Alternators and Generators

An electrical generator is a device that converts mechanical energy into electrical energy using electromagnetic induction.

Generators do not create energy. Instead, they transform energy from another source:

mechanical energy → electrical energy

The mechanical energy may originally come from:

• falling or moving water
• wind
• steam turbines
• internal combustion engines
• human motion
• ocean currents or tides

The underlying physics is Faraday's law of electromagnetic induction:

changing magnetic flux → induced EMF

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Electromagnetic Induction in a Generator

A generator requires relative motion between:

• a conductor
• a magnetic field

This motion causes the magnetic flux linking the conductor or coil to change.

According to Faraday's law:

ε = −N ΔΦ/Δt

where:

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

The negative sign represents Lenz's law.

The faster the magnetic flux changes, the larger the induced EMF.

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Energy Conversion

A generator requires an external force to keep its moving parts rotating.

For example, a wind turbine uses:

kinetic energy of wind → rotational mechanical energy → electrical energy

A hydroelectric generator uses:

gravitational potential energy of water → kinetic energy → rotational mechanical energy → electrical energy

A fuel-powered generator uses:

chemical energy → thermal/mechanical energy → electrical energy

Energy is transformed rather than created.

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The Basic Generator Principle

A simple generator can be made from:

• a coil of conducting wire
• a magnetic field
• a rotating shaft
• a system for connecting the rotating coil to an external circuit

As the coil rotates, its orientation relative to the magnetic field changes.

Therefore, the magnetic flux through the coil changes.

This produces an induced voltage.

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Magnetic Flux Through the Coil

For a coil in a uniform magnetic field:

Φ = BA cos θ

where:

• Φ = magnetic flux through one turn
• B = magnetic field strength
• A = area of the coil
• θ = angle between the magnetic field and the normal to the coil

If the coil rotates, θ continually changes.

Therefore:

magnetic flux continually changes

and an EMF is induced.

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Rotating at Constant Speed

If the coil rotates with constant angular velocity:

θ = ωt

where:

• ω = angular velocity in rad/s
• t = time

The flux through each turn becomes:

Φ = BA cos(ωt)

For N turns, the total flux linkage is:

NΦ = NBA cos(ωt)

The flux therefore changes sinusoidally as the coil rotates.

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Generating Alternating Voltage

Faraday's law states that induced EMF depends on the rate of change of magnetic flux.

For a coil rotating at constant angular speed:

ε = ε_max sin(ωt)

where:

ε_max = NBAω

This means the voltage alternates between:

+ε_max and −ε_max

The result is an alternating voltage.

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Why the Voltage Reverses

During the first half-turn, one side of the coil moves through the magnetic field in one direction.

During the next half-turn, its motion through the field reverses.

The direction of the induced EMF therefore reverses.

As the coil continues rotating:

positive voltage → zero → negative voltage → zero → positive voltage

This produces AC.

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Maximum and Zero Generated Voltage

The generated voltage is not largest when magnetic flux is largest.

The important quantity is the rate of change of flux.

When flux is at its maximum or minimum:

rate of flux change = 0

so:

ε = 0

When flux passes through zero:

rate of flux change is maximum

so:

|ε| = ε_max

This distinction is essential when understanding generators.

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One Complete Rotation

During one complete rotation of a simple two-pole generator coil:

• magnetic flux completes one cycle
• induced voltage completes one cycle

At constant rotational speed, the output therefore has a regular repeating waveform.

For a simple ideal generator, this waveform is sinusoidal.

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Factors Affecting Generated Voltage

From:

ε_max = NBAω

the maximum generated voltage depends on:

• number of turns N
• magnetic field strength B
• coil area A
• angular speed ω

Therefore, generator output can be increased by:

• adding more turns
• using a stronger magnetic field
• increasing coil area
• rotating faster

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Worked Example: Maximum EMF

A generator has:

N = 200 turns

B = 0.40 T

A = 0.015 m²

ω = 25 rad/s

Use:

ε_max = NBAω

ε_max = (200)(0.40)(0.015)(25)

ε_max = 30 V

The ideal generated voltage varies between:

+30 V and −30 V

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Effect of Rotational Speed

Suppose the rotational speed doubles while N, B, and A remain constant.

Since:

ε_max ∝ ω

doubling ω doubles the maximum induced EMF.

Faster rotation also causes the waveform to repeat more frequently.

Therefore, rotational speed can affect both:

• generated voltage
• output frequency

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Alternators

An alternator is a generator designed to produce alternating current.

The term is commonly used for generators in:

• vehicles
• power stations
• portable generating systems
• wind turbines
• industrial equipment

In practical alternators, it is often the magnetic field that rotates, while the main output windings remain stationary.

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Major Components of an Alternator

Important alternator components include:

• rotor
• stator
• magnetic field source
• shaft
• bearings
• output terminals
• cooling system

Depending on the design, it may also contain:

• slip rings
• brushes
• rectifiers
• voltage regulators

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Rotor

The rotor is the rotating part of the machine.

In many alternators, the rotor produces the magnetic field.

The rotor may use:

• permanent magnets
• electromagnets

As the rotor turns, its magnetic field moves relative to the stationary windings.

This produces changing magnetic flux through the windings.

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Stator

The stator is the stationary part of the alternator.

It commonly contains coils of conducting wire.

As the rotor's magnetic field rotates:

• flux through the stator coils changes
• EMF is induced
• electrical energy becomes available at the output

Keeping the high-power output windings stationary can make practical generators easier to connect and cool.

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Rotor and Stator Interaction

The rotor and stator work together.

Rotor:

creates a rotating magnetic field.

Stator:

contains conductors through which the magnetic flux changes.

The sequence is:

rotor turns → magnetic field rotates → stator flux changes → EMF is induced

This is the basic operating principle of many modern alternators.

---

Slip Rings

In a simple classroom AC generator where the coil itself rotates, slip rings can connect the rotating coil to the external circuit.

Each end of the coil connects to a separate continuous conducting ring.

Stationary brushes maintain contact with the rings.

Because the connections do not swap every half-turn, the external voltage reverses naturally.

The result is:

alternating voltage

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Brushes

Brushes provide electrical contact between stationary and rotating components in some generator designs.

Traditionally, brushes may be made from carbon-based materials.

They press against:

• slip rings
• or a commutator

depending on the generator design.

Because brushes experience friction and wear, many modern systems are designed to minimize their use where possible.

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AC Generators

A basic AC generator uses:

• electromagnetic induction
• rotating mechanical motion
• a magnetic field
• a conductor or coil

Its output reverses polarity periodically.

In a simplified rotating-coil generator:

slip rings maintain the connection

while the induced voltage itself reverses.

The result is AC.

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DC Generators

A DC generator also uses electromagnetic induction.

Inside the rotating coil, the induced EMF naturally reverses every half-turn.

However, a split-ring commutator reverses the coil's connection to the external circuit every half-turn.

This causes the external output to remain in one polarity.

The result is pulsating DC rather than sinusoidal AC.

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Split-Ring Commutator

A split-ring commutator consists of two conducting segments separated electrically from each other.

As the coil rotates, the segments exchange contact with the brushes every half-turn.

At the same time that the induced EMF in the rotating coil reverses, the external connections also reverse.

The two reversals compensate for each other.

Therefore, external current continues in the same general direction.

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AC Generator vs DC Generator

Both operate using electromagnetic induction.

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AC and DC Output Graphs

An ideal simple AC generator produces a sinusoidal voltage:

positive → zero → negative → zero → repeat

A simple DC generator using a split-ring commutator produces a waveform that remains one polarity:

zero → positive maximum → zero → positive maximum → repeat

This is called pulsating DC.

Additional circuitry can smooth the output if steadier DC is required.

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Alternator vs Generator

The word generator is a broad term.

A generator can produce:

• AC
• DC

An alternator specifically produces alternating electrical output internally.

Therefore:

all alternators are generators

but:

not all generators are alternators

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Car Alternators

Cars provide a familiar example.

The engine turns the alternator mechanically.

The alternator generates AC internally.

However, a car's battery and many vehicle electrical systems require DC.

Therefore, the alternator contains rectifier diodes that convert the generated AC into DC.

The energy pathway is approximately:

fuel → engine rotation → alternator rotation → AC generation → rectification → DC electrical energy

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Rectification

Rectification is the process of converting AC into DC.

Semiconductor diodes allow current to pass primarily in one direction.

Modern automotive alternators use diodes instead of a mechanical split-ring commutator to produce DC output for the vehicle.

This is generally more practical for modern vehicle electrical systems.

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Voltage Regulation

Generator speed and electrical load can change.

Without control, this could cause undesirable changes in output voltage.

A voltage regulator helps maintain a suitable output voltage.

In an automotive alternator, the regulator can control the rotor's magnetic field.

This helps maintain appropriate output despite changes in:

• engine speed
• electrical demand
• battery condition

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Generators in Power Stations

Large power stations use generators driven by turbines.

The turbine provides mechanical rotation.

Different energy sources can drive the turbine.

Examples include:

• steam
• falling water
• moving air
• flowing gas

The generator converts the turbine's rotational mechanical energy into electrical energy.

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Fossil-Fuel Power Stations

In a conventional thermal power station:

chemical energy in fuel

↓

thermal energy

↓

steam movement

↓

turbine rotation

↓

generator rotation

↓

electrical energy

The generator itself performs the final mechanical-to-electrical conversion.

---

Nuclear Power Stations

A nuclear power station also commonly uses a steam turbine and generator.

The major difference is the original energy source.

The pathway is approximately:

nuclear energy → thermal energy → steam → turbine → generator → electrical energy

The generator operates using the same electromagnetic principles as generators in many other power stations.

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Hydroelectric Generators

In a hydroelectric power station:

gravitational potential energy of water

↓

kinetic energy of flowing water

↓

turbine rotation

↓

generator

↓

electrical energy

Large generators can therefore produce electricity without combustion.

---

Wind Turbine Generators

Wind turbines use moving air to rotate blades.

The energy pathway is:

wind kinetic energy → blade rotation → shaft rotation → generator → electrical energy

Some turbines use gearboxes to change rotational speed.

Others use direct-drive generator systems.

In either case, electromagnetic induction converts mechanical energy into electrical energy.

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Portable Generators

Portable generators usually use an internal combustion engine to rotate a generator.

The energy pathway is:

chemical energy in fuel → mechanical energy → electrical energy

They can provide temporary electrical power for:

• construction sites
• emergency backup
• outdoor activities
• remote locations

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Bicycle Generators

A small bicycle generator can use wheel rotation to move a magnet relative to a coil.

This produces changing magnetic flux and therefore an induced voltage.

The electrical energy can power lights or other small devices.

The rider provides the mechanical energy.

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Hand-Crank Generators

A hand-crank generator converts muscular work into electrical energy.

The user rotates a handle.

The rotating mechanism changes magnetic flux through a coil.

This induces an EMF.

Applications can include:

• emergency radios
• educational demonstrations
• flashlights
• portable emergency chargers

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Regenerative Braking

Electric motors can often operate in reverse as generators.

During regenerative braking in an electric vehicle:

• moving wheels rotate the motor-generator
• mechanical energy is converted into electrical energy
• electrical energy is transferred back toward the battery

The energy pathway is:

vehicle kinetic energy → mechanical rotation → electrical energy → stored battery energy

This reduces the amount of energy that must be dissipated entirely as heat by friction brakes.

---

Lenz's Law in Generators

Lenz's law explains why generating electricity requires mechanical work.

As current is drawn from a generator, the induced current produces magnetic effects that oppose the change responsible for the induction.

This creates an opposing torque.

Therefore, the driving system must continue doing mechanical work.

More electrical power demanded from the generator generally requires more mechanical power supplied to it.

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Why a Generator Becomes Harder to Turn

Suppose a hand-crank generator is disconnected from a load.

Only a small current may flow.

Now connect a lamp.

More current flows.

The resulting magnetic effects produce a stronger opposing torque.

The user must push harder to maintain the same rotational speed.

This demonstrates conservation of energy:

greater electrical energy output requires greater mechanical energy input

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Generator Power

Electrical power is:

P = VI

where:

• P = electrical power
• V = voltage
• I = current

If a generator supplies:

V = 120 V

and:

I = 5.0 A

then:

P = VI

P = (120)(5.0)

P = 600 W

The mechanical input power must be at least this large and, in a real generator, must be larger because of losses.

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Generator Efficiency

Generator efficiency can be calculated using:

efficiency = useful electrical power output / mechanical power input × 100%

For example, suppose:

mechanical input power = 5000 W

electrical output power = 4500 W

Then:

efficiency = 4500/5000 × 100%

efficiency = 90%

The remaining energy is transferred mainly into unwanted forms such as thermal energy.

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Energy Losses

Real generators are not perfectly efficient.

Energy can be lost through:

• electrical resistance in windings
• eddy currents
• magnetic hysteresis
• bearing friction
• air resistance
• vibration
• sound
• heating

Engineering design attempts to minimize these losses.

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Reducing Eddy-Current Losses

Changing magnetic fields can produce unwanted eddy currents in conducting core materials.

These currents produce heating.

Generators and transformers can use specially designed magnetic cores to reduce these losses.

Methods can include:

• laminated magnetic materials
• electrically resistive magnetic materials
• appropriate core geometry

Reducing unwanted eddy currents improves efficiency.

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Three-Phase Alternators

Large electrical generators commonly produce three-phase AC rather than a single alternating voltage.

Three separate alternating voltages are produced with phase differences between them.

Three-phase systems provide important advantages for:

• power transmission
• industrial machinery
• large electric motors
• efficient power generation

The underlying principle remains electromagnetic induction.

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Generator Frequency

The frequency of generated AC depends on:

• rotational speed
• number of magnetic poles

Faster rotation generally increases the frequency if the pole arrangement remains unchanged.

Large power-grid generators must maintain carefully controlled frequency.

This requires precise control of their rotational operation.

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Comparing a Generator and a Motor

Generators and motors use closely related electromagnetic principles but perform opposite energy transformations.

Many machines can operate as either a motor or generator under appropriate conditions.

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Alternators in Modern Energy Systems

Alternators and generators remain essential because they allow many forms of mechanical energy to become useful electrical energy.

They connect electricity generation with:

• wind energy
• hydroelectricity
• thermal power stations
• nuclear power
• vehicle systems
• backup power
• regenerative braking

Although the machines vary enormously in size, their operation ultimately depends on the same physical principle:

changing magnetic flux produces EMF

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A Reliable Strategy for Generator Problems

For rotating-coil generator problems, first identify:

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

For maximum EMF use:

ε_max = NBAω

For frequency and period:

f = 1/T

and:

T = 1/f

For electrical power:

P = VI

For efficiency:

efficiency = output/input × 100%

Always check whether the problem asks for:

• instantaneous voltage
• maximum voltage
• RMS voltage
• power
• frequency
• efficiency

These quantities are not interchangeable.

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Worked Example: Changing Generator Design

A generator initially has:

N = 100 turns

B = 0.20 T

A = 0.010 m²

ω = 40 rad/s

Maximum EMF:

ε_max = NBAω

ε_max = (100)(0.20)(0.010)(40)

ε_max = 8.0 V

Suppose the number of turns doubles while everything else remains constant.

Since:

ε_max ∝ N

the new maximum EMF becomes:

16 V

If the angular speed also doubles:

ε_max ∝ Nω

so the output becomes:

32 V

compared with the original 8.0 V.

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Worked Example: Generator Efficiency

A generator receives:

12.0 kW

of mechanical power and produces:

10.2 kW

of electrical power.

Efficiency:

efficiency = 10.2/12.0 × 100%

efficiency = 85%

Power lost:

12.0 − 10.2

power lost = 1.8 kW

This energy is mainly transferred to unwanted thermal, sound, and mechanical effects.

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Common Mistakes

Thinking generators create energy

Generators convert mechanical energy into electrical energy.

Thinking magnetic flux alone produces voltage

The magnetic flux must change.

Thinking maximum flux produces maximum EMF

Maximum EMF occurs when flux is changing most rapidly.

Confusing alternators with all generators

An alternator is specifically associated with alternating electrical generation.

Confusing slip rings and split-ring commutators

Slip rings are associated with simple AC generator models. Split-ring commutators mechanically rectify the output of simple DC generators.

Thinking a car alternator sends AC directly to the battery

The alternator generates AC internally, which is rectified to DC before charging the battery.

Thinking faster rotation only affects frequency

Faster rotation can also increase maximum induced EMF.

Ignoring energy losses

Real generators are less than 100% efficient.

Thinking the generator turns itself

An external energy source must provide mechanical input.

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Key Terms

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

Alternator — generator that produces alternating electrical output.

Electromagnetic induction — production of EMF caused by changing magnetic flux.

Rotor — rotating part of an electrical machine.

Stator — stationary part of an electrical machine.

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

Induced EMF — voltage produced by changing magnetic flux.

Slip rings — continuous conducting rings used to maintain electrical contact with rotating components in some AC generator designs.

Brushes — stationary electrical contacts that touch rotating rings or commutator segments.

Split-ring commutator — mechanical device that reverses external connections every half-turn in a simple DC generator.

Rectifier — device or circuit that converts AC into DC.

Diode — semiconductor component that conducts preferentially in one direction and can be used for rectification.

Voltage regulator — system used to control generator output voltage.

Turbine — rotating machine that extracts mechanical energy from moving fluid.

Alternating voltage — voltage that periodically reverses polarity.

Pulsating DC — unidirectional voltage or current whose magnitude changes with time.

Lenz's law — principle stating that induced effects oppose the change that produced them.

Regenerative braking — process that uses a motor as a generator to recover some of a moving vehicle's kinetic energy.

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Key Takeaways

• Generators convert mechanical energy into electrical energy.
• They operate using electromagnetic induction.
• Changing magnetic flux through a conductor produces an induced EMF.
• Rotating a coil or magnetic field can produce continuous changes in magnetic flux.
• A rotating coil in a uniform magnetic field can produce sinusoidal alternating voltage.
• For an ideal rotating coil, ε_max = NBAω.
• Increasing turns, magnetic field strength, coil area, or rotational speed can increase maximum generated EMF.
• An alternator is a generator that produces alternating electrical output.
• The rotor is the rotating part of an alternator.
• The stator is the stationary part.
• Many practical alternators rotate a magnetic field inside stationary output windings.
• Simple AC generator models use slip rings.
• Simple DC generators use a split-ring commutator to produce unidirectional external output.
• Modern vehicle alternators generate AC internally and use diodes to rectify it into DC.
• Generators require continuous mechanical energy input.
• Lenz's law explains the opposing torque that appears when electrical energy is drawn from a generator.
• More electrical output requires more mechanical input.
• Generators are used in wind turbines, hydroelectric stations, thermal and nuclear power stations, vehicles, portable generators, and emergency equipment.
• Motors can sometimes operate as generators, allowing regenerative braking.
• Real generators lose some energy through resistance, friction, eddy currents, hysteresis, and heating.
• Alternators and generators are fundamental technologies for converting many different energy resources into usable electrical energy.

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Check Your Understanding

1. What energy transformation occurs in an electrical generator?
2. Explain how electromagnetic induction allows a generator to produce an EMF.
3. Why does rotating a coil in a magnetic field produce an alternating voltage?
4. State four factors that affect the maximum EMF generated by a rotating coil.
5. A generator has N = 250 turns, B = 0.30 T, A = 0.020 m², and ω = 40 rad/s. Calculate its maximum EMF.
6. Explain why the induced EMF is zero when magnetic flux through a rotating coil is at its maximum.
7. Identify the roles of the rotor and stator in a typical alternator.
8. Explain the difference between slip rings and a split-ring commutator.
9. Compare the electrical outputs of a simple AC generator and a simple DC generator.
10. Explain why a modern car alternator requires a rectifier before its output can charge the battery.
11. A generator supplies 240 V at 15 A. Calculate its electrical output power.
12. A generator receives 20 kW of mechanical power and produces 17 kW of electrical power. Calculate its efficiency.
13. Explain why a generator becomes more difficult to turn when a larger electrical load is connected.
14. Describe the complete energy transformation that occurs when a wind turbine generates electricity.
15. Compare an electric motor and a generator, explaining how the direction of energy conversion differs between them.

Transformers

A transformer is an electrical device that transfers electrical energy from one circuit to another using mutual induction.

Transformers can change alternating voltage without requiring a direct electrical connection between the input and output circuits.

The basic process is:

AC in primary coil → changing magnetic field → changing magnetic flux in core → induced AC voltage in secondary coil

Transformers are essential in electrical power systems because they allow voltages to be increased or decreased efficiently.

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Main Parts of a Transformer

A basic transformer contains:

• primary coil
• secondary coil
• magnetic core

The primary coil is connected to the input AC supply.

The secondary coil provides the output voltage.

The magnetic core helps guide magnetic flux from the primary coil through the secondary coil.

The coils are normally electrically separate but magnetically coupled.

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The Primary Coil

When alternating current flows through the primary coil:

• current continuously changes
• the magnetic field changes
• the direction of the magnetic field periodically reverses
• magnetic flux through the core changes

This changing magnetic flux is necessary for transformer operation.

A steady magnetic field would not continuously induce a voltage in the secondary.

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The Magnetic Core

The transformer core provides a low-reluctance path for magnetic flux.

It helps concentrate the changing magnetic field so that a large fraction of the magnetic flux produced by the primary links the secondary.

This improves magnetic coupling between the coils.

Common transformer core materials are selected to have useful magnetic properties while minimizing energy losses.

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The Secondary Coil

The changing magnetic flux from the primary passes through the secondary coil.

According to Faraday's law:

ε = −N ΔΦ/Δt

Therefore, changing flux through the secondary produces an induced EMF.

Because the magnetic flux repeatedly changes direction and magnitude, the induced voltage in the secondary is also alternating.

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Mutual Induction

Transformer operation is an application of mutual induction.

Mutual induction occurs when:

changing current in one circuit produces an induced EMF in another circuit

The primary current creates the changing magnetic field.

The secondary experiences the resulting changing magnetic flux.

Therefore:

primary circuit → magnetic coupling → secondary circuit

No direct conducting connection between the two windings is required in an ordinary transformer.

---

Why Transformers Require Changing Flux

Suppose steady DC is connected to a transformer.

Immediately after switching:

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

Once the current becomes steady:

ΔI/Δt = 0

The magnetic field becomes approximately constant.

Therefore:

ΔΦ/Δt = 0

and continuous induced secondary voltage stops.

Transformers therefore require changing magnetic flux.

AC naturally provides this.

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AC Transformer Operation

During one part of the AC cycle:

• primary current increases in one direction
• magnetic flux increases
• voltage is induced in the secondary

Later:

• primary current decreases
• magnetic flux changes
• the secondary voltage changes

During the opposite half-cycle:

• primary current reverses
• magnetic field reverses
• induced secondary voltage reverses

The output is therefore also AC.

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Transformer Turns Ratio

The voltage produced in each winding depends on the number of turns.

For an ideal transformer:

V_s/V_p = N_s/N_p

where:

• V_s = secondary voltage
• V_p = primary voltage
• N_s = number of turns in secondary
• N_p = number of turns in primary

The ratio:

N_s/N_p

is called the turns ratio.

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Understanding the Turns Ratio

If:

N_s > N_p

then:

V_s > V_p

The transformer increases voltage.

If:

N_s < N_p

then:

V_s < V_p

The transformer decreases voltage.

If:

N_s = N_p

then ideally:

V_s = V_p

The voltage is unchanged.

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Step-Up Transformers

A step-up transformer increases voltage.

Therefore:

V_s > V_p

and:

N_s > N_p

The secondary has more turns than the primary.

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Worked Example: Step-Up Transformer

A transformer has:

N_p = 200 turns

N_s = 1000 turns

V_p = 24 V

Use:

V_s/V_p = N_s/N_p

V_s/24 = 1000/200

V_s/24 = 5

V_s = 120 V

V_s = 120 V

The transformer increases the voltage from 24 V to 120 V.

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Step-Down Transformers

A step-down transformer decreases voltage.

Therefore:

V_s < V_p

and:

N_s < N_p

The secondary has fewer turns than the primary.

---

Worked Example: Step-Down Transformer

A transformer has:

N_p = 1200 turns

N_s = 200 turns

V_p = 240 V

Use:

V_s/V_p = N_s/N_p

V_s/240 = 200/1200

V_s/240 = 1/6

V_s = 40 V

V_s = 40 V

The transformer reduces the voltage from 240 V to 40 V.

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Finding the Number of Turns

The transformer equation can be rearranged to find an unknown number of turns.

Suppose:

V_p = 120 V

V_s = 12 V

N_p = 600 turns

Find N_s.

Use:

V_s/V_p = N_s/N_p

12/120 = N_s/600

0.10 = N_s/600

N_s = 60

N_s = 60 turns

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Finding Primary Turns

Suppose:

V_p = 50 V

V_s = 200 V

N_s = 800 turns

Use:

V_s/V_p = N_s/N_p

200/50 = 800/N_p

4 = 800/N_p

N_p = 200

N_p = 200 turns

This is a step-up transformer.

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Identifying the Transformer Type

A quick way to identify transformer type is to compare the number of turns.

More secondary turns than primary turns:

N_s > N_p

→ step-up

Fewer secondary turns than primary turns:

N_s < N_p

→ step-down

You can also compare the voltages.

V_s > V_p → step-up

V_s < V_p → step-down

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Voltage Ratio and Turns Ratio

Suppose:

N_s/N_p = 4

Then:

V_s/V_p = 4

The secondary voltage is four times the primary voltage.

If:

N_s/N_p = 0.25

then:

V_s/V_p = 0.25

The secondary voltage is one-quarter of the primary voltage.

The voltage changes by the same factor as the number of turns.

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Energy Conservation in an Ideal Transformer

An ideal transformer has no energy losses.

Therefore:

input power = output power

Electrical power is:

P = VI

So for an ideal transformer:

V_pI_p = V_sI_s

where:

• V_p = primary voltage
• I_p = primary current
• V_s = secondary voltage
• I_s = secondary current

This relationship follows from conservation of energy.

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Voltage and Current Trade-Off

A transformer does not create additional electrical power.

If voltage increases, current must decrease.

If voltage decreases, current can increase.

Therefore:

step-up voltage → step-down current

and:

step-down voltage → step-up current

for an ideal transformer supplying a load.

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Current Ratio

From:

V_pI_p = V_sI_s

and:

V_s/V_p = N_s/N_p

we obtain:

I_s/I_p = N_p/N_s

Notice that the current ratio is the inverse of the turns ratio.

This is important.

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Step-Up Transformer Current

For a step-up transformer:

V_s > V_p

but:

I_s < I_p

For example, if voltage increases by a factor of 10, the ideal current decreases by a factor of 10.

Power remains unchanged.

---

Step-Down Transformer Current

For a step-down transformer:

V_s < V_p

but:

I_s > I_p

For example, if voltage decreases by a factor of 5, the ideal current can increase by a factor of 5.

Again, the transformer does not create energy.

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Worked Example: Current in an Ideal Transformer

A transformer has:

V_p = 120 V

I_p = 2.0 A

V_s = 24 V

Use:

V_pI_p = V_sI_s

(120)(2.0) = (24)I_s

240 = 24I_s

I_s = 10 A

I_s = 10 A

The voltage decreased by a factor of 5.

The current increased by a factor of 5.

---

Worked Example: Step-Up Voltage and Current

A transformer receives:

V_p = 50 V

I_p = 4.0 A

and produces:

V_s = 200 V

Input power:

P_p = V_pI_p

P_p = (50)(4.0)

P_p = 200 W

For an ideal transformer:

P_s = 200 W

Therefore:

I_s = P_s/V_s

I_s = 200/200

I_s = 1.0 A

Voltage increased by a factor of 4 while current decreased by a factor of 4.

---

Why Transformers Cannot Create Energy

Suppose a transformer increases voltage from:

100 V → 1000 V

This does not mean the transformer produces ten times as much energy.

If the ideal voltage increases by a factor of 10, current decreases by a factor of 10.

For example:

Primary:

100 V × 10 A = 1000 W

Secondary:

1000 V × 1 A = 1000 W

The power remains:

1000 W

Energy has been transferred and transformed, not created.

---

Ideal vs Real Transformers

An ideal transformer assumes:

• no electrical resistance
• no magnetic flux leakage
• no eddy-current losses
• no hysteresis losses
• perfect magnetic coupling

Therefore:

P_input = P_output

Real transformers are not perfect.

Therefore:

P_output < P_input

---

Transformer Efficiency

Efficiency can be calculated using:

efficiency = P_output/P_input × 100%

A well-designed transformer can have high efficiency, especially at its intended operating conditions.

---

Worked Example: Transformer Efficiency

A transformer receives:

P_input = 500 W

and supplies:

P_output = 450 W

Efficiency:

efficiency = 450/500 × 100%

efficiency = 90%

Power lost:

500 − 450

P_loss = 50 W

This lost energy is mostly transferred into thermal energy.

---

Copper Losses

Transformer windings are made from conducting wire.

Real wire has resistance.

Therefore, some electrical energy becomes thermal energy.

The power loss in a winding can be described by:

P_loss = I²R

These losses are often called copper losses.

Reducing winding resistance can improve efficiency.

---

Eddy-Current Losses

The changing magnetic flux in the core can induce circulating currents inside conducting core material.

These are eddy currents.

Eddy currents produce unwanted heating.

One way to reduce them is to construct the core from thin insulated layers called laminations.

---

Why Laminations Reduce Eddy Currents

A solid conducting core can allow relatively large circulating current loops.

Laminating the core:

• divides it into thin sections
• interrupts large current paths
• increases resistance to circulating currents
• reduces eddy-current heating

This improves transformer efficiency.

---

Hysteresis Losses

The transformer core is repeatedly magnetized in opposite directions as AC changes direction.

The magnetic domains inside the material repeatedly rearrange.

Some energy is lost during this process.

This is called hysteresis loss.

Suitable magnetic materials are chosen to reduce this loss.

---

Flux Leakage

Ideally, all magnetic flux produced by the primary would link the secondary.

In reality, some magnetic flux does not pass through both windings.

This is called leakage flux.

Flux leakage reduces effective magnetic coupling.

Transformer designs attempt to keep the primary and secondary strongly magnetically coupled.

---

Heating

Copper losses, eddy currents, and hysteresis all contribute to heating.

Large transformers therefore require effective cooling.

Cooling methods may include:

• natural air cooling
• forced-air cooling
• insulating cooling fluids
• radiators
• heat exchangers

Managing heat is an important part of transformer design.

---

Why Transformers Are Important for Power Transmission

Transformers are extremely important in electrical power grids.

Electrical power is:

P = VI

For a fixed amount of transmitted power:

higher V → lower I

Transmission-line heating is approximately:

P_loss = I²R

Therefore, reducing current greatly reduces energy loss in transmission wires.

---

Example: Why High Voltage Helps

Suppose:

P = 100,000 W

If electricity is transmitted at:

V = 1000 V

then:

I = P/V

I = 100,000/1000

I = 100 A

Suppose instead it is transmitted at:

V = 10,000 V

I = 100,000/10,000

I = 10 A

Current has decreased by a factor of 10.

Since:

P_loss = I²R

the resistive loss for the same wire resistance decreases by a factor of:

10² = 100

This demonstrates why high-voltage transmission is so useful.

---

Transformers in the Electrical Grid

Electrical power systems use multiple transformers.

A simplified sequence is:

generator → step-up transformer → high-voltage transmission → step-down transformers → consumers

Near the generator, voltage is increased.

Near users, voltage is reduced to levels appropriate for distribution and equipment.

---

Step-Up Transformers in Power Stations

Generators produce electrical power at a particular voltage.

A step-up transformer increases this voltage before long-distance transmission.

This allows the same power to be transmitted with lower current.

Lower current reduces:

I²R heating losses

in transmission lines.

---

Step-Down Transformers

High transmission voltages are unsuitable for direct use by most equipment.

Step-down transformers progressively reduce voltage during electrical distribution.

This makes the electrical supply appropriate for:

• homes
• businesses
• schools
• factories
• electronic devices

Different systems require different voltage levels.

---

Transformers in Chargers and Power Supplies

Many electrical devices require voltages different from the available supply.

Transformers can be part of systems that:

• reduce voltage
• provide electrical isolation
• transfer power
• prepare voltage for further conversion

Modern electronic chargers often use high-frequency switching circuits and relatively small transformers.

---

Why High-Frequency Transformers Can Be Small

The induced voltage depends on how rapidly magnetic flux changes.

At higher operating frequencies, the flux changes more rapidly.

This can allow transformers designed for high-frequency switching power supplies to be much smaller than traditional low-frequency transformers handling comparable power.

This is one reason modern chargers can be compact.

---

Electrical Isolation

A transformer can transfer energy without a direct electrical connection between the primary and secondary windings.

This can provide galvanic isolation.

Isolation can be useful for:

• electrical safety
• protecting sensitive electronics
• separating circuit grounds
• measurement equipment
• medical and industrial systems

Not every transformer configuration provides isolation; for example, an autotransformer uses a shared winding.

---

Isolation Transformers

An isolation transformer may have approximately equal primary and secondary voltages.

For example:

N_s ≈ N_p

so:

V_s ≈ V_p

Its main purpose is therefore not necessarily to change voltage.

Instead, it provides electrical separation between circuits.

This shows that transformers can have functions beyond simply stepping voltage up or down.

---

Audio and Signal Transformers

Transformers can also transfer changing electrical signals.

They may be used for:

• impedance matching
• signal isolation
• audio equipment
• communication circuits

These transformers operate using the same principle of mutual induction.

---

Instrument Transformers

High-power electrical systems can contain voltages and currents that are too large to measure directly with ordinary instruments.

Special transformers can produce smaller proportional quantities for:

• measurement
• monitoring
• protection systems

Examples include current transformers and voltage transformers.

---

Welding Transformers

Some welding equipment requires:

• relatively low voltage
• very high current

A step-down transformer can reduce voltage while increasing available current.

The large current produces intense heating at the welding point.

This is an excellent example of the voltage-current trade-off in transformer operation.

---

Doorbells and Low-Voltage Systems

Some low-voltage devices use transformers to reduce supply voltage.

Applications can include:

• doorbells
• control circuits
• lighting systems
• older low-voltage power adapters

The transformer allows equipment to operate at a more suitable voltage.

---

Microwave Oven Transformers

Traditional microwave ovens may use transformers to produce a much higher voltage required by the magnetron circuitry.

This is an example of a step-up application.

Such circuits can involve dangerously high voltages and are not suitable for experimentation.

---

Transformers and Wireless Charging

Traditional transformers use a magnetic core to strongly couple nearby windings.

Wireless charging uses a related principle with physically separated coils.

The transmitter coil produces changing magnetic flux.

The receiver coil experiences this changing flux and develops an induced EMF.

Both technologies rely on mutual induction.

---

Transformer Frequency

Transformers are designed for particular operating frequencies or frequency ranges.

Changing frequency affects:

• magnetic flux
• core behaviour
• losses
• heating
• efficiency

Using a transformer far outside its designed frequency can cause poor performance or damage.

---

Magnetic Saturation

Transformer cores can only support a limited magnetic flux density before approaching saturation.

When saturation occurs:

• the core no longer responds proportionally
• effective inductive behaviour decreases
• primary current can increase greatly
• overheating can occur

Transformer design therefore considers voltage, frequency, turns, and core properties together.

---

A Complete Transformer Problem

A transformer has:

V_p = 240 V

N_p = 1200 turns

N_s = 100 turns

The primary current is:

I_p = 0.50 A

First calculate secondary voltage:

V_s/V_p = N_s/N_p

V_s/240 = 100/1200

V_s = 20 V

V_s = 20 V

For an ideal transformer:

V_pI_p = V_sI_s

(240)(0.50) = (20)I_s

120 = 20I_s

I_s = 6.0 A

I_s = 6.0 A

The transformer reduces voltage by a factor of 12 and increases current by a factor of 12.

Input power:

P_p = (240)(0.50)

P_p = 120 W

Output power:

P_s = (20)(6.0)

P_s = 120 W

Energy is conserved in the ideal model.

---

A Real Transformer Problem

Suppose the transformer above actually produces:

V_s = 20 V

I_s = 5.4 A

Output power:

P_s = V_sI_s

P_s = (20)(5.4)

P_s = 108 W

Input power was:

120 W

Efficiency:

efficiency = 108/120 × 100%

efficiency = 90%

Power lost:

120 − 108

P_loss = 12 W

---

A Reliable Strategy for Transformer Problems

First identify:

• V_p
• V_s
• N_p
• N_s
• I_p
• I_s

For voltage and turns:

V_s/V_p = N_s/N_p

For an ideal transformer's power:

V_pI_p = V_sI_s

For current ratio:

I_s/I_p = N_p/N_s

For a real transformer's efficiency:

efficiency = P_output/P_input × 100%

Then determine whether the transformer is:

step-up: V_s > V_p and N_s > N_p

or:

step-down: V_s < V_p and N_s < N_p

Finally, check that the result makes physical sense.

If voltage increases in an ideal transformer, current should decrease.

---

Common Mistakes

Thinking transformers create energy

Transformers transfer energy. An ideal transformer conserves energy.

Confusing primary and secondary

The primary receives input. The secondary provides output.

Reversing the turns ratio

Use:

V_s/V_p = N_s/N_p

Keep secondary quantities together and primary quantities together.

Thinking step-up means more current

A step-up transformer increases voltage but decreases current in the ideal loaded case.

Thinking step-down means less power

An ideal step-down transformer has the same input and output power; voltage decreases while current increases.

Thinking transformers work continuously with steady DC

Continuous transformer action requires changing magnetic flux.

Assuming real transformers are 100% efficient

Real transformers lose energy through resistance, eddy currents, hysteresis, leakage, and other effects.

Thinking laminations increase eddy currents

Laminations are used to reduce unwanted eddy currents.

Ignoring units

Use volts for voltage, amperes for current, watts for power, and turns as a simple count.

---

Key Terms

Transformer — device that transfers electrical energy between circuits through mutual induction.

Primary coil — winding connected to the input electrical supply.

Secondary coil — winding in which the output voltage is induced.

Mutual induction — induction of an EMF in one circuit caused by changing current and magnetic flux from another circuit.

Magnetic core — material that provides a path for magnetic flux and improves coupling between windings.

Turns ratio — ratio of the number of turns in the secondary to the number in the primary.

Step-up transformer — transformer that produces a higher secondary voltage than primary voltage.

Step-down transformer — transformer that produces a lower secondary voltage than primary voltage.

Ideal transformer — theoretical transformer with no energy losses.

Efficiency — fraction or percentage of input power converted into useful output power.

Copper loss — energy loss caused by electrical resistance in transformer windings.

Eddy current — circulating current induced in conducting material by changing magnetic flux.

Lamination — thin insulated core layer used to reduce eddy currents.

Hysteresis loss — energy loss caused by repeated magnetization and demagnetization of a magnetic material.

Leakage flux — magnetic flux that does not effectively link both windings.

Galvanic isolation — electrical separation between circuits with no direct conductive path.

Magnetic saturation — condition where increasing magnetizing influence produces relatively little additional magnetization in a core.

---

Key Takeaways

• Transformers operate using mutual induction.
• Alternating primary current creates changing magnetic flux.
• Changing magnetic flux induces an alternating voltage in the secondary.
• A magnetic core helps couple flux between the windings.
• Transformer voltage depends on the turns ratio.
• The ideal transformer equation is V_s/V_p = N_s/N_p.
• A step-up transformer has more secondary turns and produces higher secondary voltage.
• A step-down transformer has fewer secondary turns and produces lower secondary voltage.
• An ideal transformer conserves power: V_pI_p = V_sI_s.
• Increasing voltage requires decreasing current in an ideal transformer.
• Decreasing voltage allows current to increase.
• Transformers do not create electrical energy.
• Real transformers lose some energy through resistance, eddy currents, hysteresis, leakage, and heating.
• Laminated cores help reduce eddy-current losses.
• Transformers make efficient high-voltage electrical transmission possible.
• High transmission voltage allows lower current for the same power.
• Lower current greatly reduces I²R transmission losses.
• Transformers are used in power grids, chargers, power supplies, welding equipment, measurement systems, isolation systems, and electronic equipment.
• High-frequency transformer operation allows compact transformers in many modern switching power supplies.
• Transformers are one of the most important practical applications of electromagnetic induction.

---

Check Your Understanding

1. Explain how mutual induction allows a transformer to transfer electrical energy from the primary to the secondary.
2. Why must the magnetic flux in a transformer change with time?
3. Identify the primary coil, secondary coil, and magnetic core, and describe the function of each.
4. A transformer has N_p = 500 turns and N_s = 2000 turns. Is it a step-up or step-down transformer? Explain.
5. A transformer has V_p = 120 V, N_p = 600 turns, and N_s = 150 turns. Calculate V_s.
6. A transformer changes 20 V to 100 V. If the primary contains 200 turns, calculate the required number of secondary turns.
7. An ideal transformer receives 240 V at 2.0 A and produces 24 V. Calculate the secondary current.
8. Explain why the secondary current of an ideal step-up transformer is lower than its primary current.
9. A transformer receives 800 W and produces 720 W. Calculate its efficiency.
10. Identify three major sources of energy loss in a real transformer and explain how at least one can be reduced.
11. Explain why transformer cores are often laminated.
12. A power line must transmit the same amount of power at either 10 kV or 100 kV. Compare the currents and explain why the higher transmission voltage reduces energy loss.
13. Explain why connecting steady DC directly to a conventional transformer does not produce continuous secondary voltage and may cause problems.
14. Compare a step-up and step-down transformer in terms of turns, voltage, and current.
15. Describe three practical applications of transformers and explain why a transformer is useful in each application.

Resonance in LC Circuits

An LC circuit contains an inductor (L) and a capacitor (C) connected together. These two components can repeatedly exchange stored energy.

• A capacitor stores energy in an electric field.
• An inductor stores energy in a magnetic field.

Under suitable conditions, energy moves back and forth between these two fields, producing electrical oscillations.

At the circuit's resonant frequency, this exchange occurs naturally and efficiently. LC resonance is extremely important in radio, television, wireless communication, filters, oscillators, and many other electronic systems.

---

The LC Circuit

The two main components are:

Capacitor (C)

Stores energy in an electric field.

Inductor (L)

Stores energy in a magnetic field.

The capacitor's stored energy is:

E_C = ½CV²

where:

• C = capacitance
• V = voltage across the capacitor

The inductor's stored energy is:

E_L = ½LI²

where:

• L = inductance
• I = current through the inductor

These two forms of stored energy can continually change into one another.

---

Beginning with a Charged Capacitor

Imagine that the capacitor is initially fully charged.

At this moment:

• capacitor voltage is maximum
• charge on the capacitor is maximum
• current is zero
• magnetic field around the inductor is zero

Most or all of the circuit's stored energy is in the capacitor's electric field.

Therefore:

electric energy = maximum

magnetic energy = zero

---

The Capacitor Begins to Discharge

When the capacitor is connected to the inductor, it begins to discharge.

Current starts flowing through the inductor.

As current increases:

• capacitor charge decreases
• capacitor voltage decreases
• electric-field energy decreases
• magnetic field around the inductor becomes stronger
• magnetic-field energy increases

Energy is being transferred:

electric field → magnetic field

---

Maximum Current

Eventually, the capacitor becomes momentarily uncharged.

At this instant:

V = 0

so:

E_C = ½CV² = 0

However, current through the inductor is now at its maximum.

Therefore:

E_L = ½LI²

is maximum.

At this stage:

• electric-field energy is minimum
• magnetic-field energy is maximum
• current is maximum

The circuit's stored energy has moved from the capacitor into the inductor.

---

Why the Current Does Not Immediately Stop

When the capacitor reaches zero charge, current does not simply stop.

The inductor opposes changes in current.

Its collapsing magnetic field produces an induced EMF that keeps current flowing.

As a result, the capacitor begins charging again — but with the opposite polarity.

Energy now moves:

magnetic field → electric field

---

The Capacitor Charges in Reverse

As the inductor's magnetic field decreases:

• current decreases
• magnetic energy decreases
• charge builds on the capacitor
• capacitor voltage increases in the opposite direction
• electric-field energy increases

Eventually:

• current becomes zero
• magnetic field becomes zero
• capacitor charge reaches maximum again

But the capacitor's polarity is now opposite to its original polarity.

---

The Cycle Repeats

The capacitor then begins discharging again in the opposite direction.

The complete process is:

charged capacitor

↓

current increases

↓

magnetic field strengthens

↓

capacitor becomes uncharged

↓

current reaches maximum

↓

magnetic field collapses

↓

capacitor charges with opposite polarity

↓

current reaches zero

↓

cycle repeats

This produces an electrical oscillation.

---

Energy During One Complete Cycle

A useful way to visualize an ideal LC cycle is:

The process then repeats in the opposite direction.

---

Conservation of Energy

In an ideal LC circuit, there is no resistance.

Therefore, no energy is lost.

Total energy remains constant:

E_total = E_C + E_L

so:

E_total = ½CV² + ½LI²

At different moments, the energy is divided differently between the electric and magnetic fields.

But the total remains constant.

---

Maximum Electric Energy

When the capacitor is fully charged:

I = 0

Therefore:

E_L = 0

and:

E_total = ½CV_max²

All stored energy is electric-field energy.

---

Maximum Magnetic Energy

When the capacitor is momentarily uncharged:

V = 0

Therefore:

E_C = 0

and:

E_total = ½LI_max²

All stored energy is magnetic-field energy.

Because total energy is conserved:

½CV_max² = ½LI_max²

---

Worked Example: Energy Transfer

A capacitor has:

C = 20 μF

and is initially charged to:

V = 10 V

Convert:

20 μF = 20 × 10⁻⁶ F

Initial energy:

E = ½CV²

E = ½(20 × 10⁻⁶)(10²)

E = 1.0 × 10⁻³ J

E = 0.0010 J

In an ideal LC circuit, when the capacitor becomes completely discharged, this same energy is stored in the inductor's magnetic field.

---

Worked Example: Maximum Current

Suppose the circuit from the previous example has:

L = 0.050 H

At maximum current:

½LI_max² = 0.0010 J

So:

I_max² = 2(0.0010)/0.050

I_max² = 0.040

I_max = √0.040

I_max = 0.20 A

---

Oscillation and Simple Harmonic Motion

The behaviour of an ideal LC circuit is mathematically similar to simple harmonic motion.

Mechanical systems can exchange energy between:

• kinetic energy
• potential energy

An LC circuit exchanges energy between:

• magnetic-field energy
• electric-field energy

A mass-spring system and an LC circuit therefore have important mathematical similarities.

---

Mechanical Analogy

Consider a mass attached to a spring.

At maximum displacement:

• speed = 0
• elastic potential energy = maximum
• kinetic energy = zero

At equilibrium:

• speed = maximum
• kinetic energy = maximum
• elastic potential energy = minimum

Similarly, in an LC circuit:

maximum capacitor charge → current = 0

and:

zero capacitor charge → current = maximum

---

Resonance

Every ideal LC circuit has a natural frequency at which it tends to oscillate.

This is called its resonant frequency.

The resonant frequency is:

f₀ = 1/(2π√(LC))

where:

• f₀ = resonant frequency in hertz
• L = inductance in henries
• C = capacitance in farads

The values of L and C determine the frequency.

---

Effect of Inductance

From:

f₀ = 1/(2π√(LC))

increasing L increases the denominator.

Therefore:

larger L → lower resonant frequency

Similarly:

smaller L → higher resonant frequency

The relationship is:

f₀ ∝ 1/√L

when C remains constant.

---

Effect of Capacitance

Increasing C also increases the denominator.

Therefore:

larger C → lower resonant frequency

and:

smaller C → higher resonant frequency

The relationship is:

f₀ ∝ 1/√C

when L remains constant.

---

Worked Example: Resonant Frequency

An LC circuit contains:

L = 0.020 H

C = 5.0 μF

Convert:

C = 5.0 × 10⁻⁶ F

Use:

f₀ = 1/(2π√(LC))

f₀ = 1/[2π√((0.020)(5.0 × 10⁻⁶))]

LC = 1.0 × 10⁻⁷

√LC ≈ 3.16 × 10⁻⁴

Therefore:

f₀ ≈ 1/[2π(3.16 × 10⁻⁴)]

f₀ ≈ 503 Hz

---

Worked Example: Changing Capacitance

Suppose the capacitance in an LC circuit increases by a factor of 4 while L remains unchanged.

Since:

f₀ ∝ 1/√C

then:

f_new/f_old = 1/√4

f_new/f_old = 1/2

Therefore:

f_new = ½f_old

Quadrupling the capacitance halves the resonant frequency.

---

Worked Example: Changing Inductance

Suppose inductance increases by a factor of 9.

Since:

f₀ ∝ 1/√L

then:

f_new/f_old = 1/√9

f_new/f_old = 1/3

Therefore:

f_new = ⅓f_old

---

Finding Capacitance from Resonant Frequency

Starting with:

f₀ = 1/(2π√(LC))

the equation can be rearranged:

C = 1/(4π²f₀²L)

Suppose:

f₀ = 1000 Hz

L = 0.010 H

Then:

C = 1/[4π²(1000²)(0.010)]

C ≈ 2.53 × 10⁻⁶ F

or:

C ≈ 2.53 μF

---

Finding Inductance

The resonant-frequency equation can also be rearranged:

L = 1/(4π²f₀²C)

This allows engineers to select an inductor that produces a desired resonant frequency with a particular capacitor.

---

Resonance in a Driven Circuit

An LC circuit can also be driven by an external alternating voltage.

If the frequency of the external source is close to the circuit's natural frequency, resonance occurs.

At resonance, energy transfer between the source and circuit can become particularly effective.

The circuit can respond much more strongly near its resonant frequency than at frequencies far from resonance.

---

Resonance and Frequency Selection

This frequency-dependent response allows an LC circuit to distinguish between signals of different frequencies.

Suppose many frequencies are present simultaneously.

A tuned LC circuit can be designed so that:

desired signal frequency ≈ resonant frequency

The desired frequency then produces a much stronger response than many unwanted frequencies.

This is the basis of tuning.

---

Tuned Circuits

A tuned circuit is designed to respond strongly to a selected frequency or narrow range of frequencies.

LC circuits are especially useful for tuning because their resonant frequency can be controlled by changing:

• L
• C

The relationship is:

f₀ = 1/(2π√(LC))

Changing either component changes the frequency selected by the circuit.

---

Variable Capacitors

Some traditional radio tuning systems use a variable capacitor.

Turning the tuning control changes C.

Since:

f₀ = 1/(2π√(LC))

changing C changes the resonant frequency.

The receiver can therefore be tuned to different stations.

---

Example: Selecting a Radio Station

Suppose a receiver's antenna picks up signals at several frequencies.

The signals might contain frequencies such as:

• f₁
• f₂
• f₃
• f₄

The tuning circuit is adjusted so:

f₀ = f₃

The circuit responds strongly to f₃ while responding less strongly to the other frequencies.

This allows the receiver to select the desired signal.

---

Resonance in Radio Receivers

Radio waves induce electrical signals in an antenna.

The antenna may receive signals from many transmitters simultaneously.

A tuned circuit helps select the desired carrier frequency.

The process can be represented as:

many radio signals reach antenna

↓

electrical signals are produced

↓

LC circuit selects desired frequency

↓

receiver processes selected signal

↓

information is recovered

---

Resonance in Communication Systems

Modern communication systems must separate many signals sharing the electromagnetic environment.

Different signals can be assigned different frequencies.

Resonant and frequency-selective circuits help electronic systems distinguish among them.

Applications include:

• radio
• television
• wireless communication
• radar
• satellite communication
• some sensor systems

---

Transmitters

Resonant circuits are also useful in transmitters.

A transmitter needs to generate electrical oscillations at a controlled frequency.

LC circuits can form part of an oscillator that produces or stabilizes high-frequency electrical signals.

These electrical oscillations can then be used to produce electromagnetic waves.

---

Carrier Waves

Many communication systems use a high-frequency carrier wave.

Information is added to or encoded onto this carrier.

The carrier must have a carefully controlled frequency.

Resonant circuits can help generate or select particular frequencies.

The receiver can then tune to the appropriate frequency to recover the desired signal.

---

Filters

LC circuits can also be used as electrical filters.

Filters can:

• pass certain frequencies
• reduce unwanted frequencies
• separate frequency ranges
• suppress noise
• select communication channels

Different arrangements of inductors and capacitors can produce different filtering behaviour.

---

Band-Pass Behaviour

A band-pass filter allows a selected range of frequencies to pass more effectively than frequencies outside that range.

Resonant circuits can be used in band-pass systems.

This is useful when a receiver needs to accept signals near a desired frequency while rejecting others.

---

Rejecting Unwanted Frequencies

A circuit can also be designed to strongly reduce a particular unwanted frequency.

Such frequency-selective behaviour is useful for:

• reducing interference
• removing noise
• separating channels
• improving signal quality

The exact behaviour depends on how the inductor, capacitor, resistor, source, and load are arranged.

---

Real LC Circuits

A perfectly ideal LC circuit does not exist.

Real wires and components have resistance.

Therefore, some electrical energy is continually converted into thermal energy.

Instead of oscillating forever, an undriven real circuit's oscillations gradually decrease.

This is called damping.

---

Damped Oscillations

In a real circuit:

electric energy ↔ magnetic energy

still occurs.

However, during each cycle some energy becomes thermal energy.

Therefore:

total oscillation energy decreases

and the amplitude gradually becomes smaller.

---

Resistance and Resonance

Resistance affects how sharply a circuit responds around resonance.

Lower resistance generally allows:

• stronger oscillations
• less energy loss per cycle
• sharper frequency selection

Greater resistance generally causes:

• more damping
• greater energy loss
• broader and weaker resonance

This becomes particularly important in RLC circuits, which contain resistance, inductance, and capacitance.

---

Quality Factor

The sharpness of resonance is often described by the quality factor, or Q factor.

A high-Q resonant circuit generally has:

• relatively low energy loss
• narrow resonance
• strong frequency selectivity

A lower-Q circuit has:

• greater damping
• broader resonance
• less precise frequency selection

High selectivity can be valuable in communication receivers.

---

Energy at Different Points in the Cycle

Consider an ideal LC circuit beginning with a fully charged capacitor.

At t = 0:

• capacitor charge = maximum
• voltage = maximum
• current = zero
• electric energy = maximum
• magnetic energy = zero

One-quarter cycle later:

• capacitor charge = zero
• voltage = zero
• current = maximum
• electric energy = zero
• magnetic energy = maximum

One-half cycle later:

• capacitor charge = maximum in opposite direction
• voltage = maximum with opposite polarity
• current = zero
• electric energy = maximum
• magnetic energy = zero

Three-quarters of a cycle later:

• capacitor charge = zero
• current = maximum in opposite direction
• magnetic energy = maximum

One complete cycle later:

The circuit returns to its original state.

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Charge and Current Are Out of Phase

In an ideal LC circuit, capacitor charge and current do not reach their maximum values at the same time.

When charge is maximum:

I = 0

When current is maximum:

Q = 0

They are separated by one-quarter of a cycle.

This is a 90° phase difference.

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Voltage and Current

Capacitor voltage is proportional to capacitor charge:

V_C = Q/C

Therefore, capacitor voltage is also maximum when capacitor charge is maximum.

At those moments, current is zero.

When current reaches maximum magnitude, capacitor voltage passes through zero.

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Frequency and Angular Frequency

Resonance can also be described using angular frequency.

Angular frequency is:

ω₀ = 2πf₀

For an ideal LC circuit:

ω₀ = 1/√(LC)

Therefore:

f₀ = 1/(2π√(LC))

Both equations describe the same resonant behaviour.

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Worked Example: Angular Frequency

Suppose:

L = 0.025 H

C = 4.0 μF

Convert:

C = 4.0 × 10⁻⁶ F

Calculate:

ω₀ = 1/√(LC)

ω₀ = 1/√[(0.025)(4.0 × 10⁻⁶)]

ω₀ = 1/√(1.0 × 10⁻⁷)

ω₀ ≈ 3162 rad/s

Then:

f₀ = ω₀/(2π)

f₀ ≈ 503 Hz

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A Complete Resonance Problem

A tuned circuit contains:

L = 250 μH

C = 100 pF

Convert:

L = 250 × 10⁻⁶ H

C = 100 × 10⁻¹² F

Use:

f₀ = 1/(2π√(LC))

LC = (250 × 10⁻⁶)(100 × 10⁻¹²)

LC = 2.50 × 10⁻¹⁴

√LC ≈ 1.58 × 10⁻⁷

Therefore:

f₀ ≈ 1/[2π(1.58 × 10⁻⁷)]

f₀ ≈ 1.01 × 10⁶ Hz

or approximately:

f₀ ≈ 1.01 MHz

This frequency lies in the radio-frequency range, demonstrating why LC circuits are useful for radio tuning.

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Why LC Circuits Are Useful

LC circuits have several important advantages.

They can:

• produce electrical oscillations
• select particular frequencies
• reject unwanted frequencies
• store and exchange electrical energy
• form part of oscillators
• form part of filters
• tune communication equipment

Their resonant frequency can be designed by choosing suitable values of L and C.

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Applications of Tuned Circuits

Tuned circuits are used in many technologies, including:

• radio receivers
• radio transmitters
• television systems
• wireless communication equipment
• signal generators
• filters
• oscillators
• measurement equipment
• impedance-matching networks

Modern systems may also use digital and semiconductor technologies for many frequency-selection tasks, but LC resonance remains a fundamental principle in electronics.

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Resonance and Wireless Systems

Wireless systems depend on selecting and processing electromagnetic signals at particular frequencies.

Resonant circuits can help:

• select desired signals
• reject interference
• generate carrier frequencies
• filter unwanted frequencies
• improve energy transfer in some wireless systems

Resonance is therefore important far beyond traditional radio receivers.

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Resonant Wireless Power Transfer

Resonance can also improve some forms of wireless energy transfer.

If transmitting and receiving circuits are tuned to compatible resonant frequencies, energy transfer can become more effective.

This combines:

• mutual inductance
• electromagnetic induction
• LC resonance

Applications can include wireless charging and specialized contactless power systems.

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Comparing LC Energy Storage

The continuous exchange between these two energy stores produces LC oscillation.

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A Reliable Strategy for LC Resonance Problems

First identify:

• L
• C
• required resonant frequency

Convert all quantities into SI units:

• H
• F
• Hz

Common conversions include:

1 mH = 10⁻³ H

1 μH = 10⁻⁶ H

1 μF = 10⁻⁶ F

1 nF = 10⁻⁹ F

1 pF = 10⁻¹² F

Then use:

f₀ = 1/(2π√(LC))

If necessary, rearrange:

L = 1/(4π²f₀²C)

or:

C = 1/(4π²f₀²L)

Finally, check whether the answer is physically reasonable.

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Common Mistakes

Thinking the capacitor and inductor store the same type of energy

The capacitor stores electric-field energy. The inductor stores magnetic-field energy.

Thinking energy disappears when the capacitor becomes uncharged

At that moment, the energy can be stored in the inductor's magnetic field.

Thinking current is maximum when capacitor charge is maximum

Current is zero when capacitor charge reaches maximum magnitude.

Forgetting the square root in the resonance equation

The correct relationship is:

f₀ = 1/(2π√(LC))

Forgetting to convert units

Microfarads, nanofarads, picofarads, millihenries, and microhenries must be converted carefully.

Thinking larger L produces higher frequency

Increasing L decreases resonant frequency.

Thinking larger C produces higher frequency

Increasing C also decreases resonant frequency.

Assuming real LC circuits oscillate forever

Resistance causes energy loss and damping.

Thinking resonance amplifies every frequency equally

Resonant circuits respond particularly strongly near specific frequencies.

Confusing frequency with angular frequency

Frequency f is measured in hertz.

Angular frequency ω is measured in radians per second.

They are related by:

ω = 2πf

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Key Terms

LC circuit — circuit containing an inductor and capacitor that can exchange magnetic and electric field energy.

Resonance — strong response that occurs when a system is driven near its natural frequency.

Resonant frequency — natural oscillation frequency of an LC circuit.

Capacitance — ability of a capacitor to store separated electric charge and electric-field energy.

Inductance — property associated with magnetic-field energy storage and opposition to changes in current.

Electric-field energy — energy stored by a capacitor, given by E_C = ½CV².

Magnetic-field energy — energy stored by an inductor, given by E_L = ½LI².

Oscillation — repeated variation of a quantity about an equilibrium condition.

Tuned circuit — circuit designed to respond strongly to a selected frequency.

Natural frequency — frequency at which a system tends to oscillate when disturbed.

Angular frequency — frequency measured in radians per second, with ω = 2πf.

Damping — gradual reduction in oscillation amplitude caused by energy loss.

Quality factor (Q) — measure associated with the sharpness of resonance and energy loss in a resonant system.

Filter — circuit that selectively passes or reduces different frequencies.

Carrier wave — high-frequency wave used to carry information in a communication system.

Frequency selectivity — ability of a system to distinguish between signals of different frequencies.

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Key Takeaways

• An LC circuit contains an inductor and a capacitor.
• Capacitors store energy in electric fields.
• Inductors store energy in magnetic fields.
• LC oscillation involves repeated energy transfer between electric and magnetic fields.
• When capacitor charge is maximum, current is zero.
• When current is maximum, capacitor charge is zero.
• In an ideal LC circuit, total stored energy remains constant.
• Total energy can be written as E_total = ½CV² + ½LI².
• An ideal LC circuit has a natural or resonant frequency.
• Resonant frequency is f₀ = 1/(2π√(LC)).
• Increasing inductance decreases resonant frequency.
• Increasing capacitance decreases resonant frequency.
• At resonance, a driven circuit can respond strongly to an external alternating signal.
• Tuned circuits use resonance to select particular frequencies.
• LC resonance is important in radio receivers and transmitters.
• Changing L or C allows the resonant frequency to be adjusted.
• Resonant circuits can help separate desired communication signals from unwanted signals.
• Resistance causes damping in real circuits.
• High-Q circuits provide sharper frequency selection than strongly damped circuits.
• Resonance is important in communication, filtering, oscillators, measurement systems, and some wireless power-transfer systems.

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Check Your Understanding

1. What two components make up an LC circuit, and what type of energy does each store?
2. Describe what happens when a fully charged capacitor is connected to an inductor.
3. Why is the current zero when the capacitor reaches maximum charge?
4. Explain where the circuit's energy is stored when the capacitor is momentarily uncharged.
5. Write the equation for the resonant frequency of an ideal LC circuit and identify each variable.
6. Calculate the resonant frequency of a circuit with L = 0.050 H and C = 20 μF.
7. An LC circuit has a resonant frequency of 2000 Hz and an inductance of 0.010 H. Calculate the required capacitance.
8. If the capacitance of an LC circuit is increased by a factor of 4, what happens to its resonant frequency? Explain mathematically.
9. If both L and C are doubled, by what factor does the resonant frequency change?
10. A capacitor stores 0.020 J when fully charged. How much magnetic energy is stored in the inductor when the capacitor becomes completely discharged in an ideal LC circuit?
11. Explain why a real LC circuit will not continue oscillating forever without an external energy source.
12. What is meant by a tuned circuit, and how can its resonant frequency be adjusted?
13. Explain how an LC circuit can help a radio receiver select one station from many signals received by its antenna.
14. Explain the relationship between resonance, frequency selection, and communication systems.
15. A communication engineer wants a circuit that responds strongly to one particular frequency while rejecting nearby unwanted signals. Explain how an LC resonant circuit could accomplish this and discuss how resistance and Q factor would affect its performance.