Phase-locked loop: lock, hold and damping

With this VCO, phase detector and loop filter, how far can the loop lock from, how far can the resonance drift, and how fast does it settle?

K_0 = 2\pi \cdot v \cdot d \qquad \omega_n = \sqrt{\frac{K_0}{(R₁ + R₂) \cdot C₁}} \qquad ζ = \frac{\omega_n}{2} \left( R₂ \cdot C₁ + \frac{1}{K_0} \right) \qquad ℓ \approx \frac{2 \cdot ζ \cdot \omega_n}{2\pi} \qquad h = \frac{K_0}{2\pi}

Method

1. The loop gain K₀ is the phase detector gain times the VCO gain, with the VCO gain in radians per second per volt: K₀ = 2π·Kᵥ·K_d. Its unit is 1/s.
2. The lag-lead filter has two time constants: τ₁ = R₁C₁ and τ₂ = R₂C₁. The loop's natural frequency is ωₙ = √(K₀ ÷ (τ₁ + τ₂)) and its damping ratio ζ = (ωₙ/2)(τ₂ + 1/K₀).
3. The hold range is the loop gain itself, since a passive filter passes DC unchanged: Δf_H = K₀/2π. The lock range is about 2ζωₙ, in hertz. Both are the textbook second-order results.
4. The lock time is about one natural period, 2π/ωₙ. The closed-loop −3 dB bandwidth and the phase margin follow from ζ and ωₙ by the standard second-order expressions.

Assumptions

• The linearised second-order loop with a passive lag-lead filter. The 4046 with its RC filter is this; an op-amp integrator filter is a different (type 2) loop with a larger hold range.
• The phase detector is linear over its range. An XOR detector is, within ±90°; a type II phase-frequency detector also pulls in from any frequency difference, which makes its capture range wider than the lock range given here.
• The lock range formula is the usual approximation and is good to a factor of about 1.5. The pull-in range — from which the loop eventually locks after slipping cycles — is wider still and not given, because the standard estimate assumes a high-gain loop and these are not.
• Loop theory says nothing about whether the thing being locked to is the cell's resonance. That depends on the pickup coil and the phase relationship the detector sees, which are questions about the winding, not the loop.