RL Time Constant Calculator
Enter inductance and resistance to compute the RL time constant τ=L/R. L=0.1 H, R=100 Ω → τ=1 ms.
Input Data
Results
At a glance:The RL time constant (τ=L/R) is the time scale of current change when an inductor L is in series with a resistor R connected to a voltage source. Derivation: inductor voltage V=L·dI/dt, resistor voltage V_R=I·R, KVL gives V=L·dI/dt+I·R. Solving yields charging I(t)=I∞(1−e^(−t/τ)) with I∞=V/R, and discharging I(t)=I₀·e^(−t/τ). The exponential time constant τ=L/R: after 1τ the current reaches 63.2%, 2τ → 86.5%, 3τ → 95.0%, 5τ → 99.3%. Half-life t½=τ·ln2≈0.693τ. Classic examples: (1) relay L=10 mH, R=100 Ω → τ=100 µs, pull-in delay ≈0.5 ms; (2) motor L=1 mH, R=0.1 Ω → τ=10 ms, inrush current rises slowly; (3) supply filter L=100 µH, R=0.01 Ω → τ=10 ms, slow but low ripple; (4) relay snubber with large τ can arc and damage contacts. The 'time constant' concept was introduced by Heaviside in the 1880s. Applications: (1) relays — delayed pull-in/release; (2) motors — soft start; (3) supply filtering — ripple suppression; (4) signal delay — first-order RC/RL low-pass; (5) inductor snubbing — switch protection.
Formula
Time constant: τ = L/R (s)
Charging current: I(t) = I∞·(1 − e^(−t/τ))
Discharging current: I(t) = I₀·e^(−t/τ)
Half-life: t½ = τ·ln2 ≈ 0.693τ
5τ reaches ~99.3% steady state
$$\tau = \frac{L}{R}, \quad I(t) = I_\infty (1 - e^{-t/\tau}), \quad t_{1/2} = \tau \ln 2 \approx 0.693\tau$$How to Use
- Enter inductance L (H) and resistance R (Ω).
- The tool computes τ=L/R, τ(ms), t½=0.693τ and 5τ.
- Common: 10 mH+100 Ω → 100 µs; 1 H+1 kΩ → 1 ms; 100 mH+10 Ω → 10 ms.
Case Studies
Relay pull-in delay design
12 V relay coil L=10 mH, R=100 Ω → τ=L/R=100 µs.
After power-on I(t)=0.12 A·(1−e^(−t/100µs)); at 2τ=200 µs it reaches 86.5%, at 5τ=500 µs 99.3%.
Pull-in threshold 80% of rated current → delay ≈1.6τ=160 µs; a parallel diode extends release τ=L/(R+R_d) but avoids ~200 V reverse breakdown of the transistor.
Motor soft-start circuit
DC motor L=1 mH, winding R=0.1 Ω → τ=10 ms.
Direct power-on inrush I(t)=V/R·(1−e^(−t/τ)); at 5τ=50 ms reaches V/R=12/0.1=120 A, exceeding rating.
Soft-start series 0.5 Ω: τ=L/(R+r)=1.67 ms speeds build-up but limits peak to 20 A; PWM ramps duty 0→100% within 100 ms (≈6τ), avoiding mechanical and current shock.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.