Induction in AC Circuits

Alternating current (AC) is electric current that continuously changes in magnitude and periodically reverses direction.

Because electric current produces a magnetic field, an alternating current produces a magnetic field that also continuously changes in strength and direction.

This makes AC especially useful for electromagnetic induction.

The basic relationship is:

alternating current → changing magnetic field → changing magnetic flux → induced EMF

This principle is fundamental to transformers, electrical generators, power transmission, wireless charging, induction heating, and many other electrical technologies.

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Alternating Current

In a direct-current circuit, conventional current normally flows in one direction.

In an AC circuit, the current repeatedly:

• increases
• decreases
• reaches zero
• reverses direction
• increases in the opposite direction
• returns toward zero
• reverses again

This cycle repeats continuously.

A common AC waveform is sinusoidal.

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Sinusoidal Current

An ideal sinusoidal alternating current can be represented by:

I = I_max sin(2πft)

where:

• I = instantaneous current
• I_max = maximum or peak current
• f = frequency in hertz (Hz)
• t = time in seconds

The current repeatedly changes between:

+I_max and −I_max

The positive and negative signs represent opposite current directions.

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AC and Magnetic Fields

A current-carrying conductor produces a magnetic field.

Therefore, if the current changes, the magnetic field also changes.

For a coil:

larger current → stronger magnetic field

smaller current → weaker magnetic field

reversed current → reversed magnetic field

AC therefore naturally produces a continuously changing magnetic field.

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Magnetic Field Reversal

Consider an AC current passing through a solenoid.

During one half-cycle, the current flows in one direction.

The solenoid might therefore have:

north at one end and south at the other

During the next half-cycle, the current reverses.

The magnetic field also reverses:

south and north exchange ends

This repeated reversal produces a changing magnetic field around the coil.

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Changing Magnetic Flux

Electromagnetic induction depends on changing magnetic flux.

For a uniform magnetic field passing through a flat loop:

Φ = BA cos θ

where:

• Φ = magnetic flux in webers (Wb)
• B = magnetic field strength in teslas (T)
• A = area in m²
• θ = angle between the magnetic field and the normal to the loop

If B changes because of alternating current, then Φ also changes.

This changing flux can induce an EMF in another conductor or coil.

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Faraday's Law

The magnitude of the induced EMF is related to how quickly magnetic flux changes:

|ε| = N|ΔΦ/Δt|

where:

• ε = induced EMF
• N = number of turns
• ΔΦ = change in magnetic flux through each turn
• Δt = time interval

Therefore:

faster flux change → larger induced EMF

This is one reason rapidly changing currents are so useful in electromagnetic systems.

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Lenz's Law

The direction of induced EMF is described by Lenz's law.

The induced effect acts in a direction that opposes the change in magnetic flux that produced it.

Including direction, Faraday's law can be written:

ε = −N ΔΦ/Δt

The negative sign represents Lenz's law.

It is important to remember that the induced effect opposes the change, not necessarily the magnetic field itself.

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AC Producing Induced EMF

Imagine two coils placed close together.

Coil 1 is connected to an AC source.

As AC flows through Coil 1:

1. its current changes
2. its magnetic field changes
3. magnetic flux through Coil 2 changes
4. an EMF is induced in Coil 2

Because the magnetic field changes continuously, the induced EMF also changes continuously.

This is mutual induction.

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Why Steady DC Behaves Differently

Suppose a coil is connected to a steady DC supply.

When it is first switched on:

• current changes
• magnetic field changes
• induction can occur briefly

Once the current becomes constant:

• magnetic field becomes constant
• magnetic flux becomes constant
• induced EMF becomes zero

Therefore, steady DC does not continuously produce electromagnetic induction.

AC does because its current is continually changing.

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AC and Mutual Inductance

For two coils with mutual inductance M:

ε₂ = −M ΔI₁/Δt

where:

• ε₂ = induced EMF in Coil 2
• M = mutual inductance
• ΔI₁/Δt = rate of change of current in Coil 1

An AC current continually has a changing value.

Therefore:

ΔI/Δt continuously changes

and an alternating EMF can be continuously induced in the second coil.

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Frequency and Induction

The frequency of AC tells us how many complete cycles occur each second.

Frequency is measured in:

hertz (Hz)

For example:

50 Hz = 50 complete cycles per second

The period is:

T = 1/f

where T is the time for one complete cycle.

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Worked Example: Period

An alternating voltage has:

f = 50 Hz

Use:

T = 1/f

T = 1/50

T = 0.020 s

Therefore, one complete cycle takes:

20 ms

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Worked Example: Higher Frequency

An AC signal has:

f = 200 Hz

T = 1/f

T = 1/200

T = 0.0050 s

or:

T = 5.0 ms

A higher frequency means the cycle repeats more quickly.

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Why Frequency Affects Induction

For a sinusoidal current:

I = I_max sin(2πft)

A higher frequency means current changes more rapidly.

Since induced EMF depends on the rate of change:

|ε| ∝ |ΔI/Δt|

faster current changes can produce larger induced EMFs when the other relevant factors remain constant.

This is why frequency is an important consideration in transformers, inductive chargers, and electronic power converters.

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Generating Alternating Voltage

Electromagnetic induction can also be used in the opposite direction.

Instead of:

AC → changing magnetic field

we can use:

changing magnetic flux → AC voltage

This is the operating principle of an AC generator.

Mechanical energy is used to continually change magnetic flux through a coil.

Faraday's law then produces an alternating EMF.

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Rotating-Coil Generator

Consider a coil rotating in a uniform magnetic field.

The magnetic flux through the coil depends on its orientation:

Φ = BA cos θ

If the coil rotates at a constant angular speed:

θ = ωt

so:

Φ = BA cos(ωt)

For N turns:

flux linkage = NBA cos(ωt)

As the coil rotates, the magnetic flux repeatedly increases, decreases, and reverses.

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Sinusoidal Induced Voltage

Faraday's law tells us that induced EMF depends on the rate of change of flux.

For an ideal rotating coil, the resulting EMF is sinusoidal:

ε = ε_max sin(ωt)

where:

ε_max = NBAω

and:

• N = number of turns
• B = magnetic field strength
• A = area of each turn
• ω = angular speed

Therefore, increasing any of these quantities increases the maximum generated EMF.

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Factors Affecting Generator Voltage

The maximum generated voltage is:

ε_max = NBAω

Therefore, larger voltage can be produced by:

• increasing the number of turns
• increasing magnetic field strength
• increasing coil area
• rotating the coil faster

These relationships are important in generator design.

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Worked Example: Maximum Generator EMF

A coil has:

N = 200 turns

B = 0.50 T

A = 0.010 m²

ω = 20 rad/s

Use:

ε_max = NBAω

ε_max = (200)(0.50)(0.010)(20)

ε_max = 20 V

The generated voltage varies sinusoidally between:

+20 V and −20 V

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Magnetic Flux and Induced Voltage

A very important relationship exists between magnetic flux and induced voltage.

When magnetic flux is at a maximum or minimum, it is momentarily changing least rapidly.

Therefore:

induced EMF = 0

at those moments.

When magnetic flux passes through zero, it is changing most rapidly.

Therefore:

|induced EMF| is maximum

at those moments.

This creates a quarter-cycle phase difference between flux and induced EMF.

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Maximum Flux Does Not Mean Maximum EMF

This is a common source of confusion.

Faraday's law does not say:

large flux → large EMF

It says:

rapidly changing flux → large EMF

Therefore:

maximum flux → zero instantaneous EMF in steady sinusoidal rotation

and:

zero flux crossing → maximum |EMF|

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Understanding a Sinusoidal Voltage Graph

A sinusoidal voltage graph shows how instantaneous voltage changes with time.

Important features include:

• positive peak
• negative peak
• zero crossings
• amplitude
• period
• frequency

The voltage alternates between positive and negative values.

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Amplitude

The amplitude is the maximum magnitude of the voltage.

It is often written:

V_max

If a voltage varies between:

+10 V and −10 V

then:

V_max = 10 V

The peak-to-peak voltage would be:

20 V

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Period

The period, T, is the time required for one complete cycle.

It can be measured between:

• one positive peak and the next positive peak
• one negative peak and the next negative peak
• equivalent zero crossings moving in the same direction

The unit is seconds.

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Frequency

Frequency and period are related by:

f = 1/T

and:

T = 1/f

A shorter period means a higher frequency.

A longer period means a lower frequency.

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Example: Reading a Sinusoidal Graph

Suppose an AC voltage reaches:

+12 V

and:

−12 V

and one complete cycle takes:

0.025 s

Then:

V_max = 12 V

and:

T = 0.025 s

Frequency:

f = 1/T

f = 1/0.025

f = 40 Hz

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Zero Crossings

A zero crossing occurs when the voltage passes through zero.

For a sinusoidal waveform, there are two zero crossings during each complete cycle.

One occurs as the voltage changes:

negative → positive

and another occurs as it changes:

positive → negative

These crossings mark changes in voltage polarity.

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Positive and Negative Voltage

The positive and negative sections of an AC voltage graph do not mean that the circuit has "negative electricity."

Instead, the sign indicates the direction or polarity relative to the chosen reference.

When voltage changes sign, its polarity reverses.

Similarly, when AC current changes sign, its direction reverses.

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Instantaneous Voltage

The voltage at a particular moment is called the instantaneous voltage.

For a sinusoidal voltage:

V = V_max sin(2πft)

The value depends on where the system is within its cycle.

For example:

• at some moments V = 0
• at others V = +V_max
• later V = 0 again
• later V = −V_max

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Worked Example: Instantaneous Voltage

An AC source has:

V_max = 100 V

f = 50 Hz

Find the voltage at:

t = 0.0050 s

Use:

V = V_max sin(2πft)

V = 100 sin[2π(50)(0.0050)]

V = 100 sin(π/2)

Since:

sin(π/2) = 1

V = 100 V

At this moment, the voltage is at its positive maximum.

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RMS Voltage

AC voltage continually changes, so it is often useful to describe an effective voltage.

For a sinusoidal voltage:

V_rms = V_max/√2

Similarly:

I_rms = I_max/√2

RMS means root mean square.

An RMS voltage produces the same average heating effect in a resistor as an equivalent DC voltage.

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Worked Example: RMS Voltage

A sinusoidal AC source has:

V_max = 100 V

Then:

V_rms = 100/√2

V_rms ≈ 70.7 V

Conversely, if RMS voltage is known:

V_max = √2 V_rms

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Comparing Peak and RMS Values

It is important not to confuse:

• instantaneous voltage
• peak voltage
• peak-to-peak voltage
• RMS voltage

For a sinusoidal voltage:

V_peak-to-peak = 2V_max

and:

V_rms = V_max/√2

These describe different properties of the same waveform.

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Alternating Current in an Inductor

An inductor continuously experiences changing current when connected to AC.

Therefore, it continuously produces a self-induced voltage.

For an ideal inductor:

V_L = L ΔI/Δt

in magnitude terms.

A rapidly changing current produces a larger inductive voltage.

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Inductive Reactance

The opposition of an ideal inductor to sinusoidal AC is described by inductive reactance:

X_L = 2πfL

where:

• X_L = inductive reactance in ohms
• f = frequency
• L = inductance

Therefore:

higher frequency → greater inductive reactance

This occurs because higher-frequency AC changes more rapidly.

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Worked Example: Inductive Reactance

An inductor has:

L = 0.20 H

and operates at:

f = 50 Hz

Use:

X_L = 2πfL

X_L = 2π(50)(0.20)

X_L ≈ 63 Ω

If the frequency doubles while L remains constant, the inductive reactance also doubles.

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AC in Transformers

Transformers are one of the clearest examples of why AC is useful for electromagnetic induction.

An AC current in the primary winding produces:

continuously changing current

↓

continuously changing magnetic field

↓

continuously changing core flux

↓

continuously induced voltage in secondary

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Why Transformers Use Changing Current

A transformer requires changing magnetic flux.

Steady DC does not provide this continuously.

After the switching transient, steady DC produces approximately constant magnetic flux and therefore no continuous transformer action.

AC naturally provides the required changing flux.

This makes AC particularly suitable for transformer operation.

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An Important Qualification About DC

It is more accurate to say that transformers require changing magnetic flux, rather than simply saying they "only work with AC."

Pulsed or electronically switched DC can also create changing flux and operate transformer-like systems.

However:

steady DC cannot sustain normal transformer induction

because its current and magnetic field eventually stop changing.

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Transformer Voltage

For an ideal transformer:

V_s/V_p = N_s/N_p

where:

• V_s = secondary voltage
• V_p = primary voltage
• N_s = secondary turns
• N_p = primary turns

Therefore, mutual induction can be used to increase or decrease alternating voltage.

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AC and Electrical Power Transmission

Transformers make it practical to change AC voltage efficiently.

Electrical power is:

P = VI

For a given power, increasing voltage allows current to decrease.

Power loss in transmission wires is approximately:

P_loss = I²R

Lower current therefore greatly reduces heating losses.

This is one major reason transformer-based AC systems became important for electrical power distribution.

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Step-Up Transformation

A step-up transformer has:

N_s > N_p

and therefore:

V_s > V_p

The voltage increases.

For an ideal transformer, current decreases so that approximately:

V_pI_p = V_sI_s

Power is transferred rather than created.

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Step-Down Transformation

A step-down transformer has:

N_s < N_p

and therefore:

V_s < V_p

The voltage decreases.

Step-down transformers are useful when equipment requires a lower voltage than the supplied AC voltage.

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AC Generators and Transformers Together

Generators and transformers illustrate two connected uses of electromagnetic induction.

A generator:

mechanical motion → changing magnetic flux → alternating EMF

A transformer:

alternating current → changing magnetic flux → induced alternating EMF

Together, these principles form an important foundation of electrical power systems.

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Wireless Power Transfer

Alternating current is also used in many wireless charging systems.

An AC current in a transmitter coil produces a changing magnetic field.

A nearby receiver coil experiences changing magnetic flux.

This produces an induced EMF.

The electrical energy can then be converted into a form suitable for charging a battery.

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Induction Heating

An AC-powered coil can produce a rapidly changing magnetic field.

When a conducting object is placed in this field, currents can be induced inside the material.

These eddy currents can produce thermal energy.

This principle is used in:

• induction cooktops
• industrial furnaces
• metal processing
• manufacturing

Higher-frequency AC is often useful for controlling induction-heating behaviour.

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AC and Electrical Noise Filtering

Because an inductor's reactance depends on frequency:

X_L = 2πfL

inductors can be used to treat different frequencies differently.

High-frequency variations experience greater inductive opposition than lower-frequency variations.

This allows inductors to help:

• suppress electrical noise
• reduce ripple
• filter signals
• control switching interference

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Phase Relationships

In AC circuits, current and voltage do not always reach their maximum values at the same time.

For an ideal inductor, voltage reaches its peaks one-quarter cycle before current reaches its corresponding peaks.

This is often described as:

current lags voltage by 90°

or equivalently:

voltage leads current by 90°

This occurs because inductor voltage depends on the rate of change of current, not simply the current value itself.

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Why the Voltage Is Largest When Current Changes Fastest

Consider a sinusoidal current.

At the current's positive or negative peak:

• current is momentarily not changing
• ΔI/Δt = 0
• induced voltage is zero

When current passes through zero:

• current is changing most rapidly
• |ΔI/Δt| is maximum
• |induced voltage| is maximum

This is similar to the relationship between magnetic flux and induced EMF in a generator.

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Comparing Sinusoidal Quantities

For many induction systems:

This helps explain why related AC quantities can be shifted in phase.

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A Complete Generator Problem

A generator coil has:

N = 500 turns

B = 0.20 T

A = 0.015 m²

f = 50 Hz

First calculate angular speed:

ω = 2πf

ω = 2π(50)

ω ≈ 314 rad/s

Maximum EMF:

ε_max = NBAω

ε_max = (500)(0.20)(0.015)(314)

ε_max ≈ 471 V

The induced voltage therefore varies sinusoidally between approximately:

+471 V and −471 V

For an ideal sinusoid:

V_rms = V_max/√2

V_rms ≈ 471/1.414

V_rms ≈ 333 V

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Reading an AC Graph

When interpreting a sinusoidal voltage graph, use the following process.

First identify the center line:

V = 0

Then determine the maximum displacement from the center line:

V_max

Next measure the time for one complete repeating cycle:

T

Then calculate:

f = 1/T

Finally, identify:

• positive peaks
• negative peaks
• zero crossings
• direction of change
• phase relationships if multiple waves are shown

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Common Mistakes

Thinking AC simply turns on and off

AC normally reverses direction rather than merely switching between on and off.

Thinking the magnetic field stays in one direction

When current reverses, the magnetic field produced by the coil also reverses.

Thinking magnetic flux alone produces EMF

It is changing magnetic flux that produces induced EMF.

Thinking maximum flux means maximum induced voltage

For sinusoidal induction, induced voltage is zero when flux is momentarily at a maximum or minimum.

Confusing period and frequency

Period is time per cycle.

Frequency is cycles per second.

They are related by:

f = 1/T

Thinking a negative AC voltage means negative energy

The sign represents polarity relative to a chosen reference.

Confusing peak voltage with RMS voltage

For a sinusoidal voltage:

V_rms = V_max/√2

Saying transformers can only operate with AC

Transformers require changing magnetic flux. AC provides this naturally, but switched or pulsed currents can also create changing flux.

Thinking higher frequency always means higher voltage

Higher frequency can increase induced EMF when other relevant quantities, such as flux amplitude, are held constant. Real electrical systems can have additional constraints.

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Key Terms

Alternating current (AC) — current that periodically changes magnitude and reverses direction.

Alternating voltage — voltage whose magnitude and polarity change periodically.

Sinusoidal waveform — smooth repeating waveform described by a sine or cosine function.

Cycle — one complete repetition of an alternating waveform.

Frequency — number of complete cycles per second, measured in hertz.

Period — time required for one complete cycle.

Amplitude — maximum magnitude of an alternating quantity measured from zero.

Peak voltage — maximum magnitude of an alternating voltage.

Peak-to-peak voltage — difference between the maximum positive and maximum negative voltage.

RMS voltage — effective AC voltage based on equivalent heating or power in a resistor.

Magnetic flux — measure of magnetic field passing through a surface.

Electromagnetic induction — production of EMF due to changing magnetic flux.

Induced EMF — voltage produced by electromagnetic induction.

Faraday's law — relationship between changing magnetic flux and induced EMF.

Lenz's law — principle stating that the induced effect opposes the change in magnetic flux producing it.

Inductive reactance — opposition an ideal inductor provides to sinusoidal AC.

Phase — position within a repeating cycle.

Phase difference — difference in cycle position between two alternating quantities.

AC generator — device that converts mechanical energy into electrical energy using electromagnetic induction.

Transformer — device that transfers electrical energy between circuits through changing magnetic flux and mutual induction.

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Key Takeaways

• Alternating current continuously changes magnitude and periodically reverses direction.
• A changing current produces a changing magnetic field.
• Reversing current also reverses the magnetic field produced by a coil.
• Changing magnetic fields can produce changing magnetic flux through nearby conductors.
• Changing magnetic flux produces induced EMF according to Faraday's law.
• AC is especially useful for electromagnetic induction because it naturally produces continuously changing magnetic fields.
• Steady DC produces continuous induction only while it is changing, such as during switching.
• AC voltage and current are commonly represented using sinusoidal graphs.
• Frequency measures cycles per second.
• Period measures time per cycle.
• Frequency and period are related by f = 1/T.
• A rotating generator coil can produce sinusoidal alternating voltage.
• For an ideal rotating coil, ε_max = NBAω.
• Induced EMF is largest when magnetic flux changes most rapidly.
• Maximum magnetic flux does not correspond to maximum induced EMF.
• RMS values provide a useful measure of the effective size of sinusoidal AC.
• For sinusoidal voltage, V_rms = V_max/√2.
• Transformers use AC-generated changing magnetic flux to transfer electrical energy between windings.
• Transformers allow AC voltage to be increased or decreased efficiently.
• Inductors provide greater reactance at higher frequencies because X_L = 2πfL.
• In an ideal inductor, current lags voltage by 90°.
• AC induction is central to generators, transformers, wireless charging, induction heating, filters, and electrical power systems.

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Check Your Understanding

1. Define alternating current and explain how it differs from steady direct current.
2. Explain why an alternating current produces a changing magnetic field.
3. Describe how AC flowing through one coil can induce an EMF in a nearby second coil.
4. Explain why steady DC does not produce continuous mutual induction.
5. An AC waveform has a period of 0.020 s. Calculate its frequency.
6. An AC supply operates at 100 Hz. Calculate its period.
7. A sinusoidal voltage varies between +40 V and −40 V. State its peak voltage and peak-to-peak voltage.
8. A sinusoidal source has a peak voltage of 170 V. Calculate its RMS voltage.
9. Explain why the induced EMF in a rotating generator coil is zero when the magnetic flux is at a maximum.
10. A generator coil has N = 100 turns, B = 0.40 T, A = 0.020 m², and ω = 25 rad/s. Calculate its maximum induced EMF.
11. An inductor has L = 0.10 H and is connected to a 60 Hz AC source. Calculate its inductive reactance.
12. Describe the important features you would identify when interpreting a sinusoidal voltage-time graph.
13. Explain why AC is particularly well suited for transformer operation.
14. Explain the phase relationship between current and voltage in an ideal inductor and relate it to the rate of change of current.
15. Compare the roles of electromagnetic induction in an AC generator and a transformer.