634 Higher Engineering Mathematics
- Determine the Fourier series for the half
wave rectified sinusoidal voltage Vsinωt
defined by:
f(t)=
⎧
⎪⎨
⎪⎩
Vsinωt, 0 <t<
π
ω
0 ,
π
ω
<t<
2 π
ω
which is periodic of period
2 π
ω ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
f(t) =
V
π
+
V
2
sinωt
−
2 V
π
(
cos2ωt
( 1 )( 3 )
+
cos4ωt
( 3 )( 5 )
+
cos6ωt
( 5 )( 7 )
+···
)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
69.2 Half-range Fourier series for
functions defined over rangeL
(a) By making the substitution u=
πx
L
(see
Section 69.1), the rangex=0tox=L corre-
sponds to the rangeu=0tou=π. Hence a
function may be expanded in a series of either
cosine terms or sine terms only, i.e. ahalf-range
Fourier series.
(b) Ahalf-range cosine seriesin the range 0 toLcan
be expanded as:
where
f(x)=a 0 +
∑∞
n= 1
ancos
(nπx
L
)
a 0 =
1
L
∫L
0
f(x)dx and
an=
2
L
∫L
0
f(x)cos
(nπx
L
)
dx
(c) Ahalf-range sine seriesin the range 0 toLcan
be expanded as:
f(x)=
∑∞
n= 1
bnsin
(nπx
L
)
where bn=
2
L
∫L
0
f(x)sin
(nπx
L
)
dx
Problem 4. Determine the half-range Fourier
cosine series for the functionf(x)=xin the range
0 ≤x≤2. Sketch the function within and outside of
the given range.
A half-range Fourier cosine series indicates an even
function. Thus the graph off(x)=xin the range 0 to
2 is shown in Fig. 69.4 and is extended outside of this
range so as to be symmetrical about the f(x)axis as
shown by the broken lines.
f (x )
f (x ) 5 x
(^2422) x
2
0642
Figure 69.4
From para. (b), for a half-range cosine series:
f(x)=a 0 +
∑∞
n= 1
ancos
(nπx
L
)
a 0 =
1
L
∫L
0
f(x)dx=
1
2
∫ 2
0
xdx
1
2
[
x^2
2
] 2
0
= 1
an=
2
L
∫L
0
f(x)cos
(nπx
L
)
dx
2
2
∫ 2
0
xcos
(nπx
2
)
dx
⎡
⎢
⎣
xsin
(nπx
2
)
(nπ
2
) +
cos
(nπx
2
)
(nπ
2
) 2
⎤
⎥
⎦
2
0
⎡
⎢
⎣
⎛
⎜
⎝
2sinnπ
(nπ
2
) +
cosnπ
(nπ
2
) 2
⎞
⎟
⎠−
⎛
⎜
⎝^0 +
cos0
(nπ
2
) 2
⎞
⎟
⎠
⎤
⎥
⎦
⎡
⎢
⎣
cosnπ
(nπ
2
) 2 −
1
(nπ
2
) 2
⎤
⎥
⎦
(
2
πn
) 2
(cosnπ− 1 )