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To find the coecient
F
lm
of the TE expansion (1) and (2) are equated and
then the orthogonality property of the TE modes is applied.
F
lm
is given by the
following equation.
(3)
Where
J
(
k
3
r
)=
k
3
rj
l
(
k
3
r
)
and
j
l
(
k
3
r
)
[14, ch.6-1] is the spherical Bessel func-
=
√
−
1
tion,
J
l
,
μ
0
is the permeability of free space,
3
is the relative permittivity of region (3) and
Y
lm
(
is the derivative w.r.t
r
,
i
θ, ϕ
)
is the spherical harmonic
with
l
and
m
as the indices of the expansion. The wavenumber
k
3
is given by:
V
ZPHH
ZH
i
k
(
3
)
3
0
0
3
0
(4)
Where
ε
0
is the permittivity of free space,
ω
=2
πf
is the frequency in radian/sec
(
)and
σ
3
is the conductivity of region (3).
Beginning with the field in region (3) as the source, one can use the boundary
conditions presented in (5)-(10) to solve for the current density of each antenna
element given by (8).
f=500kHz
r
=
c
r
=
b
r
=
a
r
=
c
r
=
b
r
=
a
Where
E
j
represent the magnetic intensity (H-field) and electric field
in the region (j) and
H
j
and
J
i
is the current density of the i-th antenna element.
By solving the BVP by using equation (2) and (5)-(10), one can derive the
expression of the current density induced at the ith antenna. After determining
the current density it is very easy to determine the current signal induced at the
ith antenna and this is given by (11).
(11)
Where:
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