Mutual Coupling Cofficient of a Microstrip Printed
Antenna Array Mounted on Cylindrical Body for TM
Mode at 5 GHz Range Using Three Different Substrate
, Khaled Elleithy
, Hassan Bajwa
Department of Computer Science and Engineering, University of Bridgeport, Bridgeport, CT 06604, USA
2Department of Computer Science and Engineering, University of Bridgeport, Bridgeport, CT 06604, USA
3Department of 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 return loss mutual coupling
coefficient as a function of curvature for two element array antenna is introduced in this paper. These parameters are given
mode and using three different substrate materials RT/duroid-5880 PTFE, K-6098 Teflon/Glass and Epsilam-10
Keywords Fringing field, Curvature, effective dielectric constant and Return loss (S11), mutual coupling coefficient
(S12), Transverse Magnetic TM
Microstrip antenna array conformed on cylindrical
bodies is commonly used antennas in aircraft, millime-
ter-wave imaging arrays mounted on unmanned air-
borne vehicles, and antennas for medical imaging ap-
plications which may be required to conform to the
shape of the human body -. 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. Some
advantages of conformal antennas over the planer
microstrip structure include, easy installation (randome not
needed), capability of embedded structure within compo-
site aerodynamic surfaces, better angular coverage and
controlled gain, depending upon shape [5, 6].
While Conformal Antenna provide potential solution
for many applications, it has some drawbacks due to bed-
ding . The two main disadvantages of microstrip an-
tenna arrays are the narrow frequency band and the
mutual coupling between the basic elements is higher
than in the usual antenna arrays -.
Mutual coupling between array elements affects
the radiation pattern and input impedances. The radia-
tion from one element in the array induces currents on
the other elements to a nearby and scatters into the far
field. The induced current derived a voltage at the ter-
minals of other elements .
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 calcula-
tion and modeling.
Figure 1. Rectangular microstrip antenna
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.
Due to the effect of fringing, a microstrip patch an-
tenna would look electrically wider compared to its physi-
cal dimensions. As shown in Figure 2, waves travel both in
substrate and in the air. Thus an effective dielectric con-
is to be introduced. The effective dielectric con-
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
ΔL is a function of effective dielectric constant and
the width to height ratio (W/h). ΔL can be calculated ac-
cording to a practical approximate relation for the normal-
ized 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.
= L+2ΔL (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 confor-
mal antenna. The manufacturing of this antenna is easy
with respect to spherical and conical antennas.
Effect of curvature of conformal antenna on resonant
frequency been presented by Clifford M. Krowne [13, 14]
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 Figure 4.
Figurer 4. Geometry of cylindrical-rectangular patch antenna
Joseph A. et al, presented an approach to the analysis
of microstrip antennas on cylindrical surface. In this ap-
proach, 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 analy-
sis of a thin cylindrical-rectangular microstrip patch an-
tenna which includes resonant frequencies, radiation pat-
terns, input impedances and Q factors. The effect of cur-
vature on the characteristics of TM
also presented in Kwai et al. paper. The authors first ob-
tained the electric field under the curved patch using the
cavity model and then calculated the far field by consider-
ing the equivalent magnetic current radiating in the pres-
ence 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
function approach. Using Equation (4), they obtained ex-
pressions for the far zone electric field components E
as a functions of Hankel function of the second kind
. The input impedance and Q factors are also calculat-
ed 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 .
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
Nickolai Zhelev introduced a design of a small con-
formal 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 tem-
perature for TM
is introduced in , .
3. Conformal Microstrip Antenna
Array and Mutual Coupling
Conformal microstrip arrays are used to increase the
directivity of the antenna and increase the signal to noise
ratio. Better performance is achieved using arrays. The
radiation pattern is significantly affected using arrays on a
conformal surface to appear as omnidirectional pattern,
which is very useful in aerospace systems .
The equations of directivity function of the conformal
microstrip array on a cylinder and the experimental results
of pattern of array of 64 elements are given by M. Knghou
et.al. . The coupling between elements is not consid-
ered in . The authors calculated the total electric field
strength for an array of N elements using Equation (5).
represents the field strength of number i radiator
is the phase of equivalent transversal magnetic cur-
rent source of N radiators.
C. You et.al. designed and fabricated a composite an-
tenna array conformed around cylindrical structures .
The experimental results showed that the radiation pattern
is strongly dependent on the cylindrical curvature for the
transverse radiation pattern, while the array also exhibits
high side-lobes and wider beamwidth.
Problems associated with Ultra Wide Band (UWB)
antennas as phased array elements discussed in . The
authors introduced various wide bandwidth arrays of an-
tennas that can be conforming. Problems that arise de-
pending on the physical separation of antennas are dis-
cussed. Conformal placement, of an antenna, either as an
individual antenna, or as in an array configuration on any
arbitrary surface, may require very thin antenna. The au-
thors should be processed preferably on flexible substrates
so that the authors will conform to the surfaces without
changing the surface geometry.
A. Sangster and R. Jacobs developed a finite ele-
ment-boundary integral method to investigate the im-
pedance properties of a patch element array for a
microstrip printed antenna conformal on a cylindrical
body . A mutual coupling between elements is also
studied in this paper for its great effect on the imped-
ance properties. Simulation results for mutual coupling
coefficient, S12, for a planar and conformal array are
compared to a measured values and a good agreement is
A full-wave analysis of the mutual coupling be-
tween two probe-fed rectangular microstrip antennas
conformed on a cylindrical body is introduced by S. Ke
and K. Wong . The authors calculated the mutual
impedance and mutual coupling coefficient using a
moment of method technique , . The numerical
results of mutual impedance and mutual coupling coef-
ficient are compared to the measured values for the
microstrip antennas conformed on a cylindrical body
with different radius of curvatures.
A comprehensive mathematical model for mutual
impedance and mutual coupling between two rectangu-
lar patches array is introduced by A. Mohammadian et
al . The authors replaced each microstrip antenna
element in the array by an equivalent magnetic current
source distributed over a grounded dielectric slab. A
dielectric slab, using the rectangular vector wave func-
The active reflection performance and active radi-
ation pattern of two elements in the array of microstrip
antenna elements are calculated by S. Chen and R. Iwa-
ta . The authors introduced a mathematical deriva-
tion of radiation pattern and reflection performance for
each element in the array of microstrip antenna. Then by
using the introduced model, the mutual coupling be-
tween the elements in the array is easily calculated.
N. Dodov and P. Petkov explored the mutual cou-
pling between microstrip antennas provoked by the sur-
face wave . Based on the method of moments, the
authors analyze the microwave structure on the
microstrip antenna patch surface. N. Dodov and P.
Petkov conclude that, the influence of surface wave is
not significant in close neighboring resonant elements.
An accurate formula for the coupling between
patch elements is introduced by Z. Qi et al . The
classic formula for mutual coupling based on multi-port
network theory ignores the impedance mismatching
between antenna elements but on the other hand the
introduced formula consider this mismatching between
antenna array elements.
A hybrid method, based on the method of moments,
is introduced to analyze a microstrip antenna conformal
on a cylindrical body by A. Erturk et al . The au-
thors introduced three types of space-
function representations, each accurate and efficient in a
given region of space. Input impedance of various
microstrip antenna conformed on a cylindrical body and
mutual coupling between two elements of the array is
introduced and compared to some published results.
4. Input Impedance
appropriate components of the electric to magnetic fields at
pedance 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 sur-
face 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
for , we can get an expression for as shown below:
1. vanishes at the both edges for the length L:
2. vanishes at the both edges for the width W:
3. should satisfy the homogeneous wave
4. should satisfy the normalized condition:
Hence, the solution of will take the form shown be-
The coefficient A
is determined by the excitation
current. For this, substitute Equation (12) into Equation (6)
and multiply both sides of (6) by , and integrate over
area of the patch. Making use of orthonormal properties of
, one obtains:
Now, let the coaxial feed as a rectangular current
source with equivalent cross-sectional area cen-
tered at , so, the current density will satisfy the
Use of Equation (14) in (13) gives:
So, to get the input impedance, one can substitute in
the following Equation:
where is the RF voltage at the feed point and defined
By using Equations (7), (12), (14), (17) and substitute
in (16), we can obtain the input impedance for a rectangu-
lar microstrip antenna conformal in a cylindrical body as in
the following Equation:
5. Mutual Coupling
Mutual coupling between array elements affects the
radiation pattern and input impedances. The radiation
from one element in the array induces currents on the
other elements to a nearby and scatters into the far field.
The induced current derived a voltage at the terminals
of other elements .
The input terminals of the elements in an array are
represented as ports of a microwave network. The
equivalent network of two antenna array is shown in
Figure 5. Hence, the mutual coupling is represented as a
scattering matrix or S-parameters matrix as illustrated in
Figure 5. Equivalent network of two antenna array.
represent the forward and reverse volt-
age wave amplitude at the nth port respectively.
The mutual impedance formulation is shown in
Equation (20) .
Hence, the mutual coupling coefficient, S
, can be
calculated as in Equation (21) .
is the characteristic impedance of the feeding
In case of using identical array elements, same di-
mension and same feeding position, the values of Z
will give the same value and Z
Hence, the value of Z
are given by Equa-
and the value of Z
are given by Equation (23).
By using Equations (15) and (18), we can get Equa-
tion (24) for Z
and by substitute in Equation (21) the mutual coupling
coefficient can be calculated.
The dimensions of the patch are as follows: 20 mm
length, 17.5 mm width, 0.8 mm substrate height and the
position of a feed point is at 5mm from the center and par-
allel to the length. The dimensions of the ground plate are:
52 mm length and the 48 mm width. Those values are used
as constant values in this chapter for 5 GHz frequency
band. The dominant transverse magnetic mode is TM
when a height is less than a width h<<W.
Three different substrate materials RT/duroid-5880
PTFE, K-6098 Teflon/Glass and Epsilam-10 ceramic-filled
Teflon are used for verifying the new model. The dielectric
constants for the used materials are 2.2, 2.5 and 10 respec-
tively with a tangent loss 0.0015, 0.002 and 0.004 respec-
6.1 RT/duroid-5880 PTFE Substrate
RT/duroid-5880 PTFE material is a flexible material
with a dielectric constant 2.1 at low frequencies and almost
2.02 in the Giga Hertz range and tangent loss 0.0015.
Return loss (S11) is illustrated in Figure 6 . We
obtain a return loss, -15.5 dB, at frequency 5.1 GHz for
radius of curvature 20 mm, 5.18 GHz at 65 mm and
5.116 GHz for a flat antenna.
Figure 6. Return loss (S11) as a function of frequency for different radius
Figure 7. Mutual coupling coefficient, S12, as a function of resonance
frequency for different values of curvatures for TM
Figure 7 shows the mutual coupling coefficient, S12
as a function of resonance frequency for different radius of
curvature. The maximum mutual coupling is obtained at
the minimum return loss for the same resonance frequency
and the peaks are shifted to the direction of increasing fre-
quency with increasing the radius of curvature. The peaks
are almost the same at -3 dB, so changing the curvature
does not change the mutual coupling value but shift the
curve in frequency.
6.2 K-6098 Teflon/Glass Substrate
A K-6098 Teflon/Glass material is a flexible material
with a dielectric constant 2.5 at high frequency and tangent
Return loss (S11) is illustrated in Figure 8. We obtain
a very low return loss, -8.5 dB, at frequency 4.57 GHz for
radius of curvature 20 mm, 4.578 GHz at 65 mm and
4.586 GHz for a flat antenna. The return loss value,
-8.5 dB, is obtained for different radius of curvature.
Figure 8. Return loss (S11) as a function of frequency for different radius
Figure 9. Mutual coupling coefficient, S12, as a function of resonance
frequency for different values of curvatures for TM
Mutual coupling coefficient is shown in Figure 9. The
maximum mutual coupling is obtained at the minimum
return loss for the same resonance frequency.
6.3 Epsilam-10 Ceramic-filled Teflon Substrate
Epsilam-10 ceramic-filled Teflon is used as a substrate
material for verifying the new model. The dielectric
constant for the used material is 10 with a tangent loss
Return loss (S11) is illustrated in Figure 10. We obtain
a return loss, -50 dB for all values of radius of curvature,
20 mm, 65 mm and flat antenna.
Figure 10. Return loss (S11) as a function of frequency for different
radius of curvatures.
Figure 11. Mutual coupling coefficient, S12, as a function of resonance
frequency for different values of curvatures for TM
Figure 11 shows the mutual coupling coefficient as a
function of resonance frequency for different radius of
curvatures. For Epsilam-10 ceramic-filled Teflon substrate
material, the flat antenna has more flat mutual coupling
coefficient than the conformal antenna.
The effect of curvature on the performance of confor-
mal microstrip antenna on cylindrical bodies for TM
mode is very important. Curvature affects the fringing field
and fringing field affects the antenna parameters. The
Equations for return loss and mutual coupling coefficient
as a function of curvature and resonance frequency are
By using these derived equations, we introduced the
results for different dielectric conformal substrates. For the
three dielectric substrates, the decreasing in frequency due
to increasing in the curvature is the trend for all materials.
We conclude that, increasing the curvature leads to
decreasing the resonance frequency. The return loss peaks
do not change for all substrate materials, but the mutual
coupling coefficient peaks are changing according to the
substrate material used.
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