Efficiency

Prof. Dr. Thorsten Jungmann (last updated 2025-09-03)

Efficiency (German: Wirkungsgrad) describes the ratio of output power to input power and is a measure of the efficiency of a system or component. It indicates what proportion of the input energy is converted into the desired form.

1 Quantity symbol and unit

The quantity symbol for efficiency is the lower-case Greek letter $\eta$ (‘eta’).
Efficiency is a dimensionless quantity that is given as a decimal number or as a percentage:

$$[\eta ]=1\text{ or }\%$$

2 Practical and worked examples

Efficiency is defined as:

$$\eta =\frac{P_{out}}{P_{in}}=\frac{P_{in}-P_{\mathrm{loss}}}{P_{in}}=1-\frac{P_{\mathrm{loss}}}{P_{in}}$$

where:

• $P_{out}$ the output (useful) power
• $P_{in}$ the input power
• $P_{\mathrm{loss}}$ the power loss

2.1 Electric motor

An electric motor draws $P_{in}=1.2\mathrm{kW}$ of electrical power and delivers $P_{out}=1.0\mathrm{kW}$ of mechanical power:

$$\eta =\frac{P_{out}}{P_{in}}=\frac{1.0\mathrm{kW}}{1.2\mathrm{kW}}=0.833=\underline{83.3\%}$$

The electric motor has an efficiency of approx. 83 %.

Power loss: $P_{\mathrm{loss}}=P_{in}-P_{out}=1.2\mathrm{kW}-1.0\mathrm{kW}=\underline{0.2\mathrm{kW}}$

2.2 Power supply

A switch-mode power supply with $P_{in}=65\mathrm{W}$ (drawn from the mains) delivers $P_{out}=60\mathrm{W}$ (DC output):

$$\eta =\frac{60\mathrm{W}}{65\mathrm{W}}=0.923=\underline{92.3\%}$$

2.3 Light emitting diode (LED) lighting

An LED lamp draws $P_{in}=12\mathrm{W}$ and produces a luminous flux of ${\Phi }_v=1200\mathrm{lm}$.
Luminous efficacy: $\frac{1200\mathrm{lm}}{12\mathrm{W}}=100lm/W$

Compared with an incandescent bulb (15 lm/W), this is an efficiency about 6 times higher in generating light.

$${\eta }_{\mathrm{LED}}=6\cdot {\eta }_{\text{incandescent bulb}}$$

2.4 Transformer

A transformer has $P_1=500\mathrm{VA}$ on the primary side and $P_2=485\mathrm{VA}$ on the secondary side:

$$\eta =\frac{P_2}{P_1}=\frac{485\mathrm{VA}}{500\mathrm{VA}}=0.97=\underline{97.0\%}$$

Its efficiency is $0.97$.

2.5 Overall efficiency of an emergency generator

A diesel engine (${\eta }_1=35\%$) drives a generator (${\eta }_2=90\%$):

$${\eta }_{tot}={\eta }_1\cdot {\eta }_2=0.35\cdot 0.90=\underline{31.5\%}$$

Of the energy stored in the fuel, 31.5 % arrives at the output as electrical energy.

3 Typical orders of magnitude

Very high efficiencies (> 95 %):

• Transformers: $\eta =95$ to $99.5\%$
• Synchronous motors (large): $\eta =95$ to $98\%$

High efficiencies (80–95 %):

• Switch-mode power supplies: $\eta =85$ to $95\%$
• Asynchronous motors: $\eta =80$ to $95\%$
• LED drivers: $\eta =85$ to $92\%$

Medium efficiencies (50–80 %):

• Linear power supplies: $\eta =30$ to $60\%$
• Incandescent bulbs (light): $\eta \approx 5\%$
• Small electric motors: $\eta =50$ to $80\%$

Low efficiencies (< 50 %):

• Combustion engines: $\eta =25$ to $40\%$
• Steam turbines (old): $\eta =30$ to $45\%$

4 Example phrasings

• ‘The electric motor has an efficiency of $\eta =87.5\%$.’
• ‘Due to the power loss of 50 W, the efficiency drops to 83 %.’
• ‘The switch-mode power supply achieves an efficiency of over 90 %.’
• ‘The overall efficiency of the drive train is 76 %.’
• ‘At partial load, the efficiency of the transformer drops to 65 %.’

5 Related quantities

• Power (input, output, loss):

$$\eta =\frac{P_{out}}{P_{in}}=1-\frac{P_{\mathrm{loss}}}{P_{in}}$$

• Power loss:

$$P_{\mathrm{loss}}=P_{in}\cdot (1-\eta )$$

• For electrical systems:

$$\eta =\frac{U_2\cdot I_2\cdot \cos {\phi }_2}{U_1\cdot I_1\cdot \cos {\phi }_1}$$

• Overall efficiency of a chain of stages:

$${\eta }_{tot}={\eta }_1\cdot {\eta }_2\cdot {\eta }_3\cdots$$

• Energy losses:

$$E_{\mathrm{loss}}=E_{in}\cdot (1-\eta )$$

6 Related components

• Electric motor: efficiency determines operating costs and heat generation
• Transformer: high efficiency reduces grid losses
• Power supply: efficiency affects energy consumption and cooling
• Generator: efficiency determines fuel consumption