PLL Lock Time

t ≈ 3 / (ωn × ζ)

Quick answer

This calculator solves t ≈ 3 / (ωn × ζ) for Lock Time. With Natural Frequency 300 rad/s, Damping Ratio 0.5, the result is 20 ms. It solves in 3 directions, so you can enter the values you have and get the one you need.

Calculator

Lock Time (µs)
—

Formula

t ≈ 3 / (ωn × ζ)

Description

PLL lock time approximates how long it takes for a phase-locked loop to acquire and settle on the target frequency. It depends on the loop natural frequency ωn and the damping ratio ζ. Higher loop bandwidth (larger ωn) gives faster locking but also passes more reference noise to the output. Higher damping reduces overshoot and ringing during acquisition. The actual lock time also depends on the initial frequency offset, the phase detector range, and whether the PLL uses frequency-aided acquisition. This formula gives the settling time to within about 5% of final frequency.

Variables

  • t — Approximate lock time (s)
  • ωn — Loop natural frequency (rad/s)
  • ζ — Damping ratio (dimensionless)

Practical Notes

Typical PLL damping ratios: 0.5-1.0, with 0.707 being common for optimal transient response. Fractional-N PLLs achieve higher loop bandwidth than integer-N for the same reference frequency. For frequency synthesizers in communication systems, lock time must be shorter than the channel switching time requirement. Fast-locking techniques include widening the loop bandwidth during acquisition.

Symbols and units

Symbol Meaning Default unit Accepted units
t Lock Time µs ps, ns, µs, ms, s, min, h
wn Natural Frequency rad/s rad/s
zeta Damping Ratio — —

Step by step

  1. 1. What you know: Natural Frequency 300 rad/s, Damping Ratio 0.5
  2. 2. The relation: t ≈ 3 / (ωn × ζ)
  3. 3. Substitute: wn = 300 rad/s, zeta = 0.5
  4. 4. Result: Lock Time = 20 ms

Worked examples

Minimum range

Given: Natural Frequency 1 rad/s, Damping Ratio 0.025

Result: Lock Time = 2 min

Low range

Given: Natural Frequency 10 rad/s, Damping Ratio 0.25

Result: Lock Time = 1.2 s

Typical

Given: Natural Frequency 300 rad/s, Damping Ratio 0.5

Result: Lock Time = 20 ms

High range

Given: Natural Frequency 3000 rad/s, Damping Ratio 0.8

Result: Lock Time = 1.25 ms

Maximum range

Given: Natural Frequency 30000 rad/s, Damping Ratio 8

Result: Lock Time = 12.5 µs

Solving for each variable

Solve for Lock Time

Natural Frequency 1 rad/s, Damping Ratio 0.025 → 2 min

Solve for Natural Frequency

Lock Time 100 ns, Damping Ratio 0.025 → 1.2 × 10⁹ rad/s

Solve for Damping Ratio

Lock Time 100 ns, Natural Frequency 1 rad/s → 3 × 10⁷

How Lock Time changes with Natural Frequency

Natural Frequency (input) Lock Time (result)
30 rad/s 0.2 s
75 rad/s 80 ms
150 rad/s 40 ms
225 rad/s 26.67 ms
300 rad/s 20 ms
450 rad/s 13.33 ms
600 rad/s 10 ms
1200 rad/s 5 ms
2100 rad/s 2.857 ms
3000 rad/s 2 ms

The same result in other units

Unit Value
µs 20000
ms 20
s 0.02
min 0.0003333

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Frequently asked questions

How do you calculate PLL Lock Time?

Use t ≈ 3 / (ωn × ζ). With Natural Frequency 300 rad/s, Damping Ratio 0.5, Lock Time works out to 20 ms.

What is the formula for PLL Lock Time?

The formula is t ≈ 3 / (ωn × ζ). PLL lock time approximates how long it takes for a phase-locked loop to acquire and settle on the target frequency.

How do you find Natural Frequency when you know the other values?

Rearrange for Natural Frequency: with Lock Time 100 ns, Damping Ratio 0.025, Natural Frequency is 1.2 × 10⁹ rad/s.

How do you find Damping Ratio when you know the other values?

Rearrange for Damping Ratio: with Lock Time 100 ns, Natural Frequency 1 rad/s, Damping Ratio is 3 × 10⁷.

What units does PLL Lock Time use?

Lock Time is shown in µs, ms, s, min. Values are held in SI base units internally, so you can mix prefixes without converting by hand.

Related Concepts

All Timing & Oscillators formulas →
555 Timer Astable 555 Timer Monostable Crystal Load Capacitance RC Oscillator Frequency Crystal Frequency Stability PLL Lock Time VCO Tuning Sensitivity Jitter from Phase Noise

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