# ECE 252 Introduction to Electrical Engineering

## Lesson 13B.  Three-Phase Power

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 Sources

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 Vφ. Each winding also has a phase current, Iφ.  The phase voltage and phase current will produce a line voltage, VL, and a line current, IL.

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 Vφ is exactly the same as every other, keeping in mind that these variables represent rms magnitudes.  The same is true for Iφ, VL, and IL.

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)  VL = Vφ

The relationship between IL and Iφ 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)  IL = √3Iφ

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.
IL = Iφ

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)  VL = √3Vφ

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, Vφ, 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)  Pφ = Iφ2Rφ

By conservation of energy, the total power consumed by the load must be 3 times the power consumed in any phase:
(6)  PT = 3Pφ

To get the total power consumed in the load, the following equation is useful:
(7)  PT = √3VLILcosθ where cosθ is the power factor.

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

### Three-Phase Example

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:
VL = √3Vφ = 480 V

(b)  From Equation (1) for delta-connected sources or loads:
VφLoad = VL = 480 V

(c)  From Equation (7):
PT = √3VLILcosθ
IL = PT/(√3VLcosθ)
IL = 180,000/(√3 × 480 × 0.66) = 328 A

(d)  From Equation (3) for Y-connected sources or loads:
IφGen = IL = 328 A

(e)  From Equation (2) for delta-connected sources or loads:
IφLoad = IL/√3 = 328/√3 = 189 A

(f)  From Equation (6):
Pφ = PT/3 = 180 kW/3 = 60 kW

### Practical Considerations

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, VL and IL, are more useful than the phase voltage and phase current, Vφ and Iφ.  This is because VL and IL are much easier to measure.  Vφ and Iφ 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.

#### Three-Phase Power Measurement

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 Vrms × Irms.  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 P1 and Wattmeter 2 measures P2, the total power and total reactive power can be calculated:
PT = P1 + P2
QT = √3(P1 - P2)

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