168

CHAPTER 6

DESIGN OF MICROSTRIP ANTENNA AND ARRAY

CONFORMABLE ON CYLINDRICAL SURFACE AND

THEIR PERFORMANCE ANALYSIS

Analysis and design of the planar microstrip antenna have been discussed in the

preceding chapters. Planar microstrip antenna by and large can be considered to have

reached its maturity. Development of conformable antennas on non planar surfaces

lags behind vis-à-vis planar microstrip antennas. Relatively non planar or microstrip

antennas conformable to non planar surfaces are at present topic of research.

Specifically, theoretical works reported in literature pertain to conformable microstrip

antennas on non planar regular surfaces like the cylinder, the sphere and the cone.

The need for conformal antennas is more pronounced for the large-sized apertures that

are necessary for functions like military airborne surveillance radars. A conformal

microstrip antenna on a cylindrical surface with low profile has distinct advantage for

applications related to fighter and spacecraft.

Aircraft surfaces have been approximated as either cylindrical or cylindrical sectored

geometry e.g. the underbelly of an aircraft. The design approach utilizes the

application of elements of conformal mapping to transform cylindrical surface to

planar surface. There is a need to realize an antenna on planar surface with one to one

correspondence with the antenna on non planar surface for accurate analysis and

designing purpose. Transformed rectangular microstrip patch antenna has been

simulated using ADS Momentum software. The results obtained are in accordance

with the behavior of the conformal antenna.

Work has been carried out to analyze the effect of mutual coupling, due to the inter

element spacing both in E and H plane, on antenna parameters. Design and analysis of

planar array, array conformable on cylindrical surface and transformed microstrip

antenna array using conformal mapping technique has concluded that control of

antenna parameters is achieved through mutual coupling.

169

Major real world shapes can be approximated by a cylindrical surface or a cylindrical

sector. In addition uniformity in its plane helps in ease of analysis. Low profile

microstrip patch antenna conforming on the exterior of a cylindrical surface is

considered as conformable antenna on a regular surface. Microstrip antenna arrays

can also be conformed to curved aerodynamic surfaces of supersonic aircraft or

missiles and modeled approximately in the shape of a cylinder.

Design and analysis of radiating the microstrip patch antenna and array mounted on

cylindrical surface is considered in this chapter. Design procedure involves

parameters related to cylindrical microstrip elements to realize the desired resonant

frequency, input impedance, radiation patterns etc. A microstrip line has been chosen

to feed the patch antenna. The feed design is simplest in geometry, provides very

small stray radiation from the strip line. Computer aided design of microstrip antennas

and arrays and its analysis are based on efficient and accurate numerical methods.

Techniques exist to analyze conformable microstrip antennas on electrically small

cylindrical surfaces. Efficient analytical and numerical tools need to be developed for

microstrip antennas conformable to electrically large cylinders with identically shaped

substrates for theoretical analysis and practical manufacturing. The chapter is devoted

to conformal mapping to the planar surface of antenna and arrays mounted on

cylinder, full-wave analysis of cylindrical microstrip using moment-method. Design

and FEKO simulation of microstrip patch antenna and arrays on cylindrical surface,

finally analysis of the effect of mutual coupling in both planar and cylindrical surface.

Analysis of cylindrical vis-à-vis planar antenna in terms of effect on antenna

parameters due to mutual coupling has been carried out.

6.1 Necessity of Modeling Conformal Antenna for Aircraft

A modern aircraft has many antennas protruding from the aircraft surface (Figure 1.10

refers), causing considerable drag and thus increasing fuel consumption. The purpose

of the study is to build the antenna so that it becomes integrated with the structure and

does not cause extra drag. The shape can be some part of an airplane, for example

underbelly of an aircraft or the radome. They can be manufactured using modern

printed circuit techniques and can be integrated with different systems. There are

operational specific applications related to fighter aircraft that require an antenna to

conform on its surface viz. structural, aerodynamic, and space limitation compulsions.

170

Other main reasons for antenna design to conform to the aircraft surface are

particularly electromagnetic requirement such as antenna beam shape and/or angular

coverage. Microstrip antenna and arrays conformed to curved surfaces viz.

aerodynamic surfaces like supersonic aircraft or missiles, can be modeled

approximately in the shape of a cylinder.

A simpler technique is formulated to analyze microstrip antenna application on a non

planar surface conforming to the parent structure of a vehicle in use is undertaken in

this chapter. The approach involves transformation of a patch on cylindrical surface to

an equivalent planar rectangular microstrip antenna using conformal mapping

technique. The proposed approach analyzes the rectangular microstrip antenna on

cylindrical surface by taking into account the effect of curvature on the antenna

performance. Performance of realized planar patch antenna conforms to the conformal

antenna on the cylindrical surface. A relationship showing effect of curvature on the

resonant frequency has been realized. 6.2 Analysis of Cylindrical Microstrip Patch Antenna Array antennas with radiating elements on the surface of a cylinder, a sphere, and a

cone without the shape being dictated by, for example, aerodynamic or similar

reasons, are also called conformal arrays. As per IEEE Standard Definition of Terms

for Antennas (IEEE STD 145-1993), “Conformal antenna (conformal array) is

defined as an antenna [an array] that conforms to a surface whose shape is determined

by considerations other than electromagnetic; for example, aerodynamic or

hydrodynamic.” Strictly speaking, the definition includes also planar arrays, if the

planar shape is determined by considerations other than electromagnetic. This is,

however, not common practice. Usually, a conformal antenna is cylindrical, spherical,

or some other shape, with the radiating elements mounted on or integrated into the

smoothly curved surface. In the structure of the conformal microstrip antenna, the surface of the metal cylinder

is used as curved ground plane. Full wave analysis is the most accurate approach to

analyze the characteristics of cylindrical patch antennas while the cavity model

approach is suitable only for very thin substrate cases. The approach utilizing full

wave solution is computationally inefficient & time consuming [104]. The cavity

171

model though provides closed form solutions and is not time consuming, but it does

not incorporate the effect of change of curvature in the cylinder [105]. Thus, a relation

for the resonant frequency of a cylindrical rectangular microstrip patch antenna needs

to be determined analytically, that not only takes into account the variation of the

resonant frequency with variation in curvature of cylinder, but is also more

computationally efficient in comparison to the full wave analysis.

⎟⎠⎞

⎜⎝⎛

⎥⎦

⎤⎢⎣

⎡−=

bznmEE

2cos)(

2cos 1

10

πϕϕθπ

ρ ... (6.1)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

22

1 222 bn

amcf

rmn θε

... (6.2)

The rectangular patch antenna of straight edge length 2b and the curved edge (arc) of

length 2(a+h)θ1 on a cylindrical ground plane of radius ‘a’ is shown in Figure 6.1(a).

The dielectric substrate of thickness h having relative permittivity εr, subtends an

angle 2θ1 on the curved edges surface of the cylinder. Assuming that for thickness

h<<λ, only TM modes exist, the electric field under the patch in the source-free case

and the resonant frequency may be expressed as given in equations (6.1) and (6.2).

( ) ( ) ( )( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −×⎟

⎠⎞

⎜⎝⎛−= ∑

∞

=′

+−

2cos

sin

coscos

cossin2

:

0 02

1cos

0

020

10

0πϕ

θ

θεθθ

πρθ

ϕ pakH

pje

kbk

rahEE

TM

p p

pbrjk

0≈θE ... (6.3) :01TM

( )( )( )

( )⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −×

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

∞

=

+−

2cossin

sinsin

cos2

cos2

1

1

0 02

cos2

10 0πϕ

θθ

θε

θ

θεπ

πθ θ

θ pp

pakHj

er

hEjEp p

ppbrjkr

0≈ϕE ... (6.4)

The equivalent magnetic currents along edges of the curved patch are obtained

from nEM ˆˆ ×= ρρ . These magnetic currents radiate in the presence of cylindrical

surfaces. The far field can be calculated respectively for the TM10 and TM01 modes as

given by equations (6.3) and (6.4)

172

The TM10 mode vis-à-vis TM01 is very sensitive to changes in curvature of the

cylinder [106], [107]. J. S. Dahele et. al [106] have shown that hybrid modes are

generated for the thick substrate, whereas for thin substrates, modes that result in are

pure TE or TM modes.

Figure 6.1 (a) & (b): Cylindrical Rectangular Microstrip Patch Antenna

( ) ⎥⎦⎤

⎢⎣⎡

⎥⎦

⎤⎢⎣

⎡−= ∑ b

znmCjEnm

mn 2cos

2cos

,0

00

πϕϕθπωµρ … (6.5)

The modal amplitudes Cmn are defined as:

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+∆∆

−=

00022 )(4

sin2

cos2

cos)(4 θ

ππθϕπ

θ hawmc

bznm

bhakkwC ffnm

mneffmn … (6.6)

where 0,20,1 ≠=∆==∆ kforandkfor kk .

The geometry of a typical cylindrical rectangular microstrip patch antenna is shown in

Figure 6.1(a) and Figure 6.1 (b), with the dimensions of the patch defined in z andϕ

planes respectively by b2 and 02θ . Similarly in ϕ plane, 0ϕ indicates the position of

the patch. In this approach, the region underneath the patch can be modeled as a

cavity bounded by four magnetic walls and two electric walls. The E-field in the

cavity has only a ρ component. For the thin substrate that is with ah << , the E-field

is independent of ρ . When the feed is modeled by a current density, with an effective

width w , the field ρE in the cavity can be determined by a summation over all values

of cavity modes m and n [101] as given in equation (6.5).

⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=b

nha

mkmn 2)(2 0

πθ

π … (6.7)

173

( )effreff jkk δε −= 10 … (6.8)

xxxc )sin()(sin = … (6.9)

000 εµω=k … (6.10)

Further, other parameters are obtained vide equations (6.7) to (6.10),

where 0µ and 0ε are the permeability and the permittivity of free space respectively.ω

and rε respectively represent the radial frequency and relative permittivity of the

substrate. 6.3 Transformation of a Cylindrical Surface to a Planar Surface

using Conformal Mapping

Conformal Mapping preserves angles and preserves shape for infinitesimal figures.

The transformation y=f(x) will be conformal at x0 if oriented angles between curves

through x0 is preserved as well as their direction. Let the cylindrical surface be

represented by S as shown in Figure 6.1(b). A cylindrical surface is a developable

surface; the length of a curve on the cylindrical surface is the same as the length of a

curve on a plane. 222 dydxds += … (6.11)

Parameterization of the cylindrical surface may therefore be expressed by conformal

mapping of the plane with the transformation of the space 32 RR ⊂ with z = 0, can be

expressed as given in equation (6.11). Since lengths may be computed as the integral

of the square root of the first fundamental form, the first fundamental form of a

cylindrical surface is same as that of the plane. Further, for the cylinder with the

coordinates ,sin,cos ϕϕ == yx and z = 0, this isometric mapping is conformal, as

the first fundamental forms of surfaces are equal [108], [109].

Let us consider a point P = f(x, y) on the cylindrical surface S. The conformal

mapping between two surfaces S1 and S2, shown in Figure 6.2, can be defined by a

Diffeomorphism φ i.e. S1→ S2 is said to be conformal if, whenever the angle

subtended by the patch along the central axis φ takes two intersecting curves α1 (t) and

α2 (t) on S1, and maps them to curves γ1 (t) and γ2 (t) on S2 [4]. The angle of

intersection between α1 (t) and α2 (t) along with the sense of angle is equal to the

angle of intersection between γ1 (t) and γ2 (t) along with same sense of angle. In other

174

words, φ is conformal if it preserves angles. Alternatively, Diffeomorphism φ: S1→ S2

is conformal, if and only if for any surface patch δ1 on S1, the first fundamental forms

of δ1 and φ (δ1) are proportional.

Lyx ≤≤+≤≤ 0,11 ϕϕϕ … (6.12)

*For the cylindrical surface shown in Figure 6.3, S1 corresponds to a circular cylinder

of radius r2 with the transformation δ1(x, y) into rectangular patch on the cylindrical

surface defined by equation (6.12)

δ2 (x, y) = δ2 (x, y, 0) … (6.13)

With S2 being in the xy-plane, we can define δ2(x, y) as the patch upon this plane

given by the parametric equation:

Figure 6.2: Conformal Mapping

Figure 6.3: A Cylindrical Surface

Figure 6.4 shows the transformation from a cylindrical to a planar surface. The

transformed patch has axial length L and circumferential width W ′ = r2 φ. As stated

175

earlier, the patch subtends an angle of φ along the central axis. The patch edge

subtends an angle φ1 from the x axis.

*The work reported in this chapter is based on the following research paper contributions: [110] Gupta S.D., Srivastava M.C., Singh A., “Design and Performance Analysis of Cylindrical Microstrip Antenna and Array using Conformal Mapping Technique”, International Journal of Communication Engineering Applications (IJCEA), vol. 02, Issue 03, pp. 166-180, Jul. 2011.

Figure 6.4: Transformation from Cylindrical to Planar

22

22 ryx =+ … (6.14)

Width W = ( ) ϕϕϕϕ 2112 )( rr =−+ and axial length L . … (6.15)

The circumferential dimension will extend from the angle φ1 to φ1+ φ, along the x-

axis. The Cartesian equation of the cylindrical surface can thus be written as given by

equation (6.14). It is seen that the first fundamental forms of a cylindrical surface and

a plane are same hence the above theorem holds true and the mapping will be

conformal. Thus the rectangular patch on a cylindrical surface when transformed into

its planar equivalent has the dimensions as shown in equation (6.15)

( ) )16.6...(2

1

121 GGZin ±

=

176

where 2

01 90

1⎟⎟⎠

⎞⎜⎜⎝

⎛×=

λWG for 1

0

<<λW where W is the transformed patch width.

and ( ) θθθθ

θ

π

π

dLkJ

Wk

G 300

2

0

0

212 sinsincos

cos2

sin

1201

∫⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

= … (6.17)

Expressions used for calculation of resonant edge input impedance in respect of

microstrip patch antenna are as given by equation (6.16) [54], where k0 = 2π/λ0 and λ0,

the wavelength corresponds to frequency fr and Jo is Bessel function of first kind and

zero order.

6.4 Transformed Planar Patch Antenna Design

A conformal microstrip patch antenna mounted on a cylindrical surface. The

conductor thickness chosen is 8µm. The cylindrical ground plane has been considered

to be of radius r1 = 20 cm. The cylindrical ground plane is covered with a substrate

RT Duroid 5870 (with loss tangent tan δ= 0.0012). The substrate is having relative

permittivity εr = 2.32 and thickness h= 0.0795 cm. As shown in this figure, the two

axial lines of the patch antenna passing through centre of cylinder to the edges of

patch are at angles 480 and 450. Hence the angle subtended φ = (480- 450) =30.

Thus, the axial length of patch = 0472.1)20(180

3=××π cm

A single microstrip patch antenna using dielectric substrate RT Duroid 5870 of

thickness h= 0.0795 cm with loss tangent tan δ = 0.0012 has been designed to

resonate at frequency fr =10 GHz.

Based on the solution of the equation (6.16), we obtain Rin , the value of the real part

of inZ that is under the matched condition, to be 249.58 Ω. The maximum power

transfer from the source to the antenna over the frequency range is dependent not only

on the frequency response of the antenna but on the antenna, transmission line and the

feed source as a whole. To ensure the same, the antenna edge input impedance needs

to match the impedance of the source-transmission line system. Thus efficient

matching networks must be designed to match the resonant edge input impedance of

the microstrip antenna to the characteristic impedance of the source and the

transmission line. Use of a quarter wave transformer can aid in impedance match.

177

Under the condition of impedance match, the characteristic impedance of the λ/4

transformer Z0 can be computed by the following expression:

500 ×= inZZ Ω

500 ∗= inRZ = 111.712Ω

where 50 ohms is the default port impedance provided in ADS Momentum software

which has been used to simulate the patch antenna. Hence, the length and the

thickness of the feed are obtained as 1.09 cm and 0.0635 cm respectively. Figure 6.5

shows the transformed planar patch.

Condition I: For 91.890 >reZ ε , we have

282 −

= A

A

ee

hW

where⎭⎬⎫

⎩⎨⎧

++−

+⎭⎬⎫

⎩⎨⎧ +

=rr

rrZAεε

εε 11.023.011

21

60

210 … (6.18)

Condition II: For 91.890 <reZ ε , we have

( ) ( )⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−+−

−+−−−=

rr

r BBBh

Wεε

επ

61.039.01ln112ln12 where r

Bεπ 260

= … (6.19)

where ab

rrre u

−

⎟⎠⎞

⎜⎝⎛ +

−+

+=

1012

12

1 εεε … (6.20)

( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛++

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+

++=

3

4

24

1.181ln

7.181

432.052ln

4911 u

u

uua … (6.21)

and 053.0

3.09.0564.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=r

rbεε … (6.22)

The design of transmission line feed corresponding to wavelength (λ) of 3cm is as follows:

To determine the width of the feed based on its characteristics impedance of the line

should satisfy following two conditions I and II as stated above.

Design of the microstrip antenna under consideration having width W=1.164cm, with

the substrate height h=0.0795cm, the parameter u=W/h works out to be 14.64. Using

equations (6.21) and (6.22), respective values of a and b are determined as 1.023 and

0.55. Using these values of a and b, and equation (6.20), reε is found to be 1.74. The

178

condition I stated above for 15.1700 =reZ ε , that is with 91.890 >reZ ε is therefore

satisfied.

Using equation (6.18), we obtain A=2.55 and W/h=0.637. Hence, the length of the

feed and its thickness are respectively determined as 0.75cm, and 0.05 cm. The

resonant frequency of the antenna for TM10 mode can be expressed as

12

2 +=

rWcf

ε

where W = 1.164cm 2/1

1212

12

1−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ −

++

=Whrreff

rεεε = 2.15

( )

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦⎤

⎢⎣⎡ +−

⎥⎥⎤

⎢⎢⎡ ++

=∆

8.0)258.0(

264.03.00.412

effr

hW

hW

hL

effrε

ε=0.515 or 041.0515.00795.0 =×=∆L cm

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Lcf

r

1210 ε

,

The effective dielectric constant, fringing length L∆ and 10f can be determined from

the expressions given above, where L is the effective length which is determined as

1.044 cm, using the following relation: LLL ∆−=′ 2

The actual length L′of the transformed patch is thus obtained as 0.962cm by

substituting the values of ∆L and L in the above expression.

For the thin substrate, TE or TM modes are generated. When the substrate thickness is

comparable to the radius of the cylinder, hybrid modes are generated. Rectangular

microstrip antenna mounted on a cylindrical surface operating in TM10 mode is

sensitive to changes in the curvature of cylinder unlike TM01 mode which is affected

slightly due to variation in the radius of the cylinder. For a patch fed symmetrically in

the z-direction and circumferentially polarized, TM10 is the dominant mode.

Study of the effect of resonant frequency variation with the variation in the radius of

the cylinder of the rectangular microstrip patch, excited in the TM10 mode has been

carried out [111]. An effort has therefore been made to study the variation in the

resonant frequency for cylinders of larger radius.

179

The width of transformed planar patch W ′ is calculated using the following relation:

W ′ = (r1 + thickness of the substrate) * φ, where r1 is the radius of the cylindrical

ground plane and φ is the angle subtended by the patch along the central axis.

Following Balanis [54], the input impedance of the transformed planar rectangular

microstrip antenna can be determined corresponding to the condition W ′ /λ < 0.35.

Solution of equation (6.17) yields values of 1G and 12G which are used to determine

Rin , the real part of inZ using equation (6.16) as follows:

2

11

90WGλ

⎛ ⎞= ⎜ ⎟⎝ ⎠

= 0.001355 Siemens

( ) θθθθ

θ

π

π

dLkJ

Wk

G 300

20

0212 sinsin

cos

cos2

sin

1201

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

= ∫ = 0.0006483 Siemens

and ( )12121

GGRin +

= =249.58Ω

Figure 6.5: Transformed Planar Patch

Figure 6.5 shows the transformed patch with transformed width and the length of the

patch respectively computed as 1.164 cm and 0.962 cm. The length and thickness of

the microstrip feed line have respectively been computed as 0.75 cm and 0.05cm with

the quarter wave microstrip line edge feeding technique. A substrate with the

specifications εr = 2.32, tan δ=0.0012 and thickness of 0.0795 cm has been employed.

180

Figure 6.6 shows the microstrip patch antenna mounted on a cylindrical conformal

surface.

Figure 6.6: Microstrip Patch Antenna Mounted on a Cylindrical Conformal Surface

6.5 Performance Analysis of Transformed Rectangular Microstrip Antenna The performance of the transformed rectangular microstrip antenna may be analyzed

using Method of Moments, Finite Difference Time Domain Analysis or Finite

Element Method. In this section ADS Momentum based on Method of Moments

(MOM) has been used. Figure 6.7 shows the layout of the transformed rectangular

microstrip antenna using ADS Momentum. In the following subsections in addition to

the antenna performance analysis, the effect of the curvature on resonant frequency

has been discussed.

6.5.1 ADS based Antenna Performance Analysis

The current distribution of the excited patch, showing the uniform excitation of the

patch with hot zones along two radiating edges is depicted in Figure 6.8. Figure 6.9

shows the Smith chart.

.

Figure 6.7: Basic Patch Layout on Momentum

181

Figure 6.8: Current Plot of the Designed Patch

Figure 6.9: Smith Chart

Table 6.1: Antenna Parameters from Momentum (ADS)

182

Table 6.1 shows antenna parameters of the designed antenna simulated using

Momentum (ADS). It is seen radiating power of 2.3mW having directivity and gain

7.22 dB and 6.93 dB respectively. Thus the antenna radiation efficiency of 96% is

achieved.

Figure 6.10: Return loss at Resonant Frequency

Figure 6.10 shows the return loss (S11) of -20dB at resonant frequency of 10 GHz,

which may be considered to be an acceptable design. Figure 6.11 shows the field

distribution in Cartesian coordinates, indicating perfect nulls in both θ and φ plane.

Polar plots of field distributions in θ and φ planes are respectively shown in Figure

6.12(a) and Figure 6.13(a) with perfect nulls and no side lobes. Three dimensional

views of these polar plots in θ and φ planes are respectively shown in Figure 6.12(b)

and Figure 6.13(b).

Figure 6.11: Cartesian plot of the field in θ and φ plane

183

Figure 6.12 (a): Polar plot of the field in the θ-plane Figure 6.12 (b): 3D view of the field in θ-plane

Figure 6.13 (a): Polar plot of the field in φ-plane Figure 6.13(b): 3D view of the field in φ-plane

6.5.2 Effect of Curvature on Resonant Frequency

Resonant frequencies fconformal on the rectangular microstrip antenna conformal on

cylindrical surface, are compared to resonant frequencies fplanar of the planar patch

antenna in order to confirm the validity of the commonly used assumption that

conformally mounted microstrip antennas may be treated as planar [112]. Conformal

cylindrical patch antenna transformed to the planar patch as depicted in Figure 6.7 is

obtained utilizing the transformation technique. Since the resonant frequency is

dependent on the chosen length of the patch, a curve plotted as shown in Figure 6.14

is carried out to compare resonant frequency changes in the cylindrical patch antenna

due to change in the radius r1 vis-à-vis change in the frequency due to changes in the

length L of the transformed planar antenna. Result based on a set of data which relates

to variation in the resonant frequency of the transformed planar patch vis-à-vis the

184

patch on the curved cylindrical surface, demonstrates that this assumption holds good

for the height h being small compared to the surface curvature r2. This assumption

provides excellent result when considering excitation of the antenna with no spatial

field variation normal to the surface.

Figure 6.14: Curve of fconformal / fplanar versus r1/ L′

The curve of Figure 6.14 shows the variation of the ratios fconformal/fplanar versus

Cylindrical Radius (r1)/Transformed Length ( L′ ) with r1 representing the radius of the

cylinder and L′ , the length of the transformed planar rectangular microstrip patch.

Using curve fitting, graph in this figure results in the following empirical relation:

'L1r6-103.527-

e'L1r1.332-

eplanarf

conformalf ∗∗

+

∗

= 0856.1

Smaller patch size might be preferred to reduce space requirements compared to

larger patch width as it has constraint since it results in generation of grating lobes in

antenna arrays. The patch width also affects cross polarization characteristics. The

patch width selected to obtain good radiation efficiency if space requirements or

grating lobes are not the overriding factors. The resonant frequency is dependent on

the chosen length hence W/L is chosen to be nearly equal to 1. Expression to

determine the ratio of frequency of the patch antenna on a cylindrical curved surface

185

vis-à-vis the planar surface is determined using curve fitting technique. Technique

determines the coefficients that provide the least error between the actual ordinate

values and the ordinate values predicted by the curve fitting formula. Least Square

method, also known as regression analysis, has been used in the present work to

obtain values of the frequency ratio fconformal/fplanar for the ratio r1/ L′ of the curved

surface. The best fit in the method of least squares is characterized by the sum of

squared residuals have its least value, a residual being the difference between an

observed value and the value given by the model.

6.6 Antenna Array Conformal on the Cylindrical Surface The single element radiation pattern is relatively wide and provides low value of

directivity. Certain applications demands design of antennas with high directivity

characteristics. Directive antenna can be realised by increasing the size of the antenna.

Alternatively an array can achieve the same. Performance of conformal array antennas

are based on the array shape, element pattern, and array excitation are among other

factors that must be known in order to determine its characteristics. Conformal shapes

can be classified as slightly curved (almost planar); singly curved, including ring

arrays and cylindrical arrays; and doubly curved. Slightly curved antennas behave

more or less like planar antennas and exhibit the same limitations; the design

principles are roughly the same. For other types, it is hard to make general statements,

since dimensions, shapes, element types, requirements etc. can be so different.

Early studies of conformal array and its analysis involved rudimentary simplifications

that were necessary as tools were not available to simulate and understand antenna

array behaviour. Assumptions were made based on cosine (dipole type) element

patterns or isotropic patterns. The general behavior of radiation patterns of array

elements for an initial analysis can be sufficient with such an assumption. A better

approach is to compute the element pattern in a planar environment and then use this

for all element positions in the conformal array. The ideal approach is, however, to

calculate the element patterns in the actual curved environment with the effect of

mutual coupling included. It may be worthwhile to consider simple models that can

provide results that are near accurate and easy to understand in many cases. Some of

the calculations in the array designs are based on simple models. These approaches

186

are accurate as well serve the basic purpose of illustrating fundamental characteristics

of the conformal antenna array.

Antenna arrays are expected to function as per design parameters provided important

aspects such as effects of mutual coupling between the antennas elements are

accounted for at the design stage. Analyzing radiation pattern of the antenna array

under the matched condition is an important factor especially if the array is designed

for scanning. Electromagnetic compatibility with other RF radiators that are close by

or co-located is one such issue. Feeding systems leakage and other components that

could be involved in interference with adjacent or co-located electronics subsystem

(analog and digital) needs to be considered during the design stage. One should,

therefore, check what electrical modes can be excited in the antenna structure both

internally and externally. Effect of mutual coupling can result in creation of grating

lobes. For grating-lobe suppression the element density should be sufficiently high

(roughly half-wavelength spacing) and the elements distributed evenly over the

surface. However, regular periodic-element grids can only be defined for a few

canonical shapes, such as the cylinder. Calculations of the patch dimensions of

microstrip antenna array resonant at a frequency fr = 10 GHz has already been

explained in sections 6.3 and 6.4. The transformed patch width W ′and length L′ are

found to be 1.164 cm and 0.962 cm respectively.

The following subsection discusses the design of the microstrip antenna array feed.

The array feed has been designed to be conformal on the cylindrical surface.

Transformation of the feed along with the antenna array elements is also discussed.

Subsequently the performance of the designed conformal antenna array with the feed

is considered.

6.6.1 Design Considerations for Microstrip Antenna Array Feed Conformal to

Cylindrical Surface

Conformal mapping technique has been employed to transform 2 element microstrip

patch antenna array with a corporate feeding network on a cylindrical surface as

shown in Figure 6.15. In order to achieve the mapping, there is a need to transform

array and the feed on a cylindrical surface to a corporate fed array on a planar surface

while maintaining the electromagnetic properties of the conformal array on a

cylindrical surface. Reason for the feed transformation is that the curvature changes

187

the characteristic impedance of the microstrip lines used to feed array elements. Effect

of mutual coupling between two array elements affects the radiation pattern. It may

cause deviation in the resonant frequency of the array and also lead to impedance

mismatch between elements and the feed network. All these factors lead to

unpredictability and deterioration in the array performance. Hence the effect of

mutual coupling and the curvature must also be incorporated in the transformation of

the cylindrical microstrip array to a planar microstrip array.

Figure 6.15: Two element Microstrip Patch Antenna Array with Corporate feed network

6.6.2 Design of the Feed Network using Conformal Mapping

Figure 6.16 and Figure 6.17 respectively show the feed network and the cross section

of an axially directed cylindrical microstrip line. Corresponding transformation of a

feed conformal to a cylindrical surface is shown in Figure 6.18. As shown in Figure

6.18, the strip subtends an angle ϕ2 about the central axis. The inner conducting

cylinder 1S′ has a radius r1 and is covered with dielectric substrate of permittivity εr of

thickness (r2 -r1), where r2 is the radius of outer cylinder.

Figure 6.16: Feed Network Figure 6.17: Axially directed cylindrical microstrip line

188

Figure 6.18: Transformation of Feed Conformal to Cylindrical Surface

6.6.2.1 Condition I: Narrow width microstrip line 1ln

2

1

2<

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛

rr

ϕ

( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

−⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

=πε

πεεϕ

ϕεπ4ln1

2ln

11

21ln/2

321ln4ln

12687.376

2

1

2

1

20

rr

r

r rr

rrZ

6.6.2.2 Condition 2: Wide microstrip line i.e. 1ln

2

1

2>

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛

rr

ϕ

1

1

2

1

20 94.0ln/ln451.1

211082.0441.0ln/

2687.376

−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ −++⎟⎟

⎠

⎞⎜⎜⎝

⎛=

rr

rrZ

r

r

r

r

r

ϕπε

εε

εϕε

Using the following transformation

2ln πς += jw ; where jvuw += and ηξς j+=

The width of the strip formed by the arc is conformally transformed to a planar form

as shown in Figure 6.18. Using this transformation, Z0 the characteristic impedance of

the transformed microstrip line is calculated based on either of the two conditions as

given above.

Length of the feed has been found to be 1.09cm and the thickness of the feed obtained

is 0.0635cm. Figure 6.19 shows the ADS momentum based layout of the conformal

microstrip patch antenna array with a corporate feed network to excite two elements

189

array. Figure 6.20 explicitly shows the uniform excitation of the patch with hot zones

along with two radiating edges.

Figure 6.19: Basic Patch Layout of Antenna array on Momentum

Figure 6.20: Current plot of Designed Patch

6.6.3 Performance of Microstrip Antenna Array Feed Conformal to

Cylindrical Surface

Two element rectangular microstrip antennas conformal on a cylindrical surface with

corporate feed network are simulated using ADS Momentum. Performance analysis

based on ADS Momentum results. Figure 6.21 shows the Smith chart and Table 6.2

highlights antenna parameters. The table shows that the antenna array is radiating a

power of 1.97mW, with directivity of 7.979 dB and gain of 6.988 dB. The efficiency

of the microstrip array antenna is 87.6%. When compared to antenna parameters of

the single patch antenna, the performance of the array has been found to have been

deteriorated. As seen in the case of a single patch, it is seen radiating power of

2.3mW; having a directivity of 7.22 dB, gain of 6.93 dB thus the radiation efficiency

190

obtained is 96%. Performance and the radiation pattern of the microstrip antenna

array are affected due to the mutual coupling between the two elements

Figure 6.21: Smith chart

Table 6.2: Antenna Parameters from Momentum (ADS)

.

191

Figure 6.22: Return loss at Resonant Frequency

Figure 6.22 shows the return loss (S11) of -21.8dB at resonant frequency of 10 GHz. It

can be seen that there is no deviation in the resonant frequency of the array; this

shows that impedance match exists between the elements and the feed network. In

order to improve the array performance it is therefore necessary to take into account

the effect of mutual coupling. Further, the design consideration must also take into

account the curvature of the parent structure in the transformation of the cylindrical

microstrip array to a planar microstrip array. Figure 6.23 show the field distribution in

Cartesian coordinate, indicating perfect nulls in both θ and φ plane.

Figure 6.23: Cartesian plot of Field in θ (blue) and φ (yellow) plane

192

Figure 6.24(a): 3D View of Radiations in θ Plane Figure 6.24(b): Polar plot of Field in θ plane

Figure 6.25(a): 3D View of Radiations in φ Plane Figure 6.25(b): Polar plot of Field in φ plane

Figure 6.24 (a), 6.24 (b) and Figure 6.25 (a), 6.25 (b) shows the respective polar plots

both in 3-D and cut section in θ and φ plane. Plots show perfect radiation pattern in

both θ and φ plane with no side lobes in the radiation pattern. The uniformity in both

current distribution in the patch and field distribution is indicative of the fact that the

antenna is excited for with minimum losses occurring in terms of reflection due to

mismatch. However due to the effect of mutual coupling there is drastic reduction in

the amplitude of radiation pattern in both θ and φ plane.

Figure 6.26 shows the curve of fconformal/fplanar versus Cylindrical Radius

(r1)/Transformed Length ( L′ ), where r1 is the radius of the cylinder and L′ is the

length of the transformed planar rectangular microstrip patch. This curve is similar to

193

that obtained with a single patch on the cylindrical surface. The empirical relation

obtained using curve fitting of the plot shown Figure 6.26, is obtained as:

'L1r6-103.527-

e'L1r1.332-

eplanarf

conformalf ∗∗

+

∗

= 3411.1

Figure 6.26: Curve of fconformal/fplanar versus r1/ L′

Figure 6.14 and Figure 6.26 respectively show the fconformal/fplanar versus r1/ L′plots for

conformally transformed single patch and two element array microstrip patch antenna

on the cylindrical surface transformed into planar antenna. Comparing plots for both

single element and two element array microstrip antenna it is inferred that increase in

array size results in significant changes in frequency in conformal vis-à-vis planar

antenna due to effect of the curvature and the mutual coupling on account of inter

element spacing. In the analysis it is necessary to consider the effect of mutual

coupling affecting the frequency and antenna parameters when the inter element

spacing in both H & E plane is changed. In subsequent sections this effect on antenna

parameters has been considered independently. The effect of inter element spacing for

both planar arrays and array elements conformal on cylindrical surface has also been

compared. An optimum spacing is arrived for both conformal microstrip antennas on

cylindrical surface. Thereafter a comparative study has been carried out on optimum

inter element spacing of planar microstrip antenna when compared with the

rectangular microstrip antenna conformal on a cylindrical surface.

194

6.7 Mutual Coupling among the Elements versus Surface

Geometry and Size

The singly curved surface can be used as an approximation for the shape of an aircraft

wing, fuselage or external pods. As discussed earlier, antennas mounted on singly

curved surfaces are an important class of conformal arrays for applications in which a

large (azimuthal) angular coverage is required. These types of antennas can be used in

radar and communication systems.

Figure 6.27: Configuration of two patches coupled cylindrical microstrip patch antenna array.

The focus is on the mutual coupling and its influence on the radiating characteristics.

Mutual coupling in arrays gives rise to deviations in antenna element-patterns

compared with those of corresponding isolated elements. The microstrip elements

used for these investigations employ dual patch antennas fed by two coaxial probes as

shown in Figure 6.27 and then the mutual coupling effect on combined quadratic

patch antennas are studied.

Figure 6.28 and Figure 6.29 respectively shows the E plane and the H plane layout of

a two element array with co-axial feed. A comparative study has been carried out for

isolated conformal microstrip antenna and array of conformal antennas embedded on

a cylindrical surface. The frequency of operation chosen is 10 GHz (Operating

frequency of Fire Control Radar in a Fighter Aircraft). We start by analyzing the

isolated coupling, that is, the mutual coupling between two elements only. The radius

of the cylinder is taken as λ, and the patch dimensions are Length L =0.27 λ and

Width W= 1.5×L. When changing from the E-plane to the H-plane, both patches are

rotated by 90°.

195

Figure 6.28: E plane layout of a two element array with co-axial feed

Figure 6.29: H plane layout of a two element array with co-axial feed

The mutual coupling between two elements only is referred to isolated coupling since

no other elements are involved. If all elements are present, we use the term “array

mutual coupling”. The influence of the mutual coupling is also demonstrated by

comparing radiation patterns of isolated and embedded elements. With the operating

wavelength taken as λ = 3cm (radius of the cylinder), the patch length L and width W

are respectively determined as 8.1 mm, and 12.15 mm. Substrate parameters have

196

been taken as εr = 2.5, tanδ = 0.0018, substrate thickness t = 0.787 mm. The coaxial

feed point is calculated as

( ) ( ) 6314.102.24

22

=−+

=WLpinx

Table 6. 3: Effect of Mutual Coupling on Antenna parameters with various inter element spacing

in E-Plane

E-Plane Separation

S11(dB)

Gain(dB)

Directivity(dB)

Resonant Frequency(fr)

Efficiency

0.3 λ -35.6123 5.6524 8.1053 10.076 69.73709 0.4 λ -32 5.7226 8.3922 9.957 68.18951 0.5 λ -29.8553 5.3463 8.4774 9.8537 63.06533 0.6 λ -22.683 5.5695 8.6344 10.0125 64.50361 0.7 λ -21.963 6.9295 8.9638 10.21312 77.30538 0.8 λ -18.2835 6.409 8.6472 10.3 74.11648

Effect on antenna parameters due to changes in the inter element spacing contributing

to mutual coupling, results in changes in the designed frequency of resonance. There

is a need to study the changes in array configuration in the E-plane with inter element

spacing changed with an increment of 0.1λ. Changes in the inter element spacing will

effect the resonant frequency and result in changes in antenna parameters. Table 6.3

depicts the changes with initial setting of 0.3 λ and progressively with increment of

0.1λ, reaching up to 0.8 λ.

Figure 6.30: Plot showing Return Loss (S11) variation with changes in inter element spacing in E plane

Plot in Figure 6.30 shows increase in the return loss with increase in the inter element

spacing. As the inter element spacing in E plane increases, there is deterioration in the

return loss indicative power reflection to the input port. Figure 6.31(a) and 6.31(b)

shows the effect of mutual coupling on the gain and the directivity due to changes in

197

the inter element spacing in E plane. Variation in both the gain & the directivity

shows steady increase in both these parameters up to 0.7λ, and then there is a fall in

both the gain and the directivity. The plot of efficiency in Figure 6.32 shows that there

is a dip at 0.5 λ thereafter there is a significant improvement in efficiency.

Figure 6.31(a): Variation in Gain with inter element spacing in E plane

Figure 6.31(b): Variation in Directivity with inter element spacing in E plane

Figure 6.32: Efficiency of the antenna array for changes in element spacing in E plane

The antenna array designed to resonate at 10 GHz. However as shown in Figure 6.33,

the resonating frequency is seen changing due to mutual coupling, and at 0.6 λ it is

resonating closest to the designed frequency. Figure 6.34 depicts combined

normalized plot showing effect of mutual coupling with the variation in the inter

198

element spacing to the antenna parameters viz. gain, directivity, efficiency and the

return loss. While considering the above stated parameters the shift in resonant

frequency must be kept in mind. It is observed that at 0.7 λ all antenna parameters

provide best results except poor return loss with considerable shift in the resonant

frequency from the designed frequency of 10 GHz. It can therefore be concluded that

coupling in E-plane with spacing S= 0.7λ, the return loss is -21.963 dB, whereas the

gain, directivity and efficiency of the antenna is maximum and the resonant frequency

is at 10.21 GHz. Next best results of antenna parameters are seen at 0.3 λ and 0.4 λ

spacing. Effect of the inter element spacing on the resonant frequency and antenna

parameters must be kept in mind while designing conformal arrays.

Figure 6.33: Shows effect on resonating frequency due to mutual coupling

Figure 6.34: Combined plot of antenna parameters with variation in inter element spacing in E

plane

199

Table 6.4 shows the effect on antenna parameters due to the effect of mutual coupling

on variation of the inter element spacing of antenna elements in H plane.

Table 6.4: Effect of Mutual Coupling on Antenna parameters with various inter element spacing

in H-Plane

When the inter element spacing is varied between 0.3λ and 0.8λ in the H plane, it is

observed that return loss improves at 0.5λ thereafter it deteriorates before marginally

improving at 0.8λ. Figure 6.35, shows variation in return loss. Unlike what has been

observed in E plane, where return loss consistently increases with increase in inter

element spacing, here the improvement is up to λ/2, thereafter it decays.

Figure 6.35: Plot showing Return Loss (S11) variation with changes in inter element spacing in H

plane

Variation in the gain of the antenna with inter element spacing variation in H

plane is limited but the directivity drops linearly between 0.4λ to 0.7λ (Figures

6.36(a) and 6.36 (b) refers). This linear drop in directivity results in the drop in

efficiency as shown in Figure 6.37. The maximum efficiency is observed at 0.4λ

spacing, and after consistent drop, there is a marginal increase at 0.8 λ.

200

Figure 6. 36(a): Variation in Gain with inter element spacing in H plane

Figure 6.36(b): Variation in Directivity with inter element spacing in H plane

Figure 6.37: Efficiency of the antenna array for changes in element spacing in H plane

Figure 6.38 shows the combined plot of antenna parameters with variation in the

inter element spacing in the H plane. It is observed that at 0.4λ spacing in the H

plane the most optimum result is obtained. At 0.4λ spacing, the frequency

deviation is of the order of 0.2841 GHz from the designed frequency of 10 GHz.

201

Though minimum frequency deviation is at observed 0.7λ but deterioration is seen

in the other antenna parameters. Optimum result in terms of the antenna

parameters is obtained for spacing at 0.5λ. Most importantly at 0.5λ the return loss

is seen to be the lowest. Finally in the H-plane at S= 0.5λ the resonant frequency

at 10.24 GHz, the return loss is at -26.63dB, and the antenna parameters are close

to the best result.

Figure 6.38: Combined plot of antenna parameters with variation in inter element spacing in H

plane

Figure 6.39 shows the polar plot of the radiation pattern in both E and H plane

with variation in the inter element spacing. As shown in Figure 6.28 and Figure

6.29, two patch elements both in E plane and H plane are close to each other, the

current in each patch changes in both amplitude and phase. The quantum of

variation depends on the mutual coupling between the elements. Far fields due to

changing currents in patches are expected to change. Plot shown below in Figure

6.39 shows explicitly the resultant far-field radiation pattern, due to the variation

in the phase and amplitude of the currents in these two individual patch elements

in the presence of mutual coupling. The effect of mutual impedance between two

elements placed close to each other is studied. This is shown for two patch

elements placed side-by-side as a function of separation between the centres of the

patch elements (S/λ) in E plane. Similarly the variation of mutual impedance is a

function of distance (S/λ) for patches placed in H plane. Radiation pattern for inter

202

element spacing at 0.4 λ in E plane shows polar plot ϕE and ϑE i.e. the plot in both

φ and θ plane. Plot is in compliance to the antenna parameters obtained with

frequency deviation being within the limits and the return loss is significant. In H

plane the polar plot ϕE and ϑE shows radiation pattern with minimum side lobes at

0.5λ, which again conforms to the antenna parameters with return loss at

minimum and the frequency deviation marginally higher.

203

204

205

Figure 6.39: Radiation Plot in both E and H plane for variations in inter element spacing

6.8 Behaviour of Conformal Microstrip Antenna on a Cylindrical

Surface Considering Effect of Mutual Coupling in both Planes.

For considering the design of conformal microstrip antenna on a cylindrical surface it

is necessary to study its behaviour with combined effect of mutual coupling in both

the E and the H plane. For the purpose of this study the inter element spacing in

combination of 0.5λ and above is considered. The antenna parameters obtained along

with return loss and radiation pattern in polar plot are shown in Figure 6.40. Inter

element spacing in either of plane is varied at a time while keeping the spacing in the

other plane fixed. For example, we keep in one plane inter element spacing fixed (say

E plane spacing at 0.5λ) while in the other plane the spacing is varied (say in H plane

we change the spacing starting from 0.5λ and with an increment of 0.1λ). Simulation

has been carried using FEKO-EM simulation software.

206

Figure 6.40: Radiation Plot considering mutual coupling effect in both planes for combined variations

in inter element spacing

207

The radiation pattern polar plot ϕE and ϑE , antenna parameters such as Directivity,

Gain and Return Loss is also compared for realising optimum spacing for given

antenna array design. With quadratic patches i.e. patch elements combined in E & H-

plane, effect of mutual coupling are studied. With S= 0.7λ & 0.5λ in E & H-plane

respectively, it is observed that the antenna characteristics shows best results. Antenna

parameters viz. Return Loss, Gain and Directivity are compared. Figure 6.40 also

depicts the combined far field radiation polar plots, S11 parameter and antenna

parameters for combination of inter element spacing in E and H plane.

6.9 Performance Comparisons of Identical Planar Antenna Array

with Array on Cylindrical Surface in both E & H Plane

A comparative study of the performance involving the planar array with the array

conformal to cylindrical surface is discussed in this section. Four element transformed

microstrip patch antenna has been considered for comparison. Antenna parameters for

variation in inter element spacing in the E and the H plane has been collated in Table

6.5. Effect of Sλ (S/λ) depicting the behaviour of the planar antenna in terms of return

loss, gain, directivity and efficiency in both the E and H plane is shown in this table.

Variation in the return loss with changes in the inter element spacing Sλ in both the E

and H plane is shown in Figure 6.41. It is observed that the return loss dips to a

minimum at 0.3λ and 0.5λ respectively in E and H plane. Limited variation of return

loss is found in the H plane whereas the variation in the return loss is observed to be

significant in the E plane. It is interesting to note that at 0.5λ return loss in the E plane

is marginally increased as compared to what is observed at 0.3λ.

Table 6.5: Effect of Mutual Coupling on Antenna parameters with variation in Sλ in E and H-Plane

208

Figure 6.41: Plots shows effect on Return Loss with changes in Sλ in both E and H plane

Figure 6.42: Plots depicting effect on Directivity with changes in Sλ in both E and H plane

Figure 6.42 shows the effect of variation in spacing Sλ in both E and H planes on

directivity. Increase in directivity in E plane is found to be linear with spacing.

However in H plane the directivity increases almost linearly before maximizing at

0.5λ ultimately dropping linearly with spacing. Plots in Figure 6.43 depicting the variation in gain in both the E and H plane shows

identical curves as is the case with directivity. Figure 6.44 shows the variation in

efficiency of the antenna array. The efficiency increases and decreases linearly

respectively in the E plane in the H plane with Sλ.

Figure 6.43: Plots showing effect on Gain due to changes in Sλ in both E and H plane

209

Figure 6.44: Variation in Efficiency with changes in Sλ in both E and H plane

Figure 6.45: Combined plots of Antenna Parameter with Sλ in E plane

Figure 6.46: Combined plots of Antenna Parameter with Sλ in H plane

210

Combined normalized plots for the above antenna parameters in the E plane and for

the H plane respectively are shown in Figure 6.45 and Figure 6.46. In the E plane best

results are seen at Sλ=0.8, whereas for H plane corresponding value is at Sλ=0.5.

Radiation plots in both the E and H plane are shown in Figure 6.47. The far field

radiation pattern both in ϕE and θE in the E plane is maximum at Sλ=0.4 whereas in H

plane it is maximum at Sλ=0.6. Comparing all antenna parameters along with far field

radiation pattern, ϕE and θE are found to be optimum at Sλ=0.7 and Sλ=0.5 spacing

respectively in E and H plane. It is inferred that the H-plane coupling is weaker than

the E-plane coupling (as it is in the corresponding planar case). It is therefore

concluded that effect of mutual coupling due to inter element spacing is optimum for

values of Sλ=0.7 and Sλ=0.5 in E and H plane respectively for the patch antenna array

conformal on cylindrical surface and the planar antenna array.

211

Figure 6.47: Polar plots showing radiation pattern in both E and H plane with Sλ

212

It can also be inferred that the rigour of analysis of conformal antenna on non-planar

surface like cylinder can be best understood while analysing transformed antenna in

planar domain. In addition transformed planar antenna array is suitable for ease

analysis. Hence microstrip antenna array conformable on cylindrical surface

implementation easier while fabricating on the parent body surface.

6.10 Design of Microstrip Antenna Conformal on Cylindrical

Surface based on Full Wave Analysis

Computer aided design for analysis of microstrip antennas and arrays are the

technological advancement based on efficient and accurate numerical methods.

Techniques exist to analyse conformable microstrip antennas on electrically small

cylindrical surfaces. But for those conformable to electrically large cylinders with

identically shaped substrates there is a need to have an efficient analytical and

numerical tool. Design and analysing techniques for a various applications involving conformable

microstrip antennas and arrays on cylindrical surface include cavity-model analysis

and generalized transmission-line model (GTLM) theory. These techniques are simple

and accurate for thin substrate but not suitable for many structures with thick

substrate. On the other hand full-wave analysis is more accurate and is applicable to

many structures employing method of moments (MoM) / Green’s function technique

in the spectral domain [113]. Most of the numerical results have been given for

microstrip antennas mounted on circular cylinders with electrically small radii. For

electrically large radii cylinders, solutions involving Bessel & Hankel functions and

Fourier integral involving series summation increases computational complexity. In

addition, on electrically large cylinders, the spectral-domain representation of the

Green’s function has convergence problems for electrically large separations between

source and observation points. Hence the analysis of mutual coupling between

microstrip antennas becomes difficult especially at high frequencies. Selection of

Basis functions for the expansion of the patch surface currents can lessen this problem

to a certain extent.

It is worth considering issues affecting the performance of conformal arrays.

Generally, comparisons between the radiation pattern characteristics of conformal

213

arrays with those of planar arrays shows that they are not much different from each

other. Designed performances with desired beam shaped patterns having low side-

lobes level can be realized with both types of arrays. It is necessary to develop special

methods such as full wave analysis for analyzing conformal arrays and predicting

their performance. Though this approach requires more computational effort, it can be

employed for all types of conformal antennas and arrays.

It is important to determine microstrip antenna operating frequency at which the

antenna may provide efficient radiation as per design. Full-wave analysis of

cylindrical microstrip using a moment-method is presented in this section. In this

context the Basis functions with and without edge singularity for numerical

convergence are discussed. Database is generated from a full-wave approach

incorporating a Galerkin’s moment-method calculation of a planar rectangular

microstrip patch antenna.

6.10.1 Full-Wave Analysis of a Probe-Fed Rectangular Microstrip Patch on a

Cylindrical Surface with a Superstrate.

A cylindrical structure with a rectangular microstrip antenna mounted on it is

considered for the full-wave analysis. The cylinder is shown in Figure 6.48 is of

radius a with dielectric substrate relative permittivity ε1 and relative permeability

being µ0. The patch on it is having a negligible thickness (<λ) depicted as region 1

having thickness h (= b – a). The thickness of the dielectric superstrate having a

relative permittivity of ε2 and relative permeability µ0 has been taken as t (= c – b) as

shown as region 2 in Figure 6.48. Dielectric superstrate is for protection of the patch.

Region 3 is air with free-space permittivity and permeability ε0 and µ0 respectively.

The rectangular patch on the curved substrate-superstrate interface of ρ = b is having

along the length a dimension of 2L. The angle subtended by the curved patch is equal

to 2φ0. The corresponding equivalent dimension on the curved surface is 2bφ0.

214

Figure 6.48: Rectangular Microstrip Patch with a Superstrate on a Cylindrical Surface

( ) ( )[ ] 0,, =+× zEzE PD ϕϕρ … (6.23)

( )( )

( ) ( )( )∫∑

∞

∞−

∞

∞−= ⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡

zz

zz

zjkz

q

jqDz

D

kqJkqJ

kqGedkezEzE

z

,~,~

,~

21

,, ϕϕϕ

πϕϕ

… (6.24)

Full wave analysis is carried out by replacing the patch by a surface current

distribution [113]. Feeding probe is treated as a line source with unit amplitude (coax

feed <<λ). Applying boundary condition, with the total electric field tangential to the

patch surface taken to be zero, the unknown surface current density on the patch is

given by equation (6.23), where ),( zE D ϕ is the electric field due to the patch current

and ),( zE P ϕ is the electric field due to probe with the patch being considered to be

absent. Theoretical formulation technique can be used to derive ),( zE D ϕ using (6.24),

where ( )zkqG ,~

is the Dyadic Green’s function in the spectral domain for the

grounded substrate embedded on the cylindrical surface. ( )zkqJ ,~ϕ and ( )zz kqJ ,~

are

the Fourier transform of the current density on the patch in the φ and z directions

215

respectively. ( )zE D ,ϕϕ and ( )zE Dz ,ϕ respectively are the field components in the φ

and z directions. First ),( zE P ϕ , which corresponds to the expression for the field

component due to point source in a layered medium, is determined. Considering the

boundary conditions at the interface between the ground plane and the substrate at the

cylindrical surface, an expression in form of an integral equation for ),( zE P ϕ can be

derived by adding up field contributions from point sources along the input line-

current source. Derived expression for ),( zE P ϕ and equation (6.24) is substituted in

equation (6.23) and applying Galerkin’s moment method the resulting integral

equation is solved. Thereafter, using the selected Basis functions as testing functions

and integrating over the patch area, the following homogenous matrix equation is

obtained:

( ) ( )( ) ( )

( )( )

( )( ) ,

1

1

1

1 ⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×

×

×

×

××

××

Mmz

Nn

MmZ

Nn

MMzzmkNM

znk

MNz

mlNNnl

VV

II

ZZZZ ϕϕ

ϕ

ϕϕϕ

… (6.25)

There exist nontrivial solutions for the unknown amplitudes nIϕ and mzI if the

determinant of the matrix [Z] in equation (6.25) vanishes; that is,

( ) ( )( ) ( ) 0det =

⎥⎥⎦

⎤

⎢⎢⎣

⎡

××

××

MMzzmkNM

znk

MNz

mlNNnl

ZZZZ

ϕ

ϕϕϕ

.

By solving equation (6.25), the unknown patch surface currents nIϕ and mzI are

obtained, and the input impedance, radiation pattern, and other information of interest

can be determined.

To obtain full-wave solution, numerical convergence for the moment-method

calculation needs to be tested. The numerical convergence depends strongly on the

Basis function chosen for the expansion of the patch surface current density. A good

choice of the Basis functions used in moment-method calculation satisfying the edge

condition is that the normal component of the patch surface current must vanish at the

patch edge.

6.10.2 Cylindrical Rectangular Microstrip Patch with a Superstrate

As described in section 6.10.1, Figure 6.48 shows cylindrical rectangular microstrip

structure loaded with a patch antenna protecting dielectric superstrate. The geometry

216

is described in cylindrical coordinates. The z component of the electric and magnetic

fields in each region is considered by suppressing tje ω− . The z component of the

electric and magnetic fields in each region is given by [114].

( ) ( ) ( ) ( )[ ]ρρπ

ϕρ ρρϕ

ininininzjk

zn

jnz kJBkHAedkezE z += ∫∑

∞

∞−

∞

∞−=

1

21,, … (6.26)

( ) ( ) ( ) ( )[ ]ρρπ

ϕρ ρρϕ

ininininzjk

zn

jnz kJDkHCedkezH z += ∫∑

∞

∞−

∞

∞−=

1

21,, … (6.27)

,3,2,1,, 00222 ===− ikkkk iizii εεµωρ

where Ain, Bin, Cin, and Din are unknown coefficients of the harmonic order n to be

determined by the boundary conditions at =ρ a, b and c. ( )xH n)1( is a Hankel

function of the first kind with order n, and ( )xJ n is a Bessel function of the first kind

with order n. The unknown coefficients Ain, Bin, Cin, and Din given in equations (6.26)

and (6.27) in regions 1 and 2 can be expressed in terms of A3n and C3n in region 3.

The expressions are as follows:

( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]akHbkJbkHakJ

bkJBbkHAakJA

nnnn

nnnnnn

ρρρρ

ρρρ

1)1(

11)1(

1

222)1(

211 −

+=

( )( ) n

n

nn A

akJakH

B 11

1)1(

1ρ

ρ−=

( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ] ( ) ( )[ ]bkJBbkHA

bkknkj

kakHbkJbkHakJkbkJDbkHCakJ

C nnnnz

nnnn

nnnnnn ρρ

ρρρρρρρ

ρρρρ

εεεωε

222)1(

22

1221

20

21)1(

11)1(

1

1222)1(

211 1 +⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

′−′′+′

= ′′

′

( )( ) n

n

nn C

akJakH

D 11

1)1(

1ρ

ρ

′=

′

nnn CAA 32312 αα +=

( )( )

( )( ) n

n

nn

n

nn C

ckJckH

AckJ

ckHB 3

2

2)1(

43

2

2)1(

32

1

ρ

ρ

ρ

ρ αα −+

−=

nnn CAC 34332 αα +=

( )( )

( )( ) n

n

nn

n

nn C

ckJckH

AckJ

ckHD 3

2

2)1(

43

2

2)1(

32

1

ρ

ρ

ρ

ρ αα−

−=

217

1,0 333 === εnn DB

where ( )

( ) ( ) ( ) ( )ckHckJckHckJckJ

nnnn

n

ρρρρ

ρα2

)1(22

)1(2

20 ′−= ′

( )( )

( )( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

′

ckJckJ

ckHkckHk

n

n

n

n

ρ

ρ

ρρ

ρρ

εε

αα2

2)1(

3)1(

32

3)1(

2301

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 1

2

3

223

002 ε

εωµααρρ ckk

nkj z

( )( )

( )( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

′

ckJckJ

ckHkckHk

n

n

n

n

ρ

ρ

ρρ

ρρα2

2)1(

3)1(

3

3)1(

23

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

3

2

223

304 1

εεεωεα

ρρ bkknkj z

( )( )

( )( )( )⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥⎥⎦

⎤

⎢⎢⎣

⎡

bkHCbkHA

XXXX

kJkJ

nn

nn

znz

zn

ρ

ρϕ

3)1

3

3)1(

3

2221

1211~~

… (6.28)

A relationship between the spectral- domain patch surface current density and the

field amplitudes in region 3 (air region) are then obtained and can be written as matrix

in equation (6.28).

We use the assumed value of ( )zn kJϕ~

and ( )znz kJ~ according to Galerkin’s Moment-

Method formulation. The elements Xij in the matrices of X in equation (6.28) are derived as

420

0111 β

ββ

yX

X −= … (6.29)

( )( ) 43

2

2

0

0221 β

ββ

ρ

ρ yckJbkJX

Xn

n −−= … (6.30)

218

( )( )

( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

ρρ

ρ

ρ

ρρ

ρ

ρ

ββ

β

ββε

βε

π

22

420

012

1

2

240

1

1031

2

2'

302

23012 120

kyX

kbnk

ckJbkJ

yk

XckJbkJ

yk

jkX

z

n

n

n

n

…(6.31)

( )( ) ,1

120 2

243

220

012

51401

1

3131

2

23022

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−+⎟

⎟⎠

⎞⎜⎜⎝

⎛+=

ckJbkJ

yk

Xkb

nkXyk

yk

jkX

n

nz

ρ

ρ

ρρρρ

ββ

βββ

εβ

επ

… (6.32)

where

( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( )[ ]bkJakHakJbkHbkJ

bkJakHakJbkHbkJX

nnnnn

nnnnn

ρρρρρ

ρρρρρ

11)1(

11)1(

1

11)1(

11)1(

10 ′−

−′′= ′′

′

( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( )[ ]bkJakHakJbkHbkJ

bkJakHakJbkHbkJX

nnnnn

nnnnn

ρρρρρ

ρρρρρ

11)1(

11)1(

1

11)1(

11)1(

11 ′

′′

−′′

′−=

( )( )bkJ

bkJ

n

n

ρ

ρβ1

10

′=

( )( ) ρ

ρ

ρ

ρ

ρ

ρ

ρ

ββ

πβ

2

421

2

2402

2

21

101 1

120 kyk

ckJbkJ

ykk

bkkjnk

n

nz +⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

( )( )ckJ

ckJ

n

n

ρ

ρβ2

22

′=

( ) ( )( ) ( )ckH

ckJbkJ

bkH nn

nn ρ

ρ

ρρβ 2

)1(

2

22

)1(3

′−= ′

( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ −∞

∞−

∞

−∞=∫∑

zzn

zn

n

nzjkz

n

jn

z

z

kJ

kJX

dkHkH

edkeHE

z

~)(~

)()(

21 1

)1(

)1(ϕ

ρ

ρϕ ρπ

… (6.33)

The combinations of the integer’s p, q, r and s depend on the mode numbers n and m.

For the first three modes, n = 1, 2,and 3, the values of (p, q) are (1, 0), (1, 1), and

(1,2), respectively, and the values of (r, s) are (l,0), (1,1), and (1,2) for m = 1, 2, and 3.

An Ez and Hz field in the outer region is given in equation (6.33), where

200

2zkk −= εµωρ

219

The above expressions can also be replaced by linear combinations of two other

linearly independent solutions of the Bessel equation. Once Ez and Hz and are known,

the transverse field can be obtained using the following expressions:

( ) ( )( )[ ],

///2

0

ρρ

ϕρωµρ

i

zzz

kHEkj

E∂∂+∂∂

=

( )( ) ( )[ ],

///2

0

ρϕ

ρωµϕρ

i

zzz

kHEkj

E∂∂−∂∂

=

( )( )( ) ( )[ ]2

0 ///

ρρ

ρφρεωε

i

zzzi

kHkEj

H∂∂+∂∂−

=

( ) ( )( )[ ]2

0 ///

ρϕ

ρρϕεωε

i

zzzi

kHkEj

H∂∂+∂∂

=

To solve the unknown coefficients, boundary conditions at =ρ a, b and c for the

tangential components of the electric fields is imposed, and we have the following

equations: at =ρ a,

( ) ( ) ,0111)1(

1 =+ akJBakHA nnnn ρρ

( ) ( )[ ] ( ) ( ) ( )[ ] ,01111

121

111)1(

11

0 =+−′+− ′ akJBakHA

aknkakJDakHC

kj

nnnnz

nnnn ρρρ

ρρρ

ωµ bat =ρ

)()()()( 222)1(

2111)1(

1 bkJBbkHAbkJBbkHA nnnnnnnn ρρρρ +−+

( ) ( )[ ] ( ) ( )[ ]

( ) ( )[ ] ( ) ( )[ ] ,0222)1(

222

222)1(

22

0

111)1(

121

111)1(

11

0

=++′++

+−′+−

bkJBbkHAbknkbkJDbkHC

kj

bkJBbkHAbknkbkJDbkHC

kj

nnnnz

nnnn

nnnnz

nnnn

ρρρ

ρρρ

ρρρ

ρρρ

ωµ

ωµ

and at ,c=ρ

( ) ( ) ( ) ,0222)1(

23)1(

3 =−− CkJBckHAckHA nnnnnn ρρρ

( ) ( )[ ] ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ,03

13

323

13

3

0

2221

222

222)1(

22

0

'

=++

+−′+− ′

ckHAck

nkckHCkj

ckJBckHAck

nkckJDckHCkj

nnz

nn

nnnnz

nnnn

ρρ

ρρ

ρρρ

ρρρ

ωµ

ωµ

220

Magnetic fields at c=ρ is expressed as

( ) ( ) ( ) ( ) ( ) ,02221

231

3 =−− ckJDckHCckHC nnnnnn ρρρ

These equations are solved again for the boundary condition by applying the

discontinuity at =ρ b. For the tangential components of the magnetic field on the

patch Hz and Hφ, we thus obtain

( ) ( )( ) ( ) ( )( ) ( )bkJDbkHCbkJDbkHCkJ nnnnnnnnzn ρρρρϕ 2221

21111

1~ −−+=

… (6.34)

( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( ) ( )[ ] ,

~

2221

22

22'

221

22

20

1111

11

21'

111

11

10'

bkJDbkHCbk

nkbkJBbkHAk

j

bkJDbkHCbk

nkbkJBbkHAk

jkJ

nnnnz

nnnn

nnnnz

nnnnznz

ρρρ

ρρρ

ρρρ

ρρρ

εωε

εωε

+−⎥⎦⎤

⎢⎣⎡ ++

+++−

=

′

... 6.35)

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡

z

z

HE

EE

0001

sin1

ηθϕ

θ

( )( ) ⎥

⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣

⎡−≈ −

∞

−∞=

+

∑zn n

jnnrjk

JJ

XdkH

ejr

e ϕϕ

θπηθ1

0)1(

1

0 sin001

sin1 0

… (6.36)

Far-zone radiated fields in spherical coordinates are given approximately by equation

(6.36), where Jφ and Jz are the patch surface current densities obtained in the φ and z

directions respectively. ηo is free-space intrinsic impedance.

( )( )

( )( )∫∫

∞

∞−

−

−

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥⎥⎦

⎤

⎢⎢⎣

⎡

zJzJ

edzdekJkJ

nz

nzjkjn

znz

zn z

,~,~

21

~~

ϕϕ

φπ

ϕπ

π

φϕ … (6.37)

These functions are used to set up electric-field integral equations for analyzing

microstrip patch antennas. The equations form the fundamental tools for accurate

prediction of patch antenna parameters needed for designing antenna. The integral

equations were solved using Galerkin’s method and allow for accurately predicting

the design parameters for the patch antennas is given by equation (6.37), where

( ) ( ) ( )[ ] ( )( ) ( )[ ]( ) ( ) ( )( ) ,03

13

323

13

3

30

2221

22

22'

221

22

20

'

'

=−+

+++−

ckHCck

nkckHAk

j

ckJDckHCck

nkckJBckHAkj

nnz

nn

nnnnz

nnnn

ρρ

ρρ

ρρρ

ρρρ

εωε

εωε

221

)(~zn kJϕ and )(~

znz kJ are patch surface current densities in the spectral domain (the

tilde denotes the spectral amplitude or a Fourier transform).

( )( ) ( ) ( )

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

znz

znz

znz

zn

kJkJ

knGkEkE

~~

,~

~~

ϕϕ … (6.38)

Further, the tangential components of the electric field nEϕ and nzE in the spectral

domain on the patch can be found to be related to the current density nJϕ and nzJ in

equation (6.34) and equation (6.35), and can be expressed using equation (6.38),

where

1

2212

2111

2212

2111~~~~~ −

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥⎥⎦

⎤

⎢⎢⎣

⎡=

XXXX

YYYY

GGGG

Gzzz

z

ϕ

ϕϕϕ ... (6.39)

( )( )

( )( )∫∫

∞

∞−

−

−

−⎥⎦

⎤⎢⎣

⎡=⎥⎥⎦

⎤

⎢⎢⎣

⎡zEzE

edzdekEkE

nz

nzjkjn

znz

zn z

,,

21

~~

ϕϕ

ϕπ

ϕπ

π

ϕϕ

G~

in equation (6.39) is a dyadic Green’s function in the spectral domain; zGϕ~ denotes

the ϕ -directed tangential electric field at b=ρ due to a unit-amplitude −z directed

patch surface current density. zzz GandGG ~,~,~ϕϕϕ have similar meanings. The elements

Xij and Yij in the matrices of YandX respectively, Xij are derived in equation (6.29)

to equation (6.32), Yij are derived below.

where

( )( ) ,

120

1

01

2

2402

111

ρρ

ρ

ρ

πββ

kkj

ckJbkJ

ybk

nkY

n

nz −⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−=

,120

1

052

1

4121

ρρ

πββk

kjbkynkY z −

−=

( )( )ckJ

bkJyY

n

n

ρ

ρβ2

24012 +=

4122 βyY =

( ) ( )( ) ( )ckH

ckJbkJ

bkH nn

nn ρ

ρ

ρρβ 2

)1(

2

22

)1(4

′−=

( )( ) ρ

ρ

ρ

ρ

ρ

ρ

ρ

βπ

ββ

2

1

2

2332

2

21

10

415 1

120 kk

ckJbkJ

ykk

bkkjynk

n

nz

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ′++⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

222

( )( )ckH

ckH

n

n

ρ

ρβ2

)1(2

)1(

6

′

=

( )( )ckH

ckH

n

n

ρ

ρβ3

)1(3

)1(

7

′

=

3

232

3

131 ,

εεε

εεε ==

( ) ( ) ,1

323

22

2)1(

620 ⎟

⎟⎠

⎞⎜⎜⎝

⎛−

−=

εβ

ββ ρ

ρ

ρ kk

ckHy

n

( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= 11201

23

22

232021

621

ρ

ρ

ρρ επ

ββ kk

ckjknk

ckHy z

n

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= 2

3

22

202)1(

622 1

1201

ρ

ρ

ρρ πββ kk

ckjknk

ckHy z

n

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

ρ

ρ

ρ

ββ

ββ 3

272

2)1(

623

1kk

ckHy

n

( )( ) ( ) ( )

( ) ,00

,~,~

,~

21

,,

⎥⎦⎤

⎢⎣⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡∫∑∞

∞−

∞

∞−= zJzJ

knGedkezEzE

nz

nz

zjkz

n

jn

z

z

ϕϕ

πϕϕ ϕϕϕ … (6.40)

( )( )

( )( ) ,0

0~~

21

,,

⎥⎦⎤

⎢⎣⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡∫∑∞

∞−

∞

∞−=

−

znz

znzjkz

n

jn

z kJkJ

edkezJzJ

z ϕϕϕ

πϕϕ

… (6.41)

Finally, by imposing the boundary conditions on the patch and outside the patch, the

following integral equations can be obtained: On the patch, as shown in equation

(6.40) and outside the patch, as given by equation (6.41). Applying Galerkin’s

moment method by first expanding the unknown surface current density in terms of

linear combinations of the known basis function then we solve integral equations

(6.40) and (6.41)

( ) ( ) ( )∑ ∑= =

+=N

n

M

mmzmznn zJIzJIzJ

1 1

,,, ϕϕϕ ϕϕ

where nIϕ and mzI are unknown coefficients for the basis functions nJϕ and mzJ in

the ϕ and z-directions, respectively [115]. A convenient choice of Basis functions is

the cavity- model function of

223

( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ +⎥

⎦

⎤⎢⎣

⎡+= Lz

LqpzJ n 2

cos2

sinˆ, 0'

0

πϕϕϕπϕϕϕ … (6.42)

( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡+⎥⎦

⎤⎢⎣⎡ += 0

'

02cos

2sinˆ, ϕϕ

ϕππϕ sLz

LrzzJ mz … (6.43)

( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ +⎥

⎦

⎤⎢⎣

⎡+

−= Lz

Lqp

zLzJ n 2

cos2

sin1ˆ, 0'

022

πϕϕϕπϕϕϕ … (6.44)

( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡+⎥⎦

⎤⎢⎣⎡ +

−= 0

'

02'2

02

cos2

sin1ˆ, ϕϕϕππ

ϕϕϕ sLz

LrzzJ mz … (6.45)

where 2L is the patch length and 2 0ϕ is the angle subtended by the curved

patch; 2/' πϕϕ −= . The sinusoidal Basis functions of equation (6.44) and equation

(6.45) consider the edge-singularity condition for the tangential component of the

surface current at the edge of the patch. While the Basis function in equation (6.42)

and equation (6.43) do not consider the edge-singularity condition. The combination

of the integers p, q, r, and s depend on the mode numbers n and m. for the first three

modes, n = 1, 2, and 3, the values of (p, q) are (1, 0), (1, 1), and (1, 2), respectively,

and the values of (r, s) are (1, 0), (1, 1), and (1, 2) for m = 1, 2, and 3.

Next, by taking the spectral amplitude of the selected Basis functions and substituting

into (6.40), we have

( )

( )∑ ∫

∑

∑∞

∞−=

∞

∞−

=

=

⎥⎦⎤

⎢⎣⎡=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

uM

mzumzmz

N

nzunn

zkjz

uj

kJI

kJIGedke z

00

~

~~~

1,

1,ϕϕ

ϕ … (6.46)

where,

( ) ( )∫ ∫− −

−−=0

0

,21~

,

ϕ

ϕϕ

ϕϕ ϕϕ

π

L

Ln

zkjujzun zJedzedkJ z … (6.47)

( ) ( )∫ ∫− −

−−=0

0

,21~

,

ϕ

ϕ

ϕ ϕϕπ

L

Lmz

zkjujzumz zJedzedkJ z … (6.48)

From equation (6.47) and equation (6.48), the spectral amplitudes of the Basis

functions of equations (6.42) and (6.43) are expressed as

224

( ) ( )[ ]( )

( )[ ]( )222

02

0

0

1

, 2/2/sin

2/2/sin~

LqkLkq

puuppkjkJ

z

zzqp

zun ππ

ϕπϕπ

ϕϕ−

−−

−=

−+

… (6.49)

( ) ( )[ ]( )

( )[ ]( )222

02

01

, 2/2/sin

2/2/sin~

LrkLkr

suus

LurjkJ

z

zsr

zumz ππ

ϕπϕπ

−

−

−

−=

−+

… (6.50)

As for the basis functions of (6.44)-(6.45), the spectral amplitudes are written as

( ) ( )[ ]( )

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ −

−

−=

−+

LkqJLkqJpu

uppjkJ zq

z

qp

zun 21

22/2/sin

4~

0020

20

0

1

,ππ

ϕπϕπ

ϕπ

ϕ …

(6.51)

( ) ( )[ ]( )

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ −

−−

=++

000022

1

, 21

22/2/sin

4~ ϕπϕπ

πππ usJusJ

LrkLkr

LrjkJ s

z

zsr

zumz …

(6.52)

where ( )xJ 0 is a Bessel function of the first kind with order zero.

Then, using the selected Basis functions as testing functions and integrating over the

patch area, we can have the following homogeneous matrix equation:

( ) ( )( ) ( )

( )( ) ⎥⎦

⎤⎢⎣⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×

×

××

××

00

1

1

Mmz

Nn

MMzzmkNM

znk

MNz

mlNNnl

II

ZZZZ ϕ

ϕ

ϕϕϕ

… (6.53)

( ) ( )∑∫∞

∞−=

∗−

∞

∞−

=u

zunzulznl kJGkJdkZ ,,~~~ϕϕϕϕ

ϕϕ … (6.54)

( ) ( )∑∫∞

∞−=

∗−

∞

∞−

=u

zumzzzulzz

ml kJGkJdkZ ,,~~~

ϕϕϕ … (6.55)

( ) ( )∑∫∞

∞−=

∗−

∞

∞−

=u

zunzzukzzznk kJGkJdkZ ,,

~~~ϕϕ

ϕ … (6.56)

( ) ( )∑∫∞

∞−=

∗−

∞

∞−

=u

zumzzzzukzzzzmk kJGkJdkZ ,,

~~~ … (6.57)

k, m = 1, 2, …, M , l, n = 1, 2, …, N .

There exist nontrivial solutions for the unknown amplitudes nIφ and mzI if the

determinant of the [Z] matrix in (6.53) vanishes; that is,

225

( ) ( )( ) ( ) 0det =

⎥⎥⎦

⎤

⎢⎢⎣

⎡

××

××

MMzzmkNM

znk

MNz

mlNNnl

ZZZZ

ϕ

ϕϕϕ

… (6.58)

The solutions to (6.58) are found to be satisfied by complex frequencies for a

particular mode. This complex frequency, f = f’ + jf’’ gives the resonant frequency f’

and the quality factor f’/2f’’ for the microstrip patch. The imaginary part of the

complex resonant frequency also represent the radiation loss (the loss includes the

surface-wave loss) of the microstrip structure. Since the microstrip patch is a resonant

structure, the inverse of the quality factor also represents the half-power operating

bandwidth for the microstrip patch as a radiator.

It is advantageous to study the current distribution ( )zJ z ,ϕ and ( )zJ ,ϕϕ on the

patch, which may allow one to express the current in terms of more appropriate Basis

functions [116]. The current distribution obtained after solving equation (6.40) is

plotted as a function of z at ϕ = 0 and also as a function of ϕ at z = 0 (i.e., along

the centre lines of the patch). For analysis using Galerkin’s Moment Method,

parameters of cylindrical microstrip antenna chosen are ε1= 2.2, ε2 = 3 and ε3= 1,

Figure (6.48) refers. Resonant frequency chosen is 1.2GHz, with a = 20 cm, h = 4 cm,

W = 16.8 cm, k0= 25.1327, 3πϕ = and θ = 0.360 radians. Using Basis function, both

triangular and impulse for exciting the cylindrical surface Jz and Jφ respectively as

shown in Figure 6.49 and Figure 6.50, we obtain radiation pattern in both θ and φ

plane viz. Eθ and Eφ shown respectively in Figure 6.51 and Figure 6.52.

Jz Jφ

z/Wz ϕ / ϕL Figure 6.49: Impulse Basis function exciting Figure 6.50: Ttriangular Basis function exciting the cylindrical surface the cylindrical surface

226

Eθ

Figure 6.51: Radiation pattern in θ plane

Eφ

Figure 6.52: Radiation pattern in φ plane

227

Excitation of cylindrical patch using Basis function with impulse function in φ plane

with singularity as shown in Figure 6.53 and cosine function in z plane again with

singularity shown in Figure 6.54.

Jφ

ϕ / ϕL

Figure 6.53: Impulse Basis function with singularity exciting the cylindrical surface Jz

z/Wz

Figure 6.54: Cosine Basis function with singularity exciting the cylindrical surface

Figure 6.55 and Figure 6.56 respectively shows the corresponding field plots Eθ and

Eφ. The plot in the φ plane is identical as without edge singularity, however in the θ

plane we observe that the radiation pattern shows significant side lobe unlike without

228

singularity has directional beam which is symmetrical in all the four quadrants. The

singularity in both the z and φ planes attributes to change in the field pattern and may

at times be detrimental.

Eθ

Figure 6.55: Eθ Plot

Eφ

Figure 6.56: ϕE Plot

229

6.11 Conclusion

An effort has been made to design microstrip antenna to be conformal on the surface

of an aircraft. Various restrictions are imposed by the aerodynamic design of

structural surfaces viz. fuselage, wings, tailfin etc. An aircraft surfaces has been

approximated as either cylindrical or cylindrical sectored geometry e.g. the underbelly

of an aircraft. The design approach utilizes the application of elements of conformal

mapping to transform cylindrical surface to planar surface. There is a need to realize

an antenna on planar surface with one to one correspondence with the antenna on non

planar surface for accurate analysis and designing purpose. Transformed rectangular

microstrip patch antenna has been simulated using ADS Momentum software. The

results obtained are in accordance with the behavior of the conformal antenna.

A simple analytic technique has been devised to analyze cylindrical rectangular

microstrip antenna. Based on this technique and analyzing the structure using cavity

model, a relation has been developed determining the effect of curvature on resonant

frequency. These types of antennas, which account for large (azimuthal) angular

coverage, can be used in radar and communication systems.

The mutual coupling in arrays gives rise to deviations in antenna element-patterns

compared with those of corresponding isolated elements. An effort has been made to

focus on the mutual coupling and its influence on the antenna array radiating

characteristics. The microstrip elements used for these investigations employ dual

patch antennas fed by two coaxial probes and then mutual coupling effect on

combined quadratic patch antennas are studied. A comparison has been carried out for

the performance of isolated conformal microstrip antennas and array-embedded

conformal antennas operating at frequency of 10 GHz.

Study of the current distribution ( )zJ z ,ϕ and ( )zJ ,ϕϕ on the patch, expresses the

current in terms of Basis functions. The current distribution is plotted along the center

lines of the patch. Using Basis function, both triangular and impulse function for

exciting the cylindrical surface, radiation pattern in both θ and φ plane viz. Eθ and Eφ

shown respectively are obtained.

230

Excitation of cylindrical patch using Basis function with impulse function in φ plane

with singularity and cosine function in z plane again with singularity shows the

corresponding field plots Eθ and Eφ. In the φ plane the plot is identical as without edge

singularity, however in the θ plane, the radiation pattern shows significant side lobe

unlike without singularity has directional beam which is symmetrical in all the four

quadrants. In both the z and φ planes the singularity contributes to change in the field

pattern.