ECE 329
Homework 12
Due: Tue, April 15, 2008, 5PM
1.
a)
Smith Chart derivation:
Since
Γ =
z

1
z
+1
=
r
+
jx

1
r
+
jx
+1
=
[(
r

1) +
jx
][(
r
+ 1)

jx
]
(
r
+ 1)
2
+
x
2
=
(
r
2
+
x
2

1) +
j
2
x
(
r
+ 1)
2
+
x
2
≡
Γ
r
+
j
Γ
i
,
it follows that
Γ
r
=
(
r
2
+
x
2

1)
(
r
+ 1)
2
+
x
2
and
Γ
i
=
2
x
(
r
+ 1)
2
+
x
2
,
where
r
and
x
are normalized line resistance and reactance, respectively, and
Γ
r
and
Γ
i
denote
the real and imaginary parts of reFection coe±cient
Γ
.
Using the above expressions for
Γ
r
and
Γ
i
, verify that the following relations are valid:
(Γ
r

r
r
+1
)
2
+ Γ
2
i
=(
1
r
+1
)
2
constant
r
circles
(Γ
r

1)
2
+ (Γ
i

1
x
)
2
=(
1
x
)
2
constant
x
circles
b)
Smith Chart construction:
The equations veri²ed above, describe circles on the complex
Γ
plane with
r
and
x
dependent centers and radii, respectively. Using a compass, draw constant
r
circles (on a plane with
Γ
r
and
Γ
i
axes) for the values of
r
=0
, 1, 2,
∞
(this one, you can
draw without the compass!) — set the scales of your axes such that your circles comfortably
²ll a 8.5”X11” page, and mark each circle by the
r
value. Also draw and mark portions of
constant
x
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 Spring '08
 FRANKE
 Complex number, Transmission line, Impedance matching, Standing wave

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