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# fourier_4pp

DisciplinaIntrodução Aos Circuitos Elétricos30 materiais191 seguidores
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```s
de
Fo
ur
ie
r
S
ér
ie
s
de
Fo
ur
ie
r
\ufffd
E
xe
m
pl
o:
O
nd
a
tr
ia
ng
ul
ar
:
a n
=
2 2
\u222b 3/
2
\u22121
/2
x(
t)
co
s
n\u3c0
td
t=
\u222b 1/
2
\u22121
/2
2A
tc
os
n\u3c0
td
t+
\u222b 3/
2
1/
2
2A
(1
\u2212t
)
co
s
n\u3c0
td
t=
0
b n
=
\u222b 1/
2
\u22121
/2
2A
ts
en
n\u3c0
td
t+
\u222b 3/
2
1/
2
2A
(1
\u2212t
)
se
n
n\u3c0
td
t
b n
=
8A n2
\u3c0
2
se
n
( n\u3c0 2
) b n
=
\u23a7 \u23aa \u23aa \u23aa \u23a8 \u23aa \u23aa \u23aa \u23a90
n
pa
r
8A n2
\u3c0
2
n
=
1,
5,
9,
13
,.
..
\u2212
8A n2
\u3c0
2
n
=
3,
7,
11
,1
5,
..
.
A
ss
im
,
x(
t)
=
8A \u3c0
2
[ sen
\u3c0
t\u2212
1 9
se
n
3\u3c0
t+
1 25
se
n
5\u3c0
t\u2212
1 49
se
n
7\u3c0
t+
..
.]
30
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
S
ér
ie
s
de
Fo
ur
ie
r
\ufffd
N
a
fo
rm
a
co
m
pa
ct
a,
co
m
o
se
n
kt
=
co
s
(k
t\u2212
90
\u25e6 )
e
\u2212s
en
kt
=
co
s
(k
t+
90
\u25e6 )
:
x(
t)
=
8A \u3c0
2
[ cos
(\u3c0
t\u2212
90
\u25e6 )
+
1 9
co
s
(3
\u3c0
t+
90
\u25e6 )
+
1 25
co
s
(5
\u3c0
t\u2212
90
\u25e6 )
+
1 49
co
s
(7
\u3c0
t+
90
\u25e6 )
+
..
.]
\ufffd
R
ep
re
se
nt
aç
ão
gr
á\ufb01
ca
da
fo
rm
a
co
m
pa
ct
a:
31
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
S
ér
ie
s
de
Fo
ur
ie
r
\ufffd
E
xe
m
pl
o:
E
xp
re
ss
e
a
se
gu
in
te
sé
rie
co
m
o
um
a
sé
rie
tr
ig
on
om
ét
ric
a
de
Fo
ur
ie
re
tra
ce
o
es
pe
ct
ro
de
am
pl
itu
de
e
fa
se
de
x(
t)
:
x(
t)
=
2
+
3
co
s
2t
+
4
se
n
2t
+
2
se
n
(3
t+
30
\u25e6 )
\u2212c
os
(7
t+
15
0\u25e6
)
\ufffd
N
a
sé
rie
tr
ig
on
om
ét
ric
a
co
m
pa
ct
a
de
Fo
ur
ie
r,
os
te
rm
os
em
se
no
e
co
ss
en
o
de
m
es
m
a
fre
qu
ên
ci
a
sã
o
co
m
bi
na
do
s
em
um
ún
ic
o
te
rm
o
e
os
te
rm
os
sã
o
de
sc
rit
os
em
co
ss
en
os
co
m
am
pl
itu
de
s
po
si
tiv
as
.
A
ss
im
,
3
co
s
2t
+
4
se
n
2t
=
5
co
s
(2
t\u2212
53
,1
3\u25e6
)
se
n
(3
t+
30
\u25e6 )
=
co
s
(3
t+
30
\u25e6 \u2212
90
)
=
co
s
(3
t\u2212
60
\u25e6 )
\u2212c
os
(7
t+
15
0\u25e6
)
=
co
s
(7
t+
15
0\u25e6
\u22121
80
\u25e6 )
=
7t
\u22123
0\u25e6
Po
rt
an
to
,
x(
t)
=
2
+
5
co
s
(2
t\u2212
53
,1
3\u25e6
)
+
2
co
s
(3
t\u2212
60
\u25e6 )
+
co
s
(7
t\u2212
30
\u25e6 )
32
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
S
ér
ie
s
de
Fo
ur
ie
r
\ufffd
N
a
fo
rm
a
co
m
pa
ct
a,
x(
t)
=
2
+
5
co
s
(2
t\u2212
53
,1
3\u25e6
)
+
2
co
s
(3
t\u2212
60
\u25e6 )
+
co
s
(7
t\u2212
30
\u25e6 )
\ufffd
R
ep
re
se
nt
aç
ão
gr
á\ufb01
ca
da
fo
rm
a
co
m
pa
ct
a:
33
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
S
ér
ie
s
de
Fo
ur
ie
rn
a
Fo
rm
a
E
xp
on
en
ci
al
\ufffd
A
té
ag
or
a
fo
ie
st
ud
a
a
fo
rm
a
tr
ig
on
om
ét
ric
a
da
sé
rie
de
Fo
ur
ie
r:
x(
t)
=
a 0
+
\u221e \u2211 n=1a
n
co
s
n\u3c9
t+
\u221e \u2211 n=1b
n
se
n
n\u3c9
t
\ufffd
U
sa
nd
o
as
eq
ua
çõ
es
de
E
ul
er
,
co
s
\u3b8
=
( ej\u3b8
+
e\u2212
j\u3b8
) /2
se
n
\u3b8
=
( ej\u3b8
\u2212e
\u2212j\u3b8
) /j2
po
de
m
os
re
es
cr
ev
er
a
fo
rm
a
tr
ig
on
om
ét
ric
a
na
fo
rm
a
ex
po
ne
nc
ia
l:
x(
t)
=
a 0
+
\u221e \u2211 n=1a
n
( ejn\u3c9
t
+
e\u2212
jn
\u3c9
t)
2
+
\u221e \u2211 n=1\u2212
jb
n
( ejn\u3c9
t
\u2212e
\u2212jn
\u3c9
t)
2
34
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
S
ér
ie
s
de
Fo
ur
ie
rn
a
Fo
rm
a
E
xp
on
en
ci
al
\ufffd
C
on
tin
ua
nd
o:
x(
t)
=
a 0
+
\u221e \u2211 n=1
a n
\u2212j
b n
2
ej
n\u3c9
t
+
\u221e \u2211 n=1
a n
+
jb
n
2
e\u2212
jn
\u3c9
t
x(
t)
=
a 0
+
\u221e \u2211 n=1
a n
\u2212j
b n
2
ej
n\u3c9
t
+
\u2212\u221e \u2211 n=\u22121
a n
\u2212j
b n
2
ej
n\u3c9
t
x(
t)
=
\u221e \u2211 n=\u2212\u221e
a n
\u2212j
b n
2
ej
n\u3c9
t
=
\u221e \u2211 n=\u2212\u221e
C
n
ej
n\u3c9
t
em
qu
e,
C
n
=
a n
\u2212j
b n
2
ou
C
n
=
1 T 0
\u222b T 0
x(
t)
e\u2212
jn
\u3c9
t
dt
35
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
S
ér
ie
s
de
Fo
ur
ie
rn
a
Fo
rm
a
E
xp
on
en
ci
al
\ufffd
E
xe
m
pl
o:
D
et
er
m
in
e
a
sé
rie
ex
po
ne
nc
ia
ld
e
Fo
ur
ie
rd
o
si
na
l
pe
rió
di
co
:
N
es
se
ca
so
,o
pe
río
do
é
T 0
=
\u3c0
e
\u3c9
0
=
2\u3c0 T
0
=
2r
/s
.
Po
rt
an
to
,
x(
t)
=
\u221e \u2211 n=\u2212\u221e
C
n
ej
2n
t
C
n
=
1 T 0
\u222b T 0
x(
t)
e\u2212
j2
nt
dt
=
1 \u3c0
\u222b \u3c0 0
e\u2212
t/
2
e\u2212
j2
nt
dt
=
1 \u3c0
\u222b \u3c0 0
e\u2212
(1
/2
+
j2
nt
)t
dt
,
C
n
=
\u22121
\u3c0
( 1 2+
j2
n) e
\u2212(
1/
2+
j2
n)
t\u2223 \u2223 \u2223 \u2223 \u2223 \u2223 \u2223\u3c0 0=
0,
50
4
1
+
j4
n
36
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
S
ér
ie
s
de
Fo
ur
ie
r
x(
t)
=
0,
50
4
\u221e \u2211 n=\u2212\u221e
1
1
+
j4
n
ej
2n
t ,
x(
t)
=
0,
50
4
[ 1+
1
1
+
j4
ej
2t
+
1
1
+
j8
ej
4t
+
1
1
+
j1
2
ej
6t
+
..
.
+
1
1
\u2212j
4
e\u2212
j2
t
+
1
1
\u2212j
8
e\u2212
j4
t
+
1
1
\u2212j
12
e\u2212
j6
t
+
..
.]
\ufffd
R
ep
re
se
nt
aç
ão
gr
á\ufb01
ca
da
fo
rm
a
co
m
pa
ct
a:
37
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
A
Tr
an
sf
or
m
a
de
Fo
ur
ie
r
\ufffd
A
té
ag
or
a,
ap
en
as
si
na
is
pe
rió
di
co
s
fo
ra
m
ab
or
da
do
s,
o
qu
e
fa
ze
rc
om
si
na
is
nã
o-
pe
rió
di
co
s?
\ufffd
C
on
si
de
re
um
si
na
lp
er
ió
di
co
x T
0
(t
)
fo
rm
o
pe
la
re
pe
tiç
ão
de
um
si
na
ln
ão
-p
er
ió
di
co
x(
t)
em
in
te
rv
al
os
de
T 0
se
gu
nd
os
.
O
si
na
lp
er
ió
di
co
x T
0
(t
)
po
de
se
rr
ep
re
se
nt
o
po
ru
m
sé
rie
ex
po
ne
nc
ia
ld
e
Fo
ur
ie
r.
\ufffd
S
e
\ufb01z
er
m
os
T 0
\u2192
\u221e:
lim T 0
\u2192
\u221e
x T
0
(t
)
=
x(
t)
\ufffd
A
sé
rie
de
Fo
ur
ie
rq
ue
re
pr
es
en
ta
x T
0
(t
)
ta
m
bé
m
irá
re
pr
es
en
ta
r
x(
t)
no
lim
ite
de
T 0
\u2192
\u221e.
A
sé
rie
ex
po
ne
nc
ia
ld
e
Fo
ur
ie
rp
ar
a
x T
0
(t
)
é
da
da
po
r:
x T
0
(t
)
=
\u221e \u2211 n=\u2212\u221e
C
n
ej
n\u3c9
0
t
e
C
n
=
1 T 0
\u222b T 0
/2
\u2212T
0/
2
x T
0
(t
)e
\u2212jn
\u3c9
0t
dt
38
/4
7
R
ep
re
se
nt
aç
õe
s
de
Fo
ur
ie
r
A
Tr
an
sf
or
m
a
de
Fo
ur
ie
r
\ufffd
U
m
a
ve
z
qu
e
x T
0
(t
)
=
x(
t)
no
in
te
rv
al
o
(\u2212
T 0
/2
,
T 0
/2
)
e
x(
t)
=
0
fo
ra
de
ss
e
in
te
rv
al
o:
C
n
=
1 T 0
\u222b T
0
/2
\u2212T
0
/2
x(
t)
e\u2212
jn
\u3c9
0
t
dt
=
1 T 0
\u222b \u221e \u2212\u221e
x(
t)
e\u2212
jn
\u3c9
0
t
dt
\ufffd
D
e\ufb01
ne
-s
e
um
a
fu
nç
ão
co
nt
ín```