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

756 Chapter 23 Fourier series
5 Theoutput from ahalf-wave rectifier isgiven by
{
Isinωt 0<t <T/2
i(t) =
0 T/2<t<T
and isperiodic with periodT = 2π ω .
Find its Fourier seriesrepresentation.
6 Find the complex Fourier series representation ofthe
functionwith periodT = 0.02defined by
{
V(constant) 0 t < 0.01
v(t) =
0 0.01 t < 0.02
7 Find the Fourier seriesrepresentation ofthe function
with period8given by
{
2 −t 0<t<4
f(t)=
t−6 4<t<8
8 Ther.m.s.voltage, v r.m.s. ,ofaperiodicwaveform,
v(t),with periodT, isgiven by
√
∫
1 T
v r.m.s. = (v(t))
T
2 dt
0
If v(t)has Fourier coefficientsa n andb n show, using
Parseval’stheorem,that
√
1
v r.m.s. =
4 a2 0 + 1 ∞∑
(an 2 2
+b2 n )
n=1
9 If f (t)has Fourier series
f(t)= a ∞ (
0
2 + ∑
a n cos 2nπt
T
n=1
)
+b n sin 2nπt
T
prove Parseval’stheorem.
[Hint: multiply both sidesby f (t)to obtain
(f(t)) 2 = a 0 f(t)
(
∞∑
+ a
2 n f(t)cos 2nπt
T
n=1
)
+b n f(t)sin 2nπt
T
and integrate both sidesover the interval[0,T]using
Equations(23.3)--(23.5).]
10 Findthe half-rangecosine seriesandthe half-range
sineseriesforthe function
f(t)=sinhπt 0<t <1
Solutions
1
2
π
∞ 2 − ∑ 4n(1 +cosnπ)
π(n +1) 2 (n −1) 2 sin(nt)
2
2
π
3 (a)
∞∑
1
n(1 −cosnπ)
(n 2 −4)
32 ∑ ∞
3π 2
1
sin(nt)
sin(nπ/4)sin(nπt/τ)
n 2
5
6
7
I
π + I 2 sin ωt − I π
v
2π
∞∑
−∞
8 ∑ ∞
π 2
1
j cosnπ−1
n
1−cosnπ
n 2
∞∑
2
cosnπ+1
n 2 −1
e 100nπjt
cos
(
)
nπt
4
cosnωt
4
(b)
1
2 + 8 π 2
×
v
3 + 2v π
∞∑
1
∞∑
1
4cos(nπ/4) −cosnπ −3
3n 2
sin(nπ/3)cos(2nπt/T)
n
cos
(
)
nπt
τ
10 cosineseries: 1 (coshπ −1)
π
+
∞∑
n=1
2
π(1+n 2 ) ((−1)n coshπ −1)cosnπt
sineseries:
∞∑ 2n
−
π(1+n 2 ) (−1)n sinh πsinnπt
n=1