Three-phase systems

In a single-phase ac circuit, instantaneous power to a load is of a pulsating nature. Even at the unity power factor (i.e. when the voltage and the current are in phase with respect to each other), the instantaneous power is less than unity. The instantaneous power is not only zero four times in each cycle but it is also negative twice in each cycle. Therefore, because of economy and performance, almost all electrical power is produced by polyphase sources.
A phase is one of three branches making up a three-phase circuit.

For example, in a wye connection, a phase is made up of those circuit elements connected between one line and neutral; however, in a delta connection, a phase consists of those circuit elements connected between two lines.

A polyphase generator has two or more single phases connected so that they provide loads with voltages of equal magnitudes and equal phase differences, hence the power delivered is steadier and there is less vibration in the rotating machinery, which, in turn, performs more efficiently. For example, in a balanced n-phase system, there are n voltage sources connected together. Each phase voltage (or source) alternates sinusoidally, has the same magnitudes, and has a phase difference of 360/n◦ (where n is the number of phases) from its adjacent voltage phasors, except in the case of two-phase systems. In a two-phase generator, the two equal voltages differ in phase by 90◦, but in a three-phase generator, the three equal voltages have a phase-angle difference of 120◦. However, the use of two-phase systems is very uncommon. Today, virtually all the power generated in the world is three-phase power with a frequency of 50 or 60 Hz.

Most of the generation, transmission, distribution, and heavy-power utilization of electrical energy are done using three-phase systems. A three-phase system is supplied by a three-phase generator (i.e. alternator), which consists essentially of three single-phase systems displaced in time phase from each other by one-third of a period, or 120 electrical degrees. The advantages of three-phase systems over single-phase systems are:

1. Less conductor material is required in the three-phase transmission of power, hence it is more economical.
2. Constant rotor torque and therefore steady machine output can be achieved.
3. Three-phase machines (generators or motors) have higher efficiencies.
4. Three-phase generators may be connected in parallel to supply greater power more easily than single-phase generators.

Figure above shows the structure of an elementary three-phase and two-pole ac generator (alternator).
Its structure has two parts: the stationary outside part which is called the stator and the rotating inside part which is called the rotor. The field winding is located on the rotor and is excited by a direct current source through slip rings located on the common shaft. Thus, an alternator has a rotating electromagnetic field; however, its stator windings are stationary.
The elementary generator shown above has three identical stator coils (aa’, bb’, and cc’), of one or more turns, displaced by 120◦ in space with respect to each other. If the rotor is driven counterclockwise at a constant speed ω, voltages will be generated in the three phases according to Faraday’s law.
The stator windings constitute the armature of the generator (unlike dc machines, where the armature is the rotor). Thus, the field rotates inside the armature.
Each of the three stator coils makes up one phase in this single generator.

Above figures display generated voltage waveforms in the time domain, and corresponding phasors of the three voltages.
The stator phase windings can be connected in either wye or delta. In wye configuration, if a neutral conductor is brought out, the system is defined as a four-wire three-phase system; otherwise, it is a three-wire, three-phase system. In a delta connection, no neutral exists and therefore it is a three-wire three-phase system.

Ideal three-phase power sources

An ideal and balanced three-phase voltage source has three equal voltages that are 120◦ out of phase with respect to one another. Furthermore, in balanced three-phase systems, each phase has equal impedance so that the resulting phase currents are equal and phase-displaced from each other by 120◦.
The term balanced describes three-phase voltages or currents, which are equal in magnitude and are 120◦ out of phase with respect to each other and form a symmetrical three-phase set. If the three-phase system is balanced, then equal real and reactive power flow in each phase. On the other hand, if the three-phase system is not balanced, it may lack some or all of these characteristics.

The line-to-neutral voltages or line-to-ground voltages are defined as phase voltages. The line-to-line voltages or phase-to-phase voltages are defined as line voltages.

For further reading, refer to...    wye and delta connected ideal three-phase source

Balanced three-phase loads

Three-phase loads may be connected either in wye connections or delta connections.
In a wye connection, the line voltages are 30◦ ahead of the corresponding phase voltages. However, the line currents and the corresponding phase currents are the same. The magnitudes of line voltages are √3 times those for phase voltages.
In a delta connection, the line currents are 30◦ behind the corresponding phase currents. Here, the magnitudes of line currents lag the line-to-neutral voltages.

When the impedances in all three phases are identical, the load is said to be balanced. If a balanced three-phase source is connected to a balanced load over an inherently balanced transmission or distribution line, then the total system is balanced. Table above provides a summary comparison of the basic variables of delta and wye-connected, balanced, three-phase loads. We can see that the connection type does not affect the power calculations.

The virtue of working with balanced systems is that they can be analyzed on a single-phase basis. The current in any phase is always the phase-to-neutral voltage divided by the per-phase load impedance. Therefore, it is not necessary to calculate the currents in the remaining phase separately. Calculating the current, voltage, and power in only one phase is sufficient in an analysis because of the complete symmetry that exists between the three phases. The knowledge of these variables in one phase, which is referred to as the reference phase, directly provides information about all phases. This type of analysis is called per-phase analysis.

Unbalanced three-phase loads

If an unbalanced three-phase load connected in delta or wye is present in an otherwise balanced three-phase system, the method of symmetrical components is normally used to analyze the system. However, in simple situations, the direct application of conventional circuit theory can be used without much difficulty.
Here we will not go any further through some examples, but we have mentioned this situation to keep in mind in the future...