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Return Loss and Mutual Coupling Cofficient of a

Microstrip Printed Antenna Array Conformed on a

Cylindrical Body for TM

01

Mode

Ali Elrashidi

1

, Khaled Elleithy

2

, Hassan Bajwa

3

1

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

(aelrashi@bridgeport.edu)

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

(elleithy@bridgeport.edu)

3Department of 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 return loss and mutual cou-

pling coefficient as a function of curvature for two element array antenna is introduced in this paper. These parameters are

given for TM

01

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 and Return loss (S11), mutual coupling coefficient

(S12), Transverse Magnetic TM

01

mode.

1. Introduction

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 [1]-[3]. Low profile, low

weight, low cost and its ability of conforming to curve

surfaces [4], 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 [7]. 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 [8]-[10].

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 [11].

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 [12], 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

L

W

ɛ

r

y

x

L

r

R

d

d

d

s

s

d

z

substrate and in the air. Thus an effective dielectric con-

stant ε

reff

is to be introduced. The effective dielectric con-

stant ε

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 [12], as shown in Equation 1.

(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 [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

= 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

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 [13]. 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]

as:

(4)

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[9]

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[15]. Kwai et al. [16]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

10

and TM

01

modes is

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

θ

and

E

φ

as a functions of Hankel function of the second kind

H

p

(2)

. 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

h

W

ΔL

L

ΔL

is designed and implemented by using perturbation theory

is introduced by Sun L., Zhu J., Zhang H. and Peng X [17].

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 con-

formal microstrip GPS patch antenna [18]. 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 [19]. Also, the effect of curvature on the

performance of a microstrip antenna as a function of tem-

perature for TM

01

and TM

10

is introduced in [21], [21].

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 [22].

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. [22]. The coupling between elements is not consid-

ered in [22]. The authors calculated the total electric field

strength for an array of N elements using Equation (5).

(5)

where, E

i

represents the field strength of number i radiator

and φ

i

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 [23].

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 [54]. 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 [25]. 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

obtained.

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 [26]. The authors calculated the mutual

impedance and mutual coupling coefficient using a

moment of method technique [27], [28]. 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 [29]. 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-

tions.

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 [30]. 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 [31]. 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 [32]. 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 [33]. 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

The input impedance is defined

appropriate components of the electric to magnetic fields at

e is a function of the feeding

position as we will see in the next few lines [19].

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:

(6)

By solving this Equation, the electric field at the sur-

face can be expressed in terms of various modes of the

cavity as [19]:

(7)

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:

(8)

2. vanishes at the both edges for the width W:

(9)

3. should satisfy the homogeneous wave

Equation :

(10)

4. should satisfy the normalized condition:

(11)

Hence, the solution of will take the form shown be-

low:

(12)

with

The coefficient A

mn

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:

(13)

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

Equation below:

(14)

Use of Equation (14) in (13) gives:

(15)

So, to get the input impedance, one can substitute in

the following Equation:

(16)

where is the RF voltage at the feed point and defined

as:

(17)

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:

(18)

Z

s

Z

L

b

1

a

1

b

2

a

2

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 [11].

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

Equation 19.

Figure 5. Equivalent network of two antenna array.

where a

n

and b

n

represent the forward and reverse volt-

age wave amplitude at the nth port respectively.

The mutual impedance formulation is shown in

Equation (20) [11].

(19)

(20)

Hence, the mutual coupling coefficient, S

12

, can be

calculated as in Equation (21) [26].

(21)

where Z

0

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

11

and Z

22

will give the same value and Z

12

and Z

21

are the

same.

Hence, the value of Z

12

and Z

21

are given by Equa-

tion (22).

(22)

and the value of Z

11

and Z

22

are given by Equation (23).

(23)

By using Equations (15) and (18), we can get Equa-

tion (24) for Z

21

as follow:

(24)

and by substitute in Equation (21) the mutual coupling

coefficient can be calculated.

6. Results

For the range of GHz, the dominant mode is TM

01

for

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

the range 2.5 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 con-

stant 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 respec-

tively with a tangent loss 0.0015, 0.002 and 0.004 respec-

tively.

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 [34]. We

obtain a return loss, -36 dB, at frequency 2.1563 GHz for

radius of curvature 20 mm, 2.158 GHz at 65 mm and

2.1595 GHz for a flat antenna.

Figure 6. Return loss (S11) as a function of frequency for different radius

of curvatures..

Network

Figure 7. Mutual coupling coefficient, S12, as a function of resonance

frequency for different values of curvatures for TM

01

mode

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 -8 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

loss 0.002.

Return loss (S11) is illustrated in Figure 8. We obtain

a very low return loss, -50 dB, at frequency 1.935 GHz for

radius of curvature 20 mm, 1.937 GHz at 65 mm and 1.938

GHz for a flat antenna. The return loss value, -50 dB, is

obtained for different radius of curvature.

Figure 8. Return loss (S11) as a function of frequency for different radius

of curvatures.

Figure 9. Mutual coupling coefficient, S12, as a function of resonance

frequency for different values of curvatures for TM

01

mode.

Mutual coupling coefficient is shown in Figure 9. The

maximum mutual coupling is obtained at the minimum

return loss for the same resonance frequency. Mutual cou-

pling coefficients are increased in the frequency with de-

creasing the radius of curvature and the peaks are -47.5

dB at radius of curvature 20 mm, -48.5 dB at 65 mm and

-46 dB for the flat antenna.

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

0.004.

Return loss (S11) is illustrated in Figure 10. We obtain

a return loss, -4.3 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

01

mode

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.

7. Conclusion

The effect of curvature on the performance of confor-

mal microstrip antenna on cylindrical bodies for TM

01

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

derived.

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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