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Electrical
Inductor current after a switching event
Current in an inductor after a switch changes a circuit with a constant source: it moves smoothly from its starting value towards its final value with time constant L/R.
- Ref
- ELE-34
- Rev
- v1
- Field
- Electrical
- Result
- i(t) [A]
After a switch changes, the current in an inductor moves exponentially from its initial value I₀ to its final value I∞: i(t) = I∞ + (I₀ − I∞)e−t/τ, with the time constant τ = L/Req and Req the resistance seen by the inductor. The current cannot jump, so it starts at I₀ and reaches 63 % of the change after one τ and 99 % after about five. Inductance in mH, resistance in Ω and time in ms give a time constant in ms.
Current in an inductor after a switch changes a circuit with a constant source: it moves smoothly from its starting value towards its final value with time constant L/R. The inductor current cannot jump.
Calculator
- Time constant, L/R10 ms
- Fraction of the change completed63.21 %
- Rate of change of current, (I∞ − i)/τ73.58 A/s
- Time to complete 99 % of the change, 4.6τ46.05 ms
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.
- After 10 ms (t = 1 τ) the current has gone from 0 A to 1.264 A, 63.2 % of the way to 2 A.
- At 5 time constants (50 ms) the current is within 1 % of its final value; interrupting the current in an inductor quickly causes a large voltage spike, so a diode or snubber is added across a relay coil.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (A) | Local step | Change in result |
|---|---|---|---|---|---|---|
| I_∞ Final current | 0 to 20 A | 0 | 12.64 | 12.64 | +10% | +10 % |
| I0 Initial current | 0 to 20 A | 1.264 | 8.622 | 7.358 | +10% | 0 % |
| t Time since the switch changed | 0 to 100 ms | 0 | 2 | 2 | +10% | +5.54 % |
| Req Resistance seen by the inductor | 0.1 to 100 Ω | 0.0199 | 2 | 1.98 | +10% | +5.54 % |
| L Inductance | 1 to 1,000 mH | 0.1903 | 2 | 1.81 | +10% | −5.54 % |
Sweep of Time since the switch changed
| Time since the switch changed (ms) | Inductor current (A) |
|---|---|
| 0 | 0 |
| 12.5 | 1.427 |
| 25 | 1.836 |
| 37.5 | 1.953 |
| 50 | 1.987 |
| 62.5 | 1.996 |
| 75 | 1.999 |
| 87.5 | 2 |
| 100 | 2 |
Sweep one variable
Variables and how each one contributes
Inductor current
What the equation gives you. Everything else feeds it.
Final current
The final (steady-state) current, in amps, that the circuit settles at. Raises the current at time t, because the current heads for a higher final value. Raising it by 0.25 A (to 2.25 A) moves the result by +12.5 %, from 1.264 to 1.422 A.
Initial current
The initial current just after the switch changes, in amps. Raises the current at time t, because the starting value still contributes through the decaying term. Raising it by 0.25 A (to 0.25 A) moves the result by +7.27 %, from 1.264 to 1.356 A.
Inductance
The inductance, in mH. Lowers the current at time t here (the current is still rising): a larger inductance makes the time constant longer, so the current changes more slowly. Raising it by 10 mH (to 110 mH) moves the result by −5.54 %, from 1.264 to 1.194 A.
Resistance seen by the inductor
The resistance seen by the inductor after the switch changes, in ohms. Raises the current at time t here: a larger resistance shortens the time constant, so the current reaches its final value sooner. (The final current is entered separately.) Raising it by 1 Ω (to 11 Ω) moves the result by +5.54 %, from 1.264 to 1.334 A.
Time since the switch changed
The time since the switch changed, in ms. Raises the current at time t, as it moves further towards the final value. Raising it by 5 ms (to 15 ms) moves the result by +22.9 %, from 1.264 to 1.554 A.
Worked example with the values above
- Time constant: τ = L/R = 100 mH / 10 Ω = 10 ms.
- Exponential term: e−t/τ = e−10/10 = 0.3679.
- i = 2 + (0 − 2) × 0.3679 = 1.264 A.
Assumptions and limits
- A linear inductor with constant inductance (no saturation) and a single equivalent resistance.
- A single switching event, with the circuit having reached the initial state.
Common mistakes
- Ignoring the winding resistance of the coil, which is part of Req.
- Using the capacitor formula for the voltage: for an inductor it is the current that varies smoothly.
Statistics and uncertainty
The time constant depends on the ratio of two components with ±10 % tolerance (including the winding resistance of the coil), so τ is uncertain by about 15 %. The current at a given time is most uncertain around t = τ.
Where it is used
- The rise and decay of current in relays, solenoids and motor windings.
- Choosing the delay or switching time for coil circuits.
Maths you need
- Time constant: time for the change to complete 63 %
- Inductor: stores energy in a magnetic field; its current cannot jump