Library / Electrical / ELE-35
Electrical
Time constant of an RL circuit
Time for the current in an inductor to cover about 63 % of the way to its final value after switching.
- Ref
- ELE-35
- Rev
- v1
- Field
- Electrical
- Result
- τ [ms]
The time constant of a series circuit with inductance L and resistance Req is τ = L/Req. After a switching event the current completes 63.2 % of its change in one τ, 86.5 % in two and 99.3 % in five. A large inductance or small resistance means slow changes. Here L is in mH and R in ohms, giving τ in ms.
Time for the current in an inductor to cover about 63 % of the way to its final value after switching. After five time constants it is within 1 % of its final value.
Calculator
- Time to complete 63.2 % of the change (one τ)10 ms
- Time to complete 99 % of the change, 4.6τ46.05 ms
- Corner frequency of the equivalent RL filter, R/(2πL)15.92 Hz
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 100 mH coil in a 10 Ω circuit has τ = 10 ms: the current is 63 % of the way to its final value after 10 ms and within 1 % after 50 ms.
- The same τ sets the corner frequency of an RL filter, 15.9 Hz here, linking the time and frequency pictures.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (ms) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Req Resistance seen by the inductor | 0.1 to 100 Ω | 1 | 1,000 | 999 | +10% | −9.09 % |
| L Inductance | 1 to 1,000 mH | 0.1 | 100 | 99.9 | +10% | +10 % |
Sweep of Inductance
| Inductance (mH) | Time constant (ms) |
|---|---|
| 1 | 0.1 |
| 2.371 | 0.2371 |
| 5.623 | 0.5623 |
| 13.34 | 1.334 |
| 31.62 | 3.162 |
| 74.99 | 7.499 |
| 177.8 | 17.78 |
| 421.7 | 42.17 |
| 1,000 | 100 |
Sweep one variable
Variables and how each one contributes
Time constant
What the equation gives you. Everything else feeds it.
Inductance
The inductance, in mH. Raises the time constant in direct proportion: a larger inductance opposes changes of current more. Raising it by 10 mH (to 110 mH) moves the result by +10 %, from 10 to 11 ms.
Resistance seen by the inductor
The resistance seen by the inductor, in ohms (include the coil’s winding resistance). Lowers the time constant in inverse proportion: more resistance lets the current reach its final value sooner. Raising it by 1 Ω (to 11 Ω) moves the result by −9.09 %, from 10 to 9.091 ms.
Worked example with the values above
- Inductance in henries: L = 100 mH = 0.1 H.
- Divide by the resistance: 0.1 H / 10 Ω = 0.01 s.
- τ = 10 ms.
Assumptions and limits
- A single linear inductor with constant inductance and one equivalent resistance.
- No coupling or saturation effects.
Common mistakes
- Forgetting the winding resistance.
- Using R in kΩ with L in mH without checking the unit of the answer.
Statistics and uncertainty
The time constant is a ratio of two toleranced quantities, so its relative error is the quadrature sum of the two. The coil’s DC resistance and its temperature rise change R: a copper winding increases its resistance by about 0.4 % per kelvin, so τ falls as the coil warms.
Where it is used
- Switching times of relays, solenoids and fuel-injector coils.
- Choosing the time to wait before the current in a coil has settled.
Maths you need
- Inductor: stores energy in a magnetic field
- Time constant: time for the change to complete 63 %