Power factor

Prof. Dr. Thorsten Jungmann (last updated 2025-11-16)

Power factor: How much of the apparent power is really usable? – The closer to 1, the better!

The power factor (German: Leistungsfaktor) describes the ratio of active power to apparent power in AC circuits. It indicates what proportion of the apparent power provided by the grid is actually converted into usable work. The power factor is a measure of the efficiency of energy use in the AC grid.

In the fundamentals of electrical engineering (EE1) only for completeness

This quantity appears in the EE1 script only for the sake of completeness, because it belongs to the components inductor and capacitor. Calculations and the deeper understanding are part of AC theory (EE2).

1 Quantity symbol and unit

The usual quantity symbol of the power factor is $\cos \phi$ (cosine of the phase angle) or $\lambda$.
The power factor is a dimensionless quantity with values between 0 and 1:

$$[\cos \phi ]=1$$

2 Practical and worked examples

The power factor is defined as:

$$\cos \phi =\frac{P}{S}=\frac{P}{U\cdot I}$$

where:

• $P$ the active power in watts (W)
• $S$ the apparent power in volt-amperes (VA)
• $U$ the voltage in volts (V)
• $I$ the current in amperes (A)
• $\phi$ the Phase angle between voltage and current

2.1 Electric motor without compensation

At $U=400\mathrm{V}$, an asynchronous motor draws a current of $I=10\mathrm{A}$ and delivers an active power of $P=2.8\mathrm{kW}$:

$$\cos \phi =\frac{P}{U\cdot I}=\frac{2.8\mathrm{kW}}{400\mathrm{V}\cdot 10\mathrm{A}}=\frac{2800\mathrm{W}}{4000\mathrm{VA}}=\underline{0.70}$$

The electric motor has a power factor of $0.70$, which is typical of inductive loads.

Apparent power: $S=U\cdot I=400\mathrm{V}\cdot 10\mathrm{A}=4.0\mathrm{kVA}$

Reactive power: $Q=\sqrt{S^2-P^2}=\sqrt{4.0^2-2.8^2}\mathrm{kvar}=\sqrt{16-7.84}\mathrm{kvar}=\underline{2.86\mathrm{kvar}}$

2.2 Reactive power compensation in the AC grid

To improve the power factor of the electric motor above from $\cos {\phi }_1=0.70$ to $\cos {\phi }_2=0.95$, a compensating capacitor is connected in parallel.

Required capacitive reactive power of the capacitor:

$$Q_C=P\cdot (\tan {\phi }_1-\tan {\phi }_2)$$

With ${\phi }_1=\arccos (0.70)=45.6°$ and ${\phi }_2=\arccos (0.95)=18.2°$:

$$Q_C=2.8\mathrm{kW}\cdot (\tan (45.6°)-\tan (18.2°))=2.8\mathrm{kW}\cdot (1.02-0.33)$$ $$Q_C=2.8\mathrm{kW}\cdot 0.69=\underline{1.93\mathrm{kvar}}$$

After compensation:

• New apparent power: $S_2=\frac{P}{\cos {\phi }_2}=\frac{2.8\mathrm{kW}}{0.95}=2.95\mathrm{kVA}$
• New current: $I_2=\frac{S_2}{U}=\frac{2.95\mathrm{kVA}}{400\mathrm{V}}=7.4\mathrm{A}$
• Current saving: $\Delta I=10\mathrm{A}-7.4\mathrm{A}=\underline{2.6\mathrm{A}}$ (26 %)

The compensation reduces the current drawn from the grid by $26\%$, which leads to lower conductor losses and better utilization of the grid. In the electric lines of the electric circuit, however, the higher currents still flow even after compensation, which you have to take into account when dimensioning the electric lines.

2.3 Household

A typical household with predominantly resistive loads (heating, lighting) has a power factor of $\cos \phi \approx 0.95$ to $1.0$.
Halogen transformers, fluorescent lamps and induction hobs, for example, worsen the power factor if they do not have internal compensating capacitors.

2.4 Industrial plant

An industrial plant with many electric motors and transformers without compensation can have a power factor of $\cos \phi =0.6$ to $0.8$. Targeted reactive power compensation raises this to at least $\cos \phi =0.9$ in order to avoid penalty payments to the energy supplier and to increase energy efficiency.
Why penalty payments? The poor power factor harms the power grid; therefore, following the polluter-pays principle, the energy suppliers have the necessary countermeasures paid for by those who cause them.

3 Typical orders of magnitude

Very good values (0.95 – 1.0):

• Heating devices, kettles: $\cos \phi \approx 1.0$ (purely resistive)
• Light emitting diode lighting (with compensation): $\cos \phi =0.95$ to $0.99$
• Compensated industrial installations: $\cos \phi =0.95$ to $0.98$

Good values (0.85 – 0.95):

• Synchronous motors: $\cos \phi =0.85$ to $1.0$
• Modern asynchronous motors (full load): $\cos \phi =0.85$ to $0.9$
• Households: $\cos \phi =0.9$ to $0.95$

Medium values (0.70 – 0.85):

• Asynchronous motors (partial load): $\cos \phi =0.70$ to $0.85$
• Fluorescent lamps (without compensation): $\cos \phi =0.5$ to $0.7$
• Transformers (idle operation): $\cos \phi =0.2$ to $0.3$

Poor values (< 0.70):

• Uncompensated welding machines: $\cos \phi =0.5$ to $0.7$
• Electric motors in idle operation: $\cos \phi =0.2$ to $0.4$
• Old ballasts: $\cos \phi =0.4$ to $0.6$

4 Example phrasings

• ‘The electric motor has a power factor of $\cos \phi =0.87$.’
• ‘Reactive power compensation improved the power factor from $0.75$ to $0.92$.’
• ‘The energy supplier requires a minimum power factor of $0.9$.’
• ‘With a power factor below $0.9$, surcharges for reactive current are charged.’
• ‘The resistive load operates with an ideal power factor of $\cos \phi =1.0$.’

5 Related quantities

• Active power $P$:

$$P=S\cdot \cos \phi =U\cdot I\cdot \cos \phi$$

• Apparent power $S$:

$$S=U\cdot I=\frac{P}{\cos \phi }$$

• Reactive power $Q$:

$$Q=S\cdot \sin \phi =U\cdot I\cdot \sin \phi$$

• Relationship between the powers:

$$S^2=P^2+Q^2$$

• Phase angle $\phi$:

$$\phi =\arccos (\cos \phi )$$

• Impedance $Z$:

$$\cos \phi =\frac{R}{|Z|}=\frac{R}{\sqrt{R^2+X^2}}$$

• Capacitor for compensation:

$$C=\frac{Q_C}{\omega \cdot U^2}=\frac{Q_C}{2\pi f\cdot U^2}$$

6 Related components

• Compensating capacitor: Supplies capacitive reactive power to compensate for inductive loads
• Electric motor: Main cause of inductive reactive power in industry
• Transformer: Requires magnetizing reactive power
• Inductor: Causes inductive reactive power
• Energy meter: Records active and reactive energy separately