Three-Phase Power

Power Definitions

3-Phase Instantaneous Power

Irrespective of the connection style type, the instantaneous 3-phase power can be derived as:

$$p_{\text{3-phase}}(t)=3V_{\text{phase}}{I}_{\text{phase}}\cos \theta$$

Even though the instantaneous powers of individual phases are fluctuating, the total 3-phase instantaneous power is constant. This is one of the advantages of 3-phase systems over single-phase systems.

3-Phase Average Power

Since the total 3-phase instantaneous power is constant, it is just the average power:

$$P_{\text{3-phase}}=3V_{\text{phase}}{I}_{\text{phase}}\cos \theta =3\times P_{1-\text{phase}}$$

3-Phase Reactive Power

3-phase reactive power is given by:

$$Q_{\text{3-phase}}=3V_{\text{phase}}{I}_{\text{phase}}\sin \theta =3\times Q_{\text{1-phase}}$$

3-Phase Apparent Power

3-phase apparent power is given by:

$$S_{\text{3-phase}}=3V_{\text{phase}}{I}_{\text{phase}}\sin \theta =3\times S_{\text{1-phase}}$$

3-Phase Complex Power

3-phase complex power is given by:

$$S_{\text{3-phase, complex}}=3{\vec{V}}_{\text{phase}}{\vec{I}}_{\text{phase}}^{\ast }$$

3-Phase Power Factor

The power factor, as in the case of single-phase circuits, is the cosine of the angle between the phase voltage and phase current or is given by the ratio of the 3-phase real power to 3-phase apparent power.

Relations based on Line-to-Line Voltage

In a 3-phase system, line-to-line voltage and line current are usually used to describe the system. Thus, it’s useful to know how 3-phase power is expressed in terms of line-to-line voltage and line current.

If the load is Y-connected, we have

$$\{\begin{array}{l}{V}_{LL}=\sqrt{3}{V}_{\text{phase}}\\ I_L=I_{\text{phase}}\end{array}$$

If the load is $\Delta$-connected, we have

$$\{\begin{array}{l}{V}_{\text{LL}}=V_{\text{phase}}\\ I_{\text{L}}=\sqrt{3}{I}_{\text{phase}}\end{array}$$

Thus, the above relations can be written as:

$$\begin{array}{rl}{P}_{\text{3-phase}}& =3V_{\text{phase}}{I}_{\text{phase}}\cos \theta =\sqrt{3}{V}_{LL}{I}_L\cos \theta \\ Q_{\text{3-phase}}& =3V_{\text{phase}}{I}_{\text{phase}}\sin \theta =\sqrt{3}{V}_{LL}{I}_L\sin \theta \\ S_{\text{3-phase}}& =3V_{\text{phase}}{I}_{\text{phase}}=\sqrt{3}{V}_{LL}{I}_L\end{array}$$

Note that $\theta$ is the phase impedance angle (the angle between the phase voltage and current). Also written with $\varphi$ sometimes.