A balanced three-phase load receives power from three equal phase systems shifted by $120^\circ$. Adding the phase powers gives a constant total average power.

For each phase,

$$P_{\mathrm{ph}}=V_{\mathrm{ph}}I_{\mathrm{ph}}\cos\phi.$$

Therefore

$$P_{3\phi}=3V_{\mathrm{ph}}I_{\mathrm{ph}}\cos\phi.$$

Line quantities

Using the star or delta line-to-phase relationships, the same balanced power can be written

$$P_{3\phi}=\sqrt3,V_LI_L\cos\phi.$$

This expression is valid for balanced star and balanced delta loads when $V_L$ and $I_L$ are RMS line quantities.

Reactive and apparent power

Similarly,

$$Q_{3\phi}=\sqrt3,V_LI_L\sin\phi,$$

and

$$|S_{3\phi}|=\sqrt3,V_LI_L.$$

The power factor remains

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

Why the total power is smooth

Although each phase's instantaneous power varies, the three variations are phase-shifted so that they cancel in a balanced system. This nearly constant power transfer is one reason three-phase supplies are especially suitable for motors and large power systems.