Three-phase AC is not hard to understand if you use a phasor diagram

Most alternating-current (AC) generation and transmission, and a good part of use, take place through three-phase circuits. If you want to understand electric power, you must know something about three-phase. It is rather simple if you go at it the right way, though it has a reputation for difficulty.

*Phase* is a frequently-used term around AC. The word comes from Greek fasis, "appearance," from fanein, "to appear." It originally referred to the eternally regular changing appearance of the moon through each month, and then was applied to the periodic changes of some quantity, such as the voltage in an AC circuit. Electrical phase is measured in degrees, with 360° corresponding to a complete cycle. A sinusoidal voltage is proportional to the cosine or sine of the phase.

*Three-phase*, abbreviated 3φ, refers to three voltages or currents that that differ by a third of a cycle, or 120 electrical degrees, from each other. They go through their maxima in a regular order, called the *phase sequence*. The three phases could be supplied over six wires, with two wires reserved for the exclusive use of each phase. However, they are generally supplied over only three wires, and the phase or line voltages are the voltages between the three possible pairs of wires. The phase or line currents are the currents in each wire. Voltages and currents are usually expressed as rms or effective values, as in single-phase analysis.

When you connect a load to the three wires, it should be done in such a way that it does not destroy the symmetry. This means that you need three equal loads connected across the three pairs of wires. This looks like an equilateral triangle, or delta, and is called a delta load. Another symmetrical connection would result if you connected one side of each load together, and then the three other ends to the three wires. This looks like a Y, and is called a wye load. These are the only possibilities for a symmetrical load. The center of the Y connection is, in a way, equidistant from each of the three line voltages, and will remain at a constant potential. It is called the *neutral*, and may be furnished along with the three phase voltages. The benefits of three-phase are realized best for such a symmetrical connection, which is called *balanced*. If the load is not balanced, the problem is a complicated one one whose solution gives little insight, just numbers. Such problems are best left to computer circuit analysis. Three-phase systems that are roughly balanced (the practical case) can be analyzed profitably by a method called *symmetrical components*. Here, let us consider only balanced three-phase circuits, which are the most important anyway.

The key to understanding three-phase is to understand the phasor diagram for the voltages or currents. In the diagram at the right, a, b and c represent the three lines, and o represents the neutral. The red phasors are the line or delta voltages, the voltages between the wires. The blue phasors are the wye voltages, the voltages to neutral. They correspond to the two different ways a symmetrical load can be connected. The vectors can be imagined rotating anticlockwise with time with angular velocity ω = 2πf, their projections on the horizontal axis representing the voltages as functions of time. Note how the subscripts on the V's give the points between which the voltage is measured, and the sign of the voltage. V_{ab} is the voltage at point a relative to point b, for example. The same phasor diagram holds for the currents. In this case, the line currents are the blue vectors, and the red vectors are the currents through a delta load. The blue and red vectors differ in phase by 30°, and in magnitude by a factor of √3, as is marked in the diagram.

Suppose we want to take two phase wires and neutral to make a three-wire household service supplying 120 V between each hot wire and ground. The neutral will become the grounded conductor, the two phases the hot conductors. Then, the wye voltage is 120, so the delta voltage will be √3 x 120 = 208 V. This is the three-phase line voltage necessary in this case. Note that the two 120 V sources are not opposite in phase, and will not give 240 V between them. On the other hand, suppose we do want a 240 V service. Then this must be the line voltage, and the voltages to neutral will be 139 V, not 120 V. A 120 V three-phase service will give only 69 V from line to neutral. Note that √3 appears everywhere, and that the differences in phase explain the unexpected results.

If the load consists of general impedances Z, the situation is described by current and voltage phasors connected by V = IZ, both in magnitude and phase. The diagrams are similar in shape, and rotated by the phase angle between voltage and current in each impedance. Remember that the line voltages are the red vectors, while the line currents are the blue vectors. Z relates either the line voltages and delta currents, or the wye voltages and the line currents, depending on the connection. Z does not relate the line current and line voltage, which are different in phase by 30° even for unity power factor (pure resistance load).

This comes out more clearly when we consider the power P delivered to the load. For a resistive delta load, P = 3 V_{line}I_{delta} = √3 V_{line}I_{line}, since I_{delta} = √3 I_{line}. For a wye load, P = 3 V_{wye}I_{line} = √3 V_{line}I_{line}. This is, of course, the same expression. For other than unity power factor, this must be multiplied by cos θ, which is the angle of Z, not the phase difference between the line voltage and line current. This means, most emphatically, that our usual rule for finding the power from phasors does not apply to three-phase!

If you write out the three phase currents as explicit functions of time, I_{max}cos ωt, I_{max}cos (ωt - 120°) and I_{max}cos (ωt + 120°), square them, multiply by the resistance R, and add, the result is the constant (3/2)I_{max}^{2}R = 3 I^{2}R. The power is applied steadily as in DC circuits, not in pulses as in single-phase AC circuits. This is a great advantage, giving three-phase machines 48% greater capacity than identical single-phase machines.

In Germany and Switzerland, where three-phase power was originated and developed, it is known as *Drehstrom*, "rotating current" for this property of constant power. Ordinary AC is called *Wechselstrom*, or "change current." Nikola Tesla, the discoverer of polyphase currents and inventor of the induction motor, employed two-phase current, where the phase difference is 90°. This also can be used to create a rotating magnetic field, and is more efficient than single-phase, but is not quite as advantageous as three-phase. Two-phase power was once rather common in the United States, where Tesla was important in the introduction of AC, but has now gone completely out of use.

Two-phase can be supplied over three wires, but there is no true neutral, since the phases are not symmetrical. However, it is always easy to double the number of phases in a transformer secondary by making two secondary windings and connecting them in opposing phases. Four-phase does have a neutral, like three-phase, but requires four wires. In fact, three-phase is more economical than any other number of phases. For applications like rectifiers and synchronous converters where DC is produced, it is most efficient to use six-phase AC input, which is easily produced from three-phase in a transformer.

If you are transmitting a certain amount of power single-phase, adding one more conductor operated at the same line voltage and current and using three-phase will increase the power transmitted by 72% with only a 50% increase in the amount of copper and losses. The advantage is obvious. Under certain conditions, transmitting a certain amount of power by three-phase only requires 75% of the copper of single-phase transmission. This is not the major advantage of three-phase, but it does play a factor.

Three wires are usually seen in high-voltage transmission lines, whether on towers or poles, with pin or suspension insulators. Some high-voltage lines are now DC, since solid state devices make it easier to convert to and from AC. The DC lines are free of the problems created by phase, as well as eliminating the skin effect that reduces the effective area of the conductors. It is not nearly as easy to manage long-distance electrical transmission as might be thought.

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Composed by J. B. Calvert

Created 11 February 2001

Last revised