It should be noted that the model problem of exciting the conical slotted
antenna manufactured of material with high conductivity by a linear element of electric
current is reduced to the boundary problem considering in this case [2,9]. Taking into
account continuity of the potential (2) in slots we come to such representation for
function
()
1
,
mi
U
(3):
()
(
)
(
)
(
)
1
(1)
1/2
,,
1/2
(cos)
()
,
(cos)
nN
in
N
i
mi
mn
nN
n
i
P
U
x
e
P
+
+
+
−+
+
=−
−+
=
=
(5)
where the upper signs in arguments of Legendre function correspond to the area
0
and the lower areas to
. Using boundary condition (4) and condition
of a continuity of partial derivative of potential (2) in slots leads to such system of the
functional equations with respect to
,
mn
x
:
0
(1)
,
imN
inN
mn
n
x
e
e
+
=−
=
, :
S
dl
N
;
(6)
(
)
(1)
(1)
,,
(
)
1
0
inN
mn
mn
n
n
Nn
x
e
n
+
=−
+
−
=
,
:
CS
d
N
l
;
(7)
(
)
(
)
(
)
(
)
(
)
()
(1)
12
12
||
(1)
(1/2
(
))
(
)
(1
)
(1/2
(
))
1
.
cos
cos
nN
n
n
N
n
N
ii
n
ch
i
n
N
Nn
n
i
n
N
PP
+
++
−
+
−
+
−
++
+
+
−
=
+−+
−
(8)
For matrix coefficients
(1)
,
mn
(8) such estimation takes place at
(
)
1
Nn
+
:
2
(1)
,
22
sin
()
mn
O
Nn
=
+
.
We introduce into consideration the coefficients
,
mn
y
related with required
coefficients
,
mn
x
:
(
)
(1)
(1)
(1)
,
,
,
(
)
1
mn
mn
mn
n
y
Nn
x
n
=
+
−
, (9)
Then the system of the equations (6), (7) can be represented in the form
(
)
0
(1)
(1)
,,
1
1
()
im
in
mn
mn
n
n
y
e
e
Nn
n
+
=−
−=
+
,
:
S
dl
, (10)
(1)
,
0
in
mn
n
ye
+
=−
=,
:
d
CS
l
, (11)
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