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```PARALLEL-PLANES APPROXIMATION
FOR PROPAGATION IN RECTANGULAR
TUNNELS
Considering, for example, only vertically polarised waves,
we shall assume that kx and ky are given by the transverse
wave numbers of two infinite slot waveguides of slot widths a
and b, respectively, one supporting a TE mode and the other a
TM mode as shown in Fig. 1.
Indexing terms: Waveguides, Propagation
EH
Natural propagation of electromagnetic waves in rectangular
tunnels is analysed using parallel lossy planes approximations with no restrictions on the electrical properties of
the walls. The analysis provides a better understanding of the
physical characteristics of the low-order modes as a function
of frequency. Numerical results are compared satisfactorily
with published measurements.
An exact analytical solution for a rectangular guide with lossy
walls is not possible because of the difficulty in matching the
boundary conditions. Approximate analysis usually makes use
of some kind of perturbation theory.
The attenuation constant a of any mode in the lossy guide
above cutoff is normally obtained using the power loss
method. 1 Below cutoff, a is given by the propagation constant
of the lossless guide.
A better approximation for the propagation constant in imperfect rectangular waveguides has been obtained assuming an
arbitrary, but small, surface impedance in the walls, which
couples the TE and TM modes, making necessary a definite
combination to satisfy the boundary conditions. 1 ' 2 The propagation constant y for the dominant (perturbed TE 10 ) mode in
a rectangular guide predicted by this 'surface impedance'
method is given by
(1)
where
a, b = width and height of the guide
k0 = free-space wave number
y0 = TE 1 0 propagation constant in the loss-free waveguide
Zs = surface impedance of the walls
In the case of road or mine tunnels eqn. 1 fails to predict the
attenuation constant at high frequencies since Zs will no
longer be small. On the other hand, at low frequencies it may
predict a too highly attenuated wave.
At high frequencies, the surrounding medium of the tunnel
will behave more like a low-loss dielectric than as an imperfectly conducting material if the electrical parameters a and e
are assumed to be frequency independent. Under this condition the attenuation rates of the waveguide modes will
depend almost entirely on refraction loss, rather than on
ohmic loss. For the dominant vertically polarised mode identified as a hybrid E H U , the attenuation constant has been
determined supposing that the height and the width of the
tunnel are much greater than the wavelength and that the
losses are small.3-4 The obtained result is
TE €
TM
0
1777777777777/
V///////////7
'-k2
c.t
2-1/2
y 1?-k
K
"Kx
0
Fig. 1 Geometry of waveguides
This 'parallel-planes' approach has the merit of being exact
when the guide walls are perfectly conducting and also yields
the approximate solutions given by eqns. 1 and 2 when the
corresponding restrictions are observed.
Applying symmetry and boundary conditions to the fields
shown in Fig. 1 the following transcendental equations for kx
and ky are found:
(4)
(5)
where
(6)
is the propagation constant of a plane wave in the lossy
medium.
Eqns. 4 and 5 provide an approximate method for finding
the propagation constant in a rectangular tunnel without any
restriction on size, frequency or electrical properties of the
walls.
At very low frequencies it can be shown that kx —> ym and
ky—*• 0. Therefore y—> ym, which means that the attenuation in
the tunnel will be the same as through the lossy medium, and
the propagating mode will have no variations in the transverse
section.
If now the conditions |y m | P | y t | and \ym\ P \y2\ hold,
eqns. 4 and 5 lead to the same propagation constant predicted
by the surface impedance method (eqn. 1), and the propagating mode will be a hybrid mode similar to the TE 1 0 mode in a
perfectly conducting guide.
At high frequencies, where the lossy medium will behave as
a low-loss dielectric we can assume that yf. « y2. ~ —founder these conditions eqns. 4 and 5 lead to the same attenuation constant given by eqn. 2, and the propagating mode will
be similar to a hybrid E ^ x mode.
In order to test the validity of the proposed method, numerical solutions of eqns. 4 and 5 were obtained using the
parameters of two tunnels analysed by Deryck5 and assuming
that e = 10e0. The calculated results for the attenuation constant as a function of frequency for a concrete and a rock
tunnel are shown in Figs. 2 and 3, where it can be seen a
better agreement with experimental results than that obtained
(er - I) 1 ' 2
where er is the complex relative permittivity of the walls (er =
[e — 7'(ff/w)]/e0]) and X is the free-space wavelength.
The restrictions inherent in the above methods may preclude the application of eqns. 1 and 2 in a real tunnel of
general characteristics when natural propagation is considered
in a wide frequency range.
In this letter we propose a simple approach in order to
determine the propagation constant in a lossy rectangular
tunnel in the most general case.
A general EH mode in a rectangular guide will have a
propagation constant y given by
(3)
where kx and kv are complex transverse wave numbers.
ELECTRONICS LETTERS 11th November 1982
Vol.18
f r e q u e n c y , MHz
B5Z771
Fig. 2 Attenuation against frequency in a concrete tunnel
I7m,b = 4-9 m, a - 0-1 s/m, e = l(te0
O experimental
eqns. 4 and 5
- eqn. 1
- - eqn. 2
No. 23
1011
by Deryck using the power loss method. The approximate
solutions for a obtained from eqns. 1, 2 and 6 are also included in Figs. 2 and 3, where their range of validity is clearly
appreciated.
equatorial and tropical climates is derived from an indirect
method by using measurements of attenuation due to rain in
the People's Republic of Congo (equatorial and tropical
climates). Attenuations calculated from this distribution are
compared with experimental data gathered on the Ivory Coast
(tropical climate) at a different frequency.
Raindrop size distribution: The size distribution of raindrops
depends on precipitation rate R (mm/h), and is generally represented by an exponential function:1
n(a) = No exp (—
102
frequency, MHz
Fig. 3 Attenuation against frequency in a rock tunnel
d = 4m, b = 5 m, a = 0-01 s/m, e •= 10eo
O experimental
eqns. 4 and 5
eqn. 1
eqn. 2
eqn. 6
Basic integral equation: Considering single scattering, the propagation constant Kv H for vertical or horizontal polarisation
is given by the Van de Hulst equation: 1
+ 00
I
Acknowledgments: This work was supported by the University
of Chile under grant DDI 1448 8213.
12th October 1982
References
1 COLLIN, R. E.: 'Field theory of guided waves' (McGraw-Hill, 1960)
2 LEWIN, L. : 'Theory of waveguides' (Halsted Press, 1975)
3
EMSLIE, A. c , LAGACE, R. L., and STRONG, P. F.: 'Theory of the
propagation of UHF radio waves in coal mine tunnels', IEEE
Trans., 1975, AP-23, pp. 192-205
4
(1)
where a is the equivolumetric radius of the oblate spheroidal
drop, and No (cm" 4 ) and A (cm" 1 ) parameters, the value of
which vary, respectively, with rainfall rates and with regional
characteristics of raindrops.
W3
B. JACARD
A. VALENZUELA
Department of Electrical Engineering
University of Chile
PO Box 5037, Santiago, Chile
= A o i?
LAAKMANN, K. D., and STEIER, w. H.: 'Characteristic modes of
hollow rectangular dielectric waveguides', Appl. Opt., 1976, 5, pp.
1334-1340
5 DERYCK, L.: 'Natural propagation of electromagnetic waves in tunnels', IEEE Trans., 1978, VT-27, pp. 145-150
(2)
where n(a) d(a) is the number per unit volume (cm3) having
radius between a and a + da, k0 (cm" 1 ) is the propagation
constant in free space, and fViH(6, <x0, a) is the forward scattering amplitude (with 6 = 0° and a 0 = 90° for terrestrial
links) of an oblate spheroidal raindrop for vertical or horizontal polarisation, respectively.
Attenuation and phase shift: Using eqns. 2 and 1, one may
calculate the attenuation A (dB/km) and phase shift <p
(deg/km) of the electromagnetic wave after passing through
the rainy medium for vertical or horizontal polarisation as 1
AY
H
= 8-686 Im (KVH - k0) x 105
<pv<H = (180/TT)
Re'(KVJ1-k0)
x 10
dB/km
5
deg/km
(3)
These two quantities can be determined experimentally.
Then, setting
FvA",
0013-5194/82/231011-02\$ 1.50/0
da
f 8-686 x X x \mfVH
5
A) = 10 x j ( 1 8 O / 7 r ) x k x R e ^ H
for attenuation
for p h a s e
shift
(4)
with X = 2n/k0, one can write
+ OO
J
RAINDROP SIZE DISTRIBUTION FROM
MICROWAVE SCATTERING
MEASUREMENTS IN EQUATORIAL AND
TROPICAL CLIMATES
= No
Indexing terms: Radiowave propagation, Microwave measurements, Rain attenuation
A raindrop size time averaged distribution is inferred from
an indirect method by using measurements of attenuation
due to rain in Congo (equatorial and tropical climates).
Attenuations calculated from this distribution are compared
with experimental data gathered on the Ivory Coast (tropical
climate) at a different frequency.
Introduction: The influence of rainfall on radiowave propagation through the atmosphere has been extensively investigated
in European, North American and Japanese hydrometeorological areas, but very little is known for African
equatorial and tropical zones.
In this letter, a raindrop size time averaged distribution for
1012
FVH(a, X) exp ( - Ao R ~ pa) da
(5)
where ^/Rv H are measured data (attenuation or phase shift for
each given rainfall data).
From eqn. 5, the three parameters No, A o and /? may be
determined. This has been done by fitting with a nonlinear
minimum-least-square method the experimental data on rain
attenuation gathered by Moupfouma 2 on a 33-5 km, 7 GHz
during 16 months at Brazzaville (Congo).
We obtain the following expressions:
n(a) = 0-052e~Aa (cm" 4 ),
A = 70-4R- 0 2 3 (cm" 1 )
(6a)
or, alternatively, using the equivolumetric diameter of the
drops,
n{D) = 0026e " AD (cm " 4 ),
A = 35-2R
ELECTRONICS LETTERS 11 th November 1982 Vol.18
No. 23
```
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