Definition of Balanced Three-Phase Voltages in AC Circuits

Three-phase voltage may have different voltages in each phase. There will be some cases where we need the balanced three-phase voltages.

There are four types of three phase circuit:

1. Balanced three phase voltage
2. Balanced three phase power
3. Unbalanced three phase power
4. Three phase power measurement

Three-phase voltages are often produced with a three-phase ac generator (or alternator) whose cross-sectional view is shown in Figure.(1). The generator basically consists of a rotating magnet (called rotor) surrounded by a stationary winding (called the stator).

Figure 1. A three-phase generator

Make sure to read what is three-phase ac circuit first.

Figure 2. The generated voltages are 120o apart from each other

Balanced Three-Phase Voltages

Three separate windings or coils with terminals a-a’, b-b’, and c-c’ are physically placed 120^o apart around the stator.

Terminals a and a’, for example, stand for one of the ends of coils going into and the other end coming out of the page.

As the rotor rotates, its magnetic field “cuts” the flux from the three coils and induces voltages in the coils.

Because the coils are placed 120^o apart, the induced voltage in the coils are equal in magnitude but out of phase by 120^o like shown in Figure.(2).

Since each coil can be regarded as a single-phase generator by itself, the three-phase generator can supply power to both single-phase and three-phase loads.

Figure 3. Three-phase voltage sources: (a) Y-connected source, (b) delta-connected source

A typical three-phase system consists of three voltage sources connected to loads by three or four wires (or transmission lines). (Three-phase current sources are very scarce.)

A three-phase system is equivalent to three single-phase circuits. The voltage sources can be either wye-connected as shown in Figure.(3a) or delta-connected as in Figure.(3b).

Let us consider the wye-connected voltages in Figure.(3a) for now. The voltages V_an, V_bn, and V_cn are respectively between lines a, b, and c, and the neutral line n.

These voltages are called phase voltages.

If the voltage sources have the same amplitude and frequency ω and are out of phase with each other by 120^o, the voltages are said to be balanced. This implies that

(1)

(2)

Thus,

Balanced phase voltages are equal in magnitude and are out of phase with each other by 120^o.

Since the three-phase voltages are 120^o out of phase with each other, there are two possible combinations. One possibility is shown in Figure.(4a)

Figure 4. Phase sequences: (a) abc or positive sequence, (b) acb or negative sequence

and expressed mathematically as

(3)

where V_p is the effective or rms value. This is known as the abc sequence or positive sequence.

In this phase sequence, V_an leads V_bn, which in turn leads V_cn.

This sequence is produced when the rotor in Figure.(1) rotates counterclockwise. The other possibility is shown in Figure.(4b)

and is given by

(4)

This is called the acb sequence or negative sequence. For this phase sequence, V_an leads V_cn, which in turn leads V_bn.

The acb sequence is produced when the rotor in Figure.(1) rotates in the clockwise direction.

It is easy to show that the voltages in Equations.(3) or (4) satisfy Equations.(1) and (2). For example, from Equation.(3),

(5)

The phase sequence is the time order in which the voltages pass through their respective maximum values.

The phase sequence is determined by the order in which the phasors pass through a fixed point in the phase diagram.

In Figure.(4a), as the phasors rotate in the counterclockwise direction with frequency ω, they pass through the horizontal axis in a sequence abcabca . . . .

Thus, the sequence is abc or bca or cab. Similarly, for the phasors in Figure.(4b), as they rotate in the counterclockwise direction, they pass the horizontal axis in a sequence acbacba . . . .

This describes the acb sequence. The phase sequence is important in three-phase power distribution.

It determines the direction of the rotation of a motor connected to the power source, for example.

Like the generator connections, a three-phase load can be either wye-connected or delta-connected, depending on the end application.

Figure.(5a) shows a wye-connected load, and Figure.(5b) shows a delta connected load.

Figure 5. Two possible three-phase load configurations: (a) a Y-connected load, (b) a Δ-connected load

The neutral line in Figure.(5a) may or may not be there, depending on whether the system is four- or three-wire. (And, of course, a neutral connection is topologically impossible for a delta connection.)

A wye- or delta-connected load is said to be unbalanced if the phase impedances are not equal in magnitude or phase.

A balanced load is one in which the phase impedances are equal in magnitude and in phase.

For a balanced wye-connected load,

(6)

where Z_Y is the load impedance per phase. For a balanced delta-connected load,

(7)

where Z_Δ is the load impedance per phase in this case.

(8)

so we know that a wye-connected load can be transformed into a delta-connected load, or vice versa, using Equation.(8).

The table below shows us the summary of phase and line voltages/ currents for a balanced three-phase system:

Since both the three-phase source and the three-phase load can be either wye- or delta-connected, we have four possible connection :

• Balanced wye-wye connection (i.e., Y-connected source with a Y-connected load).
• Balanced wye-delta connection.
• Balanced delta-delta connection.
• Balanced delta-wye connection.

In subsequent sections, we will consider each of these possible configurations.

It is appropriate to mention here that a balanced delta-connected load is more common than a balanced wye-connected load.

This is due to the ease with which loads may be added or removed from each phase of a delta-connected load.

This is very difficult with a wye-connected load because the neutral may not be accessible.

On the other hand, delta-connected sources are not common in practice because of the circulating current that will result in the delta-mesh if the three-phase voltages are slightly unbalanced.

Balanced Three-Phase Voltages Example

Determine the phase sequence of the set of voltages

Solution :
The voltages can be expressed in phasor form as

We notice that V_an leads V_cn by 120^o and V_cn, in turn, leads V_bn by 120^o. Hence, we have an acb sequence.