RL Time Constant
The time constant of an RL circuit determines how quickly the current changes after a voltage is applied or removed. Because an inductor opposes sudden changes in current, the current does not instantly reach its maximum value when the circuit is switched on, nor does it instantly fall to zero when switched off. The RL time constant is one of the most important concepts in power electronics, relay circuits, motor control and switching applications.
What Is the Time Constant?
The time constant, represented by the Greek letter τ (tau), is the time required for the current in an RL circuit to reach approximately 63.2% of its final value after the circuit is energised. When the supply is removed, one time constant is also the time required for the current to decrease to approximately 36.8% of its initial value.
Time Constant Formula
The RL time constant depends on the inductance and the total resistance in the circuit.
τ = L / R
| Symbol | Description | Unit |
|---|---|---|
| τ | Time Constant | Seconds (s) |
| L | Inductance | Henries (H) |
| R | Total Resistance | Ohms (Ω) |
Current Rise After Switching On
The current increases exponentially rather than instantly.
| Time | Current Reached |
|---|---|
| 1τ | 63.2% |
| 2τ | 86.5% |
| 3τ | 95.0% |
| 4τ | 98.2% |
| 5τ | 99.3% |
After approximately five time constants, the circuit is generally considered to have reached its steady-state current.
Current Decay After Switching Off
When the supply is disconnected, the magnetic field collapses and the stored energy keeps the current flowing briefly. The current decreases exponentially.
| Time | Remaining Current |
|---|---|
| 1τ | 36.8% |
| 2τ | 13.5% |
| 3τ | 5.0% |
| 4τ | 1.8% |
| 5τ | 0.7% |
Worked Example
An RL circuit contains:
Inductance = 200 mH Resistance = 40 Ω
Calculate the time constant.
L = 0.2 H τ = L / R τ = 0.2 / 40 τ = 0.005 s τ = 5 ms
The current reaches approximately 63% of its final value after 5 milliseconds, and more than 99% after approximately 25 milliseconds.
Factors Affecting the Time Constant
| Factor | Effect |
|---|---|
| Higher Inductance | Longer time constant. |
| Lower Inductance | Shorter time constant. |
| Higher Resistance | Shorter time constant. |
| Lower Resistance | Longer time constant. |
Applications
| Application | Purpose |
|---|---|
| Relay Coils | Determines pull-in and release time. |
| Solenoids | Controls magnetic field build-up. |
| Motor Controllers | Current regulation. |
| Switch-Mode Power Supplies | Energy storage and filtering. |
| Industrial Automation | Timing and switching control. |
Measuring the Time Constant
- Observe the current waveform using an oscilloscope.
- Measure the inductance using an LCR meter.
- Measure the circuit resistance with a multimeter.
- Calculate τ using the formula.
- Compare the measured response with the calculated value.
Common Mistakes
- Confusing the RL time constant with the RC time constant.
- Using only the resistor value instead of the total circuit resistance.
- Assuming the current changes instantly.
- Ignoring the winding resistance of the inductor.
- Using incorrect unit conversions between millihenries and henries.
Interesting Facts
- The Greek letter τ (tau) is commonly used to represent time constants.
- After five time constants, an RL circuit is considered to be at steady state for most practical purposes.
- The larger the inductance, the more slowly the current changes.
- The larger the resistance, the more quickly the current reaches its final value.
- Relay and solenoid operation depends heavily on the RL time constant.
Key Points
- The RL time constant determines how quickly current rises and falls.
- The formula is τ = L / R.
- Current reaches approximately 63.2% of its final value after one time constant.
- After five time constants, the circuit is effectively in steady-state.
- The time constant depends on both inductance and total circuit resistance.