CHAP. 17]

FOURIER METHOD OF WAVEFORM ANALYSIS

Fig. 17-33

441

Fig. 17-34

17.10 Find the exponential Fourier series for the half-wave rectiﬁed sine wave shown in Figs. 17-26 and 17-34, and sketch the line spectrum. In the interval 0 < !t < , f ðtÞ ¼ V sin !t; and from  to 2, f ðtÞ ¼ 0. Then ð 1  An ¼ V sin !t ejn!t dð!tÞ 2 0   V ejn!t Vðejn þ 1Þ ¼ ¼ ðjn sin !t  cos !tÞ 2 2 ð1  n Þ 2ð1  n2 Þ 0 For even n, An ¼ V=ð1  n2 Þ; for odd n, An ¼ 0. However, for n ¼ 1, the expression for An becomes indeterminate. L’Hoˆpital’s rule may be applied; in other words, the numerator and denominator are separately diﬀerentiated with respect to n, after which n is allowed to approach 1, with the result that A1 ¼ jðV=4Þ. The average value is ð i V 1  V h  cos !t ¼ A0 ¼ V sin !t dð!tÞ ¼ 0 2 0 2  Then the exponential Fourier series is f ðtÞ ¼    

V j4!t V j2!t V j!t V V j!t V j2!t V j4!t  þj þ j    e e e e  e e 15 3 4  4 3 15

The harmonic amplitudes, V c0 ¼ A0 ¼ 

8 < 2V=ðn2  1Þ cn ¼ 2jAn j ¼ V=2 : 0

ðn ¼ 2; 4; 6; . . .Þ ðn ¼ 1Þ ðn ¼ 3; 5; 7; . . .Þ

are exactly as plotted in Fig. 17-27.

17.11 Find the average power in a resistance R ¼ 10 , if the current in Fourier series form is i ¼ 10 sin !t þ 5 sin 3!t þ 2 sin 5!t (A). The current has an eﬀective value Ieff ¼ 2 power is P ¼ Ieff R ¼ ð64:5Þ10 ¼ 645 W.

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 1 1 1 64:5 ¼ 8:03 A. Then the average 2 ð10Þ þ 2 ð5Þ þ 2 ð2Þ ¼

Another Method The total power is the sum of the harmonic powers, which are given by 12 Vmax Imax cos . voltage across the resistor and the current are in phase for all harmonics, and n ¼ 0. Then,

But the

vR ¼ Ri ¼ 100 sin !t þ 50 sin 3!t þ 20 sin 5!t and P ¼ 12 ð100Þð10Þ þ 12 ð50Þð5Þ þ 12 ð20Þð2Þ ¼ 645 W.

17.12 Find the average power supplied to a network if the applied voltage and resulting current are v ¼ 50 þ 50 sin 5  103 t þ 30 sin 104 t þ 20 sin 2  104 t 3

4

ðVÞ

i ¼ 11:2 sin ð5  10 t þ 63:48Þ þ 10:6 sin ð10 t þ 458Þ þ 8:97 sin ð2  104 t þ 26:68Þ ðAÞ

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An