Lock-In Time Constant Calculator

• Single-pole relationship between cutoff frequency and time constant: f_c = 1 / (2π τ)
• First-order step response (normalized): y(t) = 1 - e^{-t/τ}
• Time to reach a fraction p of final value: t = -τ ln(1 - p)
• Time to settle within an error band ε (as a fraction): t = -τ ln(ε)
• Equivalent Noise Bandwidth for a single-pole low-pass: ENBW = 1 / (4τ)

1. Select a mode to compute or analyze settings.
2. Enter either cutoff frequency or time constant.
3. Choose a target definition and value for timing.
4. Press Calculate to show results above the form.
5. Use Download CSV or Download PDF for exports.

1) What the time constant controls

A lock-in output filter behaves like a low-pass stage that smooths the demodulated signal. The time constant τ sets how quickly the displayed value responds to real changes. Larger τ reduces random noise and ripple, but it also slows response to modulation amplitude changes, phase shifts, and scan motion.

2) Converting between τ and cutoff frequency

For a single-pole model, the -3 dB cutoff is f_c=1/(2π τ). That means τ=100 ms corresponds to about 1.59 Hz, while τ=10 ms corresponds to about 15.9 Hz. This conversion helps you compare settings across instruments that report bandwidth instead of τ.

3) Settling time and display stability

After a step change, the output approaches its final value exponentially. A common planning rule is that 1% settling occurs near 4.6τ, and 0.1% settling occurs near 6.9τ. With τ=100 ms, that is roughly 0.46 s to 1% and 0.69 s to 0.1%.

4) Equivalent noise bandwidth and SNR

The filter also sets the noise passed to the output. For a one-pole response, the equivalent noise bandwidth is ENBW=1/(4τ). With τ=100 ms, ENBW is about 2.5 Hz. If your noise is roughly white near the demodulated output, RMS noise scales with the square root of ENBW. This is why long time constants often produce calmer, repeatable numerical readouts.

5) Matching τ to modulation frequency

Choose τ based on how fast the demodulated amplitude changes, not the reference frequency. Even with 1 kHz modulation, slow drifts may suit τ=100 ms–1 s. For fast sweeps, small dwell times, or rapid chopping, choose a smaller τ.

6) Impact of multi-pole filter settings

Many instruments offer 2-pole or 4-pole slopes. Higher order suppresses noise more, but can overshoot and extend effective settling. Use these numbers as a baseline, then check your instrument’s step response to set a safe wait time.

7) Choosing τ for scanning and mapping

When scanning or sweeping, τ interacts with scan speed and resolution. If you move too quickly, the output lags and smears features. Set τ so the 10–90% rise time (≈2.2τ) is below your dwell time per point.

8) Practical checklist before recording data

Start with a τ that stabilizes the reading, then confirm you are not averaging away real variation. After changes, wait several τ before logging. Validate by repeating at two τ values; if the mean shifts, τ is too large.

1) Is the time constant the same as averaging time?

It is related, but not identical. A first-order filter averages exponentially, weighting recent samples more strongly. τ sets the exponential decay rate, which determines both smoothing and how fast the output responds to changes.

2) How long should I wait after changing settings?

A common rule is 5–7 time constants for a stable reading after a step change. For tighter stability, use the settling estimates: about 4.6τ to 1% and about 6.9τ to 0.1%.

3) Why does a larger τ reduce noise?

Larger τ lowers the effective bandwidth of the low-pass filter, so less noise power passes through. For approximately white noise, RMS noise decreases as the square root of bandwidth, improving the displayed signal-to-noise ratio.

4) What target should I use: error band or fraction reached?

Error band is best for “how long until stable.” Fraction reached is useful for “how fast does it move.” Choose 1% or 0.1% error for metrology, and 90% reached for response-speed comparisons.

5) How does ENBW relate to measured noise?

ENBW converts your filter into an equivalent rectangular bandwidth. If the noise spectral density is flat near the output, RMS noise is approximately proportional to sqrt(ENBW). Halving ENBW reduces RMS noise by about 29%.

6) Can I use this for other output filters?

This calculator assumes a single-pole low-pass response. It can still guide intuition for similar smoothing stages, but high-pass or band-pass paths have different step behavior and bandwidth definitions, so treat results as approximate.

7) My instrument uses 2-pole or 4-pole filters. What then?

Use τ conversion and settling as a baseline, then increase waiting time based on filter order and observed response. Higher order typically narrows noise more for the same τ but can lengthen effective settling and add overshoot.