RL Circuits

An RL circuit is a circuit containing both a resistor (R) and an inductor (L). One of the most important features of an RL circuit is that its current cannot change instantly.

When a voltage is applied, the current rises gradually. When the voltage source is removed, the current falls gradually.

This happens because the changing current produces a changing magnetic field in the inductor. The changing magnetic flux creates a self-induced EMF that opposes the change in current.

The basic idea is:

changing current → changing magnetic field → self-induced EMF → opposition to the change in current

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The Components of an RL Circuit

A simple series RL circuit contains:

• a voltage source
• a resistor
• an inductor
• connecting wires
• often a switch

The same current flows through the resistor and inductor because they are connected in series.

The resistor obeys:

V_R = IR

The inductor produces an induced voltage when current changes:

V_L = L ΔI/Δt

The inductor's induced effect opposes the change in current.

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What Happens When the Switch Is Closed?

Imagine an RL circuit connected to a DC battery.

Before the switch closes:

I = 0

When the switch closes, the battery attempts to establish a current.

However, as the current begins increasing:

• the inductor's magnetic field begins increasing
• magnetic flux changes
• self-induced EMF is produced
• the induced EMF opposes the increase in current

Therefore, the current does not immediately reach its final value.

Instead:

current rises gradually

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Why Current Cannot Change Instantly

The induced EMF of an inductor is related to the rate of current change:

|ε| = L|ΔI/Δt|

Imagine the current changed instantly.

Then:

Δt → 0

which would make:

ΔI/Δt

extremely large.

The corresponding induced voltage would also become extremely large.

Therefore, an ideal inductor prevents instantaneous changes in current.

An important circuit rule is:

current through an ideal inductor must change continuously.

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Current Growth

When a DC voltage is first applied to a series RL circuit, current begins at zero and gradually approaches a maximum value.

The current is described by:

I = I_max(1 − e^(−t/τ))

where:

• I = current at time t
• I_max = final steady current
• t = time
• τ = RL time constant
• e ≈ 2.718

The curve rises rapidly at first and then gradually levels off.

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Interpreting the Current-Growth Graph

At:

t = 0

the current is:

I = 0

Initially, the current increases rapidly.

As time passes, the rate of increase becomes smaller.

Eventually:

I → I_max

The curve approaches the final value asymptotically.

In practical calculations, the current is considered very close to its final value after several time constants.

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Final Current

Once the current becomes steady:

ΔI/Δt = 0

Therefore, the self-induced EMF becomes zero.

For an ideal inductor in a DC circuit, the final current is determined by the resistance:

I_max = V/R

where:

• V = supply voltage
• R = total circuit resistance

The inductor strongly affects how quickly the current reaches this value, but an ideal inductor does not change the final steady-state current.

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Example: Final Current

An RL circuit contains:

V = 12 V

R = 6.0 Ω

L = 2.0 H

Once the current becomes steady:

I_max = V/R

I_max = 12/6.0

I_max = 2.0 A

The inductance affects how quickly the current approaches 2.0 A.

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The RL Time Constant

The rate at which current changes is described using the time constant.

For an RL circuit:

τ = L/R

where:

• τ = time constant in seconds (s)
• L = inductance in henries (H)
• R = resistance in ohms (Ω)

The time constant tells us approximately how quickly the circuit responds to a change.

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Meaning of the Time Constant

During current growth, after one time constant:

t = τ

the current has reached approximately:

63% of its final value

Therefore:

I ≈ 0.63I_max

After two time constants:

I ≈ 0.86I_max

After three:

I ≈ 0.95I_max

After four:

I ≈ 0.98I_max

After five:

I ≈ 0.99I_max

After about 5τ, the current is normally treated as essentially steady.

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Current Growth Table

This pattern is useful for interpreting RL current-time graphs without performing detailed exponential calculations.

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Worked Example: Finding the Time Constant

An RL circuit contains:

L = 3.0 H

R = 6.0 Ω

Use:

τ = L/R

τ = 3.0/6.0

τ = 0.50 s

After:

0.50 s

the current has reached approximately 63% of its final value.

After approximately:

5τ = 5(0.50)

5τ = 2.5 s

the current is essentially at its steady value.

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Worked Example: Current After One Time Constant

An RL circuit has a final current:

I_max = 8.0 A

After one time constant:

I ≈ 0.63I_max

I ≈ (0.63)(8.0)

I ≈ 5.0 A

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Effect of Inductance on the Time Constant

From:

τ = L/R

increasing inductance increases the time constant.

Therefore:

larger L → larger τ → slower current change

A larger inductor produces a greater self-induced EMF for the same rate of current change.

It therefore takes longer for the current to reach its steady value.

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Effect of Resistance on the Time Constant

From:

τ = L/R

increasing resistance decreases the time constant.

Therefore:

larger R → smaller τ

However, increasing resistance also reduces the final current because:

I_max = V/R

So resistance affects both:

• the final current
• the time constant

These effects should not be confused.

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Comparing Two RL Circuits

Circuit A:

L = 2.0 H

R = 4.0 Ω

Therefore:

τ_A = 2.0/4.0

τ_A = 0.50 s

Circuit B:

L = 4.0 H

R = 4.0 Ω

Therefore:

τ_B = 4.0/4.0

τ_B = 1.0 s

Circuit B has the greater inductance and therefore responds more slowly.

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Current at Any Time During Growth

The full current-growth equation is:

I = I_max(1 − e^(−t/τ))

This allows the current to be calculated at any time.

Remember:

I_max = V/R

and:

τ = L/R

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Worked Example: Current During Growth

An RL circuit has:

V = 12 V

R = 4.0 Ω

L = 2.0 H

Find the current after:

t = 0.50 s

First calculate the final current:

I_max = V/R

I_max = 12/4.0

I_max = 3.0 A

Now calculate the time constant:

τ = L/R

τ = 2.0/4.0

τ = 0.50 s

Therefore:

t = τ

After one time constant, the current is approximately 63% of its final value:

I ≈ 0.63(3.0)

I ≈ 1.9 A

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Using the Exponential Equation

For the same circuit:

I = I_max(1 − e^(−t/τ))

I = 3.0(1 − e^(−0.50/0.50))

I = 3.0(1 − e⁻¹)

Since:

e⁻¹ ≈ 0.368

I = 3.0(1 − 0.368)

I = 3.0(0.632)

I ≈ 1.90 A

This agrees with the 63% time-constant rule.

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What Happens to the Inductor Voltage?

When the circuit is first switched on, the current is changing most rapidly.

Therefore, the inductor's induced voltage is largest.

As the current approaches its steady value:

• current changes more slowly
• induced voltage decreases

Eventually:

ΔI/Δt = 0

and:

V_L = 0

for an ideal inductor under steady DC conditions.

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Voltage Across the Resistor

The resistor voltage is:

V_R = IR

Immediately after switching on:

I = 0

so:

V_R = 0

As current increases:

V_R increases

Eventually:

I = V/R

so:

V_R = V

for the ideal series RL circuit.

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Voltage Changes During Current Growth

The behaviour can be summarized:

This demonstrates how the inductor initially opposes the establishment of current.

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Kirchhoff's Voltage Law in an RL Circuit

At any moment during current growth:

V = V_R + V_L

Therefore:

V = IR + L ΔI/Δt

The supply voltage is shared between the resistor and the inductor.

Initially, most of the voltage appears across the inductor.

Later, as current becomes steady, most or all of the supply voltage appears across the resistor in the ideal model.

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Energy During Current Growth

While current increases, the inductor stores energy in its magnetic field.

The stored energy is:

E = ½LI²

As current increases:

• magnetic field strengthens
• stored magnetic energy increases

At the final steady current:

E_max = ½LI_max²

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Worked Example: Final Stored Energy

An RL circuit contains:

L = 2.0 H

V = 12 V

R = 6.0 Ω

Final current:

I_max = V/R

I_max = 12/6.0

I_max = 2.0 A

Stored energy:

E = ½LI²

E = ½(2.0)(2.0²)

E = 4.0 J

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Switching the Circuit Off

Suppose an inductor is carrying a steady current and the supply is removed while a closed path remains available for the current.

The current attempts to decrease.

The magnetic field begins collapsing.

This produces a self-induced EMF that acts to maintain the original current.

Therefore:

the current decreases gradually rather than instantly

This process is called current decay.

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Current Decay

For a simple RL decay circuit:

I = I₀e^(−t/τ)

where:

• I = current at time t
• I₀ = initial current
• τ = L/R
• t = time

The current decreases exponentially.

It approaches zero but does not mathematically reach exactly zero in the ideal exponential model.

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Interpreting the Current-Decay Graph

At:

t = 0

the current is:

I = I₀

After one time constant:

I ≈ 0.37I₀

After two:

I ≈ 0.14I₀

After three:

I ≈ 0.050I₀

After four:

I ≈ 0.018I₀

After five:

I ≈ 0.007I₀

So after approximately five time constants, the current is usually treated as essentially zero.

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Current Decay Table

Notice the connection:

During growth after one τ:

63% reached

During decay after one τ:

37% remains

because:

63% + 37% = 100%

approximately.

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Worked Example: Current Decay

An inductor initially carries:

I₀ = 6.0 A

The circuit has:

L = 2.0 H

R = 4.0 Ω

Calculate the time constant:

τ = L/R

τ = 2.0/4.0

τ = 0.50 s

After one time constant:

I ≈ 0.37I₀

I ≈ 0.37(6.0)

I ≈ 2.2 A

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Current After Two Time Constants

Using the same circuit:

t = 2τ

Approximately 14% of the original current remains.

I ≈ 0.14(6.0)

I ≈ 0.84 A

Using the exponential equation gives approximately the same result.

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Growth vs Decay Graphs

The two graphs have different shapes.

Current growth:

• begins at zero
• rises rapidly at first
• gradually levels off
• approaches I_max

Current decay:

• begins at I₀
• falls rapidly at first
• gradually levels off
• approaches zero

Both are exponential curves.

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Reading the Time Constant from a Graph

For a current-growth graph:

1. identify the final current I_max
2. calculate 63% of I_max
3. find where the curve reaches this value
4. read the corresponding time

That time is approximately:

τ

For a current-decay graph:

1. identify the initial current I₀
2. calculate 37% of I₀
3. find where the curve reaches this value
4. read the corresponding time

That time is approximately:

τ

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Example: Finding τ from a Growth Graph

Suppose a graph approaches a final current of:

10 A

Calculate 63%:

0.63(10)

6.3 A

If the graph reaches 6.3 A at:

0.40 s

then:

τ = 0.40 s

If R = 5.0 Ω, then:

τ = L/R

so:

L = τR

L = (0.40)(5.0)

L = 2.0 H

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Example: Finding τ from a Decay Graph

Suppose current begins at:

8.0 A

After one time constant, the current should be:

0.37(8.0)

≈ 3.0 A

If the graph reaches approximately 3.0 A at:

0.25 s

then:

τ ≈ 0.25 s

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Comparing Different Time Constants

A circuit with a small time constant changes current quickly.

Its current-time graph is relatively steep.

A circuit with a large time constant changes current more slowly.

Its graph is more gradual.

Therefore:

larger τ → slower response

smaller τ → faster response

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Effect of Increasing Inductance on the Graph

If resistance remains constant and inductance increases:

τ = L/R

so τ increases.

The current-growth graph becomes more gradual.

The circuit takes longer to approach its final current.

Similarly, during decay, the current remains significant for a longer time.

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Effect of Increasing Resistance on the Graph

If inductance remains constant and resistance increases:

τ = L/R

so τ decreases.

The current responds more quickly.

However, during current growth from a fixed-voltage supply:

I_max = V/R

also decreases.

Therefore, increasing resistance changes both:

• the time scale
• the final current

This is an important point when comparing graphs.

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Complete Growth Problem

A series RL circuit contains:

V = 24 V

R = 8.0 Ω

L = 4.0 H

Find the final current, time constant, and current after one time constant.

Final current:

I_max = V/R

I_max = 24/8.0

I_max = 3.0 A

Time constant:

τ = L/R

τ = 4.0/8.0

τ = 0.50 s

After one time constant:

I ≈ 0.63I_max

I ≈ 0.63(3.0)

I ≈ 1.9 A

After about:

5τ = 2.5 s

the current is essentially at its final value.

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Complete Decay Problem

An RL circuit initially carries:

I₀ = 5.0 A

The circuit has:

L = 1.5 H

R = 3.0 Ω

Time constant:

τ = L/R

τ = 1.5/3.0

τ = 0.50 s

Find the current after:

t = 1.0 s

Since:

1.0 s = 2τ

approximately 14% remains:

I ≈ 0.14(5.0)

I ≈ 0.70 A

Using the exact equation:

I = I₀e^(−t/τ)

I = 5.0e^(−1.0/0.50)

I = 5.0e⁻²

I ≈ 5.0(0.135)

I ≈ 0.68 A

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Finding Time Using the Exponential Equation

Sometimes the unknown is time.

For decay:

I = I₀e^(−t/τ)

Suppose the current falls to half its initial value:

I/I₀ = 0.50

Then:

0.50 = e^(−t/τ)

Taking the natural logarithm:

ln(0.50) = −t/τ

Therefore:

t = −τ ln(0.50)

Since:

ln(0.50) ≈ −0.693

then:

t ≈ 0.693τ

So an RL decay does not take one full time constant to fall to half its initial current.

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Energy During Current Decay

As current decreases, the magnetic energy:

E = ½LI²

also decreases.

The collapsing magnetic field transfers energy back into the circuit.

In a simple resistor-inductor decay circuit, much of this energy eventually becomes thermal energy in the resistor.

Therefore:

magnetic field energy → electrical energy → thermal energy

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Why Switching an Inductor Can Produce a Large Voltage

If the circuit is opened suddenly without a suitable current path, the current attempts to fall very rapidly.

From:

|ε| = L|ΔI/Δt|

a very large rate of change of current can produce a very large induced voltage.

This can cause:

• sparks across switches
• damage to electronic components
• electrical interference

Practical circuits often provide a safe path for the current to decay.

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Flyback Protection

Coils in relays, motors, and solenoids can behave as inductive loads.

When they are switched off, their stored magnetic energy must be released.

A protective component such as a flyback diode can provide a temporary current path.

The current then decreases gradually instead of producing a damaging voltage spike.

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Practical Uses of RL Behavior

RL circuits appear wherever coils and changing currents are present.

Examples include:

• electromagnets
• relays
• solenoids
• electric motors
• transformers
• power supplies
• electronic filters
• switching circuits

In each case, inductance affects how rapidly current can change.

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RL Circuits in Electromagnets

An electromagnet contains a coil with inductance.

When switched on:

• current gradually increases
• magnetic field gradually strengthens

When switched off:

• current gradually decreases if a path remains
• magnetic field collapses
• stored energy is released

The time constant therefore influences how quickly the electromagnet responds.

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RL Circuits in Motors

Motor windings have both:

• resistance
• inductance

They therefore behave partly like RL circuits.

The inductance prevents motor current from changing instantly.

This affects:

• starting behaviour
• current control
• switching
• motor speed control
• electronic drive design

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RL Circuits and Filters

Inductors oppose rapid current changes.

This makes them useful in electrical filters.

Rapidly changing signals can experience stronger inductive effects than slowly changing signals.

RL networks can therefore help:

• reduce electrical noise
• separate signals by frequency
• smooth current
• shape electrical signals

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RL Circuits vs Purely Resistive Circuits

A purely resistive DC circuit responds almost immediately in the ideal model.

Its current is:

I = V/R

An RL circuit behaves differently during switching because the inductor produces self-induced EMF.

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RL Circuits and RC Circuits

RL and RC circuits both show exponential behaviour, but they involve different quantities.

Both systems are important examples of circuits with time-dependent behaviour.

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A Reliable Strategy for RL Problems

First identify:

• V = supply voltage
• R = resistance
• L = inductance
• t = time

Calculate the time constant:

τ = L/R

For current growth, calculate the final current:

I_max = V/R

Then use either the approximate time-constant percentages or:

I = I_max(1 − e^(−t/τ))

For current decay use:

I = I₀e^(−t/τ)

Remember:

growth at 1τ → 63%

decay at 1τ → 37% remaining

After approximately:

5τ

the change is essentially complete.

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Common Mistakes

Thinking current changes instantly in an RL circuit

Inductance opposes changes in current.

Thinking the inductor stops current permanently

The inductor mainly affects changing current. With steady DC, an ideal inductor has zero induced voltage.

Using τ = R/L

For an RL circuit:

τ = L/R

Confusing 63% and 37%

During growth after one τ:

63% of final current has been reached.

During decay after one τ:

37% of initial current remains.

Thinking the time constant is the time required to complete the change

The exponential process is not mathematically complete after one τ. One τ describes a characteristic fraction of the change.

Forgetting to calculate the final current

For a simple DC series RL circuit:

I_max = V/R

Using the growth equation for decay

Growth:

I = I_max(1 − e^(−t/τ))

Decay:

I = I₀e^(−t/τ)

Thinking resistance only affects the time constant

Resistance also affects the final current.

Ignoring stored magnetic energy

During current decay, the magnetic field releases energy back into the circuit.

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Key Terms

RL circuit — circuit containing resistance and inductance.

Resistor — component that opposes current and dissipates electrical energy, commonly as thermal energy.

Inductor — component that stores energy in a magnetic field and opposes changes in current.

Inductance — measure of an inductor's ability to produce self-induced EMF.

Self-induction — production of an EMF caused by a circuit's own changing current.

Self-induced EMF — induced voltage produced when a circuit's own magnetic flux changes.

Steady state — condition in which current has stopped changing with time.

Transient — temporary period during which current or voltage changes from one steady condition toward another.

Time constant — characteristic time describing how quickly an RL circuit responds.

Current growth — exponential increase of current toward its final value after a DC source is applied.

Current decay — exponential decrease of current after the source is removed while a suitable current path remains.

Exponential change — change whose rate depends on how far the system remains from its final state.

Magnetic energy — energy stored in the magnetic field of an inductor.

Flyback diode — protective component providing a current path for an inductive load when it is switched off.

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Key Takeaways

• An RL circuit contains both resistance and inductance.
• Current cannot change instantly through an ideal inductor.
• Self-induction causes the inductor to oppose changes in current.
• When a DC source is connected, current rises gradually toward its final value.
• The final current in a simple ideal series RL circuit is I_max = V/R.
• The RL time constant is τ = L/R.
• Larger inductance produces a larger time constant and slower current response.
• Larger resistance produces a smaller time constant when inductance remains fixed.
• After one time constant during growth, current reaches about 63% of its final value.
• After one time constant during decay, about 37% of the original current remains.
• After approximately five time constants, the transition is normally considered essentially complete.
• Current growth follows I = I_max(1 − e^(−t/τ)).
• Current decay follows I = I₀e^(−t/τ).
• Current-time graphs for RL circuits are exponential rather than straight lines.
• During switch-on, inductor voltage is initially large and decreases toward zero.
• During switch-on, resistor voltage increases as current increases.
• Inductors store energy according to E = ½LI².
• During current decay, stored magnetic energy is transferred back into the circuit.
• Rapidly interrupting an inductive current can produce a large voltage.
• RL behaviour is important in motors, relays, electromagnets, power supplies, filters, and switching circuits.

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Check Your Understanding

1. What is an RL circuit?
2. Explain why current cannot change instantly through an ideal inductor.
3. Describe what happens to the current immediately after a DC source is connected to an RL circuit.
4. A circuit contains L = 4.0 H and R = 8.0 Ω. Calculate its time constant.
5. Explain what a time constant of 0.50 s means during current growth.
6. An RL circuit has a final current of 10 A. Approximately what current flows after one time constant?
7. A current begins at 8.0 A and decays through an RL circuit. Approximately what current remains after one time constant?
8. A 12 V supply is connected to a series RL circuit containing R = 3.0 Ω and L = 1.5 H. Calculate the final current and time constant.
9. For the circuit in Question 8, estimate the current after one time constant.
10. An RL circuit has I_max = 6.0 A and τ = 0.40 s. Calculate the current after 0.80 s using I = I_max(1 − e^(−t/τ)).
11. An inductor initially carries 5.0 A in a circuit with τ = 0.25 s. Calculate the current remaining after 0.50 s.
12. Describe the shape of a current-time graph when an RL circuit is switched on. How would increasing inductance change the graph?
13. Explain how the voltage across the resistor and the voltage across the inductor change from the moment a DC RL circuit is switched on until steady state is reached.
14. Explain where the energy stored in an inductor goes when the current through an RL circuit decreases.
15. Two RL circuits have the same resistance, but Circuit A has twice the inductance of Circuit B. Compare their time constants and explain which circuit's current changes more slowly.