Page 243 - Numerical Analysis Using MATLAB and Excel

Chapter 6 Fourier, Taylor, and Maclaurin Series

1 ⎛ a b ⎞ ⎛ a b ⎞
2
1
1
2
ft() = --a + c ---- cos t + ----- sin t + c ----cos 2t + ----- sin 2t + …
-
2 0 1 c ⎝ 1 c 1 ⎠ 2 c ⎝ 2 c 2 ⎠
----- sin
----- cos
+ c n ⎛ a n n nt + b n n nt ⎞ ⎠
c ⎝
c
1 cos θ cos t + sin θ sin t cos θ cos 2t + sin θ sin 2t
2
2
1
1
-
= --a + c ⎛ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎞ + c ⎛ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎞ + …
2 0 1 ⎝ ( cos t – θ ) 1 ⎠ 2 ⎝ cos ( 2t θ ) – 2 ⎠
cos θ cos nt + sin θ sin nt
+ c ⎛ n ⎝ ⎧ ⎪ n ⎪ cos ( ⎪ nt – θ ) ⎪ n ⎪ n ⎪ ⎪ ⎩ ⎞ ⎠
⎨
⎪
and, in general, for ω ≠ 1 , we get
1 ∞ 1 ∞ ⎛ b n⎞
-
ft() = --a + ∑ c cos ( nωt – θ ) = ---a + c cos ⎝ ∑ nωt – atan ----- (6.67)
2 0 n n 2 0 n a ⎠
n = 1 n = 1 n
Similarly,
1 sin ϕ cos t + cos ϕ sin t
1
1
-
ft() = --a + c ⎛ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎞
2 0 1 ⎝ sin ( t + ϕ ) 1 ⎠
sin ϕ cos 2t + cos ϕ sin 2t sin ϕ cos nt + cos ϕ sin nt
c ⎛ 2 ⎝ ⎧ ⎪ 2 ⎪ sin ( 2t + ϕ ) ⎪ 2 ⎪ 2 ⎪ ⎪ ⎩ ⎞ ⎠ + … + c ⎛ ⎪ n ⎝ ⎧ ⎪ n ⎪ sin ⎪ ( nt + ϕ ) ⎪ n ⎪ n ⎪ ⎪ ⎩ ⎞ ⎠
⎨
⎪
⎨
⎪
and, in general, where ω ≠ 1 , we get
1 ∞ 1 ∞ ⎛ a n⎞
-
-
ft() = --a + ∑ c sin ( nωt + ϕ ) = --a + ∑ c sin nωt + atan ----- (6.68)
2 0 n n 2 0 n ⎝ b ⎠
n = 1 n = 1 n
The series of (6.67) and (6.68) can be expressed as phasors. Since it is customary to use the cosine
function in the time domain to phasor transformation, we choose to use the transformation of
(6.63) below.

1 ∞ ⎛ b n⎞ 1 ∞ b n
-
-
--a + c cos ⎝ ∑ nωt – atan ----- ⇔ --a + ∑ c ∠ – atan ----- (6.69)
2 0 n a ⎠ 2 0 n a
n = 1 n n = 1 n

Example 6.8
Find the first 5 terms of the alternate form of the trigonometric Fourier series for the waveform of
Figure 6.21.

6−26 Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright © Orchard Publications