# Three Phase Systems

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). 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  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 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 consumers are supplied with a single-phase supply taken between a line and neutral. This supply is often 230V and the line-to-line voltage is 400 V. Distribution practice in the USA is rather different and the 220V supply often comes into a house from a centre-tapped transformer, as shown in Figure 2.5, which in effect gives a choice of 220V (for large domestic appliances) or 110V (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 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. 2.1.1 Analysis of Simple Three-Phase Circuits

2.1.1.1 Four-Wire Systems

If the impedance voltage drops in the lines are negligible, then the voltage across each load is the source phase voltage.      2.2 Three-Phase Transformers The usual form of the three-phase transformer, that is, the core type, is shown in Figure 2.10. If the magnetic reluctances of the three limbs are equal, then the sum of the fluxes set up by the three-phase magnetizing currents is zero. In fact, the core is the magnetic equivalent of the wye-connected winding. It is apparent from the shape of Figure 2.10 that the magnetic reluctances are not exactly equal, but in an introductory treatment may be so assumed. An alternative to the three-limbed core is the use of three separate single-phase transformers. Although more expensive (about 20% extra), this has the advantage of lower weights for transportation, and  this aspect is crucial for large sizes. Also, with the installation of four single-phase units, a spare is available at reasonable cost. The wound core, as shown in Figure 2.10, is placed in a steel tank filled with insulating oil or synthetic liquid. The oil acts both as electrical insulation and as a cooling agent to remove the heat of losses from the windings and core. The low voltage windings are situated over the core limbs and the high-voltage windings are wound over the low-voltage ones. The core comprises steel laminations insulated on one side (to reduce eddy losses) and clamped together. The phasor diagram of the transformer is shown in Figure 2.11. The voltages across the two windings are related by the turns ratio (l). In this Y ! D wound transformer the secondary equivalent phase voltages are 30 out of phase with the primary phase voltages. With this winding arrangement the transformer is referred to as type Yd1 in British terminology. Y: star HV winding, d: delta LV winding: 1: one o’clock or 30 phase shift.

## 2.2.1 Autotransformers

There are several places in a power system where connections from one voltage level to another do not entail large transformer ratios, for example, 400/275, 500/345, 725/500 kV, and then the autotransformer is used (Figure 2.12). In the autotransformer only one winding is used per phase, the secondary voltage being tapped off the primary winding. There is obviously a saving in size, weight and cost over a two-windings per phase transformer. It may be shown that the ratio of the weight of conductor in an autotransformer to that in a double-wound one is given by (1N2/N1). Hence, maximum advantage is obtained with a relatively small difference between the voltages on the two sides. The effective reactance is reduced compared with the equivalent two winding transformer and this can give rise to high short-circuit currents. The general constructional features of the core and tank are similar to those of double-wound transformers, but the primary and secondary voltages are now in-phase.