Biomedical Engineering Reference
In-Depth Information
2
2
u
a
¢
w
Ê
Á
ˆ
˜
=
Ê
Ë
ˆ
¯
2
=
()
=
11
,
u
¢=
1 841
.
u
u
3 39
.
b
e
(6.6)
2
11
,
1
11
,
c
1. In the previous equations, the subscript
z
denotes
field components in axial direction, and the transverse field components are
decomposed along the two orthogonal directions,
r
and
t
s
, which represent the
radial and tangential directions, respectively.
The notation applied to a circular waveguide is shown in Figure 6.3
a
.For
any mode of transmission in a circular waveguide, there may be axial field
components (
z
), and there are the transverse field components, which may be
resolved into two directions, tangential (q) and radial (
r
). All of these compo-
nents vary periodically along a circular path concentric with the wall and vary
in a manner related to a Bessel function of order
m
along a radius. Any par-
ticular mode is identified by the notation TE
m
,
n
or TM
m
,
n
, where
m
is the order
of the Bessel function representing the field variation and
n
represents the
n
th
zero of
J
n
(·) (for TM) and
J
For water, e =
80; for air, e =
¢
n
(·) (for TE). The cutoff wavelength in a circular
waveguide for all modes depends upon the ratio of the diameter to the wave-
length. For the TE
m
,
n
wave, the cutoff wavelength is given by the formula
2
p
a
l
=
(6.7)
c
u
¢
mn
,
where
a
is the radius of the guide. The constant
u
¢
m
,
n
is the
n
th root of the equa-
tion
J
¢
(
u
)
=
0. Some of the lower values of
u
¢
m
,
n
are
u
¢=
3 832
.
u
¢=
7 016
.
01
,
02
,
u
¢=
1 841
.
u
¢=
5 332
.
11
,
12
,
u
¢=
3 054
.
u
¢=
6 706
.
21
,
22
,
u
¢=
4 201
.
u
¢=
8 016
.
31
,
32
,
For the mode TE
11
2
1 841
p
a
l
l
=
=
341
.
a
l
=
(6.8)
c
g
.
2
-(
)
1
ll
c
where l
g
is the wavelength of an air-filled hollow pipe and l is the wavelength
of a plane wave in a dielectric medium with e
80.
Similarly, the cutoff wavelength of a TM
m
,
n
wave is given by
2
p
a
l
=
(6.9)
c
u
mn
,
Some of the lower values of
u
m
,
n
are given by
u
=
2 405
.
u
=
5 520
.
01
,
02
,
u
=
3 832
.
u
=
7 016
.
11
,
12
,
u
=
5 135
.
u
=
8 417
.
12
,
32
,
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