A three-phase system uses three sinusoidal voltages of the same frequency and amplitude separated by $120^\circ$ in phase.

A balanced set can be written

$$v_a(t)=V_m\cos\omega t,$$

$$v_b(t)=V_m\cos\left(\omega t-\frac{2\pi}{3}\right),$$

$$v_c(t)=V_m\cos\left(\omega t-\frac{4\pi}{3}\right).$$

The three waveforms

At any instant, the three phase voltages are at different points of the same sinusoidal cycle. For an ideal balanced set,

$$v_a(t)+v_b(t)+v_c(t)=0.$$

The same cancellation occurs for balanced phase currents.

Balanced loads

A load is balanced when each phase presents the same electrical behavior to its corresponding phase supply. The three phase currents then have equal magnitudes and the same $120^\circ$ separation.

In a balanced four-wire system, the instantaneous neutral current is zero because the three phase currents sum to zero.

Phase sequence

The order in which the phases reach corresponding points in their cycles is the phase sequence. Reversing the sequence changes the direction of the rotating magnetic field produced by three spatially separated windings.

Why three phases are useful

A balanced three-phase system can transfer power much more smoothly than one isolated sinusoidal phase and naturally creates rotating magnetic fields.

These properties make three-phase systems fundamental to electric-power systems and rotating electrical machines.

Phasors provide a compact later representation of these phase relationships, but they are not required to understand the basic three-phase pattern.