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Copyright 2003, 1997, 1986, 1965 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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Fig. 11-1 pa t va t ia t Vp Ip cos  Vp Ip cos 2!t  pb t vb t ib t Vp Ip cos  Vp Ip cos 2!t  The total instantaneous power pT t delivered by the generator is pT t pa t pb t Vp Ip cos  Vp Ip cos 2!t  Vp Ip cos  Vp Ip cos 2!t  2Vp Ip cos  Thus, (4) pT t Pavg 2Vp Ip cos  p In the system of Fig. 11-1(a), two voltage values Vp and 2Vp are available to the load and the power ow is constant. In addition, the 908-phase shift between the two voltages may be used to produce a special rotating magnetic eld needed in some applications.
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Three-phase generators contain three sinusoidal voltage sources with voltages of the same frequency but a 1208-phase shift with respect to each other. This is realized by positioning three coils at 1208 electrical angle separations on the same rotor. Normally, the amplitudes of the three phases are also equal. The generator is then balanced. In Fig. 11-2, three coils are equally distributed about the circumference of the rotor; that is, the coils are displaced from one another by 120 mechanical degrees.
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Fig. 11-2
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Coil ends and slip rings are not shown; however, it is evident that counterclockwise rotation results in the coil sides A, B, and C passing under the pole pieces in the order . . . A-B-C-A-B-C . . . Voltage polarities reverse for each change of pole. Assuming that the pole shape and corresponding magnetic ux density are such that the induced voltages are sinusoidal, the result for the three coils is as shown in Fig. 11-3. Voltage B is 120 electrical degrees later than A, and C is 2408 later. This is referred to as the ABC sequence. Changing the direction of rotation would result in . . . A-C-B-A-C-B . . . ; which is called the CBA sequence.
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Fig. 11-3
The voltages of a balanced ABC sequence in the time and phasor domains are given in (5) and (6), respectively. The phasor diagram for the voltage is shown in Fig. 11-4.
Fig. 11-4
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POLYPHASE CIRCUITS
251 p vcn t Vp 2 cos !t 2408
p van t Vp 2 cos !t
p vbn t Vp 2 cos !t 1208 Vbn Vp 1208
5 6
Van Vp 0
Vcn Vp 2408
WYE AND DELTA SYSTEMS
The ends of the coils can be connected in wye (also designated Y; see Section 11.8), with ends A 0 , B 0 , and C 0 joined at a common point designated the neutral, N; and with ends A, B, and C brought out to become the lines A, B, and C of the three-phase system. If the neutral point is carried along with the lines, it is a three-phase, four-wire system. In Fig. 11-5, the lines are designated by lowercase a, b, and c at the supply, which could either be a transformer bank or a three-phase alternator, and by uppercase A, B, and C at the load. If line impedances must be considered, then the current direction through, for example, line aA would be IaA , and the phasor line voltage drop VaA .
Fig. 11-5
The generator coil ends can be connected as shown in Fig. 11-6, making a delta-connected (or connected), three-phase system with lines a, b, and c. A delta-connected set of coils has no neutral point to produce a four-wire system, except through the use of -Y transformers.
Fig. 11-6
PHASOR VOLTAGES
The selection of a phase angle for one voltage in a three-phase system xes the angles of all other voltages. This is tantamount to xing the t 0 point on the horizontal axis of Fig. 11-3, which can be done quite arbitrarily. In this chapter, an angle of zero will always be associated with the phasor voltage of line B with respect to line C: VBC  VL 08. p It is shown in Problem 11.4 that the line-to-line voltage VL is 3 times the line-to-neutral voltage. All ABC-sequence voltages are shown in Fig. 11-7(a) and CBA voltages in Fig. 11-7(b). These phasor
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