Phase angle

Prof. Dr. Thorsten Jungmann (last updated 2026-05-04)

Phase angle: How far does the current lag behind the voltage? – The smaller, the more efficient!

‘ELI the ICE man’: in an inductor (L) the voltage E comes before the current I; in a capacitor (C) the current I comes before the voltage E.

The phase angle (German: Phasenwinkel) $\phi$ (phi) describes the phase difference, i.e. the shift in time between voltage and current in AC circuits. It is caused by inductive and capacitive components and largely determines the power factor and thus the efficiency of the transmission of electricity. A phase angle of 0° means optimal use of energy; larger angles lead to reactive power.

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 the theory of alternating current (EE2).

1 Quantity symbol and unit

The quantity symbol of the phase angle is the lower-case Greek letter $\phi$ (phi).
The phase angle is a dimensionless quantity and is usually given in degrees (°) or radians (rad):

$$[\phi ]=°\text{ or rad}$$

Sign convention (German: Vorzeichenkonvention):

• $\phi >0$: Current lags behind the voltage (inductive, e.g. electric motor, inductor)
• $\phi =0$: Current and voltage in phase (resistive, e.g. heater)
• $\phi <0$: Current leads the voltage (capacitive, e.g. capacitor)

2 Practical and worked examples

The phase angle can be calculated from various quantities:

From the power factor:

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

From active and reactive power:

$$\phi =\arctan \left(\frac{Q}{P}\right)$$

From resistance and reactance:

$$\phi =\arctan \left(\frac{X}{R}\right)$$

From apparent, active and reactive power:

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

2.1 Resistive load (heating device)

A heating device with $R=50\mathrm{\Omega }$ is connected to $U=230\mathrm{V}$.

As the load is purely resistive, the phase angle is:

$$\phi =0°$$

Current and voltage oscillate in synchronism, maximum transmission of electricity:

$$\cos \phi =\cos (0°)=\underline{1.0}$$

2.2 Asynchronous motor

An electric motor draws $P=3.0\mathrm{kW}$ of active power and $Q=2.5\mathrm{kvar}$ of reactive power.

Phase angle:

$$\phi =\arctan \left(\frac{Q}{P}\right)=\arctan \left(\frac{2.5\mathrm{kvar}}{3.0\mathrm{kW}}\right)=\arctan (0.833)=\underline{39.8°}$$

Power factor:

$$\cos \phi =\cos (39.8°)=\underline{0.77}$$

The current lags behind the voltage by just under 40° (typical of inductive loads).

2.3 Compensation of an electric motor

The electric motor from the previous example is to be compensated from ${\phi }_1=39.8°$ to ${\phi }_2=18.2°$ (corresponds to $\cos \phi =0.95$).

Before:

• ${\phi }_1=39.8°$
• $\cos {\phi }_1=0.77$
• $\tan {\phi }_1=0.833$

After:

• ${\phi }_2=18.2°$
• $\cos {\phi }_2=0.95$
• $\tan {\phi }_2=0.329$

Required capacitive reactive power:

$$Q_C=P\cdot (\tan {\phi }_1-\tan {\phi }_2)=3.0\mathrm{kW}\cdot (0.833-0.329)=\underline{1.51\mathrm{kvar}}$$

The compensation reduces the phase angle by 21.6°.

2.4 Series RL circuit

An electric circuit consists of $R=30\mathrm{\Omega }$ and $X_L=40\mathrm{\Omega }$ in series.

Phase angle:

$$\phi =\arctan \left(\frac{X_L}{R}\right)=\arctan \left(\frac{40\mathrm{\Omega }}{30\mathrm{\Omega }}\right)=\arctan (1.33)=\underline{53.1°}$$

Impedance:

$$Z=\sqrt{R^2+X_L^2}=\sqrt{30^2+40^2}\mathrm{\Omega }=\sqrt{2500}\mathrm{\Omega }=50\mathrm{\Omega }$$

Power factor:

$$\cos \phi =\frac{R}{Z}=\frac{30\mathrm{\Omega }}{50\mathrm{\Omega }}=0.6$$

2.5 Ideal capacitor

An ideal capacitor (without ohmic losses) has a phase angle of:

$$\phi =-90°$$

The current leads the voltage by 90°. Due to losses, real capacitors have:

$$\phi \approx -85°\text{ to }-89°$$

3 Typical orders of magnitude

Resistive loads ($\phi =0°$):

• Heating devices, kettles: $\phi =0°$ ($\cos \phi =1.0$)
• Incandescent bulbs: $\phi =0°$ ($\cos \phi =1.0$)
• Electric cooker: $\phi =0°$ to $2°$ ($\cos \phi \approx 1.0$)

Slightly inductive ($0°$ < $\phi$ < $25°$):

• Compensated electric motors: $\phi =10°$ to $20°$ ($\cos \phi =0.94$ to $0.98$)
• LED drivers: $\phi =15°$ to $20°$ ($\cos \phi =0.94$ to $0.97$)
• Synchronous motor: $\phi =0°$ to $30°$ ($\cos \phi =0.87$ to $1.0$)

Inductive ($25°$ < $\phi$ < $50°$):

• Asynchronous motor (full load): $\phi =30°$ to $40°$ ($\cos \phi =0.77$ to $0.87$)
• Transformer (rated load): $\phi =2°$ to $10°$ ($\cos \phi =0.98$ to $1.0$)
• Welding transformer: $\phi =35°$ to $50°$ ($\cos \phi =0.64$ to $0.82$)

Strongly inductive ($\phi >50$):

• Asynchronous motor (idle operation): $\phi =70°$ to $80°$ ($\cos \phi =0.17$ to $0.34$)
• Transformer (idle operation): $\phi =80°$ to $85°$ ($\cos \phi =0.09$ to $0.17$)
• Inductor: $\phi =75°$ to $85°$ ($\cos \phi =0.09$ to $0.26$)
• Ideal inductor: $\phi =90°$ ($\cos \phi =0$)

Capacitive ($\phi <0$):

• Compensating capacitor: $\phi =-85°$ to $-89°$
• Ideal capacitor: $\phi =-90°$ ($\cos \phi =0$)
• Cable (open-circuit capacitance): $\phi =-80°$ to $-88°$

4 Example phrasings

• ‘The electric motor has a phase angle of $\phi =35°$.’
• ‘With a purely resistive load, the phase angle is φ = 0°. There is no phase difference’
• ‘Through compensation, the phase difference was reduced from 42° to 18°.’
• ‘The current lags behind the voltage by $\phi =28.5°$.’
• ‘At a phase angle of $\phi =20°$, the power factor is $\cos \phi =0.94$.’

5 Related quantities

• Power factor:

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

• Reactive factor:

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

• Active power:

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

• Reactive power:

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

• Tangent $\phi$:

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

• Impedance:

$$\phi =\arg (Z)=\arctan \left(\frac{\text{Im}(Z)}{\text{Re}(Z)}\right)=\arctan \left(\frac{X}{R}\right)$$

• Reactance:

$$X=R\cdot \tan \phi$$

• Active resistance:

$$R=Z\cdot \cos \phi$$

• Time shift:

$$\Delta t=\frac{\phi }{2\pi f}=\frac{\phi }{360°\cdot f}$$

6 Related components

• Inductor: Causes a positive phase difference (current lagging)
• Capacitor: Causes a negative phase difference (current leading)
• Resistor: No phase difference ($\phi =0°$)
• Electric motor: Typically $\phi =30°$ to $40°$ at rated load
• Transformer: Small under load, large in idle operation
• Phase shifter: Targeted setting of the phase angle