Calculators

RL Time Constant Calculator

Calculate the time constant of an RL circuit and determine how quickly current rises or falls through an inductor.

What Is an RL Time Constant?

An RL circuit contains a resistor and an inductor. When a DC voltage is applied, the inductor initially opposes the change in current. The current then increases gradually toward its final steady-state value. 

        R
+V ───/\/\/\────LLLL────┐
                        │
                        └──── GND

The time constant of a simple RL circuit is: 

τ = L / R

where:

  • τ = time constant in seconds
  • L = inductance in henries
  • R = resistance in ohms

RL Time Constant Formula

The RL time constant determines how quickly the current approaches its final value. 

L
τ = ─
R

Unlike an RC circuit, where the time constant is R × C, an RL circuit uses the inductance divided by resistance.

RL Time Constant Calculator

Enter inductance and resistance.

Example — 100 mH and 10 Ω

Suppose an RL circuit has a 100 mH inductor and a 10 Ω resistor. 

L = 100 mH
  = 0.1 H

R = 10 Ω

The time constant is: 

τ = L / R

τ = 0.1 / 10

τ = 0.01 s

Therefore: 

τ = 10 ms

Current During RL Charging

When a DC voltage is applied to an initially unenergized RL circuit, the current rises exponentially. 

I(t) = Ifinal × (1 - e^(-t/τ))

The final steady-state current is approximately: 

Ifinal = V / R

After one time constant, the current reaches approximately 63.2% of its final value.

Time Current
0τ 0%
1τ 63.2%
2τ 86.5%
3τ 95.0%
4τ 98.2%
5τ 99.3%

RL Current Calculator

Enter the voltage, resistance, time and time constant.

RL Discharge

When the voltage source is removed and the inductor is allowed to discharge through a resistance, the current decreases exponentially. 

I(t) = I0 × e^(-t/τ)

After one time constant, approximately 36.8% of the original current remains.

Time Current Remaining
0τ 100%
1τ 36.8%
2τ 13.5%
3τ 5.0%
4τ 1.8%
5τ 0.7%

RL Discharge Time Calculator

Enter the time constant and remaining current.

Inductor Voltage During Switching

An inductor opposes changes in current. Its voltage is related to the rate of change of current: 

V = L × di/dt

A rapid change in current can therefore produce a substantial voltage across the inductor.

This is especially important when switching inductive loads such as relays, solenoids, motors and transformers.

RL Cutoff Frequency

A simple RL network also has a characteristic frequency: 

fc = R / (2πL)

Because: 

τ = L / R

the relationship can also be written as: 

fc = 1 / (2πτ)

RL Cutoff Frequency Calculator

Enter inductance and resistance.

Example — RL Cutoff Frequency

For a 100 mH inductor and 10 Ω resistor: 

L = 0.1 H

R = 10 Ω

fc = R / (2πL)

fc = 10 / (2π × 0.1)

fc ≈ 15.92 Hz

The cutoff frequency is approximately 15.9 Hz.

Energy Stored in an Inductor

The energy stored in an inductor is: 

E = ½LI²

where:

  • E = energy in joules
  • L = inductance in henries
  • I = current in amperes

The stored magnetic energy is released when the inductor current is forced to decrease.

Inductor Energy Calculator

Enter inductance and current.

RL Circuits in Electronics

RL circuits are used in many electronic systems, particularly where inductive loads or filtering are involved.

  • Relay coils
  • Solenoids
  • Motor windings
  • Electromagnets
  • RL filters
  • Chokes
  • Power-supply inductors
  • Switch-mode converters
  • Speaker crossover networks

Relay Coil Example

Suppose a relay coil has: 

L = 200 mH

R = 40 Ω

Its time constant is: 

τ = L / R

τ = 0.2 / 40

τ = 0.005 s

τ = 5 ms

After approximately 5 ms, the coil current has reached 63.2% of its final value during energization.

After approximately five time constants, the current is very close to its final steady-state value.

RL Circuit and Flyback Voltage

When current through an inductor is interrupted rapidly, the inductor attempts to maintain current flow. This can produce a high voltage. 

V = L × di/dt

The faster the current is interrupted, the larger the resulting voltage can become.

This is why a flyback diode is commonly used across DC relay and solenoid coils. 

       +V
        │
       Coil
        │
        ├────|<|────┐
        │  Diode    │
        │           │
       Switch       │
        │           │
       GND──────────┘

The diode provides a path for the inductive current after the switch turns off and helps limit the voltage spike.

RL Circuit vs RC Circuit

Property RL Circuit RC Circuit
Energy storage Magnetic field Electric field
Time constant τ = L/R τ = RC
Stored energy E = ½LI² E = ½CV²
DC steady-state ideal component Short circuit Open circuit
Reactance XL = 2πfL XC = 1/(2πfC)

Common Mistakes

  • Using τ = R × L instead of τ = L/R.
  • Using millihenries as if they were henries.
  • Using kilohms as if they were ohms.
  • Ignoring the resistance of the inductor winding.
  • Ignoring additional circuit resistance.
  • Ignoring the voltage spike produced when current is interrupted.
  • Confusing inductive reactance with resistance.
  • Ignoring saturation in practical inductors.

Key Points

  • The RL time constant is τ = L/R.
  • After one time constant, current reaches approximately 63.2% of its final value.
  • After five time constants, current is approximately 99.3% of its final value.
  • During discharge, 36.8% of the initial current remains after one time constant.
  • The RL cutoff frequency is fc = R/(2πL).
  • An inductor stores energy according to E = ½LI².
  • Rapidly switching an inductive load can produce a high voltage spike.
  • Real circuits contain additional resistance and parasitic effects.

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