RL Time Constant Formula
Calculate the RL time constant using τ = L/R.
Understand how inductors and resistors control current rise and decay in circuits.
The Formula
The RL time constant describes how quickly current rises or falls in a circuit containing a resistor and an inductor connected in series. Just as the RC time constant governs how fast a capacitor charges through a resistor, the RL time constant governs how fast current builds up through an inductor.
When voltage is first applied to an RL circuit, the inductor opposes the change in current by generating a back-EMF (electromotive force). The current does not jump instantly to its maximum value. Instead, it rises exponentially, reaching approximately 63.2% of its final steady-state value after one time constant. After five time constants (5τ), the current is considered to have reached its full value at 99.3%.
The physics behind this behavior comes from Faraday's law of electromagnetic induction. An inductor stores energy in its magnetic field, and it takes time for this field to build up. A larger inductance means more energy storage capacity and therefore a longer time to reach full current. Conversely, a larger resistance limits the maximum current and causes the circuit to reach steady state more quickly, which is why R appears in the denominator.
When the voltage source is removed and the circuit is allowed to discharge, the current decays exponentially with the same time constant. The inductor tries to maintain the current flow by releasing its stored magnetic energy, which is why inductors can produce dangerous voltage spikes when circuits are suddenly opened. This is the reason flyback diodes are used to protect circuits that switch inductive loads like relays and motors.
The RL time constant is measured in seconds. Since inductance is in henrys (H) and resistance is in ohms (Ω), dividing H by Ω gives seconds, confirming the dimensional correctness of the formula.
Variables
| Symbol | Meaning |
|---|---|
| τ | Time constant (seconds, s) |
| L | Inductance (Henrys, H) |
| R | Resistance (Ohms, Ω) |
Current Rise and Decay Percentages
| Time | Current Rise (% of Imax) | Current Decay (% remaining) |
|---|---|---|
| 1τ | 63.2% | 36.8% |
| 2τ | 86.5% | 13.5% |
| 3τ | 95.0% | 5.0% |
| 4τ | 98.2% | 1.8% |
| 5τ | 99.3% | 0.7% |
Example 1
A 200 mH inductor is connected in series with a 50 Ω resistor and a 12 V supply. What is the time constant and the steady-state current?
τ = L / R = 0.200 H / 50 Ω = 0.004 s
Steady-state current: Imax = V / R = 12 / 50 = 0.24 A
Full current reached after 5τ = 5 × 0.004 = 0.02 s
τ = 4 ms, Imax = 240 mA, full current in 20 ms
Example 2
A relay coil has an inductance of 500 mH and a resistance of 100 Ω. How long does the current take to reach 63.2% of its final value? What is the current after 2τ if the supply is 24 V?
τ = L / R = 0.500 / 100 = 0.005 s = 5 ms
After 1τ (5 ms), current reaches 63.2% of Imax
Imax = 24 / 100 = 0.24 A
After 2τ: I = 0.865 × 0.24 = 0.2076 A
τ = 5 ms, current after 2τ ≈ 208 mA
When to Use It
Use the RL time constant when you need to:
- Calculate how quickly current builds up through an inductive load
- Design snubber circuits or flyback diode protection for switching inductors
- Determine relay or solenoid response times
- Analyse transient behavior in motor drive circuits
- Compare RL and RC circuit behavior in filter design
The full current equations for an RL circuit are:
- Current rise: I(t) = (V/R) × (1 - e-t/τ)
- Current decay: I(t) = (V/R) × e-t/τ