36 Smith Chart and VSWR
•
Consider the general phasor expressions
V
(
d
) =
V
+
e
j
β
d
(1 +
Γ
L
e

j
2
β
d
)
and
I
(
d
) =
V
+
e
j
β
d
(1

Γ
L
e

j
2
β
d
)
Z
o
describing the voltage and current variations on TL’s in sinusoidal
steadystate.
+

Wire 2
Wire 1
+

0
F
=
V
g
Z
g
Z
L
I
(
d
)
V
(
d
)
l
Transmission line
Load
Z
o
Generator
d
d
max

V
(
d
)

d
min

V
(
d
)

min

V
(
d
)

max
.2
.5
1
2
r 5
x
5
2
1
.5
.2
x 5
2
1
.5
.2
0
VSWR
SmithChart
1
Γ
(
d
)
1 +
Γ
(
d
)

1 +
Γ
(
d
)

maximizes for
d
=
d
max
Γ
(
d
max
) =

Γ
L


1 +
Γ
(
d
)

minimizes for
d
=
d
min
such that
Γ
(
d
min
) =

Γ
(
d
max
)
Complex addition displayed
graphically superposed on a
Smith Chart
z
(
d
max
)
=VSWR
–
Unless
Γ
L
= 0
, these phasors contain reflected components, which
means that voltage and current variations on the line “contain”
standing waves.
In that case the phasors go through cycles of magnitude variations as a
function of
d
, and in the voltage magnitude in particular (see margin)
varying as

V
(
d
)

=

V
+

1 +
Γ
L
e

j
2
β
d

=

V
+

1 +
Γ
(
d
)

takes maximum and minimum values of

V
(
d
)

max
=

V
+

(1 +

Γ
L

)
and

V
(
d
)

min
=

V
+

(1


Γ
L

)
at locations
d
=
d
max
and
d
min
such that
Γ
(
d
max
) =
Γ
L
e

j
2
β
d
max
=

Γ
L

and
Γ
(
d
min
) =
Γ
L
e

j
2
β
d
min
=


Γ
L

,
and
d
max

d
min
is an odd multiple of
λ
4
.
1
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–
These results can be most easily understood and verified graphi
cally on a SC as shown in the margin.
+

Wire 2
Wire 1
+

0
F
=
V
g
Z
g
Z
L
I
(
d
)
V
(
d
)
l
Transmission line
Load
Z
o
Generator
d
d
max

V
(
d
)

d
min

V
(
d
)

min

V
(
d
)

max
.2
.5
1
2
r 5
x
5
2
1
.5
.2
x 5
2
1
.5
.2
0
VSWR
SmithChart
1
Γ
(
d
)
1 +
Γ
(
d
)

1 +
Γ
(
d
)

maximizes for
d
=
d
max
Γ
(
d
max
) =

Γ
L


1 +
Γ
(
d
)

minimizes for
d
=
d
min
such that
Γ
(
d
min
) =

Γ
(
d
max
)
Complex addition displayed
graphically superposed on a
Smith Chart
z
(
d
max
)
=VSWR
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 Summer '10
 KUDEKI
 Volt, Complex number, Impedance matching, Standing wave ratio, Standing wave, Reflection coefficient, Antenna tuner

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