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Efficiency

Energy and power · level 6, lesson 1 of 6

A question to start

Perseverance's power source makes about 2,000 W of heat but only about 110 W of electricity. A good electric motor turns 90% of its electricity into motion. How do we compare machines fairly?

The idea

The efficiency of a machine is the fraction of the energy put in that comes out as useful energy:

\[\text{efficiency} = \frac{\text{useful energy out}}{\text{total energy in}} = \frac{\text{useful power out}}{\text{total power in}}.\]

It is a number between 0 and 1, often written as a percentage. By the conservation of energy, it can never be more than 100%: nothing makes energy out of nothing.

Machine Typical efficiency
Large electric motor 90 to 95%
Battery, charge then discharge 85 to 95%
Solar panel, best on spacecraft about 30%
Solar panel on a house roof about 20%
Car petrol engine 20 to 35%
Human muscles about 20 to 25%
Old filament light bulb (as light) about 2 to 5%
Perseverance's MMRTG (heat to electricity) about 6%
motor 91%MMRTG 6%heatuseful

Low efficiency is not always bad. The MMRTG's "waste" heat keeps the rover warm through freezing Martian nights; on Mars, that heat is half the point.

Worked examples

1. A motor. 500 W in, 455 W of motion out: efficiency \(= 455/500 = 0.91\), or 91%.

2. Finding the input. A solar panel 28% efficient gives 280 W. The sunlight falling on it was \(280/0.28 = 1{,}000\) W.

3. Chains multiply. Sunlight → panel (28%) → battery (90%) → motor (90%): overall \(0.28 \times 0.9 \times 0.9 = 0.23\). Of every 100 J of sunlight, 23 J ends up moving the wheels.

Try it

  1. A lamp uses 40 W and gives out 14 W of light. What is its efficiency?
  2. A motor is 85% efficient and must deliver 170 W. What power does it take in?
  3. Why can no machine be more than 100% efficient?
  4. Find the overall efficiency of a panel (25%), battery (92%) and heater (100%).
Answers
  1. \(14/40 = 35\%\).
  2. \(170/0.85 = 200\) W.
  3. It would have to give out more energy than it takes in, which would break the conservation of energy.
  4. \(0.25 \times 0.92 \times 1 = 23\%\).

Common mistake

"An efficient machine uses little energy." Efficiency is a ratio. A very efficient 3,000 W motor still uses far more energy than an inefficient 10 W one.

Where it leads