Page 266 - Numerical Analysis Using MATLAB and Excel

Summary

• The trigonometric Fourier series for the square waveform with even symmetry is

( n – 1 )
2 1
4A ⎛ 1 1 ⎞ 4A ----------------
-
–
ft() = ------- cos ωt ---cos 3ωt + -- cos 5ωt – … = ------- ∑ – ( 1 ) -- cos nωt
-
π ⎝ 3 5 ⎠ π n
n = odd
• The trigonometric Fourier series for the sawtooth waveform with odd symmetry is
2A ⎛ 1 1 1 ⎞ 2A n – 1
1
-
-
ft() = ------- sin ωt – -- sin 2ωt + -- sin 3ωt – -- sin 4ωt + … = ------- ∑ – ( 1 ) -- sin nωt
-
-
π ⎝ 2 3 4 ⎠ π n
• The trigonometric Fourier series for the triangular waveform with odd symmetry is
( n – 1 )
8A ⎛ 1 1 1 ⎞ 8A ---------------- 1
2
2 ∑
–
ft() = ------- sin ωt --sin 3ωt + ------ sin 5ωt – ------ sin 7ωt + … = ------- – ( 1 ) ----- sin nωt
-
2 ⎝
π 9 25 49 ⎠ π n 2
n = odd
• The trigonometric Fourier series for the half−wave rectification waveform with no symmetry is
A A A cos 2t cos 4t cos 6t cos 8t
ft() = ---- + ---- sin – ---- ------------- + ------------- + ------------- + ------------- + …
t
π 2 π 3 15 35 63
• The trigonometric Fourier series for the full−wave rectification waveform with even symmetry
is
∞
1
------- –
-------
ft() = 2A 4A ∑ ------------------- cos nωt
π π ( n – 1 )
2
n = 246 …
,,
,
• The Fourier series are often expressed in exponential form as
ft() = … + C e – j2ωt + C e – jωt + C + C e jωt + C e j2ωt + …
1
0
1
–
2
–
2
where the C i coefficients are related to the trigonometric form coefficients as
1 ⎛ b n⎞ 1
-
C – n = -- a – ----- = -- a +( 2 - n jb ) n
2 ⎝
⎠
n
j
1 ⎛ b n⎞ 1
-
C = -- a + ----- = --- a j– b( 2 n n )
⎠
2 ⎝
n
n
j
1
-
C = --a 0
0
2
• The C i coefficients, except C 0 , are complex, and appear as complex conjugate pairs, that is,
C – n = C ∗
n
Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−49
Copyright © Orchard Publications