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The Effect of Curvature on Resonance Frequency, Gain,

Efficiency and Quality Factor of a Microstrip Printed

Antenna Conformed on a Cylindrical Body

for TM

10

mode Using Different Substrates

Ali Elrashidi

1

, Khaled Elleithy

2

, Hassan Bajwa

3

1

Department of Computer and Electrical Engineering, University of Bridgeport, Bridgeport, CT 06604, USA

(aelrashi@bridgeport.edu)

2Department of Computer and Electrical Engineering, University of Bridgeport, Bridgeport, CT 06604, USA

(elleithy@bridgeport.edu)

3Department of Computer and Electrical Engineering, University of Bridgeport, Bridgeport, CT 06604, 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 resonance frequency, gain,

efficiency and quality factor as a function of curvature is introduced in this paper. These parameters are given for TM

10

mode and using three different substrate materials RT/duroid-5880 PTFE, K-6098 Teflon/Glass and Epsilam-10

ceramic-filled Teflon.

Keywords Fringing field, curvature, effective dielectric constant, gain, efficiency, Q-factor, and Transverse Magnetic

TM

10

mode.

1. 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.

2. Background

Conventional microstrip antenna has a metallic patch print-

ed 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 mod-

eling.

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.

L

W

ɛ

r

y

x

L

ε

r

R

d

d

d

s

s

d

z

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

reff

is to be introduced. The effective dielectric constant

reff

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.

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.

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.

Figurer 4. Geometry of cylindrical-rectangular patch antenna[9]

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 func-

tions of Hankel function of the second kind H

p

(2)

. The input

impedance and Q factors are also calculated under the same

h

W

L

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 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 Equations, 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)

3.1 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 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 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 (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 H

z

= 0:

(21)

(22)

4. Quality Factor, Gain, and

Efficiency

Quality factor, gain, and efficiency are antenna fig-

ures-of-merit, which are interrelated, and there is no com-

plete freedom to independently optimize each one [7].

4.1 Quality Factor

The quality factor is a figure-of-merit that is representative

of the antenna losses. Typically there are radiation, conduc-

tion, dielectric and surface wave losses. Therefore the total

quality factor Q

t

is influenced by all of these losses and is

written as:

(23)

where

Q

t

= total quality factor

Q

rad

= quality factor due to radiation (space wave) losses

Q

c

= quality factor due to conduction (ohmic) losses

Q

d

= quality factor due to dielectric losses

Q

sw

= quality factor due to surface wave losses

Q for any of the quantity on the right-hand side can be rep-

resented as:

(24)

where, the stored energy at resonance W

T

is the same,

independent of the mechanism of power loss. Therefore,

Equation (23) can be expressed as:

(25)

The stored energy is determined by the field under the patch,

and is expressed as [20]:

(26)

We can get the electric field under the patch using Equation

(14):

(27)

So the total energy stored is determined by substitute

Equation (27) in Equation (26) as follow:

(28)

By solving this Equation, one can get the following:

(29)

The dielectric loss is calculated from the dielectric field

under the patch as in [20]:

(30)

or as a function of total energy as:

(31)

where is the loss tangent of the dielectric material. So;

the dielectric loss is calculated using Equations (29) and

(31). The power loss due to the conduction is calculated

from the magnetic field on the patch metallization and

ground plate,

(32)

where R

s

is the surface resistivity of the conductors given

by

and σ is the conductivity of the conductor. By

using the approximation in [19] for the electric and magnetic

field

, and by using the height equal to h, one can

get the following approximate Equation:

(33)

For very thin substrate, which is the case, the losses due to

surface wave are very small and can be neglected [7]. The

power radiated from the patch P

r

can be determined by in-

tegrating the radiation field over the hemisphere above the

patch. This is,

(34)

By using Equations (13)-(14), we can get an expression for

E

and E

as shown below:

(35)

(36)

Now the radiated power will take the form:

(37)

4.2 Antenna Gain

The gain is the ratio of the output power for an an-

tenna to the total input power to the antenna. The input

power to the antenna is the total power including, radiated

power and the overall losses power. So, the gain can be

represented as follows:

(38)

4.3 Antenna Efficiency

Antenna efficiency is defined as the ratio of the total radi-

ated power to the total input power. The input power to the

antenna is the total power including, radiated power and the

overall losses power. So, the gain can be represented as

follows:

(39)

The total input power includes the power lost due to con-

duction, dielectric and surface wave losses and the radiated

power from the antenna.

5. Results

For the range of GHz, the dominant mode is TM

10

for

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

the ranges 2.49, 2.24 and 1.11 GHz for three 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.

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

respectively with a tangent loss 0.0015, 0.002 and 0.004

respectively.

5.1 RT/duroid-5880 PTFE Substrate

The mathematical and experimental results for input im-

pedance, real and imaginary parts, VSWR and return loss

for a different radius of curvatures are given in [15].

Normalized electric field for different radius of curvatures

is illustrated in Figure 5. 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 6.

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 5. Normalized electric field for radius of curvatures 20, 65 mm and

a flat antenna at θ=0:2π and φ=0

0

.

Figure 6. Normalized magnetic field for radius of curvatures 20, 65 mm

and a flat antenna at θ=0:2π and φ=0

0

.

Table 1 shows the resonance frequency, quality factor,

efficiency, and gain for TM

10

mode. The efficiency is

increased with increasing curvature to be 96.55% at radius

of curvature 10 mm. The same behavior for the gain with

radius of curvature, the gain is 9.3 dB at radius of curvature

10 mm. Increasing the quality factor with decreasing the

curvature is due to the decreasing of the radiated power

from the antenna. So, the efficiency will decreased with

increasing the quality factor, due to decreasing in the

radiated power, and the same for the gain.

Table 1. Resonance frequency, quality factor, efficiency, and gain for TM

10

for RT/duroid-5880 PTFE substrate

R

f

r

(GHz)

Q

tot

η

G(dB)

Flat

2.4962

1282.27

92.24

4.11

200 mm

2.4957

1281.902

95.07

7.82

50 mm

2.4945

1281.622

95.73

8.59

10 mm

2.4892

1281.149

96.556

9.3

5.2 K-6098 Teflon/Glass Substrate

The normalized electric field for K-6098 Teflon/Glass

substrate is given in Figure 7 at different radius of

curvatures 20, 65 mm and for a flat microstrip printed

antenna.

Normalized electric field is calculated at equal to values

from zero to 2π and 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 120

0

, and gives a high value of electric field

strength due to effective dielectric constant.

The normalized magnetic field is given in Figure 8, 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 170

0

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 7. Normalized electric field for radius of curvatures 20, 65 mm and

a flat antenna at θ=0:2π and φ=0

0

.

Figure 8. Normalized magnetic field for radius of curvatures 20, 65 mm

and a flat antenna at θ=0:2π and φ=0

0

.

As in Table 2, the resonance frequency is increased with

decreasing the curvature of microstrip printed antenna. The

total quality factor is increased by a small value for radius

of curvature 10, 50, and 200 mm. The efficiency and gain

are increased with increasing radius of curvature.

Table 2. Resonance frequency, quality factor, efficiency, and gain for TM

10

for K-6098 Teflon/Glass substrate

R

f

r

(GHz)

Q

tot

η

G(dB)

Flat

2.2404

1083.124

90.03

5.81

200 mm

2.2400

1082.8879

90.77

5.97

50 mm

2.2390

1082.76799

95.66

8.82

10 mm

2.2345

1082.348

96.169

9.3

5.3 Epsilam-10 Ceramic-Filled Teflon

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.

Figure 9. Normalized electric field for radius of curvatures 20, 65 mm and

a flat antenna at θ=0:2π and φ=0

0

.

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

and a flat antenna at θ=0:2π and φ=0

0

.

Table 3 shows the resonance frequency, quality factor,

efficiency, and gain for TM

10

mode. The efficiency is

increased with increasing the radius of curvature. The gain

has the same behavior for decreasing radius of curvature.

Table 3. Resonance frequency, quality factor, efficiency, and gain for TM

10

for Epsilam-10 ceramic-filled Teflon substrate

R

f

r

(GHz)

Q

tot

η

G(dB)

Flat

1.1149

663.795

86.739

4.81

200 mm

1.1148

663.3924

88.122

5.97

50 mm

1.1143

663.27789

90.758

8.59

10 mm

1.1126

661.3518

92.92

9.3

6. Conclusion

The effect of curvature on the performance of conformal

microstrip antenna on cylindrical bodies for TM

10

mode is

studied in this paper. Curvature affects the fringing field and

fringing field affects the antenna parameters. The Equations

for resonance frequency, gain, efficiency and quality factor

as a function of curvature is introduced in this paper. These

parameters are given for TM

10

mode and using three

different substrate materials RT/duroid-5880 PTFE, K-6098

Teflon/Glass and Epsilam-10 ceramic-filled Teflon.

For the three dielectric substrates, the decreasing in

resonance frequency due to increasing in the curvature leads

to decreasing in quality factor and on the other hand, leads

to increasing in the efficiency and the gain of the microstrip

printed antenna. The radiation pattern for electric and

magnetic fields due to increasing in curvature is also

increased and be more wider.

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