Efficiency can be written as a decimal or percentage.

For example:

efficiency = 0.80

is the same as:

efficiency = 80%

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Worked Example: Electric Motor

A motor receives:

500 J

of electrical energy.

It produces:

400 J

of useful mechanical energy.

Efficiency:

efficiency = 400/500

efficiency = 0.80

As a percentage:

0.80 × 100 = 80%

The motor is 80% efficient.

---

Finding Wasted Energy

Because energy is conserved:

input energy = useful energy + wasted energy

Therefore:

wasted energy = input energy − useful energy

For the previous motor:

wasted energy = 500 − 400

wasted energy = 100 J

So:

500 J input → 400 J useful + 100 J wasted

---

Worked Example: Lamp

A lamp receives:

200 J

of electrical energy.

It transfers:

60 J

as useful light.

Efficiency:

efficiency = 60/200 × 100

efficiency = 30%

Wasted energy:

200 − 60 = 140 J

Therefore:

200 J electrical → 60 J useful light + 140 J thermal energy

The lamp is 30% efficient at producing the desired light output.

---

Efficiency Using Power

Efficiency can also be calculated using power.

When input and output are measured over the same time interval:

efficiency = useful output power / total input power

As a percentage:

efficiency (%) = useful output power / total input power × 100

This is especially useful for machines operating continuously.

---

Worked Example: Motor Power

An electric motor receives:

2500 W

of electrical power.

It provides:

2000 W

of useful mechanical power.

Efficiency:

efficiency = 2000/2500 × 100

efficiency = 80%

Power dissipated:

2500 − 2000 = 500 W

This means energy is being dissipated at a rate of:

500 J/s

---

Why Energy and Power Give the Same Efficiency

Suppose the motor operates for 10 s.

Input energy:

E = Pt

E = 2500 × 10

E = 25,000 J

Useful output energy:

E = 2000 × 10

E = 20,000 J

Efficiency:

efficiency = 20,000/25,000

efficiency = 0.80

or:

80%

The same result is obtained using either energy or power.

---

Rearranging the Efficiency Equation

If efficiency and input energy are known:

useful output = efficiency × input

Remember to convert a percentage to a decimal.

For example:

75% = 0.75

Therefore:

useful output = 0.75 × input

If useful output and efficiency are known:

input = useful output / efficiency

---

Worked Example: Finding Useful Energy

A machine receives:

2400 J

and is:

75% efficient

Convert:

75% = 0.75

Useful energy:

useful output = 0.75 × 2400

useful output = 1800 J

Wasted energy:

2400 − 1800 = 600 J

---

Worked Example: Finding Input Energy

A machine must produce:

3600 J

of useful energy.

It is:

80% efficient

Convert:

80% = 0.80

Input energy:

input = 3600/0.80

input = 4500 J

Therefore:

900 J

is transferred into unwanted forms.

---

Worked Example: Finding Useful Power

A motor receives:

10 kW

of electrical power.

It is:

85% efficient

Useful mechanical power:

Puseful = 0.85 × 10

Puseful = 8.5 kW

Wasted power:

10 − 8.5 = 1.5 kW

---

Why Real Systems Are Not 100% Efficient

In real systems, interactions between components cause some energy to be transferred into unwanted forms.

Common causes include:

• friction
• electrical resistance
• air resistance
• sound
• vibration
• deformation
• turbulence
• heating of components

These processes transfer energy into the surroundings.

Because these transfers cannot normally be eliminated completely:

real systems are not 100% efficient

---

Friction

Friction is one of the most common causes of energy dissipation in mechanical systems.

When surfaces move against one another, mechanical energy is transferred into thermal energy.

Examples include:

• gears
• bearings
• tyres
• engine components
• conveyor systems
• bicycle chains

You may notice machinery becoming warm during operation because of these transfers.

Reducing Friction

Engineers can reduce friction using:

• lubricants
• ball bearings
• smoother surfaces
• appropriate materials
• improved alignment

Reducing friction means less mechanical energy is transferred into unwanted thermal energy.

This can increase efficiency.

---

Electrical Resistance

Electrical devices also dissipate energy because conductors have resistance.

When current passes through resistance:

electrical energy → thermal energy

This is useful in devices designed for heating.

Examples include:

• kettles
• heaters
• toasters

But it is usually unwanted in:

• motors
• power cables
• computers
• transformers
• generators

Electrical components becoming warm is evidence that some energy is being transferred thermally.

---

Useful or Wasted Depends on Purpose

An important point is that whether an energy transfer is considered useful depends on the purpose of the device.

Consider an electric heater.

Electrical energy becoming thermal energy is:

useful

But in an electric motor, excessive heating is:

wasted

Similarly, light from a lamp is useful, while light from a heating element may not be.

Efficiency must therefore always be considered in relation to the intended output.

---

Sound and Vibration

Machines often produce sound and vibration.

These transfers usually represent energy that is not contributing to the intended output.

Examples include:

• noisy engines
• vibrating motors
• rattling machinery
• tyre noise
• electrical transformers

Reducing unwanted vibration and sound can improve efficiency as well as make machinery quieter and more reliable.

---

Air Resistance

Moving objects transfer energy to the surrounding air.

This is particularly important for:

• cars
• trains
• aircraft
• bicycles

Air resistance generally increases as speed increases.

Therefore, high-speed vehicles may transfer large amounts of energy into the surrounding air.

Streamlining can reduce these unwanted energy transfers.

Improving Mechanical Efficiency

Mechanical systems can often be made more efficient by:

• reducing friction
• using lubrication
• using bearings
• reducing unnecessary mass
• improving aerodynamics
• reducing vibration
• improving gear design
• using suitable materials

However, completely eliminating unwanted transfers is usually impossible.

---

Improving Electrical Efficiency

Electrical systems can be improved through:

• lower-resistance conductors
• efficient motors
• improved insulation
• better electronic control systems
• reduced standby consumption
• efficient lighting
• improved cooling systems
• appropriate operating voltages

The goal is to increase the fraction of input energy transferred into the intended output.

---

Comparing Devices

Suppose three motors receive the same input energy:

1000 J

Motor C is the most efficient because the greatest fraction of its input energy becomes useful output.

It also dissipates the least energy.

---

Comparing Devices with Different Inputs

Efficiency is a ratio, so devices can be compared even when their input energies are different.

Device A

Input = 500 J
Useful = 400 J

efficiency = 400/500 × 100 = 80%

Device B

Input = 2000 J
Useful = 1500 J

efficiency = 1500/2000 × 100 = 75%

Device B produces more useful energy overall, but Device A is more efficient.

This is an important distinction.

---

Efficiency Does Not Mean Power

A device can be highly efficient but have low power.

Another device can be less efficient but have much greater power.

For example:

Device A:

input power = 100 W

useful power = 90 W

Efficiency:

90%

Device B:

input power = 2000 W

useful power = 1600 W

Efficiency:

80%

Device B provides much more useful power, but Device A is more efficient.

---

Sankey Diagrams

A Sankey diagram visually represents energy transfers.

The width of each arrow represents the amount of energy being transferred.

For example:

100 J input

might divide into:

70 J useful

and:

30 J wasted

Because:

70 + 30 = 100

the total width of the output arrows equals the width of the input arrow.

Sankey diagrams make it easy to compare the useful and wasted energy transfers in different systems.

---

Reading a Sankey Diagram

Suppose a motor receives:

500 J

The diagram shows:

350 J useful mechanical energy

and:

150 J thermal and sound energy

Efficiency:

efficiency = 350/500 × 100

efficiency = 70%

The wasted fraction is:

150/500 × 100

= 30%

Therefore:

70% useful + 30% wasted = 100%

---

Light Bulbs

Different lighting technologies can provide the same useful function while transferring different amounts of energy into unwanted thermal energy.

An efficient lamp transfers a greater proportion of its electrical input into useful light and a smaller proportion into unwanted heating.

This means less electrical energy is required to produce a particular useful light output.

Worked Comparison: Lamps

Suppose two lamps each receive:

100 J

of electrical energy.

Lamp A produces:

15 J useful light

Lamp B produces:

80 J useful light

Lamp A:

efficiency = 15/100 × 100

efficiency = 15%

Lamp B:

efficiency = 80/100 × 100

efficiency = 80%

Lamp B converts a much greater fraction of the electrical input into the desired output.

---

Electric Motors

Electric motors convert:

electrical energy → mechanical energy

Some energy is also transferred through:

• heating of wires
• friction in bearings
• sound
• vibration

A more efficient motor provides more mechanical output for the same electrical input.

This can reduce:

• electricity consumption
• operating costs
• unwanted heating
• environmental impacts associated with energy production

---

Vehicle Engines

Vehicle engines and motors convert stored energy into mechanical energy.

In combustion engines, significant energy can be transferred into:

• hot exhaust gases
• cooling systems
• friction
• sound

Only part of the input energy becomes useful mechanical output.

Electric motors generally have different energy-loss pathways, including electrical resistance, magnetic losses, friction, and heating.

Worked Example: Vehicle

An engine receives:

50 MJ

of chemical energy.

It transfers:

18 MJ

into useful mechanical energy.

Efficiency:

efficiency = 18/50 × 100

efficiency = 36%

Energy dissipated:

50 − 18 = 32 MJ

The remaining energy has been transferred mainly into thermal energy and other unwanted outputs.

---

Power Station Efficiency

Efficiency is also important in electricity generation.

Suppose a power station receives:

1000 MW

of energy per second from its fuel source and produces:

400 MW

of electrical power.

Efficiency:

efficiency = 400/1000 × 100

efficiency = 40%

The remaining:

600 MW

is transferred into other forms, especially thermal energy.

---

Efficiency Through Several Stages

Many real systems involve several energy conversions.

For example:

fuel → electricity → motor → motion

Suppose:

Power station efficiency = 40%

Motor efficiency = 80%

Start with:

100 J

After the power station:

100 × 0.40 = 40 J

After the motor:

40 × 0.80 = 32 J

Overall useful output:

32 J

Therefore:

overall efficiency = 32%

For several stages, the overall efficiency can be found by multiplying the decimal efficiencies:

overall efficiency = efficiency₁ × efficiency₂ × efficiency₃ ...

---

Worked Example: Multi-Stage System

A system contains:

Generator efficiency = 90%

Motor efficiency = 80%

Machine efficiency = 75%

Convert to decimals:

0.90 × 0.80 × 0.75

= 0.54

Therefore:

overall efficiency = 54%

Only 54% of the original input reaches the final useful output.

This demonstrates how losses at several stages can significantly reduce overall efficiency.

---

Efficiency and Sustainability

Improving efficiency can reduce the amount of energy required to perform a task.

This can lead to:

• lower energy consumption
• lower operating costs
• reduced fuel use
• reduced waste heat
• lower demand for energy resources
• reduced environmental impacts

For example, if two motors perform the same useful task but one is more efficient, the more efficient motor requires less input energy.

---

Worked Example: Energy Savings

Machine A is 60% efficient.

Machine B is 80% efficient.

Both must provide:

2400 J

of useful energy.

Machine A

input = 2400/0.60

input = 4000 J

Machine B

input = 2400/0.80

input = 3000 J

Machine B requires:

1000 J less input energy

to perform the same useful task.

---

Can Efficiency Be 100%?

In an idealized model, we sometimes assume:

efficiency = 100%

This simplifies calculations.

Real machines, however, always have some unwanted energy transfers.

For example:

• wires have resistance
• surfaces experience friction
• objects interact with air
• machines vibrate
• components become warm

Therefore, real energy-transfer systems do not achieve perfect efficiency.

---

Can Efficiency Be Greater Than 100%?

No.

An efficiency greater than 100% would mean:

useful output energy > input energy

That would require energy to be created.

This violates conservation of energy.

Therefore:

0% ≤ efficiency ≤ 100%

for a correctly defined system.

If a calculation gives more than 100%, check:

• the data
• the units
• which quantity is input
• which quantity is useful output

---

Efficiency Versus Energy Conservation

These ideas should not be confused.

Conservation of Energy

Total energy is always conserved.

input energy = total output energy

Efficiency

Efficiency asks what fraction of that output is useful.

For example:

100 J input → 70 J useful + 30 J thermal

Energy conservation:

100 J = 70 J + 30 J

Efficiency:

70%

The system is only 70% efficient, but 100% of the energy still exists.

---

Practical Example: Electric Elevator

An elevator motor receives:

50 kW

of electrical power.

The elevator receives:

42 kW

of useful mechanical power.

Efficiency:

efficiency = 42/50 × 100

efficiency = 84%

Power dissipated:

50 − 42 = 8 kW

This means:

8000 J

is transferred into other forms every second.

---

Practical Example: Pump

A water pump receives:

3000 W

of electrical power.

It transfers:

2250 W

usefully to the water.

Efficiency:

efficiency = 2250/3000 × 100

efficiency = 75%

Wasted power:

3000 − 2250 = 750 W

---

Improving Efficiency

Engineers try to increase efficiency by reducing unwanted energy transfers.

Possible approaches include:

Mechanical Systems

• lubrication
• bearings
• aerodynamic design
• reducing unnecessary vibration
• improving alignment
• reducing friction

Electrical Systems

• low-resistance conductors
• efficient motors
• improved electronic components
• appropriate cooling
• improved circuit design

Buildings

• insulation
• efficient lighting
• improved heating and cooling systems
• reducing unwanted air leakage

Transportation

• aerodynamic shapes
• lightweight materials
• efficient motors
• regenerative braking
• reduced rolling resistance

Efficiency improvements often involve engineering trade-offs involving cost, reliability, safety, materials, and performance.

---

Common Mistakes

Saying Energy Is "Used Up"

Energy is not destroyed.

It is transferred or dissipated.

---

Saying Wasted Energy Disappears

Wasted energy still exists.

It has been transferred into forms that are less useful for the intended purpose.

---

Dividing Input by Output

The correct relationship is:

useful output ÷ total input

not the other way around.

---

Forgetting to Multiply by 100

If the question asks for percentage efficiency:

0.75 = 75%

---

Using 80 Instead of 0.80

When using an efficiency in another calculation:

80% = 0.80

not 80.

---

Assuming the Largest Output Means the Highest Efficiency

Efficiency is a ratio.

A machine producing more useful energy may still be less efficient if it requires much more input energy.

---

Confusing Power with Energy

Efficiency can be calculated using either energy or power, but the quantities must be compared appropriately.

Use:

useful energy / input energy

or:

useful power / input power

---

Assuming Thermal Energy Is Always Wasted

Thermal energy from a heater is useful.

Thermal energy from a motor is generally unwanted.

The intended purpose of the device determines what counts as useful.

---

Assuming 100% Efficiency Is Realistic

Idealized calculations may assume 100%, but real systems experience energy dissipation.

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Check Your Understanding

1. Define efficiency.
2. Explain the difference between useful and wasted energy transfers.
3. Why is the term dissipated energy often more scientifically useful than saying energy is "lost"?
4. A machine receives 500 J and produces 400 J of useful energy. Calculate its efficiency.
5. How much energy is wasted in Question 4?
6. A motor receives 2000 J and produces 1500 J of useful mechanical energy. Calculate its efficiency.
7. A lamp receives 100 J and transfers 25 J as useful light. Calculate its efficiency.
8. A device receives 500 W and provides 400 W of useful output power. Calculate its efficiency.
9. How much power is dissipated in Question 8?
10. A motor is 80% efficient and receives 5000 J. Calculate its useful output.
11. Calculate the wasted energy in Question 10.
12. A machine produces 3000 J of useful output and is 75% efficient. Calculate its required input energy.
13. A motor receives 10 kW and is 85% efficient. Calculate its useful mechanical power.
14. Explain why friction reduces mechanical efficiency.
15. Explain why electrical resistance can reduce the efficiency of a motor.
16. Give three examples of unwanted energy transfers in machines.
17. Why can thermal energy be useful in one device but wasted in another?
18. Machine A receives 1000 J and produces 700 J usefully. Machine B receives 500 J and produces 400 J usefully. Which is more efficient?
19. Explain why the device producing the greatest useful output is not necessarily the most efficient.
20. A power station is 40% efficient and receives 2000 MW. Calculate its electrical output power.
21. A 90%-efficient motor receives 5 kW. Calculate its useful output power.
22. A system contains two stages with efficiencies of 80% and 75%. Calculate the overall efficiency.
23. A system contains three stages with efficiencies of 90%, 80%, and 70%. Calculate the overall efficiency.
24. Explain why several efficient devices connected together can have a noticeably lower overall efficiency.
25. Describe three ways engineers can improve the efficiency of a mechanical system.
26. Describe two ways engineers can reduce unwanted electrical energy transfers.
27. Explain why improving efficiency can reduce environmental impacts.
28. A machine must produce 6000 J of useful output. Compare the required input energy if it is 60% efficient and if it is 80% efficient.
29. Explain why an efficiency of 120% is impossible.
30. Explain how a system can be only 60% efficient while still obeying the law of conservation of energy.

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Key Terms

Efficiency — Fraction of the total input energy or power transferred into the intended useful output.

Useful energy — Energy transferred in the form required for the intended purpose.

Wasted energy — Energy transferred in forms that are not useful for the intended purpose.

Energy dissipation — Transfer and spreading of energy into less useful stores, often thermal energy in the surroundings.

Input energy — Total energy supplied to a system.

Output energy — Energy transferred from a system.

Useful output — Portion of the output that performs the intended function.

Input power — Rate at which energy is supplied to a system.

Useful output power — Rate at which useful energy is transferred.

Watt (W) — Unit of power equal to one joule per second.

Friction — Force opposing motion that can transfer mechanical energy into thermal energy.

Electrical resistance — Opposition to electric current that causes electrical energy to be transferred thermally.

Air resistance — Resistive force caused by motion through air.

Sankey diagram — Diagram showing energy transfers using arrows whose widths represent energy quantities.

Overall efficiency — Efficiency of a system containing several energy-transfer stages.

Conservation of energy — Principle stating that energy cannot be created or destroyed.

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Key Takeaways

• Efficiency measures the fraction of input energy transferred into the desired useful output.
• Efficiency can be expressed as a decimal or percentage.
• Efficiency can be calculated using either energy values or power values.
• Useful output is determined by the intended purpose of the device.
• Wasted energy has not disappeared; it has been transferred into less useful forms.
• Total energy is always conserved.
• Real systems are less than 100% efficient because some energy is dissipated.
• Common causes of energy dissipation include friction, electrical resistance, air resistance, sound, vibration, and deformation.
• Mechanical efficiency can often be improved by reducing friction and resistance.
• Electrical efficiency can often be improved by reducing unwanted heating and resistance.
• A more efficient device requires less input energy to produce the same useful output.
• A device with a greater useful output is not necessarily more efficient.
• Efficiency and power are different quantities.
• Several energy conversions can reduce the overall efficiency of a system.
• For multi-stage systems, decimal efficiencies can be multiplied to determine overall efficiency.
• Sankey diagrams show how input energy divides into useful and wasted outputs.
• Improving efficiency can reduce energy consumption, operating costs, resource use, and environmental impacts.
• An efficiency greater than 100% is impossible because it would violate conservation of energy.

The central idea is:

Energy is never destroyed — but not all transferred energy remains useful.

A more efficient system transfers a greater fraction of its input energy into the output we actually want.