Inductor transient response
What is an RL time constant?
The RL time constant describes how quickly current rises or decays in a resistor-inductor circuit.
It is represented by the Greek letter tau, written τ, and is measured in seconds.
Main RL relationship
RL time constant formula
Time constantτ = L ÷ R
- τ is time constant in seconds.
- L is inductance in henries.
- R is resistance in ohms.
Increasing inductance makes the current change more slowly, while increasing resistance reduces the time constant.
Current growth
RL current-rise formula
Rising currentI(t) = Imax(1 − e⁻ᵗ/τ)
After one time constant, current reaches approximately 63.2% of its final steady-state value.
Current decay
RL current-decay formula
Decaying currentI(t) = I₀e⁻ᵗ/τ
After one time constant, approximately 36.8% of the initial current remains.
Worked example
Calculate an RL time constant
Suppose an RL circuit has an inductance of 2 H and resistance of 10 Ω.
- Use τ = L ÷ R.
- Substitute L = 2 H and R = 10 Ω.
- τ = 2 ÷ 10.
- The RL time constant is 0.2 seconds.
Practical approximation
The five-time-constant rule
- After 1τ, rising current reaches about 63.2%.
- After 2τ, rising current reaches about 86.5%.
- After 3τ, rising current reaches about 95.0%.
- After 5τ, rising current exceeds 99%.
Engineers often treat five time constants as effectively complete for both current rise and current decay.
SI unit guidance
RL circuit units
- Time constant: seconds (s).
- Inductance: henries (H).
- Resistance: ohms (Ω).
- Current: amperes (A).
- Elapsed time: seconds (s).
Practical applications
Where RL time constants matter
- Relay and solenoid switching.
- Motor winding current response.
- Switching power supply inductors.
- Electromagnets and magnetic actuators.
- Transient protection and flyback circuits.
Model limitations
Assumptions and limitations
- The resistor and inductor are treated as ideal linear components.
- Inductance is assumed constant.
- Core saturation and hysteresis are not included.
- Parasitic capacitance is ignored.
- All calculations use positive magnitudes.
Common questions
RL time constant calculator FAQ
What formula does the RL time constant calculator use?
The basic RL time constant formula is τ = L/R, where inductance is measured in henries and resistance is measured in ohms.
What happens after one RL time constant?
During current rise, the inductor current reaches about 63.2% of its final value. During current decay, about 36.8% of the initial current remains.
How do I calculate current rise in an RL circuit?
Use I(t) = Imax(1 − e⁻ᵗ/τ), where Imax is the final current, t is elapsed time, and τ is the RL time constant.
How do I calculate current decay in an RL circuit?
Use I(t) = I₀e⁻ᵗ/τ, where I₀ is the initial current, t is elapsed time, and τ is the RL time constant.
How long does an RL circuit take to reach steady state?
After about five time constants, the current is commonly treated as effectively at its final steady-state value.