Reactive factor

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

Reactive factor: The counterpart of the power factor – shows the share of reactive power!

The reactive factor (German: Blindleistungsfaktor) describes the ratio of reactive power to apparent power in AC circuits. It is the counterpart of the power factor ($\cos \phi$) and indicates what share of the apparent power provided by the supply network oscillates as reactive power. The reactive factor is the sine of the phase angle $\phi$ between voltage and current.

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 a deeper understanding are part of AC circuit theory (EE2).

1 Quantity symbol and unit

The usual quantity symbol of the reactive factor is $\sin \phi$ (sine of the phase angle).
The reactive factor is a dimensionless quantity with values between 0 and 1 (or -1 for capacitive loads):

$$[\sin \phi ]=1$$

Sign convention (German: Vorzeichenkonvention):

• $\sin \phi >0$: inductive reactive power (current lags)
• $\sin \phi =0$: no reactive power (purely resistive)
• $\sin \phi <0$: capacitive reactive power (current leads)

2 Practical and worked examples

The reactive factor is defined as:

$$\sin \phi =\frac{Q}{S}$$

where:

• $Q$ is the reactive power in volt-ampere reactive (var)
• $S$ is the apparent power in volt-amperes (VA)

From the phase angle:

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

The following relationship holds between the power factor and the reactive factor:

$${\sin }^2\phi +{\cos }^2\phi =1$$ $$\sin \phi =\sqrt{1-{\cos }^2\phi }$$

2.1 Asynchronous motor

An electric motor has a power factor of $\cos \phi =0.8$ and draws an apparent power of $S=5.0,\mathrm{kVA}$.

Reactive factor:

$$\sin \phi =\sqrt{1-{\cos }^2\phi }=\sqrt{1-0.8^2}=\sqrt{1-0.64}=\sqrt{0.36}=\underline{0.6}$$

Phase angle:

$$\phi =\arcsin (0.6)=\underline{36.9°}$$

Reactive power:

$$Q=S\cdot \sin \phi =5.0,\mathrm{kVA}\cdot 0.6=\underline{3.0,\mathrm{kvar}}$$

Active power:

$$P=S\cdot \cos \phi =5.0,\mathrm{kVA}\cdot 0.8=4.0,\mathrm{kW}$$

60 % of the apparent power is reactive power, 80 % is active power.
(Yes, this really does add up to 100 %: $0.6^2+0.8^2=1$).

2.2 Welding machine

A welding machine draws $P=8.0,\mathrm{kW}$ of active power and $Q=12.0,\mathrm{kvar}$ of reactive power.

Apparent power:

$$S=\sqrt{P^2+Q^2}=\sqrt{8.0^2+12.0^2},\mathrm{kVA}=\sqrt{64+144},\mathrm{kVA}=\sqrt{208},\mathrm{kVA}=14.42,\mathrm{kVA}$$

Reactive factor:

$$\sin \phi =\frac{Q}{S}=\frac{12.0,\mathrm{kvar}}{14.42,\mathrm{kVA}}=\underline{0.832}$$

Power factor:

$$\cos \phi =\frac{P}{S}=\frac{8.0,\mathrm{kW}}{14.42,\mathrm{kVA}}=0.555$$

Phase angle:

$$\phi =\arcsin (0.832)=56.3°$$

The welding machine has a high reactive factor of 0.832 (strongly inductive).

2.3 Compensation

An electric motor with $\cos {\phi }_1=0.75$ ($\sin {\phi }_1=0.661$) is to be compensated to $\cos {\phi }_2=0.95$.

Reactive factor after compensation:

$$\sin {\phi }_2=\sqrt{1-0.95^2}=\sqrt{1-0.9025}=\sqrt{0.0975}=\underline{0.312}$$

Ratio of the reactive factors:

$$\frac{\sin {\phi }_2}{\sin {\phi }_1}=\frac{0.312}{0.661}=0.472$$

The compensation reduces the reactive factor to 47.2 % of its original value.

2.4 Resistive load

A heating device operates purely resistively with $\phi =0°$.

Reactive factor:

$$\sin \phi =\sin (0°)=\underline{0}$$

There is no reactive power ($Q=0$); the entire apparent power is active power.

2.5 Transformer in idle operation

A transformer in idle operation has $\cos {\phi }_0=0.15$.

Reactive factor:

$$\sin {\phi }_0=\sqrt{1-0.15^2}=\sqrt{1-0.0225}=\sqrt{0.9775}=\underline{0.989}$$

Almost 99 % of the apparent power drawn is magnetizing reactive power.

3 Typical orders of magnitude

Very low ($\sin \phi <0.2$):

• Resistive loads: $\sin \phi =0$ ($\phi =0°$)
• Compensated installations: $\sin \phi =0.1$ to $0.2$ ($\phi =6°$ to $12°$)
• Heating devices: $\sin \phi \approx 0$

Low ($0.2\le \sin \phi <0.4$):

• Well-compensated electric motors: $\sin \phi =0.2$ to $0.35$ ($\cos \phi =0.94$ to $0.98$)
• Synchronous motors: $\sin \phi =0$ to $0.5$ ($\cos \phi =0.87$ to $1.0$)
• Modern LED drivers: $\sin \phi =0.26$ to $0.34$

Medium ($0.4\le \sin \phi <0.7$):

• Asynchronous motor (full load): $\sin \phi =0.5$ to $0.64$ ($\cos \phi =0.77$ to $0.87$)
• Uncompensated electric motors: $\sin \phi =0.55$ to $0.66$
• Transformer (rated load): $\sin \phi =0.0$ to $0.17$

High ($0.7\le \sin \phi <0.95$):

• Welding transformers: $\sin \phi =0.64$ to $0.87$
• Inductors: $\sin \phi =0.91$ to $0.97$
• Asynchronous motor (idle operation): $\sin \phi =0.94$ to $0.98$

Very high ($\sin \phi \ge 0.95$):

• Transformer (idle operation): $\sin \phi =0.98$ to $0.99$
• Almost pure inductor/capacitor: $\sin \phi \approx 1.0$ ($\phi \approx \pm 90°$)
• Magnetizing reactive power: $\sin \phi >0.95$

4 Example phrasings

• ‘The electric motor has a reactive factor of $\sin \phi =0.62$.’
• ‘With a reactive factor of 0.3, the reactive power amounts to 30 % of the apparent power.’
• ‘Compensation reduced the reactive factor from 0.66 to 0.31.’
• ‘The transformer in idle operation shows a reactive factor of $\sin \phi =0.99$.’
• ‘A reactive factor close to 0 means optimum use of energy without any significant reactive power.’

5 Related quantities

• Power factor $\cos \phi$:

$$\cos \phi =\sqrt{1-{\sin }^2\phi }=\frac{P}{S}$$

• Trigonometric relationship:

$${\sin }^2\phi +{\cos }^2\phi =1$$

• Reactive power $Q$:

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

• Active power $P$:

$$P=S\cdot \cos \phi =S\cdot \sqrt{1-{\sin }^2\phi }$$

• Apparent power $S$:

$$S=\frac{Q}{\sin \phi }=\frac{P}{\cos \phi }$$

• Phase angle $\phi$:

$$\phi =\arcsin (\sin \phi )$$

• Tangent $\phi$:

$$\tan \phi =\frac{\sin \phi }{\cos \phi }=\frac{Q}{P}$$

• Reactance $X$:

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

6 Related components

• Inductor: High positive reactive factor (sin φ → +1)
• Capacitor: High negative reactive factor (sin φ → -1)
• Resistor: Reactive factor zero (sin φ = 0)
• Electric motor: Typically sin φ = 0.5 to 0.64 at rated load
• Transformer: Very high in idle operation, low under load
• Inductor: Very high reactive factor (sin φ > 0.9)