From VOA VAB VOB , VOA 100:7 81:088 V, and IA VOA 16:78 98:928 A ZA

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Similarly, VOC VOB VCB 95:58 18:588 V, and IC 19:12 116:48 A

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Point O is displaced from the neutral N by a phasor voltage VON , given by 150 VON VOA VAN 100:7 81:088 p 908 20:24 39:538 V 3 The phasor diagram, Fig. 11-18, shows the shift of point O from the centroid of the equilateral triangle. See Problem 11-13 for an alternate method.

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Fig. 11-18

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THREE-PHASE POWER

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The powers delivered by the three phases of a balanced generator to three identical impedances with phase angle are pa t Vp Ip cos Vp Ip cos 2!t pb t Vp Ip cos Vp Ip cos 2!t 2408 pc t Vp Ip cos Vp Ip cos 2!t 4808 pT t pa t pb t pc t 3Vp Ip cos Vp Ip cos 2!t cos 2!t 2408 cos 2!t 4808 But cos 2!t cos 2!t 2408 cos 2!t 4808 0 for all t. pT t 3Vp Ip cos P The total instantaneous power is the same as the total average power. It may be written in terms of line voltage VL and line current IL . Thus, p p In the delta system, VL Vp and IL 3Ip . Therefore, P 3VL IL cos : p p In the wye system, VL 3Vp and IL Ip . Therefore, P 3VL IL cos : p The expression 3VL IL cos gives the power in a three-phase balanced system, regardless of the connection con guration. The power factor of the three-phase system is cos . The line voltage VL in industrial systems is always known. If the load is balanced, the total power can then be computed from the line current and power factor. Therefore,

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In summary, power, reactive power, apparent power, and power factor in a three-phase system are p p p P P 3V L IL cos Q 3VL IL sin S 3VL IL pf S Of course, all voltage and currents are e ective values.

POWER MEASUREMENT AND THE TWO-WATTMETER METHOD

An ac wattmeter has a potential coil and a current coil and responds to the product of the e ective voltage, the e ective current, and the cosine of the phase angle between them. Thus, in Fig. 11-19, the wattmeter will indicate the average power supplied to the passive network, P Veff Ieff cos Re Veff I eff (see Section 10.7).

Fig. 11-19

Two wattmeters connected in any two lines of a three-phase, three-wire system will correctly indicate the total three-phase power by the sum of the two meter readings. A meter will attempt to go downscale if the phase angle between the voltage and current exceeds 908. In this event, the current-coil connections can be reversed and the upscale meter reading treated as negative in the sum. In Fig. 11-20 the meters are inserted in lines A and C, with the potential-coil reference connections in line B. Their readings will be WA Re VAB eff I eff Re VAB eff I eff Re VAB eff I eff A AB AC WC Re VCB eff I eff Re VCB eff I eff Re VCB eff I eff C CA CB in which the KCL expressions IA IAB IAC and IC ICA ICB have been used to replace line currents by phase currents. The rst term in WA is recognized as PAB , the average power in phase AB of the delta

Fig. 11-20

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[CHAP. 11

load; likewise, the second term in WC is PCB . Adding the two equations and recombining the middle terms then yields WA WC PAB Re VAB eff VCB eff I eff PCB PAB PAC PCB AC since, by KVL, VAB VCB VAC . The same reasoning establishes the analogous result for a Y-connected load. Balanced Loads When three equal impedances Z are connected in delta, the phase currents make 308 angles with their resultant line currents. Figure 11-21 corresponds to Fig. 11-20 under the assumption of ABC sequencing. It is seen that VAB leads IA by 308, while VCB leads IC by 308. Consequently, the two wattmeters will read WA VAB eff IA eff cos 308 WC VCB eff IC eff cos 308

or, since in general we do not know the relative order in the voltage sequence of the two lines chosen for the wattmeters, W1 VL eff IL eff cos 308 W2 VL eff IL eff cos 308 These expressions also hold for a balanced Y-connection.

Fig. 11-21

Elimination of VL eff IL eff between the two readings leads to p W2 W1 tan 3 W2 W1 Thus, from the two wattmeter readings, the magnitude of the impedance angle can be inferred. The sign of tan suggested by the preceding formula is meaningless, since the arbitrary subscripts 1 and 2 might just as well be interchanged. However, in the practical case, the balanced load is usually known to be inductive > 0 .