Settling Time to N-Bit Accuracy

ts = (N + 1) · ln2 · τ

Quick answer

This calculator solves ts = (N + 1) × ln(2) × τ for Settling Time. With Resolution (bits) 0.5, Time Constant 0.22 ms, the result is 0.2287 ms. It solves in 3 directions, so you can enter the values you have and get the one you need.

Calculator

Settling Time (µs)
—

Formula

ts = (N + 1) × ln(2) × τ

Description

To settle a single-pole exponential response to within half an LSB of an N-bit converter takes a fixed number of time constants—about 0.69 × (N + 1). This determines how quickly an ADC driver or multiplexed signal can be sampled at full accuracy. Higher resolution demands more settling time, trading throughput against precision.

Variables

  • ts — Settling time to ½ LSB (s)
  • N — Resolution in bits
  • τ — Dominant time constant of the response (s)

Practical Notes

Settling to 12 bits needs ~9 time constants; to 16 bits ~12. Real amplifiers add slew-rate and higher-order terms, so allow margin beyond this single-pole estimate.

Symbols and units

Symbol Meaning Default unit Accepted units
ts Settling Time µs ps, ns, µs, ms, s, min, h
N Resolution (bits) — —
tau Time Constant µs ps, ns, µs, ms, s, min, h

Step by step

  1. 1. What you know: Resolution (bits) 0.5, Time Constant 0.22 ms
  2. 2. The relation: ts = (N + 1) × ln(2) × τ
  3. 3. Substitute: N = 0.5, tau = 0.22 ms
  4. 4. Result: Settling Time = 0.2287 ms

Worked examples

Minimum range

Given: Resolution (bits) 0.025, Time Constant 0.22 µs

Result: Settling Time = 156.3 ns

Low range

Given: Resolution (bits) 0.25, Time Constant 2.2 µs

Result: Settling Time = 1.906 µs

Typical

Given: Resolution (bits) 0.5, Time Constant 0.22 ms

Result: Settling Time = 0.2287 ms

High range

Given: Resolution (bits) 0.8, Time Constant 1.1 ms

Result: Settling Time = 1.372 ms

Maximum range

Given: Resolution (bits) 8, Time Constant 11 ms

Result: Settling Time = 68.62 ms

Solving for each variable

Solve for Settling Time

Resolution (bits) 0.025, Time Constant 0.22 µs → 156.3 ns

Solve for Resolution (bits)

Settling Time 100 ns, Time Constant 0.22 µs → -0.3442

Solve for Time Constant

Settling Time 100 ns, Resolution (bits) 0.025 → 140.8 ns

How Settling Time changes with Resolution (bits)

Resolution (bits) (input) Settling Time (result)
0.05 0.1601 ms
0.125 0.1716 ms
0.25 0.1906 ms
0.375 0.2097 ms
0.5 0.2287 ms
0.75 0.2669 ms
1 0.305 ms
2 0.4575 ms
3.5 0.6862 ms
5 0.915 ms

The same result in other units

Unit Value
ns 228700
µs 228.7
ms 0.2287
s 0.0002287

Embed this calculator

Paste this into any page. It keeps working when this page updates, and credits the source.

<iframe src="https://vbatpower.com/embed/settling-time-nbits/" width="100%" height="460" style="border:1px solid #1f2a37;border-radius:12px" loading="lazy" title="Settling Time to N-Bit Accuracy"></iframe>
<p><a href="https://vbatpower.com/formulas/settling-time-nbits/">Settling Time to N-Bit Accuracy</a> Calculator by vbatpower.com</p>

Frequently asked questions

How do you calculate Settling Time to N-Bit Accuracy?

Use ts = (N + 1) × ln(2) × τ. With Resolution (bits) 0.5, Time Constant 0.22 ms, Settling Time works out to 0.2287 ms.

How do you calculate calculator the number of bits for 3.3v for 12 bit?

Rearrange for Settling Time: with Resolution (bits) 0.25, Time Constant 2.2 µs, Settling Time is 1.906 µs.

What is the formula for Settling Time to N-Bit Accuracy?

The formula is ts = (N + 1) × ln(2) × τ. To settle a single-pole exponential response to within half an LSB of an N-bit converter takes a fixed number of time constants—about 0.69 × (N + 1).

How do you find Resolution (bits) when you know the other values?

Rearrange for Resolution (bits): with Settling Time 100 ns, Time Constant 0.22 µs, Resolution (bits) is -0.3442.

How do you find Time Constant when you know the other values?

Rearrange for Time Constant: with Settling Time 100 ns, Resolution (bits) 0.025, Time Constant is 140.8 ns.

What units does Settling Time to N-Bit Accuracy use?

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

Related Concepts

All Op-Amps formulas →
Non-Inverting Amplifier Inverting Amplifier Summing Amplifier Difference Amplifier Gain-Bandwidth Product Common-Mode Rejection Ratio Op-Amp Integrator Gain Op-Amp Differentiator Gain

Need more features?

Save calculations, import telemetry data, simulate battery discharge, and collaborate with your team.

Try the App