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FOURIER METHOD OF WAVEFORM ANALYSIS
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[CHAP. 17
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EXAMPLE 17.1 Find the Fourier series for the waveform shown in Fig. 17-2.
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Fig. 17-2 The waveform is periodic, of period 2=! in t or 2 in !t. It is continuous for 0 < !t < 2 and given therein by f t 10=2 !t, with discontinuities at !t n2 where n 0; 1; 2; . . . . The Dirichlet conditions are satis ed. The average value of the function is 5, by inspection, and thus, 1 a0 5. For n > 0, (5) gives 2 an 1  2 
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  2 10 10 !t 1 !t cos n!t d !t 2 sin n!t 2 cos n!t 2 2 n n 0
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10 2 2 cos n2 cos 0 0 2 n Thus, the series contains no cosine terms. 1 bn  2 
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Using (6), we obtain
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  2 10 10 !t 1 10 !t sin n!t d !t 2 cos n!t 2 sin n!t 2 n n 2 n 0
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Using these sine-term coe cients and the average term, the series is
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1 10 10 10 10 X sin n!t sin !t sin 2!t sin 3!t 5  2 3  n 1 n
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f t 5
The sine and cosine terms of like frequency can be combined as a single sine or cosine term with a phase angle. Two alternate forms of the trigonometric series result. f t 1 a0 2 P cn cos n!t n 7
f t 1 a0 2
cn sin n!t n
p where cn a2 b2 , n tan 1 bn =an , and n tan 1 an =bn . n n amplitude, and the harmonic phase angles are n or n .
In (7) and (8), cn is the harmonic
EXPONENTIAL FOURIER SERIES
A periodic waveform f t satisfying the Dirichlet conditions can also be written as an exponential Fourier series, which is a variation of the trigonometric series. The exponential series is f t
1 X n 1
An e jn!t
To obtain the evaluation integral for the An coe cients, we multiply (9) on both sides by e jn!t and integrate over the full period:
CHAP. 17]
FOURIER METHOD OF WAVEFORM ANALYSIS
2 f t e
jn!t
2 d !t
0 2
A 2 e
j2!t jn!t
2 d !t 2
A 1 e j!t e jn!t d !t
0 2
A0 e jn!t d !t
A1 e j!t e jn!t d !t 10
An e jn!t e jn!t d !t
2 The de nite integrals on the right side of (10) are all zero except 0 An d !t , which has the value 2An . Then 1 2 1 T jn!t An f t e d !t or An f t e j2nt=T dt 11 2 0 T 0 Just as with the an and bn evaluation integrals, the limits of integration in (11) may be the endpoints of any convenient full period and not necessarily 0 to 2 or 0 to T. Note that, f t being real, A n A , n so that only positive n needed to be considered in (11). Furthermore, we have an 2 Re An bn 2 Im An 12
EXAMPLE 17.2 Derive the exponential series (9) from the trigonometric series (1). Replace the sine and cosine terms in (1) by their complex exponential equivalents. sin n!t e jn!t e jn!t 2j cos n!t e jn!t e jn!t 2
Arranging the exponential terms in order of increasing n from 1 to 1, we obtain the in nite sum (9) where A0 a0 =2 and An 1 an jbn 2 A n 1 an jbn 2 for n 1; 2; 3; . . .
EXAMPLE 17.3 Find the exponential Fourier series for the waveform shown in Fig. 17-2. Using the coe cients of this exponential series, obtain an and bn of the trigonometric series and compare with Example 17.1. In the interval 0 < !t < 2 the function is given by f t 10=2 !t. By inspection, the average value of the function is A0 5. Substituting f t in (11), we obtain the coe cients An .  2   1 2 10 10 e jn!t 10 An jn!t 1 j !te jn!t d !t 2 0 2 2n 2 2 jn 2 0 Inserting the coe cients An in (12), the exponential form of the Fourier series for the given waveform is f t j 10 j2!t 10 j!t 10 j!t 10 j2!t j 5 j e e e j e 4 2 2 4 13
The trigonometric series coe cients are, by (12), 10 n 10 10 10 sin !t sin 2!t sin 3!t f t 5  2 3 an 0 bn
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