Three-phase power distribution involves three AC voltage lines which are 120 degrees or radians out of phase with each other. The mathematics of this gives rise to several desirable physical properties, like constant power transfer and rotating magnetic fields, because of the trigonometric identity

## Cosine sum-to-product identity Edit

To derive this, we'll need a couple of other common trigonometric identities. First recall that

Add these to get the product-to-sum identity

Substitute and to finally get the sum-to-product identity

## Three-phase identity Edit

With this sum-to-product identity we can now start the derivation proper:

Simplifying,

Using the sum-to-product identity again

Which simplifies to

as the cosine of pi/2 is zero. Q.E.D.

## Constant Power Transfer Edit

A three-phase voltage supply will deliver constant instantaneous power to a star- or delta-connect resistive load. The instantaneous power is the sum of power of each of the three phases

Using the product-to-sum identity for each phases term,

The second term in the above expression is the three-phase identity, which sums to zero. So we have

which is constant.

## Rotating Magnetic Field Edit

A three-phase supply hooked up to coils offset by 120 degrees will generate a rotating magnetic field, making induction motors possible.

When each phase is distributed in a sinusoidal-distributed winding with N turns, the turn 'density' distribution by angle (for phase A) is

The magneto-motive force (m.m.f.) produced by this winding is

Phases B and C are the same, but offset by 2pi/3 and 4pi/3 respectively.

When supplied from a balanced 3-phase supply, the current in phase A is . Again, phases B and C are the same but offset by 2pi/3 and 4pi/3 respectively. The m.m.f. becomes

The total m.m.f. of the three windings is thus

Using the product-sum identity for each term, we get

The second term in the above expression is the three-phase identity, which sums to zero. So we have

This represents a revolving magnetic field (m.m.f.) rotating around the air gap at an angular speed of rad/sec.