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p
R
1
(
x
1
,
y
1
,
z
1
)
r
r
1
θ
1
r
r
-
--
--
d
1
=
r
1
•
=
r
1
cos
θ
1
(
000
,,
)
d
1
Figure 8.2 Geometry for an array antenna.
Single element
Fig. 8.2 shows the geometrical fundamentals associated with this problem.
In general, consider the radiation source located at
(
x
1
,
y
1
,
z
1
)
with respect to a
phase reference at
(
000
,,
)
. The electric field measured at far field point
P
is
jkR
1
I
0
e
E
θφ
(
,
)
=
-------------
f
θφ
(
,
)
(8.16)
R
1
where
I
0
is the complex amplitude,
k
=
2πλ
⁄
is the wave number, and
f
θφ
(
,
)
is the radiation pattern.
Now, consider the case where the radiation source is an array made of many
elements, as shown in
Fig. 8.3.
The coordinates of each radiator with respect to
the phase reference is
(
x
i
,,
y
i
z
i
)
, and the vector from the origin to the
ith
ele-
ment is given by
a
ó
a
ó
a
ó
z
i
r
i
=
x
i
++
y
i
(8.17)
The far field components that constitute the total electric field are
jkR
i
e
------------
f
θ
i
E
i
θφ
(
,
)
=
I
i
(
,
φ
i
)
(8.18)
R
i
where
)
2
)
2
)
2
R
i
=
R
i
=
r
i
=
(
xx
i
+
(
y
+
(
z
i
i
(8.19)
2
2
++
2
r
2
r
2
=
r
1
+
(
x
i
y
i
z
i
)
⁄
2
xx
i
(
++
yy
i
zz
i
)
⁄
Using spherical coordinates, where
xr
θϕ
=
sin
cos
,
yr
θϕ
=
sin
sin
, and
z
=
r
cos
θ
yields
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