filter): e′ = Δω − K sin e has a stable solution exactly when |Δω| < K (the hold-in range), then sin e = Δω/K. The PI loop has no static error, so its hold-in range is set by the VCO tuning range. The lock-in range (a frequency step handled without a cycle slip) is often quoted as 2ζω_n; the graph compares that rule of thumb with the largest step this simulation can take before |e| reaches π. The two are defined differently, so they differ by a factor that depends on ζ: the rule of thumb is conservative.

Simplifications (honest list). No ripple at 2ω₀ from the multiplier PD (a phase-domain model has none); no digital or charge-pump PD; the loop filter is an ideal PI with anti-windup at the VCO limits; the VCO is linear; the input amplitude is constant; noise is white phase noise at the PD input applied as a zero-order hold each 50 µs. The 500 Hz carrier on the scope is a display choice; loop dynamics do not depend on it. Time runs in slow motion so you can watch it.

Independent numerical checks run in the page. (1) Simulated phase step vs the closed form. (2) Closed-loop |H(jω)| measured by driving the nonlinear simulation with a small phase sine and demodulating the VCO phase, vs the formula. (3) First-order loop: 970 rad/s locks with sin e = 0.97 and 1030 rad/s slips for K = 1000 rad/s. (4) Monte-Carlo VCO phase variance for white input noise vs S·B_L.