RL Circuits and Time Constant

An RL circuit combines resistance and inductance. Its defining behavior is simple: current cannot change instantaneously. The resistor sets the final current and the inductor sets how quickly that current approaches the final value.

Learning Objectives

By the end of this lesson, you should be able to:

calculate the RL time constant tau = L / R;
predict current rise and decay in a first-order RL circuit;
estimate settling time using the five-time-constant rule;
explain why switching inductive loads requires a safe current path.

The Time Constant

For a simple series resistor and inductor:

$$
\tau=\frac{L}{R}
$$

where:

L is inductance in henries;
R is total series resistance in ohms;
tau is time in seconds.

A larger inductor slows the current change. A larger resistance speeds the current change, but also lowers the final current for a fixed supply voltage.

Worked Example

Given:

L = 10 mH = 0.01 H
R = 100 ohm
V = 12 V

$$
\tau=\frac{0.01}{100}=0.0001\ \text{s}=0.1\ \text{ms}
$$

The final current is:

$$
I_\infty=\frac{V}{R}=\frac{12}{100}=0.12\ \text{A}
$$

After one time constant, current is about 63.2% of final value. After five time constants, it is close enough to final for most practical estimates.

Current Rise

When a step voltage is applied to a series RL circuit:

$$
i(t)=\frac{V}{R}\left(1-e^{-t/\tau}\right)
$$

title "Illustrative RL current rise"
time start=0 end=5 unit=tau divisions=5

STEP: step label="applied voltage step" low=0 high=1 at=0.2 unit=norm color=#2563eb
IL: exponential label="inductor current" from=0 to=1 tau=1 delay=0.2 unit=norm color=#16a34a

marker ONE at=1.2 label="about 63%"
marker FIVE at=5 label="near settled"

This is an illustrative waveform, not a SPICE simulation. It shows the shape of a first-order response.

Current Decay

If the source is removed but a valid current path remains:

$$
i(t)=I_0e^{-t/\tau}
$$

The inductor releases stored magnetic energy. If there is no safe path, the inductor generates whatever voltage is necessary to keep current flowing until something conducts. That "something" may be a diode, a snubber, a TVS diode, transistor avalanche, or an arc.

Try It: RL Time Constant Calculator

Enter any two values to calculate the third.

Why Five Time Constants?

For a rising RL current:

Time	Percent of final current
`1 tau`	`63.2%`
`2 tau`	`86.5%`
`3 tau`	`95.0%`
`4 tau`	`98.2%`
`5 tau`	`99.3%`

The exact curve never mathematically reaches final value, but 5 tau is a useful engineering rule.

Protection Strategies

Method	Common use	Tradeoff
Flyback diode	DC relays, solenoids	low stress, slow release
Zener or TVS clamp	faster coil release	higher voltage stress
RC snubber	AC coils and contacts	dissipates energy as heat
Active clamp	power converters	efficient but more complex

Never open an inductive load and assume the current simply stops. The stored energy must be routed safely.

Common Mistakes

Using only the external resistor and forgetting coil winding resistance.
Treating tau as the time to reach exactly final current.
Omitting flyback protection because the supply voltage is low.
Ignoring the slower release caused by a simple diode clamp.
Confusing capacitor voltage transients with inductor current transients.

Summary

An RL time constant is tau = L / R. It predicts how quickly inductor current rises or decays in a first-order circuit. The same behavior that makes inductors useful in motors, relays, filters, and power converters also makes switching them hazardous without a safe discharge path.