three phase power

The following information is essential only if you want to understand how and why it works.

Ohm's Law

V = I R

Where
$V =$ voltage, in volts
$I =$ current, in Amps
and $R =$ Resistance in Ohms $(Ω)$

said another way, Resistance is the voltage loss per Amp $(R = \frac{V}{I})$ for some material.

3 phase power

Sometimes abbreviated as $3 ϕ$

Three-phase power uses between 3 and 5 wires to transmit alternating current. Each phase is a sinewave of voltage which is offset by $120 °$ (or $\frac{2 π}{3}$ radians) from either other phase.

In a $3 ϕ$ system, the current in any line is equal to the sum of currents in the other lines:

i_{1} = i_{2} + i_{3}

This means that a 3 phase system does not require a neutral wire to return current - as in a single-phase AC system - provided that all 3 phases have a balanced load.

All else being equal, a 3 phase system (without a neutral wire) supplies the same power as a single-phase AC system (with a line and a neutral wire) using 25% less wire, by mass.

3 phase power simplifies the wiring of electric motors. If all 3 lines are wired to a motor in sequence, the resulting rotating magnetic field will cause the motor to start and run at the line frequency without any additional circuity (as is typically required for a single-phase AC motor or a DC motor).

3 phase voltages

Each phase is carried on an individual wire - a line conductor - and the voltage measured between any two lines is called the line-to-line voltage or just line voltage.

In a setup that includes a neutral wire, the electric potential measured between a line and the neutral is the line-to-neutral voltage or the phase voltage.

As depicted here, the line voltage is equal to the phase voltage multiplied by $\sqrt{3}$ ,

\begin{aligned} v_{l i n e} = \sqrt{3} \cdot v_{p h a s e} & v_{p h a s e} = \frac{v_{l i n e}}{\sqrt{3}} \end{aligned}

In a typical US 480 VAC $3 ϕ$ system this equates to a line to line voltage of 480 VAC and a line to neutral voltage of 277 VAC.
Likewise, using a Wye configuration with phase voltage of 120 VAC results in a line voltage of 208 VAC. This means that a single 3 phase transformer can provide 3 sources of 120 VAC / 208 VAC from 4 wires.

3 phase wiring

Wiring for a 3 phase system requires a minimum of 1 wire per phase. A ground conductor (indicated by either bare copper or green insulation) can be added to any 3 phase system. In normal operation, the ground carries no current.

3 phase systems are wired in either a delta or a wye configuration. Because of conservation of energy, both systems will transfer the same power at different voltages and currents. With respect to line voltage $(v_{l i n e})$ and line current $(i_{l i n e})$ those equations are:

	$Δ$	Wye
voltage	$v_{l i n e} = v_{p h a s e}$	$v_{l i n e} = v_{p h a s e} \sqrt{3}$
current	$i_{l i n e} = i_{p h a s e} \sqrt{3}$	$i_{l i n e} = i_{p h a s e}$

Aside from current draw, they have other advantages and disadvantages, such as the potential for multiple output voltages or fault tolerance.

3 wire delta

A transformer wired in this manner (called a Delta or $Δ$ configuration) provides only line-to-line voltages.
$∴ v_{l i n e} = v_{p h a s e}$ for a $Δ$ .

In this configuration, current from one phase is returned by the other two phases. All else being equal, the 3 wires required to connect a $Δ$ setup will cost 75% of what a #4 wire wye will cost.

Another advantage is the fault tolerance of a $Δ$ : If one of the windings in the transformer fails (i.e. the wire breaks and it's now an open circuit) then the remaining 2 phases will provide the same voltage albeit with an increased current draw to compensate.

However, a $Δ$ is only suited for balanced 3 phase loads - where all each phase will have a symmetrical current draw - such as a 3 phase motor.
The line current for a $Δ$ configuration is greater than that of an equivalent Wye. Line voltage and phase voltage are the same, but the currents differ:

\begin{aligned} v_{l i n e} = v_{p h a s e} \\ i_{l i n e} = \sqrt{3} \cdot i_{p h a s e} & since I = \frac{P}{V} \end{aligned}

4 wire wye

By changing the wiring from the transformer windings (as shown), a neutral wire can be provided to allow for split phases, or to balance an asymmetrical load: if one of the loads is turned off or of a vastly different magnitude than the others, the return current is routed through the neutral with no change in the current draw on the other two phases.

A transformer configured in a Wye or Y configuration can provide both line-to-line and line-to-neutral voltages: a 208 VAC $3 ϕ$ system will have a phase voltage of 120 VAC (as measured from the neutral to Phase A, B, or C).

The line voltage is equal to the phase voltage multiplied by $\sqrt{3}$ and since electrical power is equal to the product of current and voltage:

\begin{aligned} v_{l i n e} = \sqrt{3} \cdot v_{p h a s e} & v_{p h a s e} = \frac{v_{l i n e}}{\sqrt{3}} \\ i_{l i n e} = i_{p h a s e} & since I = \frac{P}{V} \end{aligned}