RL Time Constant Calculator
Follow current as an ideal series resistor-inductor circuit moves from an initial current toward its DC steady state. Calculate τ, current, inductor voltage, di/dt, magnetic energy, copper loss, threshold time, and the one-through-five-tau current milestones.
Define the energized series path
Enter total series resistance, including winding resistance and other resistance in the active current path.
Current is rising from 0.000 A toward 1.200 A. At 1.600τ it has completed 79.81% of the transition while the ideal inductor still carries 4.846 V.
RL current response equations
An ideal inductor opposes a change in current. When a DC voltage is applied to a series resistor-inductor circuit, the current does not jump immediately to V/R. Instead, it moves exponentially from its initial value toward the steady-state value. Resistance dissipates energy and determines how quickly the magnetic field approaches equilibrium.
Ifinal = V ÷ R
I(t) = Ifinal + (Iinitial − Ifinal)e−t/τ
VL(t) = [V − R Iinitial]e−t/τ
t for p completion = −τ ln(1 − p)
Inductance is converted from millihenries to henries and elapsed time from milliseconds to seconds. With L in henries and R in ohms, τ is seconds. One tau completes 63.212% of the current change, leaving 36.788% of the initial-to-final difference. Five tau completes about 99.326%, although an ideal exponential only reaches the final value asymptotically.
The general equation permits a nonzero initial current. If initial current exceeds V/R, the current decays toward the new steady state and inductor voltage reverses polarity under the stated reference convention. A real current cannot be interrupted without another path: a switch opening an inductive circuit can generate a damaging voltage as the magnetic field drives current onward.
Total resistance controls the magnetic settling time
Use the resistance in the complete current path for the interval being analyzed. This can include the inductor winding’s DC resistance, external series resistance, source resistance, switch on-resistance, wiring, current-sense resistance, and a deliberate discharge or flyback path. If the path changes after a switch opens, the decay time constant must be recalculated for the new loop.
Increasing resistance reduces τ because the field’s stored energy is dissipated more quickly, but it also changes steady current when the same source voltage drives the series circuit. It is therefore misleading to discuss response speed without also checking final current, voltage drop, heating, and functional force or flux.
Winding resistance rises with temperature. That can lower steady current and shorten L/R even if inductance were constant. Copper loss creates more heating, which can continue the shift. For relays, solenoids, and electromagnets, use manufacturer coil data and thermal limits rather than treating room-temperature resistance as fixed.
Inductance may not be constant
The calculator assumes one constant inductance. Magnetic components can be nonlinear: inductance varies with current, core permeability, air gap, mechanical position, temperature, frequency, and saturation. A solenoid’s inductance may change as the armature moves. A power inductor can lose inductance near saturation, increasing current slope beyond a constant-L prediction.
Winding capacitance, core loss, skin effect, proximity effect, and leakage flux are omitted. At fast edges or high frequency, an inductor is not a pure L in series with R. Its impedance can exhibit resonance and frequency-dependent loss. Use an appropriate equivalent circuit or measured impedance data for those conditions.
DC source behavior also matters. Current-limited supplies, PWM drivers, semiconductor drops, freewheel diodes, Zener clamps, and active demagnetization create piecewise voltages rather than one constant V. Model each switching state using its actual loop voltage and resistance, or use a switching simulation.
Current and voltage milestones
| Elapsed time | Current transition completed | Inductor-voltage fraction remaining | Physical reading |
|---|---|---|---|
| 0τ | 0% | 100% | Current is continuous; the inductor initially carries the transition voltage. |
| 1τ | 63.212% | 36.788% | The defining RL time-constant point. |
| 2τ | 86.466% | 13.534% | Current is close to final but magnetic change remains measurable. |
| 3τ | 95.021% | 4.979% | Approximately 95% through the ideal transition. |
| 4τ | 98.168% | 1.832% | Within about 2% of final current. |
| 5τ | 99.326% | 0.674% | Often treated as settled for basic estimates. |
The inductor voltage and remaining current error share the same exponential factor in this ideal series model. During turn-on from zero current, most source voltage initially appears across L. As current rises, the resistor drop increases and inductor voltage falls. At DC steady state an ideal inductor is a short, so source voltage appears across resistance and current is V/R.
Stored magnetic energy is one-half L times current squared. It does not decay with the same simple percentage as current because energy depends on current squared. A flyback network must safely receive this energy repeatedly, including worst-case inductance and current. Peak semiconductor voltage and dissipation depend on the clamp path, not just τ.
Worked 250 mH coil example
A 250 mH inductor in series with 20 Ω has τ = 0.250 H ÷ 20 Ω = 0.0125 second, or 12.500 ms. With 24 V applied and zero initial current, the ideal steady current is 1.200 A. After 20.000 ms, or 1.600τ, current reaches approximately 0.958 A, completing 79.81% of the rise.
The inductor still carries about 4.846 V at that instant, and current slope is approximately 19.382 A/s. At switch-on, the slope was 24 V ÷ 0.250 H = 96.000 A/s. The present magnetic energy is about 114.674 mJ, while instantaneous resistance loss is about 18.354 W. Those are point values; energy delivered and heat accumulated require integration through the duty cycle.
Reaching 95% of the current transition takes about 37.447 ms, nearly three tau. Five tau is 62.500 ms and yields about 1.192 A. Component resistance, supply sag, inductance variation, driver voltage, and core behavior can shift real results, so measure the assembled system under intended conditions.
Switch-off and flyback safety
Opening an inductive current path does not make current disappear. The inductor changes its terminal voltage to keep current flowing, and an unprotected switch can see a high transient. A flyback diode gives current a low-voltage recirculation path but can lengthen release time. A Zener, TVS, resistor, active clamp, or snubber can permit higher demagnetization voltage and faster decay, subject to device ratings and electromagnetic compatibility.
The decay resistance and voltage path after turn-off may differ completely from the energized path entered here. A diode’s forward drop makes the decay not exactly the zero-source L/R case, and a clamped constant voltage can create a more nearly linear current decline over part of the event. Use the actual topology and device curves for switch stress, release time, and repetitive energy.
Coils can operate at hazardous voltage, temperature, and stored energy. De-energize equipment, verify absence of voltage, control stored energy, use appropriately rated probes, and follow applicable lockout and electrical safety procedures. This educational calculation does not authorize live work or select protective components.
RL time constant FAQs
Why is RL time constant L divided by R?
Inductance resists current change, while resistance dissipates magnetic energy. More inductance prolongs the transition; more loop resistance shortens it. Dimensional analysis also gives henries divided by ohms as seconds.
Does a larger resistor always make a relay faster?
It shortens L/R for a fixed inductance, but in the energized series path it also reduces final current and coil force. Release speed depends on the turn-off clamp path and mechanical behavior. Use relay or solenoid data and the actual driver topology.
What resistance should include the coil winding?
Enter total resistance in the current loop during the modeled interval: winding, external resistor, switch on-resistance, wiring, source resistance, and sense resistance as relevant. Turn-off can use a different loop and therefore a different time constant.
Why can inductor voltage exceed the supply at turn-off?
Current continuity causes the inductor to reverse terminal voltage as it releases magnetic energy. Without a controlled path, voltage rises until a switch, insulation, arc, or parasitic element conducts. A properly designed flyback or clamp network limits that stress.
Is inductance the same at every current?
Not necessarily. Core permeability, saturation, air gap, mechanical position, temperature, and frequency can change inductance. Manufacturer curves or measurements are needed when nonlinearity materially affects timing, peak current, or stored energy.
Is current fully settled after five tau?
It has completed about 99.326% of an ideal first-order transition, leaving 0.674% of the original current difference. Whether that is sufficiently settled depends on tolerance, measurement accuracy, magnetic behavior, and the functional threshold.
References
These U.S. Department of Energy resources support the inductance, L/R time-constant, stored-energy, and electrical-safety concepts used here. Component data sheets and qualified engineering govern a real coil driver.