The rotor flux of an alternating current generator induces sinusoidal e.m.f.s in the conductors forming the stator winding. In a single-phase machine these stator conductors occupy slots over most of the circumference of the stator core. The e.m.f.s that are induced in the conductors are not in phase and the net winding voltage is less than the arithmetic sum of the individual conductor voltages. If this winding is replaced by three separate identical windings, as shown in Figure 2.1(a), each occupying one-third of the available slots, then the effective contribution of all the conductors is greatly increased, yielding a greatly enhanced power capability for a given machine size. Additional reasons why three phases are invariably used in large A.C. power systems are that the use of three phases gives similarly greater effectiveness in transmission circuits and the three phases ensure that motors always run in the same direction, provided the sequence of connection of the phases is maintained.
The three windings of Figure 2.1(a) give voltages displaced in time or phase by 120°, as indicated in Figure 2.1(b). Because the voltage in the (a) phase reaches its peak 120° before the (b) phase and 240° before the (c) phase, the order of phase voltages reaching their maxima or phase sequence is a-b-c. Most countries use a, b, and c to denote the phases; however, R (Red), Y (Yellow), and B (Blue) has often been used. It is seen that the algebraic sum of the winding or phase voltages (and currents if the winding currents are equal) at every instant in time is zero. Hence, if one end of each winding is connected, then the electrical situation is unchanged and the three return lines can be dispensed with, yielding a three-phase, three-wire system, as shown in Figure 2.2(a). If the currents from the windings are not equal, then it is usual to connect a fourth wire (neutral) to the common connection or neutral point, as shown in Figure 2.2(b).
1 Throughout the book, symbols in bold type represent complex (phasor) quantities requiring complex arithmetic. Italic type is used for magnitude (scalar) quantities within the text.
Figure 2.1 (a) Synchronous machine with three separate stator windings a, b and c displaced physically by 120°. (b) Variation of e.m.f.s developed in the windings with time
Figure 2.2 (a) Wye or star connection of windings, (b) Wye connection with neutral line
Figure 2.3 (a) Phasor diagram for wye connection, (b) Alternative arrangement of line-to-line voltages. Neutral voltage is at n, geometric centre of equilateral triangle
This type of winding connection is called 'wye' or 'star' and two sets of voltages exist:
- 1. the winding, phase, or line-to-neutral voltage, that is, Van, Vbn, Vcn; and
- 2. the line-to-line voltages, Vab, Vbc, Vca. The subscripts here are important, Vab, means the voltage of line or terminal a with respect to b and Vba = —Vab.
The corresponding phasor diagram is shown in Figure 2.3(a) and it can be shown
The phase rotation of a system is very important. Consider the connection through a switch of two voltage sources of equal magnitude and both of rotation a-b-c. When the switch is closed no current flows. If, however, one source is of reversed rotation (easily obtained by reversing two wires), as shown in Figure 2.4, that is, a-c-b, a large voltage (л/3 x phase voltage) exists across the switch contacts cb' and bc', resulting in very large currents if the switch is closed. Also, with reversed phase rotation the rotating magnetic field set up by a three-phase winding is reversed in direction and a motor will rotate in the opposite direction, often with disastrous results to its mechanical load, for example, a pump.
A three-phase load is connected in the same way as the machine windings. The load is balanced when each phase takes equal currents, that is, has equal impedance. With the wye connection the phase currents are equal to the current in the lines. The four-wire system is of particular use for low-voltage distribution networks in which
Figure 2.4 (a) Two generators connected by switch; phase voltages equal for both sets of windings, (b) Phasor diagrams of voltages. Vcb = voltage across switch; Va-a = 0
consumers are supplied with a single-phase supply taken between a line and neutral. This supply is often 230 V and the line-to-line voltage is 400 V. Distribution practice in the USA is rather different and the 220 V supply often comes into a house from a centre-tapped transformer, as shown in Figure 2.5, which in effect gives a choice of 220 V (for large domestic appliances) or 110 V (for lights, etc.).
The system planner will endeavour to connect the single-phase loads such as to provide balanced (or equal) currents in the three-phase supply lines. At any instant in time it is highly unlikely that consumers will take equal loads, and at the lower distribution voltages considerable unbalance occurs, resulting in currents in the neutral line. If the neutral line has zero impedance, this unbalance does not affect the load voltages. Lower currents flow in the neutral than in the phases and it is usual to install a neutral conductor of smaller cross-sectional area than the main line conductors. The combined or statistical effect of the large number of loads on the low-voltage network is such that when the next higher distribution voltage is considered, say 11 kV (line to line), which supplies the lower voltage network, the degree of unbalance is small. This and the fact that at this higher voltage, large three-phase, balanced motor loads are supplied, allows the three-wire system to be
Figure 2.5 Tapped single-phase supply to give 220/110V (centre-tap grounded), US practice
Figure 2.6 (a) Mesh or delta-connected load-current relationships, (b) Practical connections
used. The three-wire system is used exclusively at the higher distribution and transmission voltages, resulting in much reduced line costs and environmental impact.
In a balanced three-wire system a hypothetical neutral line may be considered and the conditions in only one phase determined. This is illustrated by the phasor diagram of line-to-line voltages shown in Figure 2.3(b). As the system is balanced the magnitudes so derived will apply to the other two phases but the relative phase angles must be adjusted by 120° and 240°. This single-phase approach is very convenient and widely used in power system analysis.
An alternative method of connection is shown in Figure 2.6. The individual phases are connected (taking due cognizance of winding polarity in machines and transformers) to form a closed loop. This is known as the mesh or delta connection. Here the line-to-line voltages are identical to the phase voltages, that is.
The line currents are as follows:
For balanced currents in each phase it is readily shown from a phasor diagram that Jline = v^Iphase. Obviously a fourth or neutral line is not possible with the mesh connection. The mesh or delta connection is seldom used for rotating-machine stator windings, but is frequently used for the windings of one side of transformers. A line-to-line voltage transformation ratio of 1:л/3 is obtained when going from a primary mesh to a secondary wye connection with the same number of turns per phase. Under balanced conditions the idea of the hypothetical neutral and single-phase solution may still be used (the mesh can be converted to a wye using the A ! Y transformation).
It should be noted that three-phase systems are usually described by their line-to- line voltage (e.g. 11,132,400 kV etc.).
-  Elsewhere, throughout the book, magnitude (scalar) quantities are represented by simple italics.