Most electric power is transmitted over high voltage three-phase lines. This method of power transmission is more efficient than single phase power, as you might use in your home. As you might expect, three-phase power transmission uses three wires. The voltage between any wire and ground has the same magnitude as the voltage between any other wire and ground, but the phases are different. Each voltage will be 120° out of phase with the other voltages, as shown in the figure below.

Each color represents a different phase. At any point in time, the sum of the three voltages is exactly zero.

Three-phase power is produced by specially designed three-phase electric generators. When a wire moves through a magnetic field, a voltage is generated in the wire (Faraday's Law). For a three-phase generator, three coils of wire are placed at angles of 120° to each other, and are rotated in a magnetic field 60 times per second. This produces three voltasges which are 60 Hz sinusoids, each out of phase with the others by 120°.

There are two ways these three coils of wire can be connected together. The two connections are called delta and Y. For a delta-connected generator the coils are connected together as shown in the figure below.

The coils of wire shown do not represent inductors; rather
they represent windings of the generator. Each such winding
generates what is called a *phase voltage*, meaning the
voltage of one phase of the source, shown in the figure as V_{φ}.
Each winding also has a *phase
current*, I_{φ}. The phase
voltage and phase current will produce a *line voltage*, V_{L},
and a *line current*, I_{L}.

To back up a bit, this discussion covers only *balanced
three-phase*. This means that all loads for all phases are
exactly the same. In practice, this is (usually) very nearly
true. The analysis of unbalanced three-phase is much more
difficult and is beyond the scope of this course. Given that this
is balanced three-phase, each V_{φ}
is exactly the same as every other, keeping in mind that these
variables represent rms magnitudes. The same is true for I_{φ},
V_{L}, and I_{L}.

To return to the delta-connected generator, you can easily see
that any pair of lines is directly connected to one phase of the
generator. It should therefore be obvious that:

(1) V_{L} = V_{φ}

The relationship between I_{L} and I_{φ}
is not nearly so obvious. One
must sum the phasor currents in two phases of the source to find
the line current. The result is shown below.

(2) I_{L} = √3I_{φ}

This peculiar result occurs because the currents are 120° out of phase with each other.

A Y-connected generator has coils connected as shown in the figure below.

It should be clear from this figure that the current in any
phase must be the same as the current in its connected line.

I_{L} = I_{φ}

The line voltage can be found by taking the phasor sum of the
voltage in two phases of the source. The result is shown below.

(4) V_{L} = √3V_{φ}

The analysis of three-phase circuits reduces to little more than deciding where to put the square root of three.

The loads for three-phase power can also be either delta- or Y-connected, as shown in the figure below.

The voltage across each **Z** is a phase voltage,
V_{φ}, for the load. We can
reuse Equations (1) and (2) above for the delta-connected load. We can reuse
Equations (3) and (4) for the Y-connected load. The same equations apply for sources
and loads. For example, the current in a Y-connected load is the
same as the line current.

The analysis of a three-phase circuit is virtually the same as the analysis for a single phase AC circuit. We can still use phasor analysis, power triangles, and all the rest. There are a few specialized equations for total and phase power that may be helpful. These equations work for both delta and Y loads.

The power consumed in one phase of the load can be found if the current
resistance of the phase are known:

(5) P_{φ} = I_{φ}^{2}R_{φ}

By conservation of energy, the total power consumed by the load must be 3 times
the power consumed in any phase:

(6) P_{T} = 3P_{φ}

To get the total power consumed in the load, the following equation is useful:

(7) P_{T} = √3V_{L}I_{L}cosθ
where cosθ is the power factor.

The voltage and currents in the above equations are rms values, not peak values.

In the circuit below, a three-phase Y-connected generator is connected through power lines to a delta-connected load. Each phase of the generator produces 277 V. The load consumes 180 kW of power with a power factor of 66% lagging. The task is to find the following: (a) the line voltage, (b) the phase voltage of the load, (c) the line current, (d) the phase current of the generator, (e) the phase current of the load, and (f) the power consumed by one phase of the load.

(a) From Equation (4) for Y-connected sources or loads:

V_{L} = √3V_{φ} = 480 V

(b) From Equation (1) for delta-connected sources or loads:

V_{φLoad} = V_{L} = 480 V

(c) From Equation (7):

P_{T} = √3V_{L}I_{L}cosθ

I_{L} = P_{T}/(√3V_{L}cosθ)

I_{L} = 180,000/(√3 × 480 × 0.66) = 328 A

(d) From Equation (3) for Y-connected sources or loads:

I_{φGen} = I_{L} = 328 A

(e) From Equation (2) for delta-connected sources or loads:

I_{φLoad} = I_{L}/√3 = 328/√3 = 189 A

(f) From Equation (6):

P_{φ} = P_{T}/3 = 180 kW/3 = 60 kW

Most commercial power plants use Y-connected generators with the center tap grounded. If the phases are balanced (all phases have exactly the same current), no current will flow through the grounded center tap. When single-phase power is needed from a three-phase system, it can be extracted from any one of the three phases.

Most large motors are three-phase delta-connected motors. Three-phase motors have a distinct advantage over single-phase motors: the power is constant. In a single-phase motor, p(t) is a sinusoid, but in a three-phase motor, p(t) is a constant.

The line voltage and line current, V_{L} and I_{L}, are more useful than the phase voltage
and phase current, V_{φ} and I_{φ}. This is because V_{L} and I_{L}
are much easier to measure. V_{φ} and I_{φ} can
only be measured by opening up the machine.

Most utilities will provide a customer with only one type of electrical service, either single-phase or three-phase. (See http://www.federalpacific.com/university/transbasics/chapter6.html and http://en.wikipedia.org/wiki/Three-phase_electric_power for further explanation). If the system is single-phase, it will probably be three-wire 120 V and 240 V. For three-phase systems, more options may be available: 240 V three-wire, 480 V three-wire, 600 V three-wire, 208Y/120 V four-wire, 480Y/277 V four-wire.

Power is measured in three-phase systems using the "two-wattmeter method," as shown below.

The two wattmeters are connected to any two phases of the input, and then to the
load as shown. Each wattmeter measures the average power, the average of
v(t) × i(t), not V_{rms} × I_{rms}. Wattmeter 1
senses the current in the top wire and the voltage between the top wire and the
center wire; it calculates the average of the product of the voltage and the
current. If Wattmeter 1
measures P_{1} and Wattmeter 2 measures P_{2}, the total power and total reactive power can
be calculated:

P_{T} = P_{1} + P_{2}

Q_{T} = √3(P_{1} - P_{2})

At this time you should complete
Tutorial 13
on **three phase**.