About the RL Time Constant Calculator
This RL time constant calculator analyses a series resistor–inductor circuit connected to a DC voltage. It finds the time constant τ = L/R, the final (steady-state) current V/R, and the current, inductor voltage and resistor voltage at any moment after the switch closes. Switch to decay mode to see how the current dies away when the supply is removed and the inductor discharges through the resistance.
It is handy for estimating how fast a relay or solenoid coil pulls in, how long a motor winding takes to reach full current, sizing flyback paths, and for physics and electronics homework on first-order circuits. The chart traces the current from zero to five time constants, and the table lists the classic 63%, 86%, 95%, 98% and 99.3% milestones.
The model assumes an ideal inductor and a constant DC source. Any winding resistance of the coil should be included in R, and real cores can saturate at high current, which makes the actual current rise faster than this curve predicts.
With the default inputs, the time constant τ is 0.5 ms. Change any value above to recalculate instantly.
How to use the rl time constant calculator
- 1Choose energizing (switch closed) or de-energizing (decay through R).
- 2Enter the supply voltage, total resistance and inductance in millihenries.
- 3Enter the time after switching in milliseconds.
- 4Read τ, the current and the voltages at that instant.
- 5Use the chart and table to see when the current is effectively settled.
Formula and method
When a DC voltage is applied to a resistor and inductor in series, the inductor initially opposes any change in current, so the full supply appears across it and the current starts at zero. The current then rises exponentially toward V/R with time constant τ = L/R; with L in millihenries and R in ohms, τ comes out directly in milliseconds.
At each moment the resistor voltage is i × R and the inductor voltage is whatever remains of the supply. When the supply is removed and the inductor discharges through R, the current decays as (V/R)·e^(−t/τ) and the inductor voltage reverses to keep it flowing. After 1τ the transition is 63.2% complete, after 5τ more than 99.3%; the exact 99% point is τ·ln(100) ≈ 4.6τ. Energy stored at steady state is ½LI².
- τ
- Time constant (s)
- L
- Inductance (H)
- R
- Total series resistance (Ω)
- V
- DC supply voltage
- i(t)
- Current at time t (A)
- E
- Magnetic energy stored (J)
Worked examples
12 V across 100 Ω and 50 mH, after 0.5 ms
τ = 50 mH ÷ 100 Ω = 0.5 ms and the final current is 12 ÷ 100 = 120 mA. After one time constant the current is 0.12 × (1 − e⁻¹) ≈ 75.9 mA (63.2%), the resistor drops 7.59 V and the inductor 4.41 V. At full current the coil stores ½ × 0.05 × 0.12² = 0.36 mJ.
24 V relay coil: 400 Ω, 2 H, after 10 ms
τ = 2 H ÷ 400 Ω = 5 ms, so 10 ms is two time constants and the coil has reached 1 − e⁻² ≈ 86.5% of its 60 mA final current (51.9 mA). It takes about 23 ms to get within 1% of full current.
Decay: 0.5 A in 1 mH through 10 Ω, after 0.2 ms
τ = 1 mH ÷ 10 Ω = 0.1 ms. Starting from 5 ÷ 10 = 0.5 A, after 0.2 ms (2τ) the current has fallen to 0.5 × e⁻² ≈ 67.7 mA, 13.5% of its initial value, and the inductor holds −0.68 V to keep it flowing through R.
Frequently asked questions
What is the time constant of an RL circuit?+
The time constant is τ = L/R, in seconds when L is in henries and R in ohms. It is the time for the current to reach 63.2% of its final value when energizing, or fall to 36.8% when de-energizing.
How long does an inductor take to fully charge?+
Mathematically the current approaches V/R forever, but after 5 time constants it is within 0.7% of the final value, which engineers treat as fully settled. The 99% point is about 4.6τ.
Why does the inductor voltage start at the full supply voltage?+
Current through an inductor cannot change instantly. At t = 0 the current is zero, so the resistor drops nothing and the entire supply appears across the inductor; as current builds the inductor voltage falls exponentially.
How is an RL circuit different from an RC circuit?+
They are mirror images: in an RC circuit the capacitor voltage rises with τ = RC, while in an RL circuit the inductor current rises with τ = L/R. Increasing R lengthens an RC time constant but shortens an RL one.
What happens when the current to an inductor is suddenly interrupted?+
The inductor tries to keep the current flowing and its voltage rises sharply — often hundreds of volts — until something conducts. That is why relay and solenoid coils need a flyback diode or snubber.