Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate
Microstrip antennas are useful as antennas mounted on moving vehicles such as cars, planes, rockets, or satellites, because of their small size, light weight and low profile. Since its introduction in 1985, the features offered by this antenna element have proved to be useful in a wide variety of a...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2005
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| Cite this: | Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate / A. Boualleg, N. Merabtine // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 88-91. — Бібліогр.: 7 назв. — англ. |
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| citation_txt | Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate / A. Boualleg, N. Merabtine // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 88-91. — Бібліогр.: 7 назв. — англ. |
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| description | Microstrip antennas are useful as antennas mounted on moving vehicles such as cars, planes, rockets, or satellites, because of their small size, light weight and low profile. Since its introduction in 1985, the features offered by this antenna element have proved to be useful in a wide variety of applications, and the versatility and flexibility of the basic design have led to an extensive amount of development and design variations by workers hroughout the world.
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 88-91.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
88
PACS 84.40.Ba
Analysis of radiation patterns of rectangular microstrip antennas
with uniform substrate
A. Boualleg, N. Merabtine
Laboratory LET Electronics Department Faculty of Engineering
University of Constantine (Algeria)
bouadzdz@yahoo.fr, na_merabtine@hotmail.com
Abstract. Microstrip antennas are useful as antennas mounted on moving vehicles such
as cars, planes, rockets, or satellites, because of their small size, light weight and low
profile. Since its introduction in 1985, the features offered by this antenna element have
proved to be useful in a wide variety of applications, and the versatility and flexibility of
the basic design have led to an extensive amount of development and design variations
by workers hroughout the world.
Keywords: rectangular microstrip antennas, dielectric substrate, resonant microstrip,
radiation patterns.
Manuscript received 03.07.05; accepted for publication 25.10.05.
1. Introduction
Microstrip antennas have been the subject to study for
many years. Their analyses include the transmission line
model [1], the cavity model [2], and the method of
moments [3]. The physical size of a microstrip antenna
is small, but the electrical size measured in wavelength
λ is not so small. Much research has gone into further
reducing the microstrip antenna physical size.
Rectangular microstrip antennas have received much
attention due to their major advantage of conformability.
In this paper, we consider only rectangular patches
and discuss the aperture models for calculating the
radiation patterns of the antenna using the Fourier
integrals. The resonance problem has also been studied.
However, the excitation problem was not treated.
2. Analysis
Fig. 1 shows a rectangular microstrip antenna fed by a
microstrip line. It can also be fed by a coaxial line, with
its inner and outer conductors connected to the patch and
ground plane, respectively.
The height h of the substrate is typically of a fraction
of the wavelength, such as λ05.0=h , and the length L
is of the order of λ5.0 . The structure radiates from the
fringing fields that are exposed above the substrate at the
edges of the patch.
In the so-called cavity model, the patch acts as
resonant cavity with an electric field perpendicular to the
patch, that is, along the Z-direction. The magnetic field
has vanishing tangential components at the four edges of
the patch. The fields of the lowest resonant mode
(assuming WL ≥ ) are given by:
( )
( ) ,
22
forcos
,
22
forsin
0
0
WyW
L
x
HxH
LxL
L
x
ExE
y
z
≤≤−⎟
⎠
⎞
⎜
⎝
⎛
−=
≤≤−⎟
⎠
⎞
⎜
⎝
⎛
−=
π
π
(1)
where η/00 jEH −= . We have placed the origin at the
middle of the patch (note that ( )xEz is equivalent to
( )LxE /cos0 π for Lx ≤≤0 ).
It can be verified that Eq. (1) satisfy Maxwell’s
equations and the boundary conditions, that is,
( ) 0=xH y at 2/Lx ±= , provided the resonant
frequency is [4, 5]:
rL
c
L
cf
L
c
ε
π
ω 05.05.0 ==⇒= . (2)
Where rcc ε/0= , rεηη /0= , and rε is the
relative permittivity of the dielectric substrate. It follows
that the resonant microstrip length will be half-
wavelength:
r
L
ε
λ5.0= . (3)
Fig. 2 shows two models for calculating the radiation
patterns of the microstrip antenna. The model on the left
assumes that the fringing fields extend over a small
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 88-91.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
89
Fig. 1. Microstrip antenna and E-field pattern in substrate.
Fig. 2. Aperture models for microstrip antenna.
distance around the patch sides and can be replaced with
the fields aE that are tangential to the substrate surface
[6]. The four extended edge areas around the patch serve
as the effective radiating apertures.
The model on the right assumes that the substrate is
truncated beyond the extent of the patch [1]. The four
dielectric substrate walls serve now as the radiating
apertures.
The only tangential aperture field on these walls is
za EzE ˆ= , because the tangential magnetic fields vanish
by the boundary conditions.
For both models, the ground plane can be eliminated
using the image theory, resulting in doubling the
aperture magnetic currents, that is, ams EnJ ×−= ˆ2 . The
radiation patterns are then determined from msJ .
For the first model, the effective tangential fields can
be expressed in terms of the field zE by the relationship:
za hEaE = . This follows by requiring the vanishing of
the line integrals of E around the loops labeled ABCD
in the lower left of Fig. 2. Because 0EEz ±= at
2/Lx ±= , we obtain from the left and right such
contours:
.0
,0
0
0
0
∫
∫
=⇒=−=
=+−=
ABCD aa
ABCD a
a
hE
EaEhEEdl
aEhEEdl
In obtaining these, we assumed that the electric field
is nonzero only along the sides AD and AB. A similar
argument for the sides 2 and 4 shows that
( ) axhEE za /±= . The directions of aE at the four sides
are as shown in the figure. Thus, we have:
for sides 1 and 3 :
a
hExEa
0ˆ= .
for sides 2 and 4 :
( )
⎟
⎠
⎞
⎜
⎝
⎛=±=
L
x
a
hEy
a
xhEyE z
a
πsinˆˆ 0m . (4)
The outward normal to the aperture plane is zn ˆˆ =
for all four sides. Therefore, the surface magnetic
currents ams EnJ ×−= ˆ2 become:
for sides 1 and 3:
a
hEyJms
02ˆ= ,
for sides 2 and 4: ⎟
⎠
⎞
⎜
⎝
⎛±=
L
x
a
hExJms
πsin2ˆ 0 . (5)
3. Radiation fields
The radiated electric field is obtained by [7]
[ ]
[ ]rFFr
r
ejkH
FrFr
r
ejkE
m
jkr
m
jkr
ˆˆ
4
,ˆˆ
4
×+×−=
−××−=
−
−
η
πη
η
π
(6)
by setting 0=F and calculating mF as the sum of the
magnetic radiation vectors over the four effective
apertures:
[ ].ˆ
4
ˆ
4
4321 mmmm
jkr
m
jkr
FFFFr
r
ejk
Fr
r
ejkE
+++×=
=×=
−
−
π
π
(7)
The vectors mF are the two-dimensional Fourier
transforms over the apertures:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 88-91.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
90
The integration surfaces dxdydS = are
approximately, adydS = for 1 and 3, and adxdS = for
2 and 4. Similarly, in the phase factor yjkxjk yxe + , we
must set 2/Lx m= for sides 1 and 3, and 2/Wy m= for
sides 2 and 4. Inserting Eq. (5) into the Fourier integrals
and combining the terms for apertures 1 and 3 as well as
2 and 4, we obtain:
( ) ,
2ˆ
2/
2/
2/2/
0
13,
adyeee
a
hE
yF
yjkW
W
LjkLjk
m
yxx∫−
− +×
×=
.sin
2ˆ
2/
2/
2/2/
0
24,
adxe
L
x
ee
a
hE
xF
xjkL
L
WjkWjk
m
xyy ⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞⎜
⎝
⎛ −×
×=
∫−
− π
Using Euler’s formulae and the integrals:
( )
( )
,
1
2/cos2
sin
,
2/
2/sin
22
22/
2/
2/
2/
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=⎟
⎠
⎞
⎜
⎝
⎛
=
∫
∫
−
−
π
π
π
Lk
LkLjk
dxe
L
x
Wk
Wk
Wdye
x
xxxjkL
L
W
W y
yyjk
x
y
we find the radiation vectors:
( )
( )
( )
( ) ( ) ,sin
41
cos4
4ˆ
,
sin
cos4ˆ
2024,
013,
y
x
xx
m
y
y
xm
v
v
vv
hLExF
v
v
vhWEyF
π
π
π
π
π
π
−
=
=
(8)
where we defined the normalized wavenumbers as usual:
.sinsin
2
,cossin
2
φθ
λπ
φθ
λπ
WWk
v
LLkv
y
y
x
x
==
==
(9)
Using some trigonometric identities, we may write
the radiated fields from sides 1 and 3 in the form:
( )
[ ] ( ),,cosˆsincosˆ4
4
,
3,10 φθφθφθφ
π
φθ
FhWE
r
ejk
E
jkr
−=
=
− (10)
where we defined the function:
( ) ( ) ( )
y
y
x v
v
vF
π
π
πφθ
sin
cos,3,1 = . (11)
Similarly, we have for sides 2 and 4:
( )
[ ] ( ),,sinˆcoscosˆ4
4
,
4,20 φθφθφθφ
π
φθ
FhLE
r
ejk
E
jkr
+=
=
− (12)
( ) ( )
( ) ( )y
x
xx v
v
vvF π
π
πφθ sin
41
cos4, 24,2
−
= .
4. Radiation patterns
The normalized gain is found from Eq. (10) to be:
( ) ( )
( )
( ) ( ) .,cossincos
,
,
,
2
3,1
222
2
max
2
3,1
φθφφθ
φθ
φθ
φθ
F
E
E
g
+=
==
(13)
The corresponding expression for sides 2 and 4,
although not normalized, provides a measure for the gain
in that case:
( ) ( ) ( ) 2
4,2
222
4,2 ,sincoscos, φθφφθφθ Fg += . (14)
The E- and H-plane gains are obtained by setting
°= 0φ and °= 90φ in Eq. (13):
( ) ( )
( )
( )
.sin,
sin
cos
,sin,cos
2
3,1
2
3,1
θ
λπ
π
θθ
θ
λ
πθ
Wv
v
v
g
Lvvg
y
y
y
H
xxE
==
==
(15)
Most of the radiation from the microstrip arises from
sides 1 and 3. Indeed, ( )φθ ,3,1F has a maximum towards
broadside, 0== yx vv , whereas ( )φθ ,4,2F vanishes.
Moreover, ( )φθ ,4,2F * for all θ and 0=φ (E-plane) or
°= 90φ (H-plane).
Therefore, sides 2 and 4 contribute little to the total
radiation, and they are usually ignored.
5. Numerical results and discussion
Fig. 3 shows the E- and H-plane patterns for
λ3356.0== LW . Both patterns are fairly broad.
The choice for L comes from the resonant condition
rL ελ /5.0= . For a typical substrate with 22.2=rε ,
we find λλ 3356.022.2/5.0 ==L .
Fig. 4 shows the 3-dimensional gains computed from
Eqs (13) and (14). The field strengths (square roots of
the gains) are plotted to improve the visibility of the
graphs.
The gain from sides 2 and 4 vanishes along the xv
and yv axes, while its maximum in all directions is
1475.0=g or dB6242.16− .
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 88-91.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
91
E-Plane gain
0o
180o
90o90o
θθ 30o
150o
60o
120o
30o
150o
60o
120o
-3-6-9
dB
H-Plane gain
0o
180o
90o90o
θθ 30o
150o
60o
120o
30o
150o
60o
120o
-3-6-9
dB
Fig. 3. E- and H-plane gains of microstrip antenna.
Fig. 4. Three-dimensional gain patterns from sides 1
and 3 as well as 2 and 4.
6. Conclusion
Our study based on the analysis of radiation patterns of
the left model of Fig. 2. It found the most of the
radiation from the microstrip arises from sides 1 and 3.
On the other hand, the radiation from sides 2 and 4
vanishes almost completely.
Using the alternative aperture model shown on the
right of Fig. 2, one obtains identical expressions for the
magnetic current densities msJ along the four sides, and
therefore, identical radiation patterns. The integration
surfaces are now hdydS = for sides 1 and 3, and
hdxdS = for 2 and 4.
References
1. A.G. Demeryd, Linearly polarized microstrip anten-
nas // IEEE Trans. Antennas Propagat. AP-24, p.
846-851 (1976).
2. Y.T. Lo, D. Solomon, and W.F. Richards, Theory and
experiment on microstrip antennas // IEEE Trans.
Antennas Propagat. AP-27, p. 137-145 (1979).
3. E.H. Newman and P. Tulyathan, Analysis of micro-
strip antennas using moment methods // IEEE Trans.
Antennas Propagat. AP-29, p. 47-53 (1981).
4. S.M. Mi, T.M. Habashy, J.F. Kiang, and J.A. Kong,
Resonance in cylindrical-rectangular and wraparound
microstrip structures // IEEE Trans. Microwave
Theory Tech. 37, p. 1773-1789 (1989).
5. K.L. Won, Y.T. Cheng, and J.S. Row, Resonance in
a superstrate-loaded cylindrical-rectangular micro-
strip structure // IEEE Trans.Microwave Theory
Tech. 41, p. 814-819 (1993).
6. P. Hammer, D. Van Bouchaute, D. Verschraeven,
and A. Van de Capelle, A model for calculating the
radiation field of microstrip antennas // IEEE Trans.
Antennas Propagat. AP-27, p. 267 (1979).
7. C.A. Balanis, Antenna theory – analysis and design,
John Wiley & Sons, 2-ed., New York (1997).
|
| id | nasplib_isofts_kiev_ua-123456789-120975 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-11-30T20:50:06Z |
| publishDate | 2005 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Boualleg, A. Merabtine, N. 2017-06-13T11:38:43Z 2017-06-13T11:38:43Z 2005 Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate / A. Boualleg, N. Merabtine // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 88-91. — Бібліогр.: 7 назв. — англ. 1560-8034 PACS 84.40.Ba https://nasplib.isofts.kiev.ua/handle/123456789/120975 Microstrip antennas are useful as antennas mounted on moving vehicles such as cars, planes, rockets, or satellites, because of their small size, light weight and low profile. Since its introduction in 1985, the features offered by this antenna element have proved to be useful in a wide variety of applications, and the versatility and flexibility of the basic design have led to an extensive amount of development and design variations by workers hroughout the world. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate Article published earlier |
| spellingShingle | Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate Boualleg, A. Merabtine, N. |
| title | Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate |
| title_full | Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate |
| title_fullStr | Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate |
| title_full_unstemmed | Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate |
| title_short | Analysis of radiation patterns of rectangular microstrip antennas with uniform substrate |
| title_sort | analysis of radiation patterns of rectangular microstrip antennas with uniform substrate |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120975 |
| work_keys_str_mv | AT bouallega analysisofradiationpatternsofrectangularmicrostripantennaswithuniformsubstrate AT merabtinen analysisofradiationpatternsofrectangularmicrostripantennaswithuniformsubstrate |