ECE 450
Homework 4
Due: Tue, Sept 22, 2009, 5 PM
1. In class notes it is given that the e
ff
ective length of a
ˆ
z
directed halfwave dipole antenna is
(
θ
) =
λ
π
cos(
π
2
cos
θ
)
sin
2
θ
.
a) Use l’Hospital’s rule to determine
(0)
.
b) Plot
(
θ
)
as a function of
θ
from 0 to 180 degrees.
c) Plot
(
θ
) sin
θ
as a function of
θ
from 0 to 180 degrees.
2. Suppose an antenna has a gain function
G
(
θ
,
φ
) =
D
for
0
<
θ
<
π
2
,
D
sin
2
θ
for
π
2
<
θ
<
π
.
Determine
D
such that the solid angle integral of
G
(
θ
,
φ
)
equals
4
π
.
3. Three identical
ˆ
z
directed short dipoles are located at
(
x, y, z
) = (

d,
0
,
0)
,
(0
,
0
,
0)
, and (
d,
0
,
0)
and
have input current phasors of
1
∠
0
◦
A,
2
∠
0
◦
A, and
1
∠
0
◦
A, respectively.
Show that the radiation field of the array of dipoles at an arbitrary location on
xy
plane (i.e.,
θ
= 90
◦
plane) su
ffi
ciently far away from the origin (so that paraxial approximation can be used) is proportional
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 Fall '09
 FRANKE
 180 degrees, antenna array, Dipole antenna, input current phasors, kd cos

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