Effect of Curvature on a Microstrip Printed Antenna
Conformed on Cylindrical Body at Superhigh Frequencies
, Khaled Elleithy
, Hassan Bajwa
Department of Computer and Electrical Engineering, University of Bridgeport, Bridgeport, CT 06604, USA
2Department of Computer and Electrical Engineering, University of Bridgeport, Bridgeport, CT 06604, USA
3Department of Computer and Electrical Engineering, University of Bridgeport, Bridgeport, CT 06604, USA
Abstract- Curvature has a great effect on fringing field of a microstrip antenna and consequently fringing field affects
effective dielectric constant and then all antenna parameters. A new mathematical model for input impedance, return loss,
voltage standing wave ratio and electric and magnetic fields is introduced in this paper. These parameters are given TM
mode and RT/duroid-5880 PTFE substrate material. The introduced model is valid at superhigh frequency range
(3– 30 GHz).
Keywords: Fringing field, Curvature, effective dielectric constant and Return loss (S11), Voltage Standing Wave Ratio
(VSWR), Transverse Magnetic TM
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 cost and its
ability of conforming to curve surfaces , conformal
microstrip structures 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 . 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 . 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. Analysis tools
for conformal arrays are not mature and fully developed .
Dielectric materials suffer from cracking due to bending and
that will affect the performance of the conformal microstrip
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 Figure 1 , are preferred due to easy calculation and
Fringing fields have a great effect on the performance of a
microstrip antenna. In microstrip antennas the electric filed
in 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
Figure 2, there is no field variation along the width and
thickness. The amount of the fringing field is a function of
the dimensions of the patch and the height of the substrate.
Higher the substrate, the greater is the fringing field.
Figure. 1. Rectangular microstrip antenna
Due to the effect of fringing, a microstrip patch antenna
would look electrically wider compared to its physical
dimensions. As shown in Figure 2, waves travel both in
substrate and in the air. Thus an effective dielectric constant
εreff is to be introduced. The effective dielectric constant
εreff takes in account both the fringing and the wave
propagation in the line.
Figure. 2. Electric field lines (Side View).
The expression for the effective dielectric constant is
introduced by A. Balanis , as shown in Equation 1.
The length of the patch is extended on each end by is a
function of effective dielectric constant and the width
to height ratio (W/h). can be calculated according to a
practical approximate relation for the normalized extension
of the length , as in Equation 2.
Figure. 3. Physical and effective lengths of rectangular microstrip patch.
The effective length of the patch is L
and can be calculated
as in Equation 3.
By using the effective dielectric constant (Equation 1) and
effective length (Equation 3), we can calculate the
resonance frequency of the antenna f and all the microstrip
Cylindrical-Rectangular Patch Antenna
All the previous work for a conformal rectangular
microstrip antenna assumed that the curvature does not
affect the effective dielectric constant and the extension on
the length. The effect of curvature on the resonant frequency
has been presented previously . 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.
Figurer 4: Geometry of cylindrical-rectangular patch antenna
Effect of curvature of conformal antenna on resonant
frequency been presented by Clifford M. Krowne [9, 10] as:
Where 2b is a length of the patch antenna, a is a radius of
the cylinder, is the angle bounded the width of the patch,
represents electric permittivity and µ is the magnetic
permeability as shown in Figure 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. Kwai et al. 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
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 presence 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
as a functions of
Hankel function of the second kind H
. The input
impedance and Q factors are also calculated under the same
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 .
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
Nickolai Zhelev introduced a design of a small conformal
microstrip GPS patch antenna . 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 . Also, the effect of curvature on the
performance of a microstrip antenna as a function of
temperature for TM
is introduced in , .
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
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 , as shown in Equation
Magnetic field intensity H and electric flux density D are
related by Ampérés law as in Equation (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):
where is the magnetic permeability and is the electric
By substituting Equation (4) in Equations (2) and (3), we
where is the angular frequency and has the form of:
. In homogeneous medium, the divergence of
Equation (2) is:
From Equation (5), we can get Equation (7):
Using the fact that, any curl free vector is the gradient of the
same scalar, hence:
where is the electric scalar potential.
where A is the magnetic vector potential.
So, the Helmholtz Equation takes the form of (9):
k is the wave number and has the form of:
is Laplacian operator. The solutions of Helmholtz
Equation are called wave potentials:
3.1 Near Field Equations
By using the Equations number (10) and magnetic vector
potential in , we can get the near electric and magnetic
fields as shown below:
are also getting using Equation (7);
To get the magnetic field in all directions, we can use the
second part of Equation (10) as shown below, where H
for TM mode:
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 tends to infinity when r, in Cartesian
coordinate, tends to infinity. Also, using simple vector
analysis, one can note that, the value of k
will equal to
, and from the characteristics of Hankel
function, we can rewrite the magnetic vector potential
illustrated in Equation (12) to take the form of far field as
illustrated in Equation (17).
Hence, the electric and magnetic field can easily be
calculated as shown below:
The magnetic field intensity also obtained as shown below,
4. Input Impedance
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 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
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
By solving this Equation, the electric field at the surface can
be expressed in terms of various modes of the cavity as :
is the amplitude coefficients corresponding to the
field modes. By applying boundary conditions,
homogeneous wave Equation and normalized conditions
, we can get an expression for
as shown below:
vanishes at the both edges for the length L:
vanishes at the both edges for the width W:
should satisfy the homogeneous wave
should satisfy the normalized condition:
Hence, the solution of
will take the form shown below:
The coefficient A
is determined by the excitation current.
For this, substitute Equation (29) into Equation (23) and
multiply both sides of (23) by
, and integrate over area
of the patch. Making use of orthonormal properties of
Now, let the coaxial feed as a rectangular current source
with equivalent cross-sectional area
, so, the current density will satisfy the Equation
Use of Equation (31) in (30) gives:
So, to get the input impedance, one can substitute in the
is the RF voltage at the feed point and defined as:
By using Equations (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
5. Voltage Standing Wave Ratio and
Voltage Standing Wave Ration VSWR is defined as the
ration of the maximum to minimum voltage of the antenna.
The reflection coefficient define as a ration between
incident wave amplitude V
and reflected voltage wave
, and by using the definition of a voltage
reflection coefficient at the input terminals of the antenna ,
as shown below:
is the characteristic impedance of the antenna. If
the Equation is solved for the reflection coefficient, it is
found that, where the reflection coefficient is the absolute
vale of the magnitude of ,
The characteristic can be calculated as in ,
where : L is the inductance of the antenna, and C is the
capacitance and can be calculated as follow:
Hence, we can get the characteristic impedance as shown
The return loss s
is related through the following Equation:
For the range of GHz, the dominant mode is TM
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
RT/duroid-5880 PTFE is used as a substrate material for
verifying the new model at the 5.1 GHz range. The
dielectric constant for the used material is 2.1 with a tangent
Figure 5 shows the effect of curvature on resonance
frequency for a TM
mode mathematical and measured
data. The frequency range of resonance frequency due to
changing in curvature is from 5.07 to 5.095 GHz for a radius
of curvature from 6 mm to flat antenna. So, the frequency is
shifted by 25 MHz due to changing in curvature.
Figure 5. Resonance frequency as a function of curvature for flat, curved
antenna for mathimatical and measured data.
The mathematical results for input impedance, real and
imaginary parts for a different radius of curvatures are
shown in Figures 6 and 7. The peak value of the real part of
input impedance is almost 650 Ω at frequency 4.689 GHz
which gives a zero value for the imaginary part of input
impedance as shown in Figure 6 at 20 mm radius of
curvature. The value 4.689 GHz represents a resonance
frequency for the antenna at 20 mm radius of curvature.
VSWR is given in Figure 8. It is noted that, the value of
VSWR is almost 1.7 at frequency 4.689 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 Figure 9. We obtain a very
low return loss, -16 dB, at frequency 4.689 GHz for radius
of curvature 20 mm.
Figure 6. Real part of the input impedance as a function of frequency for
different radius of curvatures.
Normalized electric field for different radius of curvatures is
illustrated in Figure 10. Normalized electric field is plotted
for from zero to 2π and 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
Normalized magnetic field is wider than normalized electric
field, and also, it is increasing with deceasing radius of
curvature. Obtained results are at for from zero to 2π and
equal to zero and for radius of curvature 20, 65 mm and
for a flat microstrip printed antenna are shown in Figure 11.
For different radius of curvature, the resonance frequency
changes according to the change in curvature, so the given
normalized electric and magnetic fields are calculated for
different resonance frequency according to radius of
Figure 7. Imaginary part of the input impedance as a function of frequency
for different radius of curvatures.
Figure 8. VSWR versus frequency for different radius of curvatures.
Figure 9. Return loss (S11) as a function of frequency for different radius
Figure 10. Normalized electric field for radius of curvatures 20, 65 mm
Figure 11. Normalized magnetic field for radius of curvatures 20, 65 mm
and a flat
The effect of curvature on the performance of conformal
microstrip antenna on cylindrical bodies for TM
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,
VSWR 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 material. The introduced
model is valid at superhigh frequency range (3 – 30 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.
We conclude that, increasing the curvature leads to
increasing 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
25 MHz from flat to 6 mm radius of curvature.
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