ECE 329
Homework 13
Due: Thu, Dec 4, 2008, 5PM
1.
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 reflection coe
ffi
cient
Γ
. Using the above expressions for
Γ
r
and
Γ
i
, it can
be shown 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
a)
Smith Chart construction:
The equations 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,
∞
— set the scales of your axes such
that your circles comfortably fill 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, Impedance matching, ΓR, S.C., characteristic impedance Zo

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