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Effect of Curvature on a Microstrip Printed Antenna

Conformed on Cylindrical Body at Superhigh Frequencies

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 input impedance, return loss,

voltage standing wave ratio and electric and magnetic fields is introduced in this paper. These parameters are given TM

10

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

10

and TM

01

modes.

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

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.

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

L

W

ɛ

r

y

x

L

ε

r

R

d

d

d

s

s

d

z

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 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.

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

h

W

L

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 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 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)

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. 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:

(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 A

nm

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 A

mn

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)

5. 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:

(36)

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 ,

(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 s

11

is related through the following Equation:

(43)

6. 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 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

resonance frequency.

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

loss 0.0015.

Figure 5 shows the effect of curvature on resonance

frequency for a TM

01

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

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 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

curvatures.

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

of curvatures.

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

an

0

.

Figure 11. Normalized magnetic field for radius of curvatures 20, 65 mm

and a flat

0

.

7. 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 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.

REFERENCES

[1] Heckler, M.V., et al., CAD Package to Design Rectangular Probe-Fed

Microstrip Antennas Conformed on Cylindrical Structures.

roceedings of the 2003 SBMO/IEEE MTT-S International,

Microwave and Optoelectronics Conference, pp. 747-757, 2003.

[2] Q. Lu, X. Xu, and M. He, Application of Conformal FDTD Algorithm

to Analysis of Conically Conformal Microstrip Antenna. IEEE

International Conference on Microwave and Millimeter Wave

Technology, ICMMT 2008. , April 2008. 2: p. 527 – 530.

[3] Wong, K.L., Design of Nonplanar Microstrip Antennas and

Transmission Lines. 1999: John & Sons, Inc, .

[4] Josefsson, L. and P. Persson, Conformal Array Antenna Theory and

Design 1ed. 2006: Wiley-IEEE Press.

[5] Thomas, W., R.C. Hall, and D. I. Wu, Effects of curvature on the

fabrication of wraparound antennas IEEE International Symposium

on Antennas and Propagation Society,, 1997. 3: p. 1512-1515.

[6] J. Byun, B. Lee, and F.J. Harackiewicz, FDTD Analysis of Mutual

Coupling between Microstrip Patch Antennas on Curved Surfaces.

IEEE International Symposium on Antennas and Propagation Society,

1999. 2: p. 886-889.

[7] Balanis, C.A., AntennaTheory. 2005, New York: John Wiley & Sons.

[8] Pozar, D., Microstrip Antennas. IEEE Antennas and Propagation

Proceeding, 1992. 80(1).

[9] Krowne, C.M., Cylindrical-Rectangular Microstrip Antenna. IEEE

Trans. on Antenna and Propagation, 1983. AP-31: p. 194-199.

[10] Q. Wu, M. Liu, and Z. Feng, A Millimeter Wave Conformal Phased

Microstrip Antenna Array on a Cylindrical Surface. IEEE

International Symposium on Antennas and Propagation Society,

2008: p. 1-4.

[11] J. Ashkenazy, S. Shtrikman, and D. Treves, Electric Surface Current

Model for the Analysis of Microstrip Antennas on Cylindrical Bodies.

IEEE Trans. on Antenna and Propagation, 1985. AP-33: p. 295-299.

[12] K. Luk, K. Lee, and J. Dahele, Analysis of the Cylindrical-

Rectangular Patch Antenna. IEEE Trans. on Antenna and

Propagation, 1989. 37: p. 143-147.

[13] S. Lei, et al., Anti-impact and Over-loading Projectile Conformal

Antennas for GPS,. EEE 3rd International Workshop on Signal

Design and Its Applications in Communications, 2007: p. 266-269.

[14] Kolev, N.Z., Design of a Microstip Conform GPS Patch Antenna.

IEEE 17th International Conference on Applied Electromagnetic and

Communications, 2003: p. 201-204.

[15] A. Elrashidi, K. Elleithy, and Hassan Bajwa, “The Fringing Field and

Resonance Frequency of Cylindrical Microstrip Printed Antenna as a

Function of Curvature,” International Journal of Wireless

Communications and Networking (IJWCN), Jul.-Dec., 2011.

[16] A. Elrashidi, K. Elleithy, and Hassan Bajwa, “Effect of Temperature

on the Performance of a Cylindrical Microstrip Printed Antenna

for TM

01

Mode Using Different Substrates,” International Journal of

Computer Networks & Communications (IJCNC), Jul.-Dec., 2011.

[17] A. Elrashidi, K. Elleithy, and Hassan Bajwa, “The Performance of a

Cylindrical Microstrip Printed Antenna for TM

10

Mode as a Function

of Temperature for Different Substrates,” International Journal of

Next-Generation Networks (IJNGN), Jul.-Dec., 2011.

[18] S. M. Wentworth, Applied Electromagnetics, John Wiley & Sons,

Sons, New York, 2005.

[19] R. F. Richards, Time-Harmonic Electromagnetic Fields, New York:

McGraw-Hill, 1961.

[20] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna

Design Handbook, Aetech House, Boston, 2001