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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,

...,

E(Xk)

CHAP. 11]

(17)

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.

Suppose that a box contains b blue marbles and r red marbles. Let us perform n trials of an experiment in which a marble is chosen at random, its color is observed, and then the marble is put back in the box. This type of experiment is often referred to as sampling with replacement. In such a case, if X is the random variable denoting

Draw a state diagram for the simple vi-like editor. See Fig. 8-5.

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the number of blue marbles chosen (successes) in n trials, then using the binomial distribution (1) we see that the probability of exactly x successes is P(X x) n b xr n x , a b x (b r)n x 0, 1, c, n (18)

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).

since p b>(b r), q 1 p r>(b r). If we modify the above so that sampling is without replacement, i.e., the marbles are not replaced after being chosen, then b r a ba b x n x a b n r b

Fig. 11-19

max (0, n min (n, b)

r), c,

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

(19)

Fig. 11-20

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This is the hypergeometric distribution. The mean and variance for this distribution are m nb b r , s2 (b nbr(b r n) r)2 (b r 1) (20)

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