Efficiency Equation

What efficiency tells us

A device receives energy and transfers it into one or more forms. Only some of these transfers may achieve the device's intended purpose. For example, the useful transfer from a motor lifting an object increases the object's gravitational potential energy. Energy transferred by heating the motor and its surroundings is not useful for that purpose.

Efficiency compares the useful energy transferred by a device with the total energy supplied to it. It is therefore a fraction rather than an amount of energy:

$$\text{efficiency}=\frac{\text{useful energy transferred by the device}}{\text{total energy supplied to the device}}$$

Both energy quantities must refer to the same device and be measured over the same process. They must also be in the same unit, normally joules (J), so the units cancel. Efficiency consequently has no unit.

Following the energy

Energy is conserved: it is not used up or destroyed. The total energy supplied is shared between useful transfers and transfers to less useful stores. Energy that spreads into the surroundings, commonly by heating, is described as dissipated.

What counts as useful depends on the device's intended purpose. Heating the surroundings is unwanted for a lifting motor, but transferring energy to thermal stores is the intended output of an electric heater.

Total energy supplied: 100 J. Useful energy transferred: 75 J. Energy transferred in less useful ways: 25 J. Wasted energy = 100 − 75 = 25 J. Efficiency = 75 ÷ 100 × 100% = 75%. Arrow shaft widths are proportional to energy.

In the diagram, 75 J of a 100 J input is useful. The efficiency is therefore $75/100=0.75$. The remaining 25 J is transferred in less useful ways. Writing $0.75$ as a percentage gives $75\%$.

Decimal and percentage efficiencies

Efficiency can be written as a decimal between 0 and 1 or as a percentage between 0% and 100%:

$$\text{percentage efficiency}=\text{decimal efficiency}\times100$$

A perfectly efficient device would transfer all its input energy usefully, giving an efficiency of 1, or 100%. Real devices often dissipate some energy, so their useful energy transfer is smaller than their total input.

Rearranging the relationship

The equation can also be rearranged when a different quantity is unknown:

$$\text{useful energy transferred}=\text{efficiency}\times\text{total energy supplied}$$ $$\text{total energy supplied}=\frac{\text{useful energy transferred}}{\text{efficiency}}$$

Use efficiency as a decimal in these rearranged equations. For example, substitute $0.80$, not $80$, for an efficiency of 80%. After calculating, check that the useful energy does not exceed the total energy supplied.

Core equation

$$\text{efficiency}=\frac{\text{useful energy transferred}}{\text{total energy supplied}}$$

Efficiency has no unit. It is written from 0 to 1, or from 0% to 100%.