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SUMMARY OF AC POWER IN R, L, AND C
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AC power in resistors, inductors, and capacitors, is summarized in Table 10-1. We use the notation Veff and Ieff to include the phase angles. The last column of Table 10-1 is S VI where S is called apparent power. S is discussed in Section 10.7 in more detail.
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EXAMPLE 10.6 Find the power delivered from a sinusoidal source to a resistor R. Let the e ective values of voltage and current be V and I, respectively. p p pR t viR V 2 cos !t I 2 cos !t 2VI cos2 !t VI 1 cos 2!t RI 2 1 cos 2!t Thus, PR V2 1 cos 2!t R Q 0
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V2 RI 2 R
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AC POWER
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Table 10-1 p v V 2 cos !t Veff V 08 p i I 2 cos !t  Ieff I 8 P VI cos ; Q VI sin  and S VI (apparent power) Z R L C R jL! j C! i p V 2 cos !t R p V 2 cos !t 908 L! p V 2C! cos !t 908 Ieff V 08 R V 908 L! VC! 908
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p t V 1 cos 2!t R V2 sin 2!t L! V 2 C! sin 2!t
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Q 0 V2 L! V 2 C! V R
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V2 L! V 2 C!
The instantaneous power entering a resistor varies sinusoidally between zero and 2RI 2 , with twice the frequency of excitation, and with an average value of P RI 2 . v t and pR t are plotted in Fig. 10-5(a). EXAMPLE 10.7 Find the ac power entering an inductor L. p p pL t viL V 2 cos !t I 2 cos !t 908 2VI cos !t sin !t VI sin 2!t L!I 2 sin 2!t Thus, V2 sin 2!t L! P 0 Q VI V2 L!I 2 L!
The instantaneous power entering an inductor varies sinusoidally between Q and Q, with twice the frequency of the source, and an average value of zero. See Fig. 10-5(b). EXAMPLE 10.8 Find the ac power delivered in a capacitor C. p p pC t viC V 2 cos !t I 2 cos !t 908 2VI cos !t sin !t VI sin 2!t C!V 2 sin 2!t Thus, I2 sin 2!t C! P 0 Q VI I2 C!V 2 C!
Like an inductor, the instantaneous power entering a capacitor varies sinusoidally between Q and Q, with twice the frequency of the source, and an average value of zero. See Fig. 10-5(c).
EXCHANGE OF ENERGY BETWEEN AN INDUCTOR AND A CAPACITOR
If an inductor and a capacitor are fed in parallel by the same ac voltage source or in series by the same current source, the power entering the capacitor is 1808 out of phase with the power entering the inductor. This is explicitly re ected in the opposite signs of reactive power Q for the inductor and the capacitor. In such cases, the inductor and the capacitor will exchange some energy with each other, bypassing the ac source. This reduces the reactive power delivered by the source to the LC combination and consequently improves the power factor. See Sections 10.8 and 10.9.
EXAMPLE 10.9 Find the total instantaneous power p t , the average power P, and the reactive power Q, delivered p from v V 2 cos !t to a parallel RLC combination. The total instantaneous power is pT vi v iR iL iC pR pL pC
CHAP. 10]
AC POWER
Fig. 10-5
AC POWER
[CHAP. 10
Fig. 10-5 (cont.) Substituting the values of pR , pL , and pC found in Examples 10.6, 10.7, and 10.8, respectively, we get   V2 1 1 cos 2!t V 2 pT C! sin 2!t R L! The average power is PT PR V 2 =R The reactive power is QT QL QC V 2 For 1=L! C! 0, the total reactive power is zero. factor.   1 C! L! 12
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