# Equation Chapter 1 Section 1Simple Closed Form Solutions to ...

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18 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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1
?
S
IMPLE
C
LOSED
F
ORM
S
OLUTIONS
TO
S
CATTERING AND
R
S
LOTS WITH
A
PPLICATIO
N TO
W
AVEGUIDE
F
ED
S
LOT
S

Mostafa N. Issmaiel Fahmy

Essam A. Hashish

Cairo University

Faculty of Engineering

Department of Electronics and Electrical Communications

Abstract

Based on the recently introduced equivalent TEM mode theo
ry and the
relationship between slot antennas and complementary conducting strips, the main
slot
s

in large conducting screen and
generally
oriented slots on broad and narrow walls of rectangular guides have been deter
mined.
Thus, simple closed form expressions are given for radiation fields of
slots in large
conducting screen and
waveguide fed
slots of arbitrary length and orientation. The power
waveguide fed
slot is also determined as percentage of power c
arried by
propagating mode inside the guide.

1.

Introduction

The equivalent TEM mode theory has been recently introduced, together with
some applications to control scattering from conducting wires [1,2,3,4]. This theory is
simply cast in sentence
"inasmuc
h as the current distribution on the thin antenna is
concerned, the radiation process can generally be adequately and justifiably accounted for
by some transmission line model supporting a slightly lossy '
equivalent TEM mode
',
where the radiated power is j
ust equal to the lost power" and has yielded, besides an
elegant physical picture, simple closed form expressions for radiation and scattering by
short circuited multiply loaded or multiply excited wires. It goes without saying that
problems of conducting
strips with equivalent width [5,6] are immediately solved.
Moreover, in view of the duality between electric and magnetic conductors and the
subsequent relationship between slot antennas and complementary conducting strips [6,7]
simple closed form expressi
ons for slot antennas in large conducting screen may readily
be obtained. These solutions are so simple in comparison with numerical methods based
on integral equations [8]. Slots in waveguides exist in finite rather than infinite
conducting screen (with l
arge extension parallel to axis of guide) and hence they may be
treated with a view to obtain approximate or pseudo exact solutions. Comparison with
available previously published as well as quite recent results
for waveguide walls of zero
thickness
confir
med the achievement of this goal [9]. It is interesting to mention that

2
?
slotted waveguides are currently gaining new impetus in view of intended applications as
efficient feeders of dielectric resonator antennas [10]. The radiated power can be
evaluated by

integrating the radiated power density over the entire half space outside the
guide or more simply by evaluating half the input power for which closed form solutions
are easily obtained.

2.

Formulations for Voltage or Magnetic Current Distribution

2.1.

Slots i
n
L
arge
C
onducting
S
creen

2.1.1.

Slot Fed at Arbitrary Point

Fig. (1) shows
a
slot
of length L
along z direction in
a large conducting screen
with localized current source excitation at the feeding point
, which generates
tangential
magnetic field
.

The complementary conducting strip is equivalent to a
conducting cylinder of radius equal to one quarter of strip width
[5, 6]
. The magnetic field
excitation is equivalent to electric field excitation on the complementary strip, the value
of
this electric field is related to original magnetic field by the relation
[6,7]. The
fictitious electric transmission line differential equations for
V
e
, I
e

take the form

(
1
)

where

Z
,
Y

are the distributed impedance and admittance of the
fictitious
line and

is the
free space wave impedance
,

characteristic impedance of the
fictitious line
.

Solving for the electric current, one gets

(
2
)

w
here
l=L/2

is the slot half length, and

is the complex propagation constant
.

Now in view of the relationship between the voltage or magnetic current
distribution on the slot and the electric current on the complementary stri
p [6, 7], namely

3
?
,

(
3
)

t
he voltage or magnetic current distribution on the slot may be expressed by

(
4
)

immediately
be found by evaluating the vector
potential
F

and using

(
5
)

2.1.2.

Slot Scatterer

Consider a receiving slot in an infinite conducting screen subjected to incident
plane wave with mag
netic field incident in the right half space as shown in fig
.

(2). The
fictitious transmission line differential equations for complementary strip take the form

(
6
)

Solving for the current, one gets

(
7
)

The voltage or

magnetic current distribution on the waveguide slot may therefore be
expressed by

4
?

(
8
)

The
bistatic
scattered
magnetic field
H

s

in the left half space is therefore given by

(
9
)

2.2.

Slot in Broad Wall of Rectangular Wavegui
de

Fig. (
3
) shows a slot with a general orientation in the broad wall of rectangular
waveguide, the axis of the slot is chosen to be the z axis while the axis of the waveguide
is chosen to be z' axis. Assuming dominant mode propagation, the component of t
he
magnetic field along slot axis may be given by

(
10
)

where
z

is the distance measured from slot center and
H
0

is the maximum
amplitude of
H
z'
. The fictitious electric transmission line differential equations for
V
e
, I
e

take the form

(
11
)

5
?

Solving for the electric current, one gets

(
12
)

T
he voltage or magnetic current distribution
on the waveguide slot may be expressed by

(
13
)

follows and is given by

6
?

(
14
)

where

In the special case of longitudinal slot equations
(
13
)

and
(
14
)

reduce to

(
15
)

7
?

(
16
)

Similarly, in the special case of trans
versal slot equations
(
13
)

and
(
14
)

reduce to

(
17
)

(
18
)

It is worthwhile to notice that the voltage distribution on the longitudinal and
transversal slots (equations
(
15
)
,
(
17
)
) as well as the resulting radiated fields (equations
(
16
)
,
(
18
)
) comprise, each of them, just three terms. The third term in each of equations
(
15
)
,
(
17
)

bears direct correspondence to the exciting field, while the first two terms in
each equation constitute standing waves with complex propagation constant.

2.3.

Slot in Narrow Wall of Rectangular Waveguide

Fig. (
4
) shows a slot with a general orien
tation in the narrow wall of rectangular
waveguide, in this case the component of the magnetic field along slot axis may be given
by

8
?

(
19
)

The fictitious transmission line differential equations for complementary strip
take the form

(
20
)

Solving for the current, one gets

(
21
)

The voltage or magnetic current distribution
on the waveguide
-
fed
slot may therefore be
expressed by

(
22
)

The radiated magnetic field immediately follows an
d is given by

(
23
)

where

9
?

In case of longitudinal slot equations
(
22
)

and
(
23
)

reduce to

(
24
)

(
25
)

3.

Attenuation in Dominant Mode

In view

of the fact that the power radiated by the slot outside the guide is
necessarily extracted from the power of the propagating mode inside the guide, it is
necessary to assign an appropriate attenuation constant to the dominant mode when
passing nearby the

Thus, attenuation of the propagating mode has been assumed to take place
through a length equal to slot length for all slot inclination angles. While this assumption
is clearly and easily justifiable for longitudinal slots, it may be argue
d that the case of
transversal slot is not much different. Thus in view of fig (
5
), it is clear that the dominant
mode starts losing power well in the front of the “narrow” slot and that this loss exists
well behind the slot. Nevertheless, calculations pro
ved that results are almost insensitive
to reasonable variations of the assumed length

provided that the
assumed
length is not
less than 0.8 L
.

This attenuation may be effected by introducing an imaginary part in

g

in the
expressions of magnetic curren
t distribution and hence in evaluating input power to slot.

We thus have

g

=

'
g

-
j

with

'
g

equal to the old phase constant. Now the power lost
due to this attenuation may be given by the following equation

(
26
)

with the power propagating inside waveguide being given by

(
27
)

10
?
The input power to slot may be take the form

(
28
)

where
H
z
(z)

is the component of the magnetic field along slot axis. Notice that the
magnetic current is multiplied by 2 due to image effect and that the input power to the
slot, given by
(
28
)
, equals twice the power radiated by the guide.

By equating the input power to the slot given by equation
(
28
)

and power lost due
to attenua
tion, one gets the value of

.

4.

Percentage Power Radiated by Waveguide Fed Slots

4.1.

Using equation
(
28
)

and introducing the complex propagation constant

g

=

'
g

-
j

the radiated power may be expressed by

(
29
)

where

11
?

In the special case of longitudinal slot equation
(
29
)

reduce to

(
30
)

Similarly, in the

special case of transversal slot equation
(
29
)

reduce to

(
31
)

12
?
The factor
e
-
2

l

is added due to the fact that the magnetic field attenuation starts at
z' =
-
l

not at
z'=0
.

4.2.

Slots in Narrow Wall

Using equation
(
28
)

and introducing the complex propagation constant

g

=

'
g

-
j

the

radiated power may be expressed by

(
32
)

Percentage radiated power is easily obtained by dividing equations
(
29
)
,
(
30
)
,
(
31
)
,
and
(
32
)

by equation
(
27
)
.

5.

Results

5.1.

Radiation and Scattering from Slot in Large Conducting
Screen

Figures (6), (7)
, (8), and (9)

depict the magnetic current distribution on radiating and
scattering slot in large conducting screen
of length 0.5
-

with length to width of slot L/w
= 200.

Calculations using Pocklington's integral equation [8] are also depicted in figures
(6) and (7) and excellent agreement with results of closed form solutions is evident.

5.2.

Slots

Figures (
10
), and (
11
) depict the magnetic current distribution and radiated fields
from longitudinal slot in the broad wall of standard x
-
band waveguide of length 0.5
-

with length to width of slot L/w = 200. Figure (
12
) depict the magnetic current

distribution on longitudinal slot in the broad wall of standard X
-
band waveguide of
length 1
-

with length to width of slot L/w = 200. The results are compared to

(i)

Moment method solution of Pocklington equation [8].

(ii)

Recent results communicated by Isslam
Greens functions and moment method for the same L/w.

13
?
It may be stated that good agreement is manifested in both cases.

5.3.

Percentage Power Radiated by Waveguide Fed Slots

Figures (
13
), (
1
4
), and (
1
5
) depict the percentage radiated po
wer by the slot as
calculated using equations

(
30
)
,
(
31
)
, and
(
32
)
. The results are in good agreement with
comparison to percentage radiated power calculated using conductance and resistance
formulas given in [11,15]
. It should be noted that in figure (14) goo
d agreement is
noticed only when the slot is transverse (
θ
s
=90º)
,

this is because t
he given formulas
assume that the voltage or magnetic current distribution is symmetrical. This assumption
is
evident
for a transverse slot
which
is excited by the symmetrical x'
-
component of
propagating mode. However, for an incl
ined slot the
magnetic field

excitation
is not
symmetrical and the previous assumption is no longer val
id.
Figures (16) through (1
8
)
depict the percentage radiated power with the attenuation length equal to
1.5
L
.

6.

Conclusion

Simple closed form expression
s have been given for

(i)

Voltage or magnetic current distribution on and radiation or
scattering from slots in large conducting screen.

(ii)

Voltage or magnetic current distribution on and radiation from
waveguide fed slots.

(iii)

de fed slots.

(iv)

It is worthwhile to point out that closed form solutions have also
been obtained by the authors for
internal scattering of waveguide
slots with subsequent determination of resonant slot length [11].

The results of closed form solutions revea
led excellent agreement with results of
sophisticated numerical techniques.

References

[1]
M. N. I. Fahmy and I. A. Eshrah, "Pseudo
-
exact simple closed
-
form solutions for
s of the
19
th

National Radio Science Conference (NRSC), URSI
-
IEEE, Egypt, March 2002, pp.
109
-
124.

[2] M. N. I. Fahmy and I. A. Eshrah, "
Antenna and scatterer modes for thin straight wires
using the equivalent TEM mode theory
," IEEE Antennas and Propagatio
n Society
International Symposium, Ohio, USA, Volume: 4, June 22
-
27, 2003 pp. 299

302.

[3] Eshrah, I.A., “The equivalent TEM mode theory for simple closed
-
form solutions to
problems of null steering and elimination of backscattering from straight wires”,
M.Sc.
thesis, Cairo University, Giza,

Egypt
, 2002.

14
?
[4] M. N. I. Fahmy and I. A. Eshrah, "
Control of the radiation and scattering from
straight wires using the equivalent TEM mode theory
," IEEE Antennas and Propagation
Society International Symposium, Ohi
o, USA, Volume: 4, June 22
-
27, 2003 pp. 303

306.

[5] R. W. P King, Theory of Linear Antennas, Harvard University Press, 1956.

[6] R. W. P King and C. W. Harrison, Antennas and Waves, M. I. T. Press, 1969.

[7] G. H. Owyang and R. King, “Complementarity in
the study of transmission lines,”
IRE Trans., MTT
-
8, 1960, pp. 172
-
181.

[8]
A. Hadidi and M. Hamid, “A novel treatment of Pocklington equation applied to slot
apertures,”
IEEE Transactions on Antennas and Propagation, Vol. 37, No. 9, Sep. 1989,
pp. 1124
-
1
128.

[9] I. A. Eshrah, University of Mississippi USA, on leave from Cairo University, Private
communications.

[10] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, A.W. Glisson, “
Dielectric resonator
antenna excited by waveguide slot for radar applications,
” IEE
E Antennas and
Propagation Society International Symposium, Ohio, USA, Volume: 4, June 22
-
27, 2003
pp. 492

495.

15
?

Z

z = z
f

z = 0

Z

k
i

H
i

H
i

(0)

i

Fig. 1
S
lot
ant
enna
of
l
ength
L
in large conducting screen.

Fig.
2

S
lot
scatterer
of
l
ength
L
in large conducting screen.

16
?

L = 2l

Assumed region of power extraction

Fig.
3

A slot
of Length
L

in the top wall of waveguide.

Fig.
4

A slot of Length
L

in the

narrow wall of waveguide.

L

17
?

(a)

(b)

Fig.
6

Current distribution of 0.5
-

transmitting
slot

β/k

=1.07
-
j0.09,
L/w

=200,
Z
0

= 245.66

.

(a) current magnitude, (b) current phase

(a)

(b)

Fig. 7 Current distribution of 0.5
-

receiving slot

β/k

=1.07
-
j0.09,
L/w

=200,
Z
0

=
245.66

.

(a) current magnitude, (b) current phase

Fig.
9

Scattered

field
pattern corresponding to
figure 7

pattern corresponding to
figure 6

18
?

Fig.
10

Current distribution of 0.5
-

slot

s
=90

x
0
=a/2, H
0

= 1 A/m
,

β/k

=1.07
-
j0.09,
L/w

=200,
Z
0

= 245.66

.

(a) current ma
gnitude, (b) current phase

(a)

(b)

Fig.
12

Current distribution (magnitude) of 1
-

slot

s
=0

x
0
=0.4 a, H
0

= 1 A/m

β/k

=1.0
38
-
j0.
0
61
,
L/w

=200,
Z
0

= 245.66

.

Fig.
11

ttern of 0.5
-

slot

s
=90

x
0
=a/2, H
0

= 1 A/m

β/k

=1.07
-
j0.09,
L/w

=200,
Z
0

= 245.66

.

19
?

Fig.
13

by longitudinal slot in broad wall

Fig.
1
4

by inclined slot in broad wall

Fig.
1
5

by inclined slot in narrow wall

by longit

(Assumed attenuation length =
1
.5
L
)

by inclined slot in broad wall

(Assumed attenuation length =
1
.5
L
)

by inclined slot in narrow wall

(Assumed attenuation length = 1.5
L
)

This region is

physically unrealizabl
e

This region is

physically unrealizable