The voltage at a node can be described by $$ V = |V|e^{j \delta} $$ in which \$\delta\$ is the voltage angle. Voltage angles are required to calculate AC power flows.

The phase angle if often described as the angle between voltage and current.

What is the difference between the two? How can I interpret the voltage angle?


3 Answers 3


An ac voltage can be represented by the instantaneous equation:

$$v_S = V_{M_S}\ sin\ (2 \pi f t ± \theta )\ V $$

where \$\theta \$ can vary from \$-180°\ to\ 180°\$.

Specifying \$\theta \$ allows us (mathematically) to differentiate where the waveform starts, which is meaningless for one waveform, but critical for phase relationships between two or more waveforms.

To simplify calculations, we turn the ac voltage into a phasor.

$$\vec{V_S} = V_S\ ∡ ± \theta\ V $$

To simplify calculations, we put one vector on the x-axis to function as a reference, which means the phase angle is \$0°\$. As in current becomes the reference for series circuits and voltage becomes the reference for parallel or series/parallel circuits.

From your complex notation for a vector, you have \$ V = |V|e^{j \delta} \$.

As a vector:

$$\vec{V} = V\ ∡ \delta\ V $$

As an instantaneous equation:

$$v = V_M\ sin\ (2 \pi f t + \delta )\ V $$

If we have simplified the process and made current the reference, current would be:

$$i = I_M\ sin\ (2 \pi f t)\ A $$

And \$\delta\$ would be the phase angle. As in I lags V by \$\delta\$. It shows the fundamental relationship between two sine (or cosine) waves.

In a three-phase circuit: $$P_T = \sqrt {3}\ V_L\ I_L\ cos\ \delta $$

But I does not have to start at \$0°\$. If \$ I = |I|e^{j -\sigma} \$, then the phase angle would be \$ \theta = \delta - (-\sigma) \$.

$$P_T = \sqrt {3}\ V_L\ I_L\ cos\ \theta $$

where \$ pf = cos\ \theta\$.

So whether the angle is the phase angle depends upon the other waveforms, but the specified angle states where the resultant sine (or cosine) wave starts.


How can I interpret the voltage angle?

The angle that a particular voltage phasor may have is value that is relative to some reference voltage phasor (that would normally have a phase angle of zero). A good example are the line voltages of a 3 phase power feed; one voltage phasor would have an angle of 0° and the other two would be -120° and +120° relative to the reference.

The phase angle is often described as the angle between voltage and current.

Yes it can be and quite often is when talking about impedances.

  • \$\begingroup\$ Thanks for your elaboration! In case of AC power flow, how would you describe the voltage angle of a node in the network (so not of a phasor)? \$\endgroup\$
    – Martiene
    May 23, 2023 at 13:51
  • 1
    \$\begingroup\$ @Martiene, the "phase angle of the voltage at node X" is the phase angle of the phasor representing the voltage at node X. You must determine the phasor representation (or some equivalent information) before you can describe the phase angle of the voltage. \$\endgroup\$
    – The Photon
    May 23, 2023 at 14:46

Phasors can represent many things. Voltages at nodes, currents through branches, impedances of branches, outputs of generators, etc. They can in fact represent anything we measure as a complex number.

The phase angle if often described as the angle between voltage and current.

The phase angle of an impedance is the angle between the voltage across the impedance and the current through the impedance.

But an impedance is only one of many things that can be represented by a phasor or complex number. The phase angle of the phasor (or complex number) representing some other physical variable means something else.

The phase angle of a voltage just represents the phase difference between the voltage at that node and some arbitrary reference we've chosen to be "0 phase". With the caveat that we should use the same reference for measuring the phase of all voltages in a system. Then we can compare the phase angles of different nodes. For example, we might want to know how much the phase of the voltage at the output node leads or lags the phase at the input node. Or how much the phase of a feedback signal leads or lags the phase of the input signal. So even if the choice of phase reference is arbitrary, knowledge of the phase of different voltages or currents in a system can still be very useful.

  • \$\begingroup\$ Thank you! In case of a AC circuit, how would the comparison between the phase angles of different nodes go? Since there are three phases. \$\endgroup\$
    – Martiene
    May 25, 2023 at 7:09

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