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Procedia Computer Science 10 ( 2012 ) 775 – 784
1877-0509 © 2012 Published by Elsevier Ltd.
doi: 10.1016/j.procs.2012.06.099
The 9th International Conference on Mobile Web Information Systems
Performance Analysis of a Microstrip Printed
Antenna Conformed on Cylindrical Body at
Resonance Frequency 4.6 GHz for TM
01
Mode
Ali Elrashidi
1,2
, Khaled Elleithy
1
, Hassan Bajwa
1
1
Department of Computer and Electrical Engineering, University of Bridgeport, Bridgeport, CT 06604, USA
Abstra ct
Curvature has a great effect on fringing field of a microstrip antenna and consequently fringing field
affects effective d ielectric constant and then all antenna parameters. A new mathematical model for inpu t
impedance, return loss, voltage standing wave ratio and electric and magnetic fields is introduced in this
paper. These parameters are given for TM
01
mode RT/duroid-5880 PTFE substrate material. The range of
operation is around 4.7 GHz.
Keywords
Fringing field, Curvature, effective dielectric constant and Return loss (S11), Voltage Standing
Wave Ratio (VSW R), Transverse Magnetic TM
01
mode
1. Intr oduc tion
Due to the unprinted growth in wireless applications and increasing demand of low cost solutions for RF
and microwave communication systems, the microstrip flat antenna, has undergone tremendous growth
recently. Though the models used in analyzing microstrip structures have been widely accepted, the effect
of curvature on dielectric constant and antenna performance has not been studied in detail. Low profile,
low weight, low cos t and its ability of conforming to curve s urfaces [1], conforma l micros trip s tructures
have also witnessed enormous growth in the last few years. Applications of microstrip structures include
Unmanned Aerial Vehicle (UAV), planes, rocket, radars and communication industry [2]. Some
advantages of conformal antennas over the planer microstrip structure include, easy installation (randome
not needed), capability of embedded structure within composite aerodynamic surfaces, better angular
coverage and controlled gain, depending upon shape [3, 4]. While Conformal Antenna provide potential
solution for many applications, it has some drawbacks due to bedding [5].
___________
2
Corresponding author . Tel.:+1-203-449-0406.
E-mail address: aelrashi@bridgepo rt .edu.
Available online at www.sciencedirect.com
776 Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
Such drawbacks include phase, impedance, and resonance frequency errors due to the stretching and
compression of the dielectric material along the inner and outer surfaces of conformal surface. Changes in
the dielectric constant and material thickness also affect the performance of the antenna. A nalysis tools
for conformal arrays are not mature and fully developed [6]. Dielectric materials suffer from cracking due
to bending and that will affect the performance of the conformal microstrip antenna.
2. Background
Conventional microstrip antenna has a metallic patch printed on a thin, grounded dielectric substrate.
Although the patch can be of any shape, rectangular patches, as shown in Fig. 1 [7], are preferred due to
easy calculation and modeling.
Fringing fields have a great effect on the performance of a microstrip antenna. In microstrip antennas the
electric filed at the center of the patch is zero. The radiation is due to the fringing field between the
periphery of the patch and the ground plane. For the rectangular patch shown in the Fig. 2, there is no
field variation along the width and thickness. The amount of the fringing field is a functio n of the
dimensions of the patch and the height of the substrate. Higher the substrate, the greater is the fringing
field.
Fig. 1 . Rect an gula r m icro st r ip ant enn a
Due to the effect of fringing, a microstrip patch antenna would look electrically wider compared to its
physical dimensions. Waves travel both in substrate and in the air. Thus an effective dielectric constant
ε
reff
is to be introduced. The effective dielectric constant ε
reff
takes in account both the fringing and the
wave propagation in the line.
The expression for the effective dielectric constant is introduced by A. Balanis [7], as shown in Equ 1.
y
(1)
The length of the patch is extended on each end by ΔL is a function of effective dielectric constant
()
and the width to height ratio (W/ h ). ΔL can be calculated according to a practical approximate relation for
the normalized extension of the length [8], as in Equ.(2).
(2)
The effective length of the patch is L
eff
and can be calculated as in Equ. 3.
L
eff
= L+2ΔL (3)
By using the effective dielectric constant (Equ. 1) and effective length (Equ. 2), we can calculate the
resonance frequency of the antenna f and all the microstrip antenna parameters.
L
W
ɛ
r
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Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
y
x
L
ε
R
z
Cylindrical-Rectangular Patch Antenna
All the previous work for a conformal rectangular microstrip antenna assumed that the curvature does not
affect the effective d ielectric constant and the extension on the length. The effect of curvature on the
resonant frequency has been presented previously [9]. In this paper we present the effect of fringing field
on the performance of a conformal patch antenna. A mathematical model that includes the effect of
curvature on fringing field and on antenna performance is presented. The cylindrical-rectangular patch is
the most famous and popular conformal antenna. The manufacturing of this antenna is easy with respect
to spherical and conical antennas.
Fig. 2 Geometry of cylindrical-rectangular patch antenna[9]
Effect of curvature of conformal antenna on resonant frequency been presented by Clifford M. Krowne
[9, 10] as:
(4)
Where 2b is a length of the patch antenna, a is a radius of the cylinder, 2θ is the angle bounded the width
of the patch, ε represents electric permittivity and μ is the magnetic permeability as shown in Fig. 4.
Joseph A. et al, presented an approach to the analysis of microstrip antennas on cylindrical surface. In this
approach, the field in terms of surface current is calculated, while considering dielectric layer around the
cylindrical body. The assumption is only valid if radiation is smaller than stored energy [11]. Kwai et al.
[12]gave a brief analysis of a thin cylindrical-rectangular microstrip patch antenna which includes
resonant frequencies, radiation patterns, input impedances and Q factors. The effect of curvature on the
characteristics of TM
10
and TM
01
modes is also presented in Kwai et al. paper. The authors first obtained
the electric field under the curved patch using the cavity model and then calculated the far field by
considering the equivalent magnetic current radiating in the p resence of cylindrical surface. The cavity
model, used for the analysis is only valid for a very thin dielectric. Also, for much small thickness than a
wavelength and the radius of curvature, only TM modes are assumed to exist. In order to calculate the
radiation patterns of cylindrical-rectangular patch antenna. The authors introduced the exact Green’s
function approach. Using Equation (4), they obtained expressions for the far zone electric field
components E
θ
and E
φ
as a functions of Hankel function of the second kind H
p
(2)
. The input impedance
and Q factors are also calculated under the same conditions.
Based on cavity model, microstrip conformal antenna on a projectile for GPS (Global Positioning
System) device is designed and implemented by using perturbation theory is introduced by Sun L., Zhu J.,
Zhang H. and Peng X [13]. The designed antenna is emulated and analyzed by IE3D software. The
emulated results showed that the antenna could provide excellent circular hemisphere beam, better wide-
angle circular polarization and better impedance match peculiarity.
778 Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
Nickolai Zhelev introduced a design of a small conformal microstrip GPS patch antenna [14]. A
cavity model and transmission line model are used to find the initial dimensions of the antenna and then
electromagnetic simulation of the antenna model using software called FEKO is applied. The antenna is
experimentally tested and the author compared the result with the software results. It was founded that the
resonance frequency of the conformal antenna is shifted toward higher frequencies compared to the flat
one.
The effect of curvature on a fringing field and on the resonance frequency of the microstrip printed
antenna is studied in [15]. Also, the effect of curvature on the performance of a microstrip antenna as a
function of temperature for TM
01
and TM
10
is introduced in [16], [17].
3. General Expressions for Electric and Magnetic Fields Intensities
In this section, we will introduce the general expressions of electric and magnetic field intensities for a
microstrip antenna printed on a cylindrical body represented in cylindrical coordinates.
Starting from Maxwell’s Equation s, we can get the relation between electric field intensity E and
magnetic flux density B as known by Faraday’s law [18], as shown in Equation (2):
(2)
Magnetic field intensity H and electric flux density D are related by Ampérés law as in Equation (3):
(3)
where J is the electric current density.
The magnetic flux density B and electric flux density D as a function of time t can be written as in
Equation (4):
and
(4)
where μ is the magnetic permeability and ɛ is the electric permittivity.
By substituting Equation (4) in Equations (2) and (3), we can get:
and
(5)
where ω is the angular frequency and has the form of:
. In homogeneous medium, the
divergence of Equ. (2) is:
and
(6)
From Equ. (5), we can get Equ. (7):
or
(7)
Using the fact that, any curl free vector is the gradient of the same scalar, hence:
g
(8)
where φ is the electric scalar potential.
By letting:
where A is the magnetic vector potential.
So, the Helmholtz Equation takes the form of (9):
A+ -J (9)
k is the wave number and has the form of:
, and is Laplacian operator. The solutions of
Helmholtz Equation are called wave potentials:
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Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
(10)
3.1 Near Field Equations
By using the Equ. (10) and magnetic vector potential in [19], we can get the near electric and magnetic
fields as shown below:
(12)
E
φ
and E
ρ
are also getting using Equ. (7);
ggg
q
(13)
(14)
To get the magnetic field in all directions, we can use the second part of Equ. (10) as shown below, where
H
z
= 0 for TM mode:
(15)
(16)
3.2 Far field Equations
In case of far field, we need to represent the electric and magnetic field in terms of r, where r is the
distance from the center to the point that we need to calculate the field on it. By using the cylindrical
coordinate Equations, one can notice that a far field ρ tends to infinity when r, in Cartesian coordinate,
tends to infinity. Also, using simple vector analysis, one can note that, the value of k
z
will equal to
[19], and from the characteristics of Hankel function, we can rewrite the magnetic ve ctor
potential illustrated in Equ. (12) to take the form of far field as illustrated in Equ. (17).
q
(
)
(17)
Hence, the electric and magnetic field can easily be calculated as shown b elow:
(18)
(19)
(20)
The magnetic field intensity also obtained as shown below, where H
z
= 0:
(21)
(22)
4. Input Impe dance
The input impedance is defined as “the impedance presented by an antenna at its terminals” or “the ratio
of the voltage current at a pair of terminals” or “the ratio of the appropriate components of the electric to
780 Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
magnetic fields at a point”. The input impedance is a function of the feeding position as we will see in the
next few lines.
To get an expression of input impedance Z
in
for the cylindrical microstrip antenna, we need to get the
electric field at the surface of the patch. In this case, we can get the wave equation as a function of
excitation current density J as follow :
(23)
By solving this Equation, the electric field at the surface can be exp ressed in terms of various modes of
the cavity as [15]:
(24)
where A
nm
is the amplitude coefficients corresponding to the field modes. By applying boundary
conditions, homogeneous wave Equation and normalized conditions for
, we can get an expression
for
as shown below:
1. vanishes at the both edges for the length L:
(25)
2. vanishes at the both edges for the width W:
(26)
3. should satisfy the homogeneous wave Equ. :
(27)
4. should satisfy the normalized condition:
(28)
Hence, the solution of
will take the form shown below:
(29)
with
The coefficient A
mn
is determined by the excitation current. For this, substitute Equ. (29) into Equ. (23)
and multiply both sides of (23) by
y
, and integrate over area of the patch. Making use of orthonormal
properties of
, one obtains:
(30)
Now, let the coaxial feed as a rectangular current source with equivalent cross -sectional area
centered at
t
t
, so, the current density will satisfy the Equation (31) below:
yy
q
q
q
(
)
(31)
Use of Equ. (31) in (30) gives:
q
(
)
(
(32)
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Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
So, to get the input impedance, one can substitute in the following Equation:
(33)
where
is the RF voltage at the feed point and defined as:
(34)
By using Equ. (24), (29), (32), (34) and substitute in (33), we can obtain the input impedance for a
rectangular microstrip antenna conformal in a cylindrical body as in the following Equation:
(35)
5. Voltage S t an di ng W ave Rati o an d Re tur n Los s
Voltage Standing Wave Ration VS WR is defined as the ration of the maximum to minimum voltage of the
antenna. The reflection coefficient ρ define as a ration between incident wave amplitude V
i
and reflected
voltage wave amplitude V
r
, and by using the definition of a voltage reflection coefficient at the input
terminals of the antenna Γ, as shown below:
(36)
where, Z
0
is the characteristic impedance of the antenna. If the Equ. is solved for the reflection
coefficient, it is found that, where the reflection coefficient ρ is the absolute vale of the magnitude of Γ,
(37)
Consequently,
(38)
The characteristic can be calculated as in [14],
]
(39)
where : L is the inductance of the antenna, and C is the capacitance and can be calculated as follow:
(40)
(41)
Hence, we can get the characteristic impedance as shown below:
(42)
The return loss s
11
is related through the following Equation:
g
g
g
g
g
g
g
g
q
(43)
6. Res ults
For the range of GHz, the dominant mode is TM
10
and TM
01
for h<<W which is the case. Also, for
the antenna operates at the ranges 4.6 and 5.1 GHz using RT/duroid-5880 PTFE substrate material we can
use the following dimensions; the original length is 17.5 cm, the width is 20 cm and by using RT/duroid-
5880 PTFE substrate we can get the effect of curvature on the effective dielectric constant and the
resonance frequency.
782 Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
RT/duroid-5880 PTFE is used as a substrate material for verifying the new model at the 4.7 GHz
range. The dielectric constant for the used material is 2.1 with a tangent loss 0.0015.
Fig. 5 shows the effect of curvature on resonance frequency for a TM
01
mode. The frequency range of
resonance frequency due to changing in curvature is from 4.645 to 4.67 GHz for a radius of curvature
from 6 mm to flat antenna. So, the frequency is shifted by 35 MHz due to changing in curvature.
Fig. 5. Resonance frequency as a function of curvature for
flat and curved antenna.
Fig. 6. Real part of the input impedance as a function of
frequency for different radius of curvatures.
The mathematical for input impedance, real and imaginary parts for a different radius of curvatures are
shown in Fig. 6 and 7. The peak value of the real part of input impedance is almost 400 Ω at frequency
4.688 GHz which gives a zero value for the imaginary part of input impedance as shown in Fig. 7 at 20
mm radius of curvature. The value 4.688 GHz represents a resonance frequency for the antenna at 20 mm
radius of curvature.
VSWR is given in Fig. 8. It is noted that, the value of VSWR is almost 1.9 at frequency 4.688 GHz
which is very efficient in manufacturing process. It should be between 1 and 2 for radius of curvature
20 mm. The minimum VSWR we can get, the better performance we can obtain as shown clearly from
the definition of VSWR.
Return loss (S11) is illustrated in Fig. 9. We obtain a very low return loss, -6.5 dB, at frequency
4.688 GHz for radius of curvature 20 mm.
Normalized electric field for different radius of curvatures is illustrated in Fig. 10. Normalized
electric field is plotted for θ fro m zero to 2π and φ equal to zero. As the radius of curvature is decreasing,
the radiated electric field is getting wider, so electric field at 20 mm radius of curvature is wider than 65
mm and 65 mm is wider than flat antenna. Electric field strength is increasing with decreasing the radius
of curvature, because a magnitude value of the electric field is depending on the effective dielectric
constant and the effective dielectric constant depending on the radius of curvature which decreases with
increasing the radius of curvature.
Normalized magnetic field is wider than normalized electric field, and also, it is increasing with
deceasing radius of curvature. Obtained results are at for θ fro m zero to 2π and φ equal to zero and for
radius of curvature 20, 65 mm and for a flat microstrip printed antenna are shown in Fig. 11. For different
radius of curvature, the resonance frequency changes according to the change in curvature, so t he given
normalized electric and magnetic fields are calculated for different resonance frequency according to
radius of curvatures.
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Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
Fig. 7. Imaginary part of the input impedance as a function of
frequency for different radius of curvatures.
Fig. 8. VSWR versus frequency for different radius of
curvatures.
7. Conclusion
The effect of curvature on the performance of conformal microstrip antenna on cylindrical bodies
TM
01
mode is studied in this paper. Curvature affects the fringing field and fringing field affects the
antenna parameters. The Equations for real and imaginary parts of input impedance, return loss, VSW R
and electric and magnetic fields as a functions of curvature and effective dielectric constant are derived.
By using these derived equations, we introduced the results for RT/duroid-5880 PTFE substrate at TM
01
mode and resonance frequency around 4.6 GHz. For the used dielectric substrate, the decreasing in
frequency due to increasing in the curvature is the trend for both transverse magnetic modes of operations
and increasing the radiation pattern for electric and magnetic fields due to increasing in curvature is
easily noticed. All these parameters are calculated.
We conclude that, increasing the curvature leads to increase the effective dielectric constant, hence,
resonance frequency is increased. So, all parameters are shifted toward increasing the frequency with
increasing curvature. The shifting in the resonance frequency is around 35 MHz from flat to 6 mm radius
of curvature.
Fig. 9. Return loss (S11) as a function of frequency for different
radius of curvatures.
Fig. 10. Normalized electric field for radius of curvatures
20, 65 mm and a flat antenna at θ=0:2π and φ=0
0
.
784 Ali Elrashidi et al. / Procedia Computer Science 10 ( 2012 ) 775 – 784
Fig. 11. Normalized magnetic field for radius of curvatures 20, 65 mm and a flat antenna at θ=0:2π and φ=0
0
.
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