Graphics Programs Reference
In-Depth Information
p
a
ó
R
i
=
r
i
(
x
i
,,
y
i
z
i
)
a
ó
a
ó
r
a
ó
r
i
R
1
=
r
θ
1
(
000
,,
)
a
ó
a
ó
a
ó
θ
1
r
1
a
ó
a
ó
(
x
1
,
y
1
,
z
1
)
Figure 8.3 Geometry for an array antenna.
x
i
2
y
i
2
z
i
2
(
++
)
r
i
r
2
--------------------------------
=
------
«
1
(8.20)
r
2
Thus, a good approximation (using binomial expansion) for Eq. (8.19) is
R
i
=
r
x
i
θφ
(
sin
cos
+
y
i
θφ
sin
sin
+
z
i
cos
θ
)
(8.21)
It follows that the phase contribution at the far field point from the
ith
radiator
with respect to the phase reference is
jkR
i
e
jk x
i
θφ
(
sin
cos
+
y
i
θφ
sin
sin
+
z
i
cos
θ
)
jkr
e
=
e
(8.22)
Remember, however, that the unit vector
r
0
along the vector
r
is
r
r
ó
x
θφ
ó
y
θφ
ó
z
r
0
==
-
--
--
sin
cos
+
sin
sin
+
cos
θ
(8.23)
Hence, we can rewrite Eq. (8.22) as
jkR
i
e
j
Ψ
i
θφ
(
,
)
e
jk r
i
(
•
r
0
)
jkr
jkr
e
=
e
=
e
(8.24)
Finally, by virtue of superposition, the total electric field is
N
∑
I
i
e
j
Ψ
i
θφ
(
,
)
E
θφ
(
,
)
=
(8.25)
i
=
1
which is known as the array factor for an array antenna where the complex cur-
rent for the
ith
element is
I
i
.
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