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Ins$tuto Federal de Santa Catarina
Introdução a Carta de SmithEvanaska Maria Barbosa Nogueira
evanaska.nogueira@ifsc.edu.br
Ins$tuto Federal de Santa Catarina
Resumo teórico
•Recurso gráfico.
•Cálculos de linhas de transmissão.
•Idealizado por P.H. Smith em 1939.
Ins$tuto Federal de Santa Catarina
Resumo teórico
Método:
•Carta de impedâncias (Carta de Smith)
•Lugares geométricos:
•valores de resistência constante
•valores de reatância constante
•diagrama polar
Ins$tuto Federal de Santa Catarina
Funções da Carta de Smith
•Permite encontrar:
1. Como as impedâncias são transformadas ao longo de uma
L.T.
2. Como relacionar a impedância com:
• coeficiente de reflexão
• razão de onda estacionária
• posições onde ocorrem os máximos e os mínimos de onda estacionária.
Ins$tuto Federal de Santa Catarina
Funções da carta de Smith
•Determina os pontos de casamento de impedância.
•Dimensiona o elemento reativo que permite o
casamento de impedância.
Ins$tuto Federal de Santa Catarina
Carta de Smith
•Mais útil para linhas sem perdas.
•2 gráficos em 1:
1. Impedância normalizada em qualquer ponto de
uma LT.
2. Coeficiente de reflexão para qualquer ponto ao
longo da LT.
Ins$tuto Federal de Santa Catarina
Obtenção da Carta de Smith
Γ" =$" − $&$" + $&
•Normalizando a impedância da carga em função da impedância característica:
(" =$"$&
•Então:
Γ" =(" − 1(" + 1
Ins$tuto Federal de Santa Catarina
Obtenção da Carta de Smith
Substituindo a carga em conjunto com um
comprimento arbitrário de LT por uma impedância de
entrada.
Γ = Γ#$%&'( =) − 1) + 1
Ins$tuto Federal de Santa Catarina
Obtenção da Carta de Smith
Substituindo a carga em conjunto com um
comprimento arbitrário de LT por uma impedância de
entrada.
Γ = Γ#$%&'( =) − 1) + 1
Magnitude do coeficiente de reflexão na carga Fase deslocada em -./
Ins$tuto Federal de Santa Catarina
Obtenção da Carta de Smith• Expandindo Γ em parcelas reais e imaginárias:
Γ = Γ#$ + &Γ'(
) = 1 + Γ1 − Γ = , + &- = 1 + Γ#$ + &Γ'(
1 − Γ#$ − &Γ'(
, = 1 − Γ#$. − Γ'(.1 − Γ#$ . + Γ'(.
&- = &2Γ'(1 − Γ#$ . + Γ'(.
Ins$tuto Federal de Santa Catarina
Obtenção da Carta de Smith
•Equação geral para círculo de raio a centrado em x=m e y=n:
! −# $ + & − ' $ = )$
Então:
• Γ+, − --./
$+ Γ01$ = /
-./$
• Γ+, − 1 $ + Γ01 − /3$= /
3$
Círculos serão traçados nos eixos Γ01 versus Γ+,
Instituto Federal de Santa Catarina
Obtenção da Carta de Smith
! − # $ + & − ' $ = )$
Γ+, −-
- + 1$+ Γ/0$ = 1
- + 1$
• Centro = 1123 , 0
• Raio = 3123
r 0 1/3 1 3 ∞centro (0,0) (1/4, 0) (1/2, 0) (3/4, 0) (1,0)
raio 1 3/4 1/2 1/4 0
Γ+,
Γ/0
r = ∞
r = 0r = ⁄1 3r = 1
r = 3
r 0 1/3 1 3 ∞centro
raio
Instituto Federal de Santa Catarina
x 0 ±1/2 ±1 ±2 ±∞centro (1, ∞) (1, ±2) (1, ±1) (1, ±1/2) (1, 0)
raio ∞ 2 1 1/2 0
Obtenção da Carta de Smith
x 0 ±1/2 ±1 ±2 ±∞centro
raio
Γ%&
Γ'(
x = 0
x = +1
x = − ⁄1 2
x = + ⁄1 2
x = −1
x = +2
x = −2
0 − 1 2 + 3 − 4 2 = 52
Γ%& − 1 2 + Γ'( − 102= 1
02
• Centro = 1, 78
• Raio = 78 Γ%&
Γ'(
Instituto Federal de Santa Catarina
Obtenção da Carta de Smith
Γ"#
Γ$%
Instituto Federal de Santa Catarina
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1.61.6
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0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
Instituto Federal de Santa Catarina
Parte reativa
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1.81.8
1.8
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-160
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-170
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±
90-9
085
-85
80-8
0
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5
70-7
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65-6
5
60-6
0
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-45
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-40
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AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
x>0
x<0
Instituto Federal de Santa Catarina
Exercício 1
•Linha com !" = 50Ω•Localize: !'( = 10 − +35Ω
-'( = 0,2 − +0,7Ω
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60-6
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-45
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-40
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-35
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0.240.25
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0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
0,2
0,7+
-'(
Instituto Federal de Santa Catarina
0.1
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0.5
0.60.6
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0.9
0.9
1.0
1.0
1.0
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1.2
1.2
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1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
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10
10
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20
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30-30
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-50
60
-60
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-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
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0.05
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0.08
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0.1
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0.11
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0.15
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0.16
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0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
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0.46
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0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
Exercício 1
•Linha com !" = 50Ω•Localize: !'( = 100 + +75Ω
-'( = 2 + +1,5Ω
2
1,5+
012
Instituto Federal de Santa Catarina
Exercício 2
Localize as seguintes
impedâncias de carga para
uma LT de 50 Ω.
(a) ZL= 0 Ω(b) ZL= ∞(c) ZL= 100 + 100j Ω(d) ZL= 100 - 100j Ω(e) ZL= 50 Ω
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
# $%
&
'
Instituto Federal de Santa Catarina
Coeficiente de reflexão
Γ = Γ#$ + &Γ'( = Γ)*+,-. = Γ) *+/0
Γ) = distância a partir do centro da carta até o ponto
dividido pela distância do centro da carta até o
perímetro ( Γ) = 1).
Instituto Federal de Santa Catarina
Coeficiente de reflexão
Γ = Γ#$ + &Γ'( = Γ)*+,-. = Γ) *+/0
Γ) = distância a partir do centro da carta até o ponto dividido pela distância do centro da carta até o perímetro
( Γ) = 1).
! Para evitar este cálculo, uma escala para 23 é fornecida abaixo da Carta de Smith.
Instituto Federal de Santa Catarina
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
!"RFL. COEFF, E OR I
Instituto Federal de Santa Catarina
Coeficiente de reflexão
!"! Para evitar este cálculo, uma escala para !" é fornecida abaixo da Carta de Smith.
#$! O ângulo de Γ&,#$, é indicado na escala ângulo de coeficiente de reflexão, apresentada na parte externa do círculo Γ& = 1 na carta.
Instituto Federal de Santa Catarina
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
!" #$RFL. COEFF, E OR I
ANGLE OF REFLECTION COEFFICIENT IN DEGREES
Instituto Federal de Santa Catarina
Exemplo
•Linha com !" = 50Ω•!'( = 10 − +35Ω•Determine o valor de Γ'
.'( = 0,2 − +0,7Ω
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
.'(
Círculo do Γ cte
Γ
Γ =0,76∡Γ = −108,5°
Instituto Federal de Santa Catarina
Localização de !"Linha com #$ = 50Ω
#)* = 10 − -35ΩΓ)* = 0,76 ∠ −108,5°
#)6 = 100 + -75ΩΓ)6 = 0,53 ∠29,7°
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
:";
:"<
Γ)* = 0,76 ∠ −108,5°
Γ)6 = 0,53 ∠29,7°
Instituto Federal de Santa Catarina
Exercício 1
Suponha uma linha com Z0 = 50 Ω terminada em uma
carga ZL = 50 + j100 Ω. Encontre a impedância da
linha na Carta de Smith e o coeficiente de reflexão na
carga.
Instituto Federal de Santa Catarina
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
Exercício 1
Z0 = 50 Ω
ZL = 50 + j100 Ω
zL=1+2j
1
2#$%
Γ =0,7
∡Γ = 45°
Instituto Federal de Santa Catarina
Círculo de Γ" constante
•Círculo de Γ" constante.
Γ = Γ$% + 'Γ() = Γ"*+,-. = Γ" *+/0
Instituto Federal de Santa Catarina
Círculo de Γ" constante
•Círculo de Γ" constante.
Γ = Γ$% + 'Γ() = Γ"*+,-. = Γ" *+/0
Círculo de Γ" cte
! Varia-se 12 mantendo Γ" constante
Instituto Federal de Santa Catarina
Círculo de Γ" constante
•Mover ao longo do círculo Γ" cte é semelhante a se
deslocar ao longo da LT
Instituto Federal de Santa Catarina
Círculo de Γ" constante
•Mover ao longo do círculo Γ" cte é semelhante a se
deslocar ao longo da LT
Γ = Γ"$%&'( = Γ"$%)*(+
Instituto Federal de Santa Catarina
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90100
-100110
-110120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-4035
-3530
-30
25
-25
20
-20
15
-15
10
-100.0
4
0.04
0.05
0.05
0.06
0.06
0.07
0.070.08
0.080.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.320.32
0.330.33
0.340.34
0.350.35
0.360.36
0.370.37
0.380.38
0.390.39
0.40.4
0.410.41
0.420.42
0.430.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
Sentido para o
Gerador
Sentido para a Carga
Instituto Federal de Santa Catarina
Círculo de Γ" constante
Γ = Γ"$%&'()
•Devido ao fator 4+, a cada 0,5, damos uma volta
completa (período).
Instituto Federal de Santa Catarina
Círculo de Γ" cte
Linha com #$ = 50Ω
#") = 100 + ,75ΩzL2=2+1,5jΓ") = 0,53 ∠29,7°
Qual a impedância a 0,154 da
carga em direção ao gerador?
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.210.21
0.220.22
0.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
AN
GLE O
F TRAN
SMISSIO
N C
OEFFIC
IENT IN
DEG
REES
AN
GLE O
F REFLECTIO
N C
OEFFIC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RD L
OA
D <
—
IND
UCT
IVE
REAC
TAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE COMPONENT (-j
X/Zo), O
R INDUCTI
VE SU
SCEP
TAN
CE (-
jB/Y
o)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1'
12345681015203040dBS 1'
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 '
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] '
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P 0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
'0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 '
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
Black Magic DesignThe Complete Smith Chart
2
1,5,56
Γ =0,53
∡Γ = 45°
Instituto Federal de Santa Catarina
Círculo de Γ" cte
Linha com #$ = 50Ω
#") = 100 + ,75ΩzL2=2+1,5jΓ") = 0,53 ∠29,7°
Qual a impedância a 0,154 da
carga em direção ao gerador?
Instituto Federal de Santa Catarina
Círculo de Γ" cte
Linha com #$ = 50Ω
#") = 100 + ,75ΩzL2=2+1,5jΓ") = 0,53 ∠29,7°
Qual a impedância a 0,154 da
carga em direção ao gerador?
56$,789 = 0,66 − 0,98,
Instituto Federal de Santa Catarina
VSWR
•Razão de onda estacionária (VSWR):
! = # + %& = 1 + Γ1 − Γ
Instituto Federal de Santa Catarina
VSWR
•Razão de onda estacionária (VSWR):
! = # + %& = 1 + Γ1 − Γ
•Em x=0 e r>1, Γ* = Γ
! = # + %& = 1 + Γ1 − Γ = VSWR
Instituto Federal de Santa Catarina
VSWR
•Razão de onda estacionária (VSWR):
! = # + %& = 1 + Γ1 − Γ
•Em x=0 e r>1, Γ* = Γ
! = # + %& = 1 + Γ1 − Γ = VSWR
VSWR! Valor de r em /0 = 02 que cruza o círculo Γ* cte
Instituto Federal de Santa Catarina
Vmáx e Vmin
•VSWR = Ponto de Vmáx
•valor max de r em !" = 0%
•Ponto de Vmin a uma distância ⁄' ( de Vmáx
•valor min a ⁄' ( (!" = 180%) de Vmáx
Instituto Federal de Santa Catarina
Exercício 2
Para uma impedância de carga normalizada de
zL=0,86-j0,78 e uma linha de 1,8 ! de
comprimento. Determine:
a)Zin
b)VSWR
c)A qual distância da carga se encontra os valores de
tensão máximo e mínimo?
Instituto Federal de Santa Catarina
Vmáx e Vmin
Instituto Federal de Santa Catarina
Exercício 4
Uma LT com 0,334! de comprimento e Z0=50 Ω é
terminada em uma carga ZL=100 - 100j Ω. Utilize a
carta de Smith para determinar:
(a) Γ$(b) ROTE
(c) Zin
(d) A distância a partir da carga para o primeiro
mínimo de tensão
Instituto Federal de Santa Catarina
Exercício 4 - Resposta
Uma LT com 0,334! de comprimento e Z0=50 Ω é
terminada em uma carga ZL=100 - 100j Ω. Utilize a
carta de Smith para determinar:
(a) Γ$=0,61)*+,-.
(b) ROTE = 4,3
(c) Zin= 22,5 + j45
(d) A distância a partir da carga para o primeiro
mínimo de tensão = 0,208 !
Instituto Federal de Santa Catarina
Exercício 5
Uma LT com 5,2cm de comprimento, que utiliza um cabo coaxial
com dielétrico de ar e tem 100Ω de impedância característica, está
operando numa frequência de 750MHz e é interligada a uma
impedância de carga ZL=(30+j50)Ω. Considerando que as perdas são
desprezíveis e que a velocidade de propagação da onda no cabo é
aproximadamente igual à velocidade da luz, determine:
a) Γ$b) VSWR
c) Valor da impedância vista a uma distância de 2,0cm da carga
d) Valor da impedância vista na entrada da LT
Instituto Federal de Santa Catarina
Exercício 5 - RespostaUma LT com 5,2cm de comprimento, que utiliza um cabo coaxial com dielétrico de ar e tem 100Ω de impedância característica, está operando numa frequência de 750MHz e é interligada a uma impedância de carga ZL=(30+j50)Ω. Considerando que as perdas são desprezíveis e que a velocidade de propagação da onda no cabo é aproximadamente igual à velocidade da luz, determine:
zL= (30 + j 50) / 100 = 0,3 + j0,5
Γ"= 0,62 ∠123°
VSWR = 4,2
Valor da impedância vista a uma distância de 2,0cm da carga
λ = 40 cm.
2 cm = 0,05 λ
z= 0,5+0,93j
Z= 50+93j Ω
Valor da impedância vista na entrada da LT
5,2 cm = 0,130 λ
zin=2+2j
Zin=200+200j Ω
Instituto Federal de Santa Catarina
Figura 5 – Resposta do exercício proposto
Instituto Federal de Santa Catarina
Bibliografia
•Eletromagnetismo aplicado, abordagem antecipada
das linhas de transmissão. Stuart Wentworth
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