THREE-PHASE CIRCUITS

Transcription

1 THR-HAS CIRCUITS 4.1 Introduction Generation, Transmission and distribution of electricity via the National Grid system is accomplished by three-phase alternating currents. The voltage induced by a single coil when rotated in a uniform magnetic field is shown in figure (4.1) and is known as a single-phase voltage. Most consumers are fed by means of a single-phase A.C supply. Two wire are used, one called the live conductor (usually colored red) and the other is called the neutral conductor (usually colored black). Fig.(4.1) A three-phase supply is generated when three coils are placed 12 a part and the whole rotated in a uniform magnetic field as shown in figure (4.2.a). The result is three independent supplies of equal voltage which are each displaced by 12 from each other as shown in fig. (4.2.b). 94

2 Fig.(4.2) A three-phase A.C supply is carried by three conductors, called "line" which are colored Red, Yellow and Blue. The currents in these conductors are known as line currents (I L ) and the p.d's between them are known as line voltage (V L ). A fourth conductor, called the neutral (colored black, and connected through protective devices to earth). Three-phase systems are important for at least three reasons. First, nearly all electric power is generated and distributed in three-phase. Second, the instantaneous power in a three-phase system can be constant (not pulsating). This results in uniform power transmission and less vibration of three-phase machines. Third, for the same amount of power, the three-phase system is more economical than the single-phase. The amount of wire required for a three-phase system is less than that required for an equivalent single-phase system. 95

3 4.2 Basic Three-hase Relationships To keep track of voltages and current, we use the symbols and notations of figure (4.3). Capital letter subscripts are used at the source and lowercase letters at the load. As usual, is used for source voltage and v for voltage drops. Fig.(4.3) 96

4 Definitions Line voltages (also called line-to-line) are voltages between lines. Thus, AB, BC, and CA are line-to-line voltages at the generator, while v ab, v bc, and v ca are line-to-line voltages at load. hase voltage are voltages across phases. For a Y load, phase are defined from line to neutral as indicated in figure. Thus, v an, v bn, and v cn are phase voltage for a Y load. For a Δ load, phase are defined from line to line as shown in figure, thus v ab, v bc and v ca are phase voltages for a Δ. As you can see, for a Δ load, phase voltages and line voltages are the same thing. For the generator, AN, BN, and CN are phase voltages. Line currents are the currents in the line conductors. (Only a single subscript is needed). You can use either I a, I b, and I c or I A,I B, and I C as in figure (4.3). hase currents are currents through phases. For the Y load I a, I b,, and I c pass through phase impedances and are therefore phase currents. For the Δ load I ab, I bc, and I ca are phase currents. As you can seen, for a Y load, phase currents and line currents are the same thing. hase impedances for a Y load are the impedances from a-n, b-n and c-n, and are denoted by the symbols Z an, Z bn, and Z an. For a Δ load, phase impedances are Z ab, Z bc, and Z ca. In a balanced load, impedances for all phases are the same, (i.e Z an =Z bn =Z cn etc.). 4.3 Balanced Three-hase Voltages The voltage sources can be either Wye-Connected or Delta-connected as shown in figure (4.4.a). (4.4.b). Let us consider the Wye-connected voltage in figure (4.4) for now. The voltages AN, BN, and CN are respectively between lines A,B, and C, and the neutral line N. these voltages are called phase voltages. 97

5 If the voltages sources have the same amplitude and frequency and are out of phase with each other by 12, the voltages are said to be balanced. A A CN AN N CA AB BN B B BC C C Fig.(4.4) Since the three-phase voltage are 12 out of phase with each other. There are two possible combinations. One possibility is shown in fig.(4.5) and expressed mathematically as CN AN BN CN = V = V 12 = V 24 = V 12 AN BN Fig.(4.5) Where V p is the effective or r.m.s. values this is known as the (a b c)sequence or positive sequence. 98

6 The other possibility is shown in figure (4.6) and is given by BN AN = V CN = V 12 BN = V 24 = V 12 AN CN Fig.(4.6) This is called (a c b) sequence or negative 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 (4.7.a) and (4.7.b) shows a Wye-connected load, and delta-connected load. Fig.(4.7) For a balanced Wye-connected load 99

7 Where Z Y is the load impedance per phase. For a balanced Delta-connected load. Where Z Δ is the load impedance per phase in this case so that We know that a Wye-connected load can be transformed into a delta-connected load or vice versa. 4.4 Line and phase voltages for a Wye Circuit We now need the relationship between line and phase voltage for the Y circuit. Consider figure (4.8) by KVL, V ab - V an + V bn = thus, Fig.(4.8) V ab =V an -V bn Now, assume a magnitude V for each phase voltage and take V an as reference. Thus, V an =V and V bn =V -12. substitute these two into equation gets, V ab = V V 12 1

8 = V ( 1+ j) V (.5 j. 866) V ( 1.5 j.866) = 1.732V 3 + V ab = 3V 3 But V an =V. thus, quation shows that the magnitude of V ab is 3 times the magnitude of V an and that V ab leads V an by 3. This is shown in phasor diagram form in figure (4.9). They also hold at the source. Thus Fig.(4.9) Currents for a Wye circuit For a Y circuit, line currents are the same as phase currents; I L =I p 11