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Effect of Curvature on the Performance of

Cylindrical Microstrip Printed Antenna

for TM01 mode Using Two Different Substrates

Ali Elrashidi

Department of Computer and

Electrical Engineering

University of Bridgeport

Bridgeport, CT, USA

aelrashi@bridgeport.edu

Khaled Elleithy

Department of Computer and

Electrical Engineering

University of Bridgeport

Bridgeport, CT, USA

elleithy@bridgeport.edu

Hassan Bajwa

Department of Electrical

Engineering line 2: name of

University of Bridgeport

Bridgeport, CT, USA

hbjwa@bridgeport.edu

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 TM01

mode and using two different substrate materials RT/duroid-

5880 PTFE and K-6098 Teflon/Glass. Experimental results for

RT/duroid-5880 PTFE substrate are also introduced to

validate the new model.

Keywords: Fringing field, Curvature, effective dielectric constant

and Return loss (S11), Voltage Standing Wave Ratio (VSWR),

Transverse Magnetic TM01 mode.

I. 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.

II. 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.

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

L

W

ɛr

y

x

L

εr

R

d

d

d

s

s

d

z

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 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 [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 Leff and can be calculated

as in Equation 3. Leff (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.

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 TM10 and TM01 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

h

W

L

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 Hp(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 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 TM01 and TM10 is introduced in [16], [17].

III. 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 is the magnetic permeability and is the electric

permittivity.

By substituting Equation (4) in Equations (2) and (3), we

can get:

and (5)

where is the angular frequency and has the form of:

. In homogeneous medium, the divergence of

Equation (2) is:

and (6)

From Equation (5), we can get Equation (7):

or (7)

Using the fact that, any curl free vector is the gradient of the

same scalar, hence:

(8)

where 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:

(10)

A) Near Field Equations

By using the Equations number (10) and magnetic vector

potential in [19], we can get the near electric and magnetic

fields as shown below:

(12)

E and E are also getting using Equation (7);

(13)

(14)

To get the magnetic field in all directions, we can use the

second part of Equation (10) as shown below, where Hz= 0

for TM mode:

(15)

(16)

B) 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 kz 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 (17).

(17)

Hence, the electric and magnetic field can easily be

calculated as shown below:

(18)

(19)

(20)

The magnetic field intensity also obtained as shown below,

where Hz = 0:

(21)

(22)

IV. 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 Zin 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 expressed in terms of various modes of the cavity as [15]:

(24)

where Anm 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

Equation :

(27)

4. should satisfy the normalized condition:

(28)

Hence, the solution of will take the form shown below:

(29)

with

The coefficient Amn 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 ,

one obtains:

(30)

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:

(31)

Use of Equation (31) in (30) gives:

(32)

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 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

following Equation:

(35)

V. 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 Vi and reflected voltage wave

amplitude Vr, and by using the definition of a voltage

reflection coefficient at the input terminals of the antenna ,

as shown below:

(36)

where, Z0 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 ,

(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 s11 is related through the following Equation:

(43)

VI. RESULTS

For the range of GHz, the dominant mode is TM01 for

h<<W which is the case. Also, for the antenna operates at

the ranges 2.15 and 1.93 GHz for two different substrates

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.

Two different substrate materials RT/duroid-5880 PTFE and

K-6098 Teflon/Glass are used for verifying the new model.

The dielectric constants for the used materials are 2.2 and

2.5 respectively with a tangent loss 0.0015 and 0.002

respectively.

A) RT/duroid-5880 PTFE Substrate

The mathematical and experimental results 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 250 Ω at

frequency 2.156 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 2.156 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 1.4 at frequency 2.156 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, -36 dB, at frequency 2.156 GHz for radius

of curvature 20 mm.

Figure 5. Mathimatical and experimental 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 9. Normalized electric field is plotted

for θ from 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 from 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 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.

Figure 6. Mathimatical and experimental imaginary part of the input

impedance as a function of frequency for different radius of curvatures.

Figure 7. Mathimatical and experimental VSWR versus frequency for

different radius of curvatures.

Figure 8. Mathimatical and experimental 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 abd

a flat antenna at θ=0:2π and φ=00.

Figure 10. Normalized magnetic field for radius of curvatures 20, 65 mm

abd a flat antenna at θ=0:2π and φ=00.

B) K-6098 Teflon/Glass Substrate

The real part of input impedance is given in Figure 11 as a

function of curvature for 20 and 65 mm radius of curvature

compared to a flat microstrip printed antenna. The peak

value of a real part of input impedance at 20 mm radius of

curvature occurs at frequency 1.935 GHz at 330 Ω

maximum value of resistance. The imaginary part of input

impedance, Figure 12, is matching with the previous result

which gives a zero value at this frequency. The resonance

frequency at 20 mm radius of curvature is 1.935 GHz,

which gives the lowest value of a VSWR, Figure 13, and

lowest value of return loss as in Figure 14. Return loss at

this frequency is -50 dB which is a very low value that leads

a good performance for a microstrip printed antenna

regardless of input impedance at this frequency.

The normalized electric field for K-6098 Teflon/Glass

substrate is given in Figure 15 at different radius of

curvatures 20, 65 mm and for a flat microstrip printed

antenna.

Normalized electric field is calculated at θ equal to values

from 0 to 2π and φ equal to zero. At radius of curvature

20 mm, the radiation pattern of normalized electric field is

wider than 65 mm and flat antenna, radiation pattern angle

is almost 1200, and gives a high value of electric field

strength due to effective dielectric constant.

The normalized magnetic field is given in Figure 16, for the

same conditions of normalized electric field. Normalized

magnetic field is wider than normalized electric field for

20 mm radius of curvature; it is almost 1700 for 20 mm

radius of curvature. So, for normalized electric and

magnetic fields, the angle of transmission is increased as a

radius of curvature decreased.

Figure 11. Real part of the input impedance as a function of frequency for

different radius of curvatures.

Figure 12. Imaginary part of the input impedance as a function of

frequency for different radius of curvatures.

Figure 13. VSWR versus frequency for different radius of curvatures.

Figure 14. Return loss (S11) as a function of frequency for different radius

of curvatures.

Figure 15. Normalized electric field for radius of curvatures 30, 50 and 70

mm at θ=0:2π and φ=00.

Figure 16. Normalized magnetic field for radius of curvatures 20, 65 mm

abd a flat antenna at θ=0:2π and φ=00.

CONCLUSION

The effect of curvature on the performance of conformal

microstrip antenna on cylindrical bodies for TM01 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,

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

different dielectric conformal substrates. For the two

dielectric substrates, the decreasing in frequency due to

increasing in the curvature is the trend for all materials 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.

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