082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

736 Chapter 23 Fourier series
Ifnisevenb n
=0.Ifnisoddb n
= 2 . Therefore the Fourier series representation of
nπ
f(t)is
f(t)= 1 2 + 2 (
sin100πt + sin300πt + sin500πt )
+···
π 3 5
The average value of the waveform is 1 . This is the zero frequency component or d.c.
2
value. We notethat inthisexample only odd harmonics arepresent.
Example23.14 Findthe Fourier series representation of f (t) = 1 +t, −π <t π, period2π.
Solution As usual we sketch f (t) first as this often provides insight into what follows (see Figure
23.15). HereT = 2π, ω = 1, and for convenience we shall consider the period of
integration tobe[−π,π]. UsingEquation (23.3)wefind
a 0
= 1 ∫ π
π −π1+tdt= 1 [ ] π
t + t2
π 2 −π
= 1 ) ))
((π + π2
−
(−π + π2
π 2 2
= 1 π (2π)
= 2
Similarly, using Equation (23.4) wefind
a n
= 1 π
∫ π
−π
(1+t)cosntdt
Integrating by parts gives
a n
= 1 ([
(1+t) sinnt
π n
= 1 π
] π
−π
( [ ] cosnt π )
0 +
n 2 −π
= 1 (cosnπ −cos(−nπ))
πn2 ∫ π
)
sinnt
− dt
−π n
since sin±nπ = 0
f(t)
1
–p
p
t
Figure23.15
Graph forExample23.14.