sel310-A4 Linha parte 1 Sel310 612
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sel310-A4 Linha parte 1 Sel310 612


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z\u2206
),( tzi ),( tzzi \u2206+Gi Ci
zR\u2206 zL\u2206
zG\u2206 zC\u2206 zG\u2206 zC\u2206
A B
z\u2206
),( tzi ),( tzzi \u2206+Gi Ci
3/8/2012 29SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
Aplicando a lei de Kirchoff das tensões na malha compreendida no trecho z\u2206
),( tzv ),( tzzv \u2206+),( tzv ),( tzzv \u2206+
( ) ( )
( )
( )
,
, , ,
i z t
v z t i z t R z L z v z z t
t
\u2202
= \u2206 + \u2206 + +\u2206
\u2202
Equação de onda-2
zR\u2206 zL\u2206
zG\u2206 zC\u2206 zG\u2206 zC\u2206
A B
z\u2206
),( tzv ),( tzzv \u2206+
),( tzi ),( tzzi \u2206+Gi Ci
zR\u2206 zL\u2206
zG\u2206 zC\u2206 zG\u2206 zC\u2206
A B
z\u2206
),( tzv ),( tzzv \u2206+
),( tzi ),( tzzi \u2206+Gi Ci
( ),i z t\u2202
3/8/2012 30SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
( ) ( ) ( )
( )
( ) ( )
( )
( )
,
, , ,
Dividindo por e rearranjan
, ,
,
do
,
,
i z t
v z t v z z t i z t R z L z
t
z
v z z t v z t i z t
Ri z t L
z t
+\u2206 \u2212 \u2202
\u2212 =
\u2202
\u2212 +\u2206 = \u2206 + \u2206
\u2202
+
\u2206 \u2202
\u2206
Equação de onda-3
( ) ( )
( )
( )
( ) ( ) ( )
Mas,
, , ,
lim
, ,
,
,
v z z t v z t v z t
v z z t v z t i z t
Ri z t L
z t
+\u2206 \u2212 \u2202
\uf8ee \uf8f9+\u2206 \u2212 \u2202\uf8ef \uf8fa
=\uf8ef \uf8fa
\u2212 = +
\u2206 \u2202
3/8/2012 31SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
( ) ( ) ( )
( )
( )
( )
0
lim
 
Substituindo resulta em
(1) 
,
, 
,
z z
v z t i z t
z
Ri z t L
z t
\u2206 \u2192
\uf8ef \uf8fa
=\uf8ef \uf8fa
\uf8ef \uf8fa\u2206 \u2202
\uf8ef \uf8fa\uf8f0
\u2202 \u2202
\uf8fb
\u2212 = +
\u2202 \u2202
Equação de onda-4
Vamos determinar outra equação equivalente a (1).
Vamos aplicar a lei de Kirchhoff das correntes no nó B
B
Gi Ci
),( tzi ),( tzzi \u2206+B
Gi Ci
),( tzi ),( tzzi \u2206+
3/8/2012 32SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
( ) ( ), ( , ) ( , ) ,G Ci z t i z z t i z z t i z z t= +\u2206 + +\u2206 + +\u2206
Equação de onda-5
( ) ( )
( )
( )
Mas,
( , ) ,
,
( , )
, ( , ) ( , ) ,
G
G
C
i z z t v z z t G z
v
i z t i z z
z z t
i
t i z z t
z z t C z
zi z t
+\u2206 = +\u2206 \u2206
\u2202 +\u2206
= +\u2206 + +\u2206 + +
\u2206
\u2206
+ = \u2206
3/8/2012 33SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
( )
( ) ( )
( )
( )
( , )
Substituindo resulta e
,
, ,
m
,
C
i
v z z t
i z
z z t C z
t v z z t G z C z i z z t
t
t
\u2202 +\u2206
= +\u2206 \u2206 + \u2206 + +
\u2206
\u2202
\u2202
\u2206
+ = \u2206
Equação de onda-6
( ) ( ) ( )
( )
( ) ( )
( )
( ), ,
Reescrevendo e divindo por ,
,
,
,
,
, ,
v z
v z z t
v
i z z t i z t
z
z t
i z t
z
i z z t v z
G
z t
z z t C
t
G z C z
t
\u2206
\u2202
\u2212 = +
\u2202
+\u2206
+\u2206
+\u2206 \u2212
\u2202 +\u2206
\u2212 +\u2206 = +\u2206 \u2206 + \u2206
\u2206
\u2202
3/8/2012 34SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
( )
( ) ( ) ( )
( )
0
Observar que 
,
lim
: resolver (ver em e
, ,
slaides 'extras',
z
z
i z z t
v
t
i z t
z
i
z
z
z
z
t
t
\u2206 \u2192
\u2202
+
\uf8ee \uf8f9 \u2202\uf8ef \uf8fa
=\uf8ef \uf8fa
\uf8ef \uf8fa \u2202
\uf8ef \uf8fa
\u2206
\u2206
\u2206
\uf8f0 \uf8fb
+\u2206 \u2212
)
Equação de onda-7
( ) ( )
( )
( ) ( )
( )
( ),
,
,, v z z t
v z z
dv z
v z z v z
t
z
dz
G C
i z z t i z t +\u2206
+\u2206
+\u2206 = + \u2206
\u2202
\u2212 = +
+\u2206 \u2212
3/8/2012 SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP 35
( )
( )
( ) ( ) ( )
( )
( )
2
,
, , ,
,
,
,
v z z t
v z t v z t v z t
v z t z z
z t t z
v z t
v z t z
z
G C
t
G C
G C
t
z
+\u2206
\uf8ee \uf8f9\u2202 \u2202 \u2202\uf8ef \uf8fa+ \u2206 + \u2206\uf8ef \uf8fa
\u2212 = +
\u2202
\uf8ee \uf8f9
\uf8ef \uf8fa= + \uf8ef \uf8fa
\uf8ef \uf8fa
\uf8f0 \uf8fb
\uf8eb \uf8f6\u2202 \uf8f7\uf8ec \uf8f7+\uf8ec \uf8f7\uf8ec
\u2202 \u2202 \u2202 \u2202\uf8ef \uf8fa
\uf8f0 \uf8fb
\uf8ee \uf8f9\u2202\uf8ef \uf8fa= + \u2206\uf8ef \uf8fa
\u2202
\u2206
\uf8f7\uf8ec \u2202 \uf8ef
\uf8f0 \uf8fb
\uf8ed \uf8f8 \uf8fa
Equação de onda-8
( ) ( )
( )
( ), , ,
,
Para 0
i z z t i z t v z t
G G C v z t z z
z t z
z
\u3b5
+ \u2206 \u2212 \u2202\u2202
\u2212 = + + \u2206 + \u2206
\u2206 \u2202 \u2202
\uf8ee \uf8f9\uf8eb \uf8f6\uf8f7\uf8ec \uf8ef \uf8fa
\u2206 \u2192
\uf8f7\uf8ec \uf8f7\uf8ec \uf8ef \uf8fa\uf8ed \uf8f8 \uf8f0 \uf8fb
3/8/2012 SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP 36
( )
( )
( )
 (2 )
, ,
,
i z t v z t
Gv z t C
z t
\u2202 \u2202
\u2212 = +
\u2202 \u2202
Equação de onda-9
( )
( )
( ), ,
,
v z t i z t
Ri z t L
z t
\u2202 \u2202
\u2212 = +
\u2202 \u2202
(1)
3/8/2012 37SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
( )
( )
( ), ,
,
i z t v z t
Gv z t C
z t
\u2202 \u2202
\u2212 = +
\u2202 \u2202
(2)
Equações fundamentais da linha de transmissão
Equação de onda-10
( )
( )
( ), ,
,
i z t v z t
Gv z t C
z t
\u2202 \u2202
\u2212 = +
\u2202 \u2202
(2)
( )
( )
( ), ,
,
v z t i z t
Ri z t L
z t
\u2202 \u2202
\u2212 = +
\u2202 \u2202
(1)
Derivando (1) em relação a z e (2) em relação a t,
3/8/2012 38SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
Derivando (1) em relação a z e (2) em relação a t,
2 2
2
( , ) ( , ) ( , )v z t i z t i z t
R L
z t zz
\u2202 \u2202 \u2202
\u2212 = +
\u2202 \u2202 \u2202\u2202
2 2
2
( , ) ( , ) ( , )i z t v z t v z t
G C
t z t t
\u2202 \u2202 \u2202
\u2212 = +
\u2202 \u2202 \u2202 \u2202
(4)
(3)
z\u2202
\u2202
t\u2202
\u2202
Equação de onda-11
2 2
2
( , ) ( , ) ( , )v z t i z t i z t
R L
z t zz
\u2202 \u2202 \u2202
\u2212 = +
\u2202 \u2202 \u2202\u2202
2 2
2
( , ) ( , ) ( , )i z t v z t v z t
G C
t z t t
\u2202 \u2202 \u2202
= \u2212 \u2212
\u2202 \u2202 \u2202 \u2202
(4)
(3)
3/8/2012 39SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
Substituindo (4) em (3):
t z t t\u2202 \u2202 \u2202 \u2202
2 2
2 2
( , ) ( , ) ( , ) ( , )v z t i z t v z t v z t
R LG LC
z zz t
\u2202 \u2202 \u2202 \u2202
\u2212 = \u2212 \u2212
\u2202 \u2202\u2202 \u2202
Equação de onda-12
Substituindo
2 2( , ) ( , ) ( , ) ( , )v z t i z t v z t v z t\u2202 \u2202 \u2202 \u2202
( )
( )
( ), ,
,
i z t v z t
Gv z t C
z t
\u2202 \u2202
= \u2212 \u2212
\u2202 \u2202
em
3/8/2012 40SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
2 2
( , ) ( , ) ( , ) ( , )v z t i z t v z t v z t
R LG LC
z zz t
\u2202 \u2202 \u2202 \u2202
\u2212 = \u2212 \u2212
\u2202 \u2202\u2202 \u2202
2 2
2 2
( , ) ( , ) ( , ) ( , )
( , )
v z t v z t v z t v z t
R Gv z t C LG LC
t tz t
\uf8ee \uf8f9\u2202 \u2202 \u2202 \u2202\uf8ef \uf8fa\u2212 = \u2212 \u2212 \u2212 \u2212\uf8ef \uf8fa
\u2202 \u2202\u2202 \u2202\uf8ef \uf8fa\uf8f0 \uf8fb
resulta em
Equação de onda-13
( )
2 2
2 2
( , ) ( , ) ( , )
( , )
v z t v z t v z t
RGv z t RC LG LC
tz t
\u2202 \u2202 \u2202
= + + +
\u2202\u2202 \u2202
Equação de onda para tensão
Equação de onda para corrente
3/8/2012 41SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
( )
2 2
2 2
( , ) ( , ) ( , )
( , )
i z t i z t i z t
RGi z t RC LG LC
tz t
\u2202 \u2202 \u2202
= + + +
\u2202\u2202 \u2202
Sistemas que obedecem a estas equações podem ser utilizados 
para transmitir informações sob a forma de onda eletromagnética 
Equação de onda-14
2 2
2 2
( , ) ( , )v z t v z t
LC
z t
\u2202 \u2202
=
\u2202 \u2202
Linha sem perdas, R=G=0
Equação de onda para tensão
Equação de onda para corrente
3/8/2012 42SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
2 2
2 2
( , ) ( , )i z t i z t
LC
z t
\u2202 \u2202
=
\u2202 \u2202
Equação de onda para corrente
Sistemas que obedecem a estas equações podem ser utilizados 
para transmitir informações sob a forma de onda eletromagnética 
Solução da equação de onda para tensão-1
( , )
z z
v z t v t v t+ \u2212
\uf8eb \uf8f6 \uf8eb \uf8f6\uf8f7 \uf8f7\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec= \u2212 + +\uf8f7 \uf8f7\uf8ec \uf8ec
2 2
2 2
( , ) ( , )v z t v z t
LC
z t
\u2202 \u2202
=
\u2202 \u2202
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( , )
f f
z z
v z t v t v t
v v
+ \u2212\uf8f7 \uf8f7\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec= \u2212 + +\uf8f7 \uf8f7\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec\uf8ed \uf8f8 \uf8ed \uf8f8
1
f
v
LC
= Velocidade de propagação da onda (velocidade de fase)
Solução da equação de onda para tensão-2
( , )
f f
z z
v z t v t v t
v v
+ \u2212
\uf8eb \uf8f6 \uf8eb \uf8f6\uf8f7 \uf8f7\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec= \u2212 + +\uf8f7 \uf8f7\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec\uf8ed \uf8f8 \uf8ed \uf8f8
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f f
v v\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec\uf8ed \uf8f8 \uf8ed \uf8f8
Onda propagando na 
direção positiva de z
Onda propagando na 
direção negativa de z
Solução da equação de onda para tensão-3
f
z
v t
v
+
\uf8eb \uf8f6\uf8f7\uf8ec \uf8f7\uf8ec \u2212 \uf8f7\uf8ec \uf8f7\uf8ec \uf8f7\uf8ec\uf8ed \uf8f8
z=z
1
z=z
2
>z
1
movimento
3/8/2012 45SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
2 1
f
z z
t
v
\u2212
\u2206 =
t
Solução da equação de onda para tensão-4
f
z
v t
v
+
\uf8eb \uf8f6\uf8f7\uf8ec \uf8f7\uf8ec \u2212 \uf8f7\uf8ec \uf8f7\uf8ec \uf8f7\uf8ec\uf8ed \uf8f8
t=t
1
t=t
2
>t
1
movimento
3/8/2012 46SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
( )2 1fz v t t\u2206 = \u2212
z
Solução da equação de onda para tensão-5
f
z
v t
v
+
\uf8eb \uf8f6\uf8f7\uf8ec \uf8f7\uf8ec \u2212 \uf8f7\uf8ec \uf8f7\uf8ec \uf8f7\uf8ec\uf8ed \uf8f8 v-
v+
f
z
v t
v
\u2212
\uf8eb \uf8f6\uf8f7\uf8ec \uf8f7\uf8ec + \uf8f7\uf8ec \uf8f7\uf8ec \uf8f7\uf8ec\uf8ed \uf8f8
3/8/2012 47SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
z
movimentomovimento
Solução da equação de onda para tensão-6
0 0
( , ) cos cos
f f
z z
v z t V t V t
v v
\u3c9 \u3c6 \u3c9 \u3c6+ + \u2212 \u2212
\uf8ee \uf8f9 \uf8ee \uf8f9\uf8eb \uf8f6 \uf8eb \uf8f6\uf8f7 \uf8f7\uf8ec \uf8ec\uf8ef \uf8fa \uf8ef \uf8fa\uf8f7 \uf8f7\uf8ec \uf8ec= \u2212 + + + +\uf8f7 \uf8f7\uf8ef \uf8fa \uf8ef \uf8fa\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec\uf8f7 \uf8f7\uf8ec \uf8ec\uf8ef \uf8fa \uf8ef \uf8fa\uf8ed \uf8f8 \uf8ed \uf8f8\uf8f0 \uf8fb \uf8f0 \uf8fb
( ) ( )0 0( , ) cos cosz zv z t V t k z V t k z\u3c9 \u3c6 \u3c9 \u3c6+ + \u2212 \u2212= \u2212 + + + +
3/8/2012 48SEL310/612 Ondas Eletromagnéticas A.C. Cesar SEL EESC USP
Constante de propagação
( ) ( )0 0( , ) cos cosz zv z t V t k z V t k z\u3c9 \u3c6 \u3c9 \u3c6= \u2212 + + + +
z
f
k
v
\u3c9
=
EXTRA
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Diferencial e limite-1
( )
( ) ( )
( ) ( ) ( )
( )
, ,
,
, 0 
df z
f z t f z t
df z t dz dt
z t
f z z f z df z
z dz
\u3b5 \u3b5
\u2202 \u2202
= +
\u2202 \u2202
+\u2206 \u2212
= + \u2192
\u2206
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( ) (