Research Article Electromagnetic Mathematical Modeling of ...downloads.hindawi.com/journals/mpe/2016/8130160.pdfResearch Article Electromagnetic Mathematical Modeling of 3D Supershaped

Research ArticleElectromagnetic Mathematical Modeling of3D Supershaped Dielectric Lens Antennas

L Mescia1 P Bia2 D Caratelli34 M A Chiapperino1 O Stukach4 and J Gielis5

1Dipartimento di Ingegneria Elettrica e dellrsquoInformazione Politecnico di Bari Via Edoardo Orabona 4 70125 Bari Italy2EmTeSys SRL 70122 Bari Italy3The Antenna Company Nederland BV High Tech Campus 5656 AE Eindhoven Netherlands4Institute of Cybernetics Tomsk Polytechnic University 843 Sovetskaya Street Tomsk 634050 Russia5University of Antwerp Groenenborgerlaan 171 2020 Antwerp Belgium

Correspondence should be addressed to L Mescia lucianomesciapolibait

Received 1 October 2015 Revised 14 January 2016 Accepted 7 February 2016

Academic Editor Thierry Floquet

Copyright copy 2016 L Mescia et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The electromagnetic analysis of a special class of 3D dielectric lens antennas is described in detail This new class of lens antennashas a geometrical shape defined by the three-dimensional extension of Gielisrsquo formula The analytical description of the lens shapeallows the development of a dedicated semianalytical hybrid modeling approach based on geometrical tube tracing and physicaloptic In order to increase the accuracy of the model the multiple reflections occurring within the lens are also taken into account

1 Introduction

During the last years lens antennas have attracted theattention of various researchers and companies because oftheir potential use in several application fields such as highfrequency wireless communication systems [1 2] millimeterwave imaging [3] space [4] smart antennas [5] radio-astronomy [6] and radar systems [1] In particular in[1] a flat-top radiation pattern at 60GHz is achieved byconsidering a 2D arbitrary shape lens antenna close in twometal half disks useful for high speedWi-Fi 80211ad devicesAnother application of such class of antennas is illustratedin [3] where the authors presented a lens-coupled patchantenna array for imaging applications This architectureconsists of a two-dimensional array featuring a monolithicmicrowave integrated circuit (MMIC) front-end In [1] theauthors presented a fully polarimetric integrated monopulsereceiver working in W-band frequency range The proposedstructure is composed by 2 times 2 array of slot-ring antennascovered by a hemispherical dielectric lens

Lens antennas can be easily integrated in electronic cir-cuits thanks to the relevant mechanical and thermal stabilitycombined with the possibility of shaping the radiated beam

by changing the lens geometry During the last years manyresearch activities have been devoted to the developmentof 3D dielectric lenses characterized by canonical or axial-symmetrical geometry [7 8] and only a few scientific stud-ies have been dedicated to the analysis of more complexgeometries [9ndash11] In particular in [10] the authors presenta compact lens antenna with arbitrary shape working inmillimeter wave frequency band the design being carriedout while neglecting the internal reflections occurring withinthe lens However in order to enhance the accuracy of thetheoretical model based on geometrical optic (GO) it isessential to consider the effects of the internal reflectionsespecially when high values of electric permittivity have tobe used To this end in [12 13] an accurate mathematicalmodel is illustrated implementing the effects of the second-order internal reflections for canonical lens geometry

The aim of this work is to present in detail a semiana-lytical model to design a new class of shaped dielectric lensantennas whose geometry is described by Gielisrsquo superfor-mula [14 15] The developed mathematical model is basedon the tube tracing approximation [16] The electromagneticfield inside the lens is evaluated by using the GO approxi-mation and considering the effect of the internal reflections

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8130160 10 pageshttpdxdoiorg10115520168130160

2 Mathematical Problems in Engineering

P

PFF

rd

x

rFF

r0

z

ki

120579i

120579t

y

kt

n

Figure 1 Geometrical structure of the electromagnetic systembased on dielectric lens antenna

In this way an improvement of the numerical accuracy canbe obtained Moreover the evaluation of the electromagneticfield radiated from the lens has been carried out applying thephysical optics (PO) approximation In particular accordingto the equivalence principle the radiation outgoing the lensis evaluated by the radiation of the equivalent electric andmagnetic currents on the lens surface The current distribu-tion is calculated by making use of the Fresnel transmissioncoefficient on the lens surface in accordance with the GOprocedure

Thanks to Gielisrsquo formulation it is possible to generate awide range of 3D shapes in a simple and analytical way bychanging a reduced number of parameters The use of thisformulation allows also an easy description of the geometri-cal characteristic of the antenna in terms of lens volume areacurvature radius and aspect ratio Moreover the analyticalrepresentation of the lens shape provides a number of benefitspertaining to the evaluation of the main physical quantitiesinvolved in the electromagnetic propagation equations Inthis way a reduced computation effort is required and moreaccurate results can be obtained

2 Mathematical Model

The geometrical structure considered in the proposed elec-tromagneticmodel consists of a dielectric lens antenna placedon a perfectly electric conductor (PEC) circular plate havingradius 119903119889 (see Figure 1)

The metal plate acts as a ground plane and mechan-ical support of the radiating structure while reducing thebackscattered radiation level The lens is illuminated by theelectromagnetic field emitted by a primary radio source suchas open ended waveguide patch antenna horn antennaor a coaxial probe integrated in the metal plate The tubetracing approach based on the GO approximation is used tomodel the electromagnetic propagation inside the homoge-neous dielectric lensThis approximation allows a significantsimplification of the mathematical model since it makes itpossible to reduce the solution of Maxwellrsquos equations toa geometrical problem As a result the adoption of suchmethod allows the simulation of electrically large structureswith a lower computational effort in comparison to full-wavenumerical methods such as Finite-Difference-Time-Domain(FDTD) and Finite-Element-Method (FEM)

z

y

x

Pmminus1A

Pmminus1B

Pmminus1CAmminus1

PmA

PmB

Am PmC

Figure 2 Area changes from119860119898minus1 to119860119898 of the tube surface duringthe propagation inside the lens

The applicability condition of such method requires thatthe lens surface is placed within the far field zone of thefeeding source and that the geometrical shape of the lens has acurvature radius larger than the operating wavelength In thisway the travelling electromagneticwave can be approximatedby a set of tubes propagating over a rectilinear path insidethe lens Each tube has a triangular cross section with givensurface area and is defined by three rays departing from eachof the triangle vertices The reflected and transmitted raytubes are determined by using GO principles As a result thepropagation path determines a change of the triangle areathat leads a variation of the electromagnetic field amplitude inaccordance with the power conservation law In Figure 2 thegeometrical change of the tube during the propagation insidethe lens is illustrated In particular the surface area variationfrom 119860119898minus1 to 119860119898 is given by triangle vertices intersectionwith the lens surface in points 119875119860 119875119861 and 119875119862

The surface area of each tube is evaluated by consideringthe overlap between the triangular surface with vertices 119875

119898

119860

119875119898

119861 and 119875

119898

119862 and the elements of the tessellation of the lens

surface The amplitude of the electromagnetic field insidethe tube surface is evaluated by using the Inverse DistanceWeighting (IDW) algorithm In particular considering point119875 inside of the triangle on the lens surface the electric fieldamplitude is given by

E119898119894

(119875) =

E119898119894

(119875119898

119860) 119906119860 + E119898

119894(119875119898

119861) 119906119861 + E119898

119894(119875119898

119862) 119906119862

119906119860 + 119906119861 + 119906119862

(1)

where

E119898119894

(119875119898

119896) = radic

119860119898

119860119898minus1

E119898minus1119894

(119875119898minus1

119896) 119896 = 119860 119861 119862 (2)

119860119898minus1 and 119860119898 denote the area of the triangular sectionof the tube on the lens surface referring to (119898 minus 1)thand 119898th reflection order respectively However in order toaccount for the changes of the tube cross section surfaceand to enforce the energy conservation law the correctioncoefficient radic119860119898119860119898minus1 is introduced In (1) 119875

119898

119860 119875119898

119861 and

119875119898

119862denote the vertices of the surface intersected by the

Mathematical Problems in Engineering 3

tube on the lens surface corresponding to 119898th reflectionorder The coefficients 119906119860 119906119861 and 119906119862 are the inverse of theEuclidean distance between point 119875 and points 119875

119898

119860 119875119898119861 and

119875119898

119862 respectivelyAccording to the GO approach the electric field trans-

mitted outside the lens due to the direct ray and the internalreflections occurring inside the lens is given by the followingequation

E119905 = sum

119898

E119898119905 (3)

where the transmitted field outside the lens due to the 119898thorder internal reflection is

E119898119905

= 119864119898

119905

(n timesk119898

119905) times

k119898

119905

100381710038171003817100381710038171003817(n times

k119898

119905) times

k119898

119905

100381710038171003817100381710038171003817

+ 119864119898

119905perp

n timesk119898

119905

100381710038171003817100381710038171003817n times

k119898

119905

100381710038171003817100381710038171003817

(4)

In (4) n is the unit vector normal to the lens surfacek119898

119905= k119898119905k119898119905 is the transmitted unit wave vector due to

the 119898th internal reflection with k119898119905 = 212058711989901205820 where

1198990 is the refractive index of the external medium and1205820 is the free space wavelength The orthogonal 119864

119898

119905perpand

parallel components 119864119898

119905of the transmitted electric field are

determined bymultiplying the orthogonal119864119898119894perpand parallel119864119898

119894

components of the incident electric field E119898119894in such point by

the proper Fresnel transmission coefficientThe evaluation ofthe incident components of the electric field is carried out bymeans of the following equations

119864119898

119894perp= E119898119894

sdot

n timesk119898

119894

100381710038171003817100381710038171003817n times

k119898

119894

100381710038171003817100381710038171003817

(5)

119864119898

119894= E119898119894

sdot

(n timesk119898

119894) times

k119898

119894

100381710038171003817100381710038171003817(n times

k119898

119894) times

k119898

119894

100381710038171003817100381710038171003817

(6)

where k119898

119894= k119898119894k119898119894 is the incident unit wave vector due

to 119898th internal reflection with k119898119894 = 21205871198991198891205820 where 119899119889 is

the refractive index of the homogeneous dielectric materialforming the lens

In case of direct tube 119898 = 1 the incident field E119898119894(119875119898) at

point 119875119898 is calculated using the far field pattern of the sourceand instead for 119898 gt 1 E119898

119894(119875119898) is derived from (119898 minus 1)th

reflected wave contribution as

E119898119894

(119875119898) = E119898minus1119903

(119875119898minus1) 119890119895k119898119894119889119898

(7)

where 119889119898 is the path length between observation point 119875119898

and point119875119898minus1 where the reflection takes placeThe reflectedfield E119898minus1

119903(119875119898minus1) appearing in (7) is given by the following

equation

E119898minus1119903

(119875119898minus1) = 119864119898minus1

119903

[k119898minus1

119903times (minusn)] times

k119898minus1

119903

100381710038171003817100381710038171003817[k119898minus1


k119898minus1

119903

100381710038171003817100381710038171003817

+ 119864119898minus1

119903perp

k119898minus1

119903times (minusn)

100381710038171003817100381710038171003817

k119898minus1


100381710038171003817100381710038171003817

(8)

where parallel 119864119898minus1

119903and orthogonal 119864

119898minus1

119903perpelectric field

components of (119898 minus 1)th reflected wave are obtained bymultiplying the corresponding components with the Fresnelreflection coefficient k

119898

119894= k119898119894k119898119894 term is the reflected unit

vector referring to (119898minus1)th internal reflection at point 119875119898minus1In order to evaluate the transmitted and reflected unit

wave vector Snellrsquos law at the interface between the dielectriclens and the outside medium has to be considered Inparticular the transmission and reflection phenomena aremodeled by the following relations

119899119889k119894 times n = 1198990

k119905 times n

k119894 times n =k119903 times (minusn)

(9)

where the transmitted unit wave vector k119905 the reflected unitwave vector k119903 the tangential unit vector t and the normalunit vector n to the lens surface are given by

k119905 (120579 120601) = cos (120579119905) n + sin (120579119905) t

= [cos (120579119905) 119899119903 + sin (120579119905) 119905119903] r

+ [cos (120579119905) 119899120579 + sin (120579119905) 119905120579]120579

+ [cos (120579119905) 119899120593 + sin (120579119905) 119905120593]

(10)

k119903 times n = (sin 120579119894 minus cos 120579119894) n (11)

t =

minusn times (n timesk119894)

10038171003817100381710038171003817n times (n times

k119894)10038171003817100381710038171003817

(12)

n =

120597r120597120579 times 120597r1205971205931003817100381710038171003817120597r120597120579 times 120597r1205971205931003817

100381710038171003817

(13)

where the normal unit vector n = 119899119903r + 119899120579120579 + 119899120593 and

tangential unit vector t = 119905119903r + 119905120579120579 + 119905120593 to the lens surface

in spherical coordinates are introduced The terms 120579119905 and 120579119894

denote the transmitted and incident angle formed by thewavepropagation vector and the normal unit vector to the surfacein a given point respectively

120579119894 = tanminus1 (k119894 sdot tk119894 sdot n

)

120579119905 = sinminus1 (119899119889 sin 120579119894)

(14)

Once the transmitted and incident angles are computed theFresnel coefficients for the transmitted wave component canbe evaluated as

119905perp =

2119899119889 cos 120579119894119899119889 cos 120579119894 + 1198990 cos 120579119905

119905 =

2119899119889 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119894

(15)


and the reflected component of the electric field is

119903perp =

119899119889 cos 120579119905 minus 1198990 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119905

119903 =


(16)

Vector r = 119903r appearing in (13) is the radius vector describingthe lens surface defined byGielisrsquo superformula as follows [1718]

r = rradic1199112+ 1199102+ 1199092 (17)

with

119909 (120595 120577) = 119877120595 cos (120595) 119877120577 cos (120577)

119910 (120595 120577) = 119877120595 sin (120595) 119877120577 cos (120577)

119911 (120595 120577) = 119877120577 sin (120577)

119877120595 = [

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus11198871

119877120577 = [

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994

]

minus11198872

(18)

where 119899119901 119886119901 119898119901 119901 = 1 sdot sdot sdot 4 and 119887119902 119902 = 1 2 are real-valuedparameters selected in such a way that the surface of the lensis actually closed and characterized at any point by curvatureradius larger than theworkingwavelength in accordancewiththe GO approximation The parameters 120577 isin [0 1205872] and 120595 isin

[minus120587 120587] denote real-valued parameters appearing in Gielisrsquoformula whereas the spherical angles 120579 and 120593 are obtainedby means of the following relations

120579 = cosminus1 (119911

119903

)

120593 = tanminus1 (119910

119909

)

(19)

The derivatives of the radius vector r appearing in (13) aregiven by

120597r120597120579

=

120597119903

120597120579

r + 119903120579

120597r120597120593

=

120597119903

120597120593

r + 119903 sin (120579)

(20)

where

120597119903

120597120579

=

120597119903

120597120577

120597120577

120597120579

(21)

120597119903

120597120593

=

120597119903

120597120595

minus tan 120574

120597119903

120597120579

(22)

120574 = tanminus1 ( 120597120579

120597120595

) (23)

According to Gielisrsquo description of the lens surface thederivatives of r with respect to 120577 appearing in (21) can beevaluated as follows

120597119903

120597120577

=

1

119903

(119909

120597119909

120597120577

+ 119910

120597119910

120597120577

+ 119911

120597119911

120597120577

)

120597120577

120597120579

=[[

[

119903 (119889119911119889120577) minus 119911 (119889119903119889120577)

1199032radic1 minus (119911119903)

2

]]

]

minus1 (24)

with

119889119911

119889120577

=

119889119877120577

119889120577

sin 120577 + 119877120577 cos 120577

119889119910

119889120577

= 119877120595

119889119877120577

119889120577

cos 120577 sin120595 minus 119877120595119877120577 sin 120577 sin120595

119889119909

119889120577

= 119877120595

119889119877120577

119889120577

cos 120577 cos120595 minus 119877120595119877120577 sin 120577 cos120595

120597119877120577

120597120577

= minus

1

1198872

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994

]

minus(1+1198872)1198872

sdot [1198993

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

+ 1198994

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

= minus

1198983

41198863

sin(

1198983120577

4

) [2119867 (1198863) minus 1]

sdot [2119867(

10038161003816100381610038161003816100381610038161003816

1198983120577

4

100381610038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

10038161003816100381610038161003816100381610038161003816

1198983120577

4

100381610038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

=

1198984

41198864

cos(1198984120577

4

) [2119867 (1198864) minus 1] [1

minus 2119867(

10038161003816100381610038161003816100381610038161003816

1198984120577

4

100381610038161003816100381610038161003816100381610038162120587

minus 120587)]

(25)

where 119867(sdot) denotes the Heaviside functionThe derivative of rwith respect to120595 appearing in (22) can

be evaluated as follows

120597119903

120597120595

=

1

119903

(119909

120597119909

120597120595

+ 119910

120597119910

120597120595

) (26)


005

0

minus005

y (m)

x (m)minus002

0

002

0y (m)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0

005

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0 002minus002

x (m)

minus005

0

005

y(m

)

(b)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0 minus005005

y (m)

0

002

004

z(m

)

(c)

minus001minus002 001 0020

x (m)

0

001

002

003

004

005

z(m

)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

(d)

Figure 3 Densities current distribution on the lens surface considering 119898 = 2 internal reflections (a) prospective view (b) top view and (cd) lateral views

where

119889119910

119889120595

= 119877120577

119889119877120595

119889120595

cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595

119889119909

119889120595

= 119877120577

119889119877120595

119889120595

cos 120577 cos120595 minus 119877120595119877120577 cos 120577 sin120595

120597119877120595

120597120595

= minus

1

1198871

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus(1+1198871)1198871

sdot [1198991

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

+ 1198992

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

= minus

1198981

41198861

sin(

1198981120595

4

) [2119867 (1198861) minus 1]

sdot [2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

=

1198982

41198862

cos(1198982120595

4

) [2119867 (1198862) minus 1] [1

minus 2119867(

1003816100381610038161003816100381610038161003816

1198982120595

4

10038161003816100381610038161003816100381610038162120587

minus 120587)]

120597120579

120597120595

=

119911

1199032[radic1 minus (

119911

119903

)]

minus12

120597119903

120597120595

(27)

Once the contribution of all internal reflections has beenevaluated by the GO method the equivalent electric J119904 andmagnetic M119904 current densities excited along the surface ofthe lens can be easily determined According to the PO


20

2000

minus20minus20

y

x

z

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

(a)

01020

30

210

60

240

90

270

120

300

150

330

180 0minus10

100∘

120579

120601 = 0∘

120601 = 90∘

(b)

Figure 4 Radiation solid (a) and polar sections (b) with 120601 = 0∘ solid line and 120601 = 90

∘ dashed line of the considered lens antenna

x (m)y (m)

005

0

minus005minus004

minus0020

002 02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0

001

002

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

minus002 0 002minus004

x (m)

minus005

0

005

y(m

)

(b)

0

001

002

z(m

)

0 005minus005

y (m)

020406081

Nor

mal

ized

curr

ent

dens

ities

(c)

0 002minus002minus004

x (m)

0

001

002

z(m

)

020406081

Nor

mal

ized

curr

ent

dens

ities

(d)

Figure 5 Densities current distribution on the lens surface considering 119898 = 3 internal reflection (a) prospective view (b) top view and (cd) lateral views

approximation the electromagnetic field radiated in thespatial domain outside the lens at the general observationpoint 119875FF = (rFF 120579FF 120593FF) can be computed by the followingintegral expression

EFF (119875FF)

= j 119890minusj1198960rFF

21205820rFFint

119878

[1205780J119878 times u0 minus M119878] times u0119890j1198960r0sdotu0

119889119878

(28)


z

2020

0 0

minus20 minus20

yx

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

minus10

0

10

20

30

(a)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0120579

120601 = 0∘

120601 = 90∘

minus10

70∘

(b)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0

120601 = 0∘

120601 = 60∘

120601 = 120∘

120579

minus10

(c)

Figure 6 (a) Radiation solid (b) polar sections with 120601 = 0∘ (solid line) and 120601 = 90

∘ (dashed line) and (c) polar sections of the three mainlobes in 120601 = 0

∘ (solid line) 120601 = 60∘ (dashed line) and 120601 = 120

∘ (dotted line)

where r0 is the vector pointing from point 119875 on the lenssurface and observation point 119875FF 1205780 is the characteristicimpedance of the vacuum rFF is the distance vector betweenthe origin of the coordinate system and observation point119875FFand u0 is the unit vector corresponding to r In such way thedirectivity of the antenna can be obtained by the followingexpression

ℎ (120579FF 120593FF) =

4120587r2FF1003817100381710038171003817EFF

1003817100381710038171003817

2

1205780119875tot (29)

where 119875tot is the total electromagnetic power radiated by thelens

3 Numerical Results

Gielisrsquo formulation allows the design of the proper lensshape by changing a few equation parameters In this wayit is possible to analytically define the lens geometry usefulfor meeting size constraints due to the packaging round-ness aspect ratio or any sort of shape-related requirement


30

210

60

240

90

270

120

300

150

330

180 0

Full waveGOPO approach

0

minus10

minus20

minus30

minus40

(a)

30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(b)

Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90

∘

Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach

The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2

internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution

Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture

at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi

Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =

100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =

27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account

Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0

∘ 120601 = 120∘ and

120601 = 240∘ with a directivity of about 11 dBi and an elevation

angle of 120579 = 45∘

Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be


30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)

Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60

∘ and (c)120601 = 120

beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula

4 Model Validation

In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out

Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in


GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation

5 Conclusion

A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998

[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014

[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992

[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and

Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014

[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014

[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993

[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993

[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989

[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005

[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012

[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014

[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006

[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998

[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003

[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011

[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005

[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013

[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


P

PFF

rd

x

rFF

r0

z

ki

120579i

120579t

y

kt

n

Figure 1 Geometrical structure of the electromagnetic systembased on dielectric lens antenna

In this way an improvement of the numerical accuracy canbe obtained Moreover the evaluation of the electromagneticfield radiated from the lens has been carried out applying thephysical optics (PO) approximation In particular accordingto the equivalence principle the radiation outgoing the lensis evaluated by the radiation of the equivalent electric andmagnetic currents on the lens surface The current distribu-tion is calculated by making use of the Fresnel transmissioncoefficient on the lens surface in accordance with the GOprocedure

Thanks to Gielisrsquo formulation it is possible to generate awide range of 3D shapes in a simple and analytical way bychanging a reduced number of parameters The use of thisformulation allows also an easy description of the geometri-cal characteristic of the antenna in terms of lens volume areacurvature radius and aspect ratio Moreover the analyticalrepresentation of the lens shape provides a number of benefitspertaining to the evaluation of the main physical quantitiesinvolved in the electromagnetic propagation equations Inthis way a reduced computation effort is required and moreaccurate results can be obtained

2 Mathematical Model

The geometrical structure considered in the proposed elec-tromagneticmodel consists of a dielectric lens antenna placedon a perfectly electric conductor (PEC) circular plate havingradius 119903119889 (see Figure 1)

The metal plate acts as a ground plane and mechan-ical support of the radiating structure while reducing thebackscattered radiation level The lens is illuminated by theelectromagnetic field emitted by a primary radio source suchas open ended waveguide patch antenna horn antennaor a coaxial probe integrated in the metal plate The tubetracing approach based on the GO approximation is used tomodel the electromagnetic propagation inside the homoge-neous dielectric lensThis approximation allows a significantsimplification of the mathematical model since it makes itpossible to reduce the solution of Maxwellrsquos equations toa geometrical problem As a result the adoption of suchmethod allows the simulation of electrically large structureswith a lower computational effort in comparison to full-wavenumerical methods such as Finite-Difference-Time-Domain(FDTD) and Finite-Element-Method (FEM)

z

y

x

Pmminus1A

Pmminus1B

Pmminus1CAmminus1

PmA

PmB

Am PmC

Figure 2 Area changes from119860119898minus1 to119860119898 of the tube surface duringthe propagation inside the lens

The applicability condition of such method requires thatthe lens surface is placed within the far field zone of thefeeding source and that the geometrical shape of the lens has acurvature radius larger than the operating wavelength In thisway the travelling electromagneticwave can be approximatedby a set of tubes propagating over a rectilinear path insidethe lens Each tube has a triangular cross section with givensurface area and is defined by three rays departing from eachof the triangle vertices The reflected and transmitted raytubes are determined by using GO principles As a result thepropagation path determines a change of the triangle areathat leads a variation of the electromagnetic field amplitude inaccordance with the power conservation law In Figure 2 thegeometrical change of the tube during the propagation insidethe lens is illustrated In particular the surface area variationfrom 119860119898minus1 to 119860119898 is given by triangle vertices intersectionwith the lens surface in points 119875119860 119875119861 and 119875119862

The surface area of each tube is evaluated by consideringthe overlap between the triangular surface with vertices 119875

119898

119860

119875119898

119861 and 119875

119898

119862 and the elements of the tessellation of the lens

surface The amplitude of the electromagnetic field insidethe tube surface is evaluated by using the Inverse DistanceWeighting (IDW) algorithm In particular considering point119875 inside of the triangle on the lens surface the electric fieldamplitude is given by

E119898119894

(119875) =

E119898119894

(119875119898

119860) 119906119860 + E119898

119894(119875119898

119861) 119906119861 + E119898

119894(119875119898

119862) 119906119862

119906119860 + 119906119861 + 119906119862

(1)

where

E119898119894

(119875119898

119896) = radic

119860119898

119860119898minus1

E119898minus1119894

(119875119898minus1

119896) 119896 = 119860 119861 119862 (2)

119860119898minus1 and 119860119898 denote the area of the triangular sectionof the tube on the lens surface referring to (119898 minus 1)thand 119898th reflection order respectively However in order toaccount for the changes of the tube cross section surfaceand to enforce the energy conservation law the correctioncoefficient radic119860119898119860119898minus1 is introduced In (1) 119875

119898

119860 119875119898

119861 and

119875119898

119862denote the vertices of the surface intersected by the



119898

119860 119875119898119861 and

119875119898



E119905 = sum

119898

E119898119905 (3)


E119898119905

= 119864119898

119905

(n timesk119898

119905) times

k119898

119905

100381710038171003817100381710038171003817(n times

k119898

119905) times

k119898

119905

100381710038171003817100381710038171003817

+ 119864119898

119905perp

n timesk119898

119905

100381710038171003817100381710038171003817n times

k119898

119905

100381710038171003817100381710038171003817

(4)





119898

119905perpand




119894



119864119898

119894perp= E119898119894

sdot

n timesk119898

119894

100381710038171003817100381710038171003817n times

k119898

119894

100381710038171003817100381710038171003817

(5)

119864119898

119894= E119898119894

sdot

(n timesk119898

119894) times

k119898

119894

100381710038171003817100381710038171003817(n times

k119898

119894) times

k119898

119894

100381710038171003817100381710038171003817

(6)

where k119898








E119898119894

(119875119898) = E119898minus1119903

(119875119898minus1) 119890119895k119898119894119889119898

(7)




equation

E119898minus1119903

(119875119898minus1) = 119864119898minus1

119903

[k119898minus1


k119898minus1

119903

100381710038171003817100381710038171003817[k119898minus1


k119898minus1

119903

100381710038171003817100381710038171003817

+ 119864119898minus1

119903perp

k119898minus1


100381710038171003817100381710038171003817

k119898minus1


100381710038171003817100381710038171003817

(8)



119898minus1



119898




119899119889k119894 times n = 1198990

k119905 times n


(9)


k119905 (120579 120601) = cos (120579119905) n + sin (120579119905) t

= [cos (120579119905) 119899119903 + sin (120579119905) 119905119903] r

+ [cos (120579119905) 119899120579 + sin (120579119905) 119905120579]120579

+ [cos (120579119905) 119899120593 + sin (120579119905) 119905120593]

(10)


t =


10038171003817100381710038171003817n times (n times

k119894)10038171003817100381710038171003817

(12)

n =

120597r120597120579 times 120597r1205971205931003817100381710038171003817120597r120597120579 times 120597r1205971205931003817

100381710038171003817

(13)






)

120579119905 = sinminus1 (119899119889 sin 120579119894)

(14)


119905perp =

2119899119889 cos 120579119894119899119889 cos 120579119894 + 1198990 cos 120579119905

119905 =

2119899119889 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119894

(15)



119903perp =


119903 =


(16)


r = rradic1199112+ 1199102+ 1199092 (17)

with

119909 (120595 120577) = 119877120595 cos (120595) 119877120577 cos (120577)

119910 (120595 120577) = 119877120595 sin (120595) 119877120577 cos (120577)

119911 (120595 120577) = 119877120577 sin (120577)

119877120595 = [

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus11198871

119877120577 = [

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994

]

minus11198872

(18)



120579 = cosminus1 (119911

119903

)

120593 = tanminus1 (119910

119909

)

(19)


120597r120597120579

=

120597119903

120597120579

r + 119903120579

120597r120597120593

=

120597119903

120597120593

r + 119903 sin (120579)

(20)

where

120597119903

120597120579

=

120597119903

120597120577

120597120577

120597120579

(21)

120597119903

120597120593

=

120597119903

120597120595

minus tan 120574

120597119903

120597120579

(22)

120574 = tanminus1 ( 120597120579

120597120595

) (23)


120597119903

120597120577

=

1

119903

(119909

120597119909

120597120577

+ 119910

120597119910

120597120577

+ 119911

120597119911

120597120577

)

120597120577

120597120579

=[[

[

119903 (119889119911119889120577) minus 119911 (119889119903119889120577)

1199032radic1 minus (119911119903)

2

]]

]

minus1 (24)

with

119889119911

119889120577

=

119889119877120577

119889120577

sin 120577 + 119877120577 cos 120577

119889119910

119889120577

= 119877120595

119889119877120577

119889120577


119889119909

119889120577

= 119877120595

119889119877120577

119889120577


120597119877120577

120597120577

= minus

1

1198872

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994

]

minus(1+1198872)1198872

sdot [1198993

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

+ 1198994

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

= minus

1198983

41198863

sin(

1198983120577

4

) [2119867 (1198863) minus 1]

sdot [2119867(

10038161003816100381610038161003816100381610038161003816

1198983120577

4

100381610038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

10038161003816100381610038161003816100381610038161003816

1198983120577

4

100381610038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

=

1198984

41198864

cos(1198984120577

4

) [2119867 (1198864) minus 1] [1

minus 2119867(

10038161003816100381610038161003816100381610038161003816

1198984120577

4

100381610038161003816100381610038161003816100381610038162120587

minus 120587)]

(25)



120597119903

120597120595

=

1

119903

(119909

120597119909

120597120595

+ 119910

120597119910

120597120595

) (26)


005

0

minus005

y (m)

x (m)minus002

0

002

0y (m)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0

005

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0 002minus002

x (m)

minus005

0

005

y(m

)

(b)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0 minus005005

y (m)

0

002

004

z(m

)

(c)


x (m)

0

001

002

003

004

005

z(m

)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

(d)


where

119889119910

119889120595

= 119877120577

119889119877120595

119889120595

cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595

119889119909

119889120595

= 119877120577

119889119877120595

119889120595


120597119877120595

120597120595

= minus

1

1198871

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus(1+1198871)1198871

sdot [1198991

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

+ 1198992

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

= minus

1198981

41198861

sin(

1198981120595

4

) [2119867 (1198861) minus 1]

sdot [2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

=

1198982

41198862

cos(1198982120595

4

) [2119867 (1198862) minus 1] [1

minus 2119867(

1003816100381610038161003816100381610038161003816

1198982120595

4

10038161003816100381610038161003816100381610038162120587

minus 120587)]

120597120579

120597120595

=

119911


119911

119903

)]

minus12

120597119903

120597120595

(27)



20

2000

minus20minus20

y

x

z

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

(a)

01020

30

210

60

240

90

270

120

300

150

330

180 0minus10

100∘

120579

120601 = 0∘

120601 = 90∘

(b)



x (m)y (m)

005

0

minus005minus004

minus0020

002 02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0

001

002

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities


x (m)

minus005

0

005

y(m

)

(b)

0

001

002

z(m

)

0 005minus005

y (m)

020406081

Nor

mal

ized

curr

ent

dens

ities

(c)


x (m)

0

001

002

z(m

)

020406081

Nor

mal

ized

curr

ent

dens

ities

(d)



EFF (119875FF)

= j 119890minusj1198960rFF

21205820rFFint

119878


119889119878

(28)


z

2020

0 0

minus20 minus20

yx

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

minus10

0

10

20

30

(a)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0120579

120601 = 0∘

120601 = 90∘

minus10

70∘

(b)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0

120601 = 0∘

120601 = 60∘

120601 = 120∘

120579

minus10

(c)




∘ (dotted line)


ℎ (120579FF 120593FF) =

4120587r2FF1003817100381710038171003817EFF

1003817100381710038171003817

2

1205780119875tot (29)


3 Numerical Results



30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(a)

30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(b)


∘










∘ 120601 = 120∘ and


angle of 120579 = 45∘



30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)


∘ and (c)120601 = 120


4 Model Validation





5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898

119860 119875119898119861 and

119875119898



E119905 = sum

119898

E119898119905 (3)


E119898119905

= 119864119898

119905

(n timesk119898

119905) times

k119898

119905

100381710038171003817100381710038171003817(n times

k119898

119905) times

k119898

119905

100381710038171003817100381710038171003817

+ 119864119898

119905perp

n timesk119898

119905

100381710038171003817100381710038171003817n times

k119898

119905

100381710038171003817100381710038171003817

(4)





119898

119905perpand




119894



119864119898

119894perp= E119898119894

sdot

n timesk119898

119894

100381710038171003817100381710038171003817n times

k119898

119894

100381710038171003817100381710038171003817

(5)

119864119898

119894= E119898119894

sdot

(n timesk119898

119894) times

k119898

119894

100381710038171003817100381710038171003817(n times

k119898

119894) times

k119898

119894

100381710038171003817100381710038171003817

(6)

where k119898








E119898119894

(119875119898) = E119898minus1119903

(119875119898minus1) 119890119895k119898119894119889119898

(7)




equation

E119898minus1119903

(119875119898minus1) = 119864119898minus1

119903

[k119898minus1


k119898minus1

119903

100381710038171003817100381710038171003817[k119898minus1


k119898minus1

119903

100381710038171003817100381710038171003817

+ 119864119898minus1

119903perp

k119898minus1


100381710038171003817100381710038171003817

k119898minus1


100381710038171003817100381710038171003817

(8)



119898minus1



119898




119899119889k119894 times n = 1198990

k119905 times n


(9)


k119905 (120579 120601) = cos (120579119905) n + sin (120579119905) t

= [cos (120579119905) 119899119903 + sin (120579119905) 119905119903] r

+ [cos (120579119905) 119899120579 + sin (120579119905) 119905120579]120579

+ [cos (120579119905) 119899120593 + sin (120579119905) 119905120593]

(10)


t =


10038171003817100381710038171003817n times (n times

k119894)10038171003817100381710038171003817

(12)

n =

120597r120597120579 times 120597r1205971205931003817100381710038171003817120597r120597120579 times 120597r1205971205931003817

100381710038171003817

(13)






)

120579119905 = sinminus1 (119899119889 sin 120579119894)

(14)


119905perp =

2119899119889 cos 120579119894119899119889 cos 120579119894 + 1198990 cos 120579119905

119905 =

2119899119889 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119894

(15)



119903perp =


119903 =


(16)


r = rradic1199112+ 1199102+ 1199092 (17)

with

119909 (120595 120577) = 119877120595 cos (120595) 119877120577 cos (120577)

119910 (120595 120577) = 119877120595 sin (120595) 119877120577 cos (120577)

119911 (120595 120577) = 119877120577 sin (120577)

119877120595 = [

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus11198871

119877120577 = [

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994

]

minus11198872

(18)



120579 = cosminus1 (119911

119903

)

120593 = tanminus1 (119910

119909

)

(19)


120597r120597120579

=

120597119903

120597120579

r + 119903120579

120597r120597120593

=

120597119903

120597120593

r + 119903 sin (120579)

(20)

where

120597119903

120597120579

=

120597119903

120597120577

120597120577

120597120579

(21)

120597119903

120597120593

=

120597119903

120597120595

minus tan 120574

120597119903

120597120579

(22)

120574 = tanminus1 ( 120597120579

120597120595

) (23)


120597119903

120597120577

=

1

119903

(119909

120597119909

120597120577

+ 119910

120597119910

120597120577

+ 119911

120597119911

120597120577

)

120597120577

120597120579

=[[

[

119903 (119889119911119889120577) minus 119911 (119889119903119889120577)

1199032radic1 minus (119911119903)

2

]]

]

minus1 (24)

with

119889119911

119889120577

=

119889119877120577

119889120577

sin 120577 + 119877120577 cos 120577

119889119910

119889120577

= 119877120595

119889119877120577

119889120577


119889119909

119889120577

= 119877120595

119889119877120577

119889120577


120597119877120577

120597120577

= minus

1

1198872

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994

]

minus(1+1198872)1198872

sdot [1198993

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

+ 1198994

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

= minus

1198983

41198863

sin(

1198983120577

4

) [2119867 (1198863) minus 1]

sdot [2119867(

10038161003816100381610038161003816100381610038161003816

1198983120577

4

100381610038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

10038161003816100381610038161003816100381610038161003816

1198983120577

4

100381610038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

=

1198984

41198864

cos(1198984120577

4

) [2119867 (1198864) minus 1] [1

minus 2119867(

10038161003816100381610038161003816100381610038161003816

1198984120577

4

100381610038161003816100381610038161003816100381610038162120587

minus 120587)]

(25)



120597119903

120597120595

=

1

119903

(119909

120597119909

120597120595

+ 119910

120597119910

120597120595

) (26)


005

0

minus005

y (m)

x (m)minus002

0

002

0y (m)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0

005

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0 002minus002

x (m)

minus005

0

005

y(m

)

(b)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0 minus005005

y (m)

0

002

004

z(m

)

(c)


x (m)

0

001

002

003

004

005

z(m

)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

(d)


where

119889119910

119889120595

= 119877120577

119889119877120595

119889120595

cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595

119889119909

119889120595

= 119877120577

119889119877120595

119889120595


120597119877120595

120597120595

= minus

1

1198871

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus(1+1198871)1198871

sdot [1198991

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

+ 1198992

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

= minus

1198981

41198861

sin(

1198981120595

4

) [2119867 (1198861) minus 1]

sdot [2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

=

1198982

41198862

cos(1198982120595

4

) [2119867 (1198862) minus 1] [1

minus 2119867(

1003816100381610038161003816100381610038161003816

1198982120595

4

10038161003816100381610038161003816100381610038162120587

minus 120587)]

120597120579

120597120595

=

119911


119911

119903

)]

minus12

120597119903

120597120595

(27)



20

2000

minus20minus20

y

x

z

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

(a)

01020

30

210

60

240

90

270

120

300

150

330

180 0minus10

100∘

120579

120601 = 0∘

120601 = 90∘

(b)



x (m)y (m)

005

0

minus005minus004

minus0020

002 02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0

001

002

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities


x (m)

minus005

0

005

y(m

)

(b)

0

001

002

z(m

)

0 005minus005

y (m)

020406081

Nor

mal

ized

curr

ent

dens

ities

(c)


x (m)

0

001

002

z(m

)

020406081

Nor

mal

ized

curr

ent

dens

ities

(d)



EFF (119875FF)

= j 119890minusj1198960rFF

21205820rFFint

119878


119889119878

(28)


z

2020

0 0

minus20 minus20

yx

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

minus10

0

10

20

30

(a)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0120579

120601 = 0∘

120601 = 90∘

minus10

70∘

(b)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0

120601 = 0∘

120601 = 60∘

120601 = 120∘

120579

minus10

(c)




∘ (dotted line)


ℎ (120579FF 120593FF) =

4120587r2FF1003817100381710038171003817EFF

1003817100381710038171003817

2

1205780119875tot (29)


3 Numerical Results



30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(a)

30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(b)


∘










∘ 120601 = 120∘ and


angle of 120579 = 45∘



30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)


∘ and (c)120601 = 120


4 Model Validation





5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903perp =


119903 =


(16)


r = rradic1199112+ 1199102+ 1199092 (17)

with

119909 (120595 120577) = 119877120595 cos (120595) 119877120577 cos (120577)

119910 (120595 120577) = 119877120595 sin (120595) 119877120577 cos (120577)

119911 (120595 120577) = 119877120577 sin (120577)

119877120595 = [

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus11198871

119877120577 = [

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994

]

minus11198872

(18)



120579 = cosminus1 (119911

119903

)

120593 = tanminus1 (119910

119909

)

(19)


120597r120597120579

=

120597119903

120597120579

r + 119903120579

120597r120597120593

=

120597119903

120597120593

r + 119903 sin (120579)

(20)

where

120597119903

120597120579

=

120597119903

120597120577

120597120577

120597120579

(21)

120597119903

120597120593

=

120597119903

120597120595

minus tan 120574

120597119903

120597120579

(22)

120574 = tanminus1 ( 120597120579

120597120595

) (23)


120597119903

120597120577

=

1

119903

(119909

120597119909

120597120577

+ 119910

120597119910

120597120577

+ 119911

120597119911

120597120577

)

120597120577

120597120579

=[[

[

119903 (119889119911119889120577) minus 119911 (119889119903119889120577)

1199032radic1 minus (119911119903)

2

]]

]

minus1 (24)

with

119889119911

119889120577

=

119889119877120577

119889120577

sin 120577 + 119877120577 cos 120577

119889119910

119889120577

= 119877120595

119889119877120577

119889120577


119889119909

119889120577

= 119877120595

119889119877120577

119889120577


120597119877120577

120597120577

= minus

1

1198872

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994

]

minus(1+1198872)1198872

sdot [1198993

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

1198993minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

+ 1198994

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

1198994minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989831205774)1198863

100381610038161003816100381610038161003816100381610038161003816

= minus

1198983

41198863

sin(

1198983120577

4

) [2119867 (1198863) minus 1]

sdot [2119867(

10038161003816100381610038161003816100381610038161003816

1198983120577

4

100381610038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

10038161003816100381610038161003816100381610038161003816

1198983120577

4

100381610038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989841205774)

1198864

100381610038161003816100381610038161003816100381610038161003816

=

1198984

41198864

cos(1198984120577

4

) [2119867 (1198864) minus 1] [1

minus 2119867(

10038161003816100381610038161003816100381610038161003816

1198984120577

4

100381610038161003816100381610038161003816100381610038162120587

minus 120587)]

(25)



120597119903

120597120595

=

1

119903

(119909

120597119909

120597120595

+ 119910

120597119910

120597120595

) (26)


005

0

minus005

y (m)

x (m)minus002

0

002

0y (m)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0

005

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0 002minus002

x (m)

minus005

0

005

y(m

)

(b)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0 minus005005

y (m)

0

002

004

z(m

)

(c)


x (m)

0

001

002

003

004

005

z(m

)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

(d)


where

119889119910

119889120595

= 119877120577

119889119877120595

119889120595

cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595

119889119909

119889120595

= 119877120577

119889119877120595

119889120595


120597119877120595

120597120595

= minus

1

1198871

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus(1+1198871)1198871

sdot [1198991

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

+ 1198992

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

= minus

1198981

41198861

sin(

1198981120595

4

) [2119867 (1198861) minus 1]

sdot [2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

=

1198982

41198862

cos(1198982120595

4

) [2119867 (1198862) minus 1] [1

minus 2119867(

1003816100381610038161003816100381610038161003816

1198982120595

4

10038161003816100381610038161003816100381610038162120587

minus 120587)]

120597120579

120597120595

=

119911


119911

119903

)]

minus12

120597119903

120597120595

(27)



20

2000

minus20minus20

y

x

z

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

(a)

01020

30

210

60

240

90

270

120

300

150

330

180 0minus10

100∘

120579

120601 = 0∘

120601 = 90∘

(b)



x (m)y (m)

005

0

minus005minus004

minus0020

002 02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0

001

002

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities


x (m)

minus005

0

005

y(m

)

(b)

0

001

002

z(m

)

0 005minus005

y (m)

020406081

Nor

mal

ized

curr

ent

dens

ities

(c)


x (m)

0

001

002

z(m

)

020406081

Nor

mal

ized

curr

ent

dens

ities

(d)



EFF (119875FF)

= j 119890minusj1198960rFF

21205820rFFint

119878


119889119878

(28)


z

2020

0 0

minus20 minus20

yx

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

minus10

0

10

20

30

(a)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0120579

120601 = 0∘

120601 = 90∘

minus10

70∘

(b)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0

120601 = 0∘

120601 = 60∘

120601 = 120∘

120579

minus10

(c)




∘ (dotted line)


ℎ (120579FF 120593FF) =

4120587r2FF1003817100381710038171003817EFF

1003817100381710038171003817

2

1205780119875tot (29)


3 Numerical Results



30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(a)

30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(b)


∘










∘ 120601 = 120∘ and


angle of 120579 = 45∘



30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)


∘ and (c)120601 = 120


4 Model Validation





5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










005

0

minus005

y (m)

x (m)minus002

0

002

0y (m)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0

005

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

0 002minus002

x (m)

minus005

0

005

y(m

)

(b)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0 minus005005

y (m)

0

002

004

z(m

)

(c)


x (m)

0

001

002

003

004

005

z(m

)

02

04

06

08

1

Nor

mal

ized

curr

ent d

ensit

ies

(d)


where

119889119910

119889120595

= 119877120577

119889119877120595

119889120595

cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595

119889119909

119889120595

= 119877120577

119889119877120595

119889120595


120597119877120595

120597120595

= minus

1

1198871

[

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991

+

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992

]

minus(1+1198871)1198871

sdot [1198991

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

1198991minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

+ 1198992

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

1198992minus1

120597

120597120577

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

cos (11989811205954)

1198861

100381610038161003816100381610038161003816100381610038161003816

= minus

1198981

41198861

sin(

1198981120595

4

) [2119867 (1198861) minus 1]

sdot [2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

3

4

120587) minus 2119867(

1003816100381610038161003816100381610038161003816

1198981120595

4

10038161003816100381610038161003816100381610038162120587

minus

120587

2

)

+ 1]

120597

120597120595

100381610038161003816100381610038161003816100381610038161003816

sin (11989821205954)

1198862

100381610038161003816100381610038161003816100381610038161003816

=

1198982

41198862

cos(1198982120595

4

) [2119867 (1198862) minus 1] [1

minus 2119867(

1003816100381610038161003816100381610038161003816

1198982120595

4

10038161003816100381610038161003816100381610038162120587

minus 120587)]

120597120579

120597120595

=

119911


119911

119903

)]

minus12

120597119903

120597120595

(27)



20

2000

minus20minus20

y

x

z

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

(a)

01020

30

210

60

240

90

270

120

300

150

330

180 0minus10

100∘

120579

120601 = 0∘

120601 = 90∘

(b)



x (m)y (m)

005

0

minus005minus004

minus0020

002 02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0

001

002

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities


x (m)

minus005

0

005

y(m

)

(b)

0

001

002

z(m

)

0 005minus005

y (m)

020406081

Nor

mal

ized

curr

ent

dens

ities

(c)


x (m)

0

001

002

z(m

)

020406081

Nor

mal

ized

curr

ent

dens

ities

(d)



EFF (119875FF)

= j 119890minusj1198960rFF

21205820rFFint

119878


119889119878

(28)


z

2020

0 0

minus20 minus20

yx

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

minus10

0

10

20

30

(a)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0120579

120601 = 0∘

120601 = 90∘

minus10

70∘

(b)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0

120601 = 0∘

120601 = 60∘

120601 = 120∘

120579

minus10

(c)




∘ (dotted line)


ℎ (120579FF 120593FF) =

4120587r2FF1003817100381710038171003817EFF

1003817100381710038171003817

2

1205780119875tot (29)


3 Numerical Results



30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(a)

30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(b)


∘










∘ 120601 = 120∘ and


angle of 120579 = 45∘



30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)


∘ and (c)120601 = 120


4 Model Validation





5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20

2000

minus20minus20

y

x

z

minus10

0

10

20

30

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

(a)

01020

30

210

60

240

90

270

120

300

150

330

180 0minus10

100∘

120579

120601 = 0∘

120601 = 90∘

(b)



x (m)y (m)

005

0

minus005minus004

minus0020

002 02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities

0

001

002

z(m

)

(a)

02

04

06

08

1

Nor

mal

ized

curr

ent

dens

ities


x (m)

minus005

0

005

y(m

)

(b)

0

001

002

z(m

)

0 005minus005

y (m)

020406081

Nor

mal

ized

curr

ent

dens

ities

(c)


x (m)

0

001

002

z(m

)

020406081

Nor

mal

ized

curr

ent

dens

ities

(d)



EFF (119875FF)

= j 119890minusj1198960rFF

21205820rFFint

119878


119889119878

(28)


z

2020

0 0

minus20 minus20

yx

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

minus10

0

10

20

30

(a)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0120579

120601 = 0∘

120601 = 90∘

minus10

70∘

(b)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0

120601 = 0∘

120601 = 60∘

120601 = 120∘

120579

minus10

(c)




∘ (dotted line)


ℎ (120579FF 120593FF) =

4120587r2FF1003817100381710038171003817EFF

1003817100381710038171003817

2

1205780119875tot (29)


3 Numerical Results



30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(a)

30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(b)


∘










∘ 120601 = 120∘ and


angle of 120579 = 45∘



30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)


∘ and (c)120601 = 120


4 Model Validation





5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

2020

0 0

minus20 minus20

yx

minus20

minus15

minus10

minus5

0

5

10

Dire

ctiv

ity (d

Bi)

minus10

0

10

20

30

(a)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0120579

120601 = 0∘

120601 = 90∘

minus10

70∘

(b)

0

10

20

30

210

60

240

90

270

120

300

150

330

180 0

120601 = 0∘

120601 = 60∘

120601 = 120∘

120579

minus10

(c)




∘ (dotted line)


ℎ (120579FF 120593FF) =

4120587r2FF1003817100381710038171003817EFF

1003817100381710038171003817

2

1205780119875tot (29)


3 Numerical Results



30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(a)

30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(b)


∘










∘ 120601 = 120∘ and


angle of 120579 = 45∘



30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)


∘ and (c)120601 = 120


4 Model Validation





5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(a)

30

210

60

240

90

270

120

300

150

330

180 0


0

minus10

minus20

minus30

minus40

(b)


∘










∘ 120601 = 120∘ and


angle of 120579 = 45∘



30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)


∘ and (c)120601 = 120


4 Model Validation





5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(a)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(b)

30

210

60

240

90

270

120

300

150

330

180 0

0

minus10

minus20

minus30

minus40


(c)


∘ and (c)120601 = 120


4 Model Validation





5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusion




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Electromagnetic Mathematical Modeling of ...downloads.hindawi.com/journals/mpe/2016/8130160.pdfResearch Article Electromagnetic Mathematical Modeling of 3D Supershaped