Three Phase AC Systems

By the end you should be able to: Describe the generation of three phase voltages and the advantages of three phase systems, and relate line and phase quantities for balanced star and delta connections.

Generation

Place three identical coils on the stator, physically 120° apart, and rotate a magnet inside. Each coil generates the same sinusoid, displaced by 120° in time:

v_R = V_m\sin\omega t

v_Y = V_m\sin(\omega t - 120°)

v_B = V_m\sin(\omega t - 240°)

The phase sequence RYB (red-yellow-blue) is the order in which they reach their maxima. Reversing it reverses the direction of rotation of any motor connected — which is why swapping any two supply leads reverses a three-phase motor, and why sequence matters when paralleling supplies.

For a balanced system the instantaneous sum is zero at every instant:

v_R + v_Y + v_B = 0

That fact does most of the work below.

Why three phase

Constant total power. In a single-phase circuit, instantaneous power pulses at twice the supply frequency, falling to zero twice per cycle. In a balanced three-phase system the three pulsating powers sum to a constant. A three-phase motor therefore produces smooth torque without the vibration inherent to a single-phase machine.

Less conductor material. For the same power delivered at the same voltage and losses, a three-phase system needs about 75% of the copper of an equivalent single-phase system. Over a national grid, that saving is enormous.

Self-starting rotating field. Three windings fed by three phases produce a rotating magnetic field with no auxiliary equipment. A single-phase motor produces only a pulsating field and needs a starting capacitor or shaded pole. This is the decisive advantage — the induction motor is the workhorse of industry and it exists because of three-phase supply.

Two voltages available. A star-connected four-wire system offers both the line voltage and the phase voltage from the same installation — 400 V for machines and 230 V for lighting and sockets.

Smaller machines. For a given frame size a three-phase machine delivers roughly 1.5 times the output of a single-phase one.

Star (Y) connection

The three winding ends are joined at a common neutral.

\boxed{\,V_L = \sqrt{3}\,V_{ph}, \qquad I_L = I_{ph}\,}

The $\sqrt3$ arises because the line voltage is the phasor difference of two phase voltages 120° apart, not their arithmetic difference:

|V_{RY}| = |V_R - V_Y| = 2V_{ph}\cos 30° = \sqrt3\,V_{ph}

This is the standard 400 V / 230 V supply: $400 = \sqrt3 \times 230$ .

In a balanced star the three line currents sum to zero, so the neutral carries no current — it can in principle be omitted. It is retained in distribution because domestic loads are never balanced, and the neutral carries the imbalance and holds the phase voltages steady.

Delta (Δ) connection

The windings form a closed triangle, each junction feeding a line.

\boxed{\,V_L = V_{ph}, \qquad I_L = \sqrt3\,I_{ph}\,}

The roles are exactly swapped. Now the line current is the phasor difference of two phase currents.

It looks alarming to short three sources in a loop, but the sum of the three phase voltages is zero at every instant, so no circulating current flows — provided the system is balanced and correctly connected. Get one winding reversed and a very large circulating current does flow, which is why phasing is checked before closing a delta.

Delta has no neutral, so it offers only one voltage. It is used for motors and for transformer windings; distribution to mixed loads uses star.

Power in a balanced three-phase system

The total power is three times the per-phase power, and in both connections this reduces to the same expression in line quantities:

\boxed{\,P = \sqrt3\,V_L I_L\cos\phi\,}

Q = \sqrt3\,V_L I_L\sin\phi, \qquad S = \sqrt3\,V_L I_L

$\phi$ is the angle between phase voltage and phase current, not between line quantities. This is the commonest error in three-phase calculations.

The formula being identical for star and delta is convenient but hides a real difference, which the second worked example makes concrete.