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2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
A New Analytical Performance Model for a
Microstrip Printed Antenna
Ali Elrashidi
1
, Khaled Elleithy
2
and Hassan Bajwa
3
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 for TM
10
mode and TM
01
mode Epsilam-10 ceramic-filled Teflon substrate material.
Keywords: Fringing field, Curvature, effective dielectric constant and Return loss (S11), Voltage Standing Wave
Ratio (VSWR), Transverse Magnetic TM
01
and TM
01
modes.
INTRODUCTION
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 [1], 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 [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]. 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 [6]. Dielectric materials suffer from cracking due to
bending and that will affect the performance of the conformal microstrip antenna.
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 Figure 1 [7], are preferred due to easy calculation and
modeling.
______________________
1 Department of Computer Engineering, University of Bridgeport, CT 06604, USA (aelrashi@bridgeport.edu)
2 Department of Computer Engineering, University of Bridgeport, CT 06604, USA (elleithy@bridgeport.edu)
3 Department of Electrical Engineering, University of Bridgeport, CT 06604, USA (hbjwa@bridgeport.edu)
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
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
ε
reff
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 [7], as shown in Equation 1.
(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 [8], as in Equation 2.
(2)
Figure. 3. Physical and effective lengths of rectangular microstrip patch.
The effective length of the patch is L
eff
and can be calculated as in Equation 3.
L
eff
(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 antenna parameters.
h
W
L
L
W
ɛ
r
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
y
x
L
ε
r
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 dielectric 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.
Figurer 4: 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, 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[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 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
and E
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.
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
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
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].
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 (5):
(5)
Magnetic field intensity H and electric flux density D are related by Ampérés law as in Equation (6):
(6)
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 (7):
and
(7)
where is the magnetic permeability and is the electric permittivity.
By substituting Equation (7) in Equations (5) and (6), we can get:
and (8)
where is the angular frequency and has the form of: . In homogeneous medium, the divergence of
Equation (8) is:
and (9)
From Equation (9), we can get Equation (10):
or
(10)
Using the fact that, any curl free vector is the gradient of the same scalar, hence:
(11)
where is the electric scalar potential.
By letting: where A is the magnetic vector potential.
So, the Helmholtz Equation takes the form of (12):
A+
-J (12)
k is the wave number and has the form of:
, and
is Laplacian operator. The solutions of Helmholtz
Equation are called wave potentials:
(13)
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
Near Field Equations
By using the Equations number (13) and magnetic vector potential in [19], we can get the near electric and magnetic
fields as shown below:
(14)
E
and E
are also getting using Equation (9);
(15)
(16)
To get the magnetic field in all directions, we can use the second part of Equation (13) as shown below, where H
z
= 0
for TM mode:
(17)
(18)
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
z
will equal to [19], 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 (19).
(19)
Hence, the electric and magnetic field can easily be calculated as shown below:
(20)
(21)
(22)
The magnetic field intensity also obtained as shown below, where H
z
= 0:
(23)
(24)
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
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:
(25)
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
By solving this Equation, the electric field at the surface can be expressed in terms of various modes of the cavity as
[15]:
(26)
where A
nm
is the amplitude coefficients corresponding to the field modes. By applying boundary conditions [7],
homogeneous wave equation and normalized conditions for
, we can get an expression for
as shown below:
The solution of
will take the form shown below:
(27)
with
The coefficient A
mn
is determined by the excitation current. For this, substitute Equation (27) into Equation (19) and
multiply both sides of (26) by
, and integrate over area of the patch. Making use of orthonormal properties of
, one obtains:
(28)
Now, let the coaxial feed as a rectangular current source with equivalent cross-sectional area
centered
at
, so, the current density will satisfy the Equation below:
(29)
Use of Equation (29) in (28) gives:
(30)
So, to get the input impedance, one can substitute in the following Equation:
(31)
where
is the RF voltage at the feed point and defined as:
(32)
By using the previous equations, we can obtain the input impedance for a rectangular microstrip antenna conformal
in a cylindrical body as in the following Equation:
(33)
VOLTAGE STANDING WAVE RATIO AND RETURN LOSS
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
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 , as
shown below:
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
(34)
where, Z
0
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 ,
(35)
Consequently,
(36)
The return loss s
11
is related through the following Equation:
(37)
RESULTS
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 0.96 and 1.12 GHz using Epsilam-10 ceramic-filled Teflon substrate material we can use
the following dimensions; the original length is 41.5 cm, the width is 50 cm and for different lossy substrate we can
get the effect of curvature on the effective dielectric constant and the resonance frequency.
Epsilam-10 ceramic-filled Teflon is used as a substrate material for verifying the new model. The dielectric
constants for the used material are 10 with a tangent loss 0.004.
Transverse Magnetic TM
10
mode
The mathematical for input impedance, real and imaginary parts for a different radius of curvatures are shown in
Figures 5 and 6. The peak value of the real part of input impedance is almost 100 Ω at frequency 1.1126 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 1.1126 GHz represents a resonance frequency for the antenna at 20 mm radius of curvature.
VSWR is given in Figure 7. It is noted that, the value of VSWR is almost 1.9 at frequency 1.1126 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 8. We obtain a very low return loss, -11 dB, at frequency 1.1126 GHz for
radius of curvature 20 mm.
Normalized electric field for different radius of curvatures is illustrated in Figure 9. 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 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 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 10. 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 curvatures.
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
Figure 5. Real part of the input impedance as a function of frequency for different radius of curvatures.
Figure 6. Imaginary part of the input impedance as a function of frequency for different
radius of curvatures.
Figure 7. VSWR versus frequency for different radius of curvatures.
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
Figure 8. Return loss (S11) as a function of frequency for different radius of curvatures.
Figure 9. Normalized electric field for radius of curvatures 20, 65 mm and a flat
0
.
Figure 10. Normalized magnetic field for radius of curvatures 20, 65 mm
an
0
.
2012 ASEE Northeast Section Conference University of Massachusetts Lowell
Reviewed Paper April 27-28, 2012
CONCLUSION
The effect of curvature on the performance of conformal microstrip antenna on cylindrical bodies for TM
10
and TM
01
modes 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, 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 Epsilam-10 ceramic-filled Teflon substrate. 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 an 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.
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