ELECTROMAGNETIC CHARACTERISTICS OF CON- FORMAL DIPOLE ANTENNAS OVER A PEC SPHERE

Transcription

1 Prgress In Electrmagnetics Research M, Vl. 26, 85 1, 212 ELECTROMAGNETIC CHARACTERISTICS OF CON- FORMAL DIPOLE ANTENNAS OVER A PEC SPHERE J. S. Meiguni *, M. Kamyab, and A. Hsseinbeig Faculty f Electrical and Cmputer Engineering, K. N. Tsi University f Technlgy, P. O. Bx , Tehran , Iran Abstract Rigrus mathematical Methd f Mments (MMs) fr analyzing varius radiating spherical structures is presented in this paper by using Dyadic Green s Functins (DGFs) in cnjunctin with Mixed Ptential Integral Equatin (MPIE) frmulatin. With the aid f linear Ra-Wiltn-Glissn (RWG) triangular basis functins and by cnverting spherical DGFs t Cartesian DGFs, a cnfrmal diple antenna in free space and ver a Perfect Electric Cnductr (PEC) sphere is analyzed. The characteristics f such antennas are cmputed by applying multilayer spherical DGFs and asympttic apprximatin methds. Mutual cuplings between elements f a cnfrmal diple antenna array in free space and ver a cnducting sphere are als investigated. Gd agreement between the cmputatinal results btained by the prpsed methds and thse btained frm cmmercial simulatr packages shws accuracy and high cnvergence speed f the presented methds. 1. INTRODUCTION By using Dyadic Green s Functins (DGFs), a field cmpnent can be expressed in terms f a current vectr cmpnent [1, 2]. Deriving apprpriate DGFs fr analyzing electrmagnetic bundary r eigenvalue prblems results in substantial simplificatin and cmpactness [1]. Althugh mst f the prblems can be slved withut DGFs, it is mre efficient t use dyadic Green s functins. It shuld be nted that DGFs have been calculated nly fr sme cannical structures and fr many cases vectr wave functins shuld be used t Received 18 August 212, Accepted 19 September 212, Scheduled 26 September 212 * Crrespnding authr: Javad Sleiman-Meiguni (meiguni@ee.kntu.ac.ir).

2 86 Meiguni, Kamyab, and Hsseinbeig btain DGFs suitable fr thse particular prblems. Electrmagnetic fields in a specific directin have been calculated fr a small electric and magnetic diple lcated inside a multilayer sphere [3]. In [4], an antenna structure ver cylindrical shell has been analyzed by using spectral dmain Green s functins. The resnance prblem f a circular micrstrip disk munted n a spherical surface has been studied theretically by utilizing Green s functin frmulatin in spectral dmain [5]. The resnant frequencies fr the lssless spherical cavity filled with a Chiral medium are btained frm the dyadic Green s functins [6]. Clsed frm equatins using DGFs cntaining a series f spherical harmnics have been used t evaluate electric and magnetic fields in such structures [2]. In [7], DGFs have been expanded asympttically resulting in a significant increase in cnvergence rate f spherical harmnics series. With this apprach, cnfrmal antenna prblems have been slved by evaluating DGFs cefficients in [8] using spherical Bessel and Hankel functins apprximatin fr large arguments. Asympttic apprximatin methd presented in [7] yields a higher cnvergence speed in the calculatin f antenna input impedance but it cannt be utilized fr radiatin pattern determinatin since field and surce pints are nt at the same distance frm the sphere center in this case. In [9], a Hybrid Finite Element-Bundary Integral (FE-BI) frmulatin is intrduced, emplying asympttic spherid DGFs fr design f dubly curved cnfrmal antennas. A cmbinatin f the Finite Difference Time Dmain (FDTD) and MMs is applied t slve the prblem f radiatin frm cnfrmal aperture and micrstrip antennas munted n arbitrarily-shaped cnducting bdies [1]. Mst f the MMs appraches mentined abve cnsider specific current distributin n antenna elements. It is knwn that mdal analysis which is a part f prcedure fr field calculatin by MMs suffers frm sme shrtcming when number f mdes increases. This is due t prir specific current distributin cnsidered. Cmplexity f meshing the structures is anther drawback encuntered in prblems where the antenna elements are segmented int curvilinear cells. In this paper, explicit frmulas are derived t analyze varius radiating structures where surce regins are divided int linear triangles. Then a spherically cnfrmal diple antenna fed at its center lcated in free space is analyzed. Thereafter using DGFs, the input admittance and radiatin pattern f such a diple antenna ver a PEC sphere are presented. In this paper, DGFs transfrmatin frm spherical t Cartesian crdinates is als accmplished, which is used t efficiently cmpute radiatin fields f varius spherical antenna structures with linear triangular mesh generatins. In rder t validate

3 Prgress In Electrmagnetics Research M, Vl. 26, the presented methd, the antenna is als analyzed by meshing the PEC sphere and cnfrmal diples int linear triangles and using free space Green s functins. Mutual cuplings between the elements f a cnfrmal diple antenna array lcated in free space r ver a cnducting sphere are investigated. Cmparisn f the results btained frm the prpsed methd with thse f CAD simulatins clearly shws the ability and accuracy f presented methd. 2. THE PROPOSED FORMULATION One f the efficient numerical methds with high preprcessing gain fr slving electrmagnetic structures is the methd f mments where the surce regin must be divided int cells. In this methd, the unknwn functins which are usually the surce currents r charges are btained via an integral equatin frmulatin with apprpriate Green s functins. Such integral equatins can be in space dmain, spectral dmain, r bth f these tw dmains. In general due t meshed finite surce regin, methds based n integral equatin frmulatins are mre accurate and require less memry and time [11]. Fr any class f integral equatins namely Electric Field Integral Equatin (EFIE), Magnetic Field Integral Equatin (MFIE) r Mixed Ptential Integral Equatin (MPIE), apprpriate type f Green s functins shuld be used. EFIE has the advantage f being applicable t bth pen and clsed structures, whereas MFIE applies nly t clsed surfaces [12]. A set f frmulatin which uses auxiliary ptentials and leads t MPIE frmulatin is weakly singular and can be easily utilized in MMs frmulatin. Many researches have been reprted t calculate input impedance and radiatin pattern f an antenna using MPIE frmulatin [13]. Dividing the surce regin int small triangles and cnsidering the cmmn edge between tw adjacent cells as a current element, and expanding the current int triangular basis functins (f n ) [12], the surce current can be defined as: J s = N I n f n (1) n=1 Here N is the number f nn-bundary edges and I n cefficients are t be btained frm MPIE frmulatin [13]. Therefre if the surce regin is segmented by linear triangular cells, unknwn current cefficients in the antenna can be determined by applying RWG basis functins and satisfying the bundary cnditins. In this case the electric field cnsidered as: E = jωa ψ, (2)

4 88 Meiguni, Kamyab, and Hsseinbeig where A is magnetic vectr ptential and ψ the electric scalar ptential as defined in [14]. Since tangential cmpnent f electric field vanishes n perfect cnducting metal, impedance matrix by applying MPIE and Galerkin s methd can be written as: [ ( Z pq = jωfp (r) ḠA f q r ) ( ( +( f p (r)) G ψ f q r ))] ds ds, (3) s s in which Ḡ A and G ψ are respectively the magnetic dyadic and the electric scalar Green s functins. These functins are presented in spherical crdinates in [14]. r(r, θ, ϕ) and r (r, θ, ϕ ) refer t field and surce psitin vectrs, respectively. f p,q (r) are triangular basis functins defined in [12]. The integratin ver the testing triangles can be avided by using the centers f field triangular cells and apprximate Galerkin s methd [12]. Therefre, the integratin is perfrmed nly ver surce triangles by applying 3-pint Gauss quadrature methd. In this case, by applying the afrementined integratin methd, n singularity will be encuntered in the integratin. Therefre, there is n need t slve the duble integratin analytically in the singular pints f the integrands as dne in [11]. Fr a structure with hmgenus media, it is expected that the impedance matrix calculated frm MMs will be symmetric and diagnally dminant. Figure 1(a) illustrates a cnfrmal diple antenna lcated in free space r in the vicinity f a PEC sphere. As shwn in Fig. 1(b), the structure can be divided int three regins. The cnfrmal antenna is lcated at the bundary f layers 1 and 2. Regin 3 may be cnsidered as free space r PEC. In the latter case, the cnducting medium is mdeled by ε, µ in rder fr the prpagatin cnstant t be finite and numerical mdeling f the antenna t be feasible [7]. Free Space r PEC Sphere Air Feed Pint (a) (b) Figure 1. (a) A cnfrmal diple antenna in free space r ver PEC sphere, (b) 3-layer sphere and the antenna feed pint mdel.

5 Prgress In Electrmagnetics Research M, Vl. 26, It shuld be nted that input impedance frmula f a cnfrmal antenna ver a spherical shell can be extracted using additin therem fr spherical Hankel functins yielding asympttic frmulatin with mre cnvergence speed [15]. Fr cnfrmal antenna ver a multilayer sphere, when bth surce and field pints are at the same distance frm the sphere center, we d nt need t cnsider Ḡ(fs) and it is (f s) nly enugh t cmpute G ψ [7]. Therefre, input impedance f antennas lcated n a multilayer sphere can be btained by using DGFs r asympttic apprximatin frmulas in MPIE Cnfrmal Diple Antenna in Free Space When antenna is lcated in an unbunded free space, scalar wave equatins can be applied instead f dyadic frms fr calculatin f impedance matrix resulting in the fllwing equatin: [ jωµ Z pq = 4π f p(r c± ) f q (r )g+ 1 ( fp (r c± ) )( f q (r ) ) ] g ds, (4) j4πωε s where g = e jkr /R and c+ r c dentes the center f psitive r negative triangles, respectively. Therefre, scalar unknwn cefficients f current distributins are determined n the cmmn edges between tw adjacent triangles by cmputing impedance and antenna excitatin matrices. Electric field vectr can be calculated frm Equatin (2) r by integrating psterir scalar prduct f DGFs and current vectrs ver the surce regin as described in the fllwing equatin: (fs) E = jωµ f ḠE J(r )ds, (5) s A in which Ḡ(fs) E defined by: = Ḡ(fs) E is DGFs in an unbunded free space and is ( Ḡ (fs) E = Ī + ) 1 kf 2 G, (6) G = e jk f R 4 π R, (7) where k f is the prpagatin cnstant in free space. In rder t simplify cmputatin prcedure, Cartesian DGFs f free space may be used when surce and field pints are bth lcated in free space. It can be

6 9 Meiguni, Kamyab, and Hsseinbeig written with the aid f Equatin (6) as fllws: R= (x x ) 2 + (y y ) 2 + (z z ) 2 (8a) ( ) Gxx G xy G xz Ḡ (fs) E = G yx G yy G yz (8b) G zx G zy G zz ( e jk f R R 2 jk f R 3 +kf 2R4 +3 (u u ) 2 k f R ( 3j+k f R) (u u ) 2) G uu = 4kf 2, (8c) πr5 ) e jk f R ( 3 3jk f R+kf 2R3 (u u )(v v ) G uv = 4kf 2π ; u v, (8d) R5 where u, v = x, y, z and G u v = G v u because f symmetrical prperties f DGFs Cnfrmal Diple Antenna in the Vicinity f a Cnducting Sphere Fr full wave analysis f a cnfrmal diple antenna ver a PEC sphere, the scattering cmpnents f DGFs (Ḡ (fs) es ) must be taken int cnsideratin. By cnsidering surce and field pints in layer 1 Ḡ (11) E = Ḡ(11) E + Ḡ(11) es in which Ḡ(11) es is as fllws: Ḡ (11) es = jk n 1 ( ) 2 δ 2n + 1 (n m)! 4π m n (n + 1) (n + m)! n=1 m= { b 11 M M(2) e (k mn 1)M (2) e (k } mn 1) + b 11 N N(2) e (k mn 1)N (2) e (k, (9a) mn 1) where δm is the Krnecker delta, and M, N vectrs are r-dependent eigenvectrs in spherical crdinate system with rthgnal prperties explained in [2]. Superscript (2) dentes spherical Hankel functin f the secnd kind. b 11 M,N are the dyadic cefficients expressed as [8]: M,N = RH,V F 2 b 11 R H,V F 2 T H,V F 1 RH,V P 1 + R H,V F 1 T H,V P 1 T H,V F 1 + T H,V P 1 (9b) Cnvergence speed f multilayer spherical DGFs is related t the radii f spheres and permittivities f layers. Mre spherical harmnic terms shuld be cnsidered fr greater radii and permittivities. Calculatin f the duble-summatin f abve DGFs is exhausting and

7 Prgress In Electrmagnetics Research M, Vl. 26, time-cnsuming since there are spherical harmnics in wave equatin slutin in spherical crdinates. The duble summatin in the spherical DGFs can be reduced t an expressin with nly a single summatin by using the fllwing relatin [13]: n ( P n (cs γ)= 2 δ )(n m)! m (n+m)! P n m (csθ)p n m (cs θ ) cs ( m(ϕ ϕ ) ), (1) m= where cs γ = cs θ cs θ + sin θ sin θ cs(ϕ ϕ ). Therefre, the cnvergence speed f DGFs cmputatin increases by cnverting the duble summatin f spherical harmnics t a single summatin. T achieve t a precise calculatin f far fields with an errr f abut 1%, using the 2 first terms in the summatin is sufficient. In the presented methd, it takes abut 7 minutes t cmpute radiatin pattern (with 6 divisins f ϕ) with a Cre GHz prcessr. The fast cmputatin time f this methd is cnsiderable in cmparisn with CAD simulatr packages. Free space Green s functins can als be used t analyze this antenna. In this case, the PEC sphere shuld als be meshed by linear triangles and current distributins n cmmn edges are cmputed by slving MPIE. Electric field cmpnents can be calculated by utilizing Equatin (2) r Equatin (5). Fig. 2 illustrates a cnfrmal diple antenna ver a meshed cnducting sphere with 84 triangles and 126 cmmn edges. Due t large number f triangular cells, this methd takes mre memry and time in cmparisn with asympttic apprximatin r multilayer spherical DGFs methds Array f Cnfrmal Diple Antennas The presented methd can be utilized t investigate mutual cuplings between antenna elements f a cnfrmal diple antenna lcated in Figure 2. A cnfrmal diple antenna ver a meshed PEC sphere.

8 92 Meiguni, Kamyab, and Hsseinbeig free space r abve a PEC sphere. Scattering matrix can be calculated as fllws [16]: [S] = (Y [I] + [Y ]) 1 (Y [I] [Y ]), (11) where I is the identity matrix and Y admittance matrix. Y is cnsidered as characteristic admittance f feed netwrk. As delta gap vltage is used t excite the electric diples, it is advantageus t cmpute scattering matrix frm admittance matrix. The cmpnents f admittance matrix are defined as: Y i j = I i (12) V j Vi =, i j Therefre, fr an array cnsisting f N elements, cupled currents due t an excited element n the center f the unexcited elements shuld be calculated Spherical t Cartesian Transfrmatin f DGF In rder t btain electrmagnetic field cmpnents, we require multiplying DGFs and current element vectr cmpnents fr the case f interest. In rder t perfrm multiplicatin, bth DGFs and current vectrs shuld be in the same crdinates. Fr the case that cnfrmal antenna area is divided int curvilinear triangles as shwn in Fig. 3(a), current cmpnents t be cnsidered are J ϕ and J θ which are in harmny with spherical cmpnents f DGFs [8]. Hwever curvilinear meshing is cmplicated as cmpared with linear triangular meshing cnsidered in this paper. As current elements n cmmn edges f linear meshes have Cartesian cmpnents, a new apprach fr dyad and vectr multiplicatin is presented in this subsectin. In rder t multiply current vectrs and DGFs, either Cartesian current vectrs shuld be (a) (b) Figure 3. (a) Curvilinear triangle, (b) Linear triangle.

9 Prgress In Electrmagnetics Research M, Vl. 26, cnverted t spherical vectrs r spherical DGFs shuld be cnverted t Cartesian nes. Cnverting a current vectr V; which cnnects tw vertices f a triangle; frm Cartesian t spherical crdinates results in a nn-unique vectr because J ϕ and J θ are different in each pint n the edge such as V x shwn in Fig. 3(b). Since there are unique transfrmatins f vectrs frm spherical t Cartesian crdinates, cnversin f a spherical dyad t a Cartesian dyad can be exactly implemented. Therefre, by emplying the centers f the field and surce triangles in the calculatin f DGFs in Cartesian crdinates the electric field vectr can be expressed as: E (Cartesian) = jωµ f Ḡ (Cartesian) J (Cartesian) (r )ds (13) s Thus by using triangular linear meshes in Cartesian crdinates nly Ḡ needs t be cnverted frm spherical t Cartesian crdinates. Fr this purpse, each unit vectr shuld be transfrmed frm spherical t Cartesian crdinates. As DGFs represent interactins between field and surce pints, the first and secnd vectrs f each dyad crrespnd t field and surce pints respectively. All spherical dyad cmpnents can be cnverted t Cartesian dyads. As an example the cnversin equatin f ˆr ˆϕ cmpnent f a dyad is extracted as fllws: ˆr ˆϕ=(sin θ f cs ϕ f ˆx+sin θ f sin ϕ f ŷ+cs θ f ẑ)( sin ϕ sˆx+cs ϕ s ŷ), (14) where subscripts f and s refer t field and surce pints, respectively. Therefre, the Cartesian DGFs cmpatible with vectr currents cmpnents can be btained by using the afrementined cnversin methd. Using this apprach is efficient when a vectr is requested as the utput f an antenna prblem slutin. Fr example t determine the near and far field radiatin patterns f an antenna all E x, E y, E z cmpnents f electric field can be calculated. 3. RESULTS T validate the present cmputatin, a spherical diple antenna is analyzed which is cnsidered first in free space and then.32 cm abve a 5 cm radius cnducting sphere. Then, cnfrmal diple antenna arrays in bth cases are analyzed. In all cases, antenna physical lengths is λ/2 with λ being the wavelength at f = 3 GHz. The medium between the antenna and the sphere is assumed t be free space. It is expected that the antenna respnse is similar t a linear thin λ/2 diple antenna which is presented in [13]. In these examples the antennas have negligible thicknesses in cmparisn with wavelength and are lcated in xy-plane as shwn in Fig. 1(b). The antenna in each case is divided

10 94 Meiguni, Kamyab, and Hsseinbeig t 12 triangular linear meshes yielding 119 cmmn edges between tw adjacent plus and minus triangles. The delta gap vltage surce is used t excite the diples. Impedance matrix and current distributin n cmmn edges are cmputed using MMs. Besides utilizing DGFs, asympttic apprximatin methd is als used Spherical Diple Antenna in Free Space As the first example a spherical diple antenna which is fed frm its center and lcated in free space is presented. Antenna input admittance is determined using MMs and slving MPIE by bth asympttic apprximatin frmulas [7] and DGFs prpsed in [14] with unit dielectric cnstants in all three layers. The results are cmpared with thse btained frm CST Micrwave Studi simulatr [17]. As illustrated in Fig. 4, gd agreement is btained between all results and little differences are due t prbe mdeling in CST simulatin. The antenna center is in xy-plane and abut 3 frm the rigin accrding t the antenna structure. The radiatin pattern as expected direct t 3 and 21. Auxiliary ptential equatins as well as electric field equatins can be utilized t cmpute radiatin pattern. Fig. 5 demnstrates radiatin patterns btained frm MMs using DGFs and CST simulatr which are in gd agreement Spherical Diple Antenna ver a PEC Sphere In this sectin a spherically cnfrmal diple antenna fed frm its center and lcated abve a PEC sphere is analyzed. T shw input admittance cnvergence speed tward spherical harmnics, the input Present Cmputatin (Asympttic Methd) Present Cmputatin (DGFs Methd) CST Micrwave Studi Present Cmputatin (Asympttic Methd) Present Cmputatin (DGFs Methd) CST Micrwave Studi -1 Cnductance (Ω ) Susceptance (Ω ) f (GHz) (a) f (GHz) (b) Figure 4. Input Admittance f a spherical diple antenna in an unbunded free space medium. (a) Cnductance, (b) susceptance.

11 Prgress In Electrmagnetics Research M, Vl. 26, db ± DGFs Methd with Frmulatin (5) DGFs Methd with Frmulatin (2) CST Micrwave Studi Simulatin Figure 5. Radiatin pattern f a spherical diple antenna in an unbunded free space medium Input Admitance ( Ω -1 ) Im Yin Re Yin Number f Terms Figure 6. Cnvergence f input admittance f a spherical diple antenna ver sphere. admittance at the center frequency f the antenna versus the number f terms in spherical harmnics series is illustrated in Fig. 6. T analyze a diple antenna lcated arbitrarily in a 3-layer dielectric sphere with the same dimensins f the presented example, at least 6 terms f spherical harmnics series are required in rder t DGFs be cnvergent. Hwever, fr the case that current elements are lcated cfrmally and layers 1 and 2 are cnsidered as free space and layer 3 as PEC, a cnvergence in antenna input admittance is achieved by cnsidering 25 terms in the series. Fr input admittance calculatin, due t cnfrmal current distributin n the diple, magnetic ptential DGFs are zer [7]. Fig. 7 illustrates the results.

12 96 Meiguni, Kamyab, and Hsseinbeig -1 Cnductance (Ω ) DGFs Methd Meshed PEC sphere -1 Susceptance (Ω ) DGFs Methd Meshed PEC sphere f (GHz) (a) f (GHz) (b) Figure 7. Input admittance f a spherical diple antenna ver PEC sphere. (a) Cnductance, (b) susceptance db ± DGFs Methd with Frmulatin (5) Meshed PEC sphere CST Micrwave Studi Simulatin -12 q -9-6 q -3 Figure 8. Radiatin pattern f a spherical diple antenna ver PEC sphere. Electric field DGFs with 2 terms f spherical harmnics is used t determine antenna radiatin pattern. Fig. 8 cmpares cmputed radiatin pattern f the antenna with the results btained frm CST sftware. As it can be seen frm this figure, radiatin pattern is directed t abut 3. It shuld be mentined that at the shadw bundary n the cnducting sphere the incident wave excites creeping waves prpagating alng the sphere surface. Due t cmparable dimensin f the sphere t the wavelength, the creeping waves radiate surface diffracted waves [18].

13 Prgress In Electrmagnetics Research M, Vl. 26, Array f Cnfrmal Diple Antennas In the fllwing, tw examples f cnfrmal diple antenna arrays lcated at the same radial distance frm the center f crdinates (Fig. 9) are analyzed and mutual cuplings between their elements are investigated d defined in degrees represents the distance between the centers f elements and can be in θ r ϕ directin. Figure 1 shws insertin lss and mutual cupling f a pair f spherical diple antennas separated by a distance f 15 in θ directin frm each ther and lcated in free space r abve a cnducting sphere. Sme changes in the results can be nticed frm Fig. 1 with and withut the presence f the cnducting sphere. Fig. 11 demnstrates mutual cupling between tw cnfrmal diples ver a PEC sphere fr different separatin distances in θ directin. Mutual cupling between a pair f spherical diples lcated in free space r abve a cnducting sphere is studied at the center frequency f the antennas and is illustrated versus d in ϕ directin in Fig. 12. Mutual cuplings between the elements f an array f five cnfrmal diples abve a PEC sphere are als investigated. Each Figure 9. A pair f cnfrmal diple antennas S 11 Diples in free space S 21 Diples in free space S 11 Diples ver PEC sphere S 21Diples ver PEC sphere -5-1 d = 5 d = 1 d = 15 S (db) Frequency (GHz) S 21 (db) Frequency (GHz) Figure 1. S-parameters f a pair f cnfrmal diple antennas. Figure 11. Mutual cupling f a pair f cnfrmal diple antennas ver a PEC sphere.

14 98 Meiguni, Kamyab, and Hsseinbeig Diples in free space (Cde) Diples ver PEC sphere (Cde) Diples in free space (HFSS Simulatin) Diples ver PEC sphere (HFSS Simulatin) S 21Present Cmputatin S 31Present Cmputatin S 21HFSS Simulatin S 31HFSS Simulatin S 21 (db) S (db) S (Degree) Frequency (GHz) Figure 12. Mutual cupling f a pair f cnfrmal diple antennas versus d in ϕ directin. Figure 13. Mutual cuplings f an array f five cnfrmal diple antennas ver a PEC sphere. antenna element is separated frm its adjacent element by a distance f 15 in θ directin and the lwest diple is cnsidered as the first element. Fig. 13 shws the mutual cuplings between the first three adjacent elements btained frm the presented methd and Ansft HFSS simulatin. As it can be nticed, gd agreement between the results is achieved. It can be nticed that the results btained frm the presented analysis methds based n DGFs r asympttic apprximatin are in gd agreement with the results btained frm HFSS and CST sftwares. Hwever, simulatr packages are highly dependent n meshing the structure, prbe mdeling and the size f radiatin bx. Therefre, mre time and memry is required in rder t btain precise and stable results frm simulatr packages. The prpsed methds have mre calculatin speed and accuracy in cmparisn with the mentined sftwares. 4. CONCLUSION In this paper, full-wave methds t analyze varius antennas in free space r ver spherical multilayer structures have been presented. In these methds, after meshing the antenna int linear triangles in Cartesian crdinates, the input admittance f a cnfrmal diple antenna ver a PEC sphere has been cmputed by using MPIE frmulatin. The infinite duble summatin has been transfrmed t a single summatin using additin therems fr Legendre plynmials and spherical Hankel functins yielding an increase in the radiatin pattern cmputatin cnvergence speed. In rder t determine electric field vectrs at entire medium, cnverting dyads frm spherical t

15 Prgress In Electrmagnetics Research M, Vl. 26, Cartesian crdinates has been presented. T validate the prpsed methds, mutual cuplings between elements f a cnfrmal diple antenna array in free space r abve a cnducting sphere have been investigated. Accuracy f the prpsed methds has been validated by cmparing the results btained by the presented methds with thse btained frm cmmercial sftwares. ACKNOWLEDGMENT The authrs wuld like t express their sincere gratitude fr material and spiritual supprt ffered by Iran Telecmmunicatin Research Center (ITRC). REFERENCES 1. Chew, W. C., Waves and Fields in Inhmgeneus Media, IEEE Press Series n Electrmagnetic Waves, Tai, C. T., Dyadic Green s Functins in Electrmagnetics Thery, IEEE Press Series n Electrmagnetic Waves, Okhmatvsk, V. I. and A. C. Cangellaris, Efficient calculatin f the electrmagnetic dyadic Green s functin in spherical layered media, IEEE Trans. Antennas Prpag., Vl. 51, N. 12, , Dec He, M. and X. Xu, Clsed-frm slutins fr analysis f cylindrically cnfrmal micrstrip antennas with arbitrary radii, IEEE Trans. Antennas Prpag., Vl. 53, N. 1, , Jan Tam, W. Y. and K. M. Luk, Resnance in spherical-circular micrstrip structures, IEEE Trans. Micrw. Thery Tech., Vl. 39, N. 4, 7 74, Apr Huia, H. T. and E. K. N. Yungb, Dyadic Green s functins f a spherical cavity filled with a Chiral medium, Jurnal f Electrmagnetic Waves and Applicatins, Vl. 15, N. 9, , Khamas, S. K., Asympttic extractin apprach fr antennas in a multilayered spherical media, IEEE Trans. Antennas Prpag., Vl. 58, N. 3, 13 18, Mar Li, L. W., P. S. Ki, M. S. Leng, and T. S. Ye, Electrmagnetic dyadic Green s functin in spherically multilayered media, IEEE Trans. Micrw. Thery Tech., Vl. 42, , Dec

16 1 Meiguni, Kamyab, and Hsseinbeig 9. Macn, C. A., K. D. Trtt, and L. C. Kempel, A practical apprach t mdeling dubly curved cnfrmal micrstrip antennas, Prgress In Electrmagnetics Research, Vl. 4, , Arakakia, D. Y., D. H. Wernerb, and R. Mittrac, A technique fr analyzing radiatin frm cnfrmal antennas munted n arbitrarily-shaped cnducting bdies, Jurnal f Electrmagnetic Waves and Applicatins, Vl. 14, N. 11, , Gibsn, W. C., The Methd f Mments in Electrmagnetics, Chapman & Hall/CRC, Taylr & Francis Grup, Ra, S. M., D. R. Wiltn, and A. W. Glissn, Electrmagnetic scattering by surfaces f arbitrary shape, IEEE Trans. Antennas Prpag., Vl. 3, , May Harringtn, R. F., Field Cmputatin by Mment Methds, IEEE Press Series n Electrmagnetic Waves, Khamas, S. K., Electrmagnetic radiatin by antennas f arbitrary shape in a layered spherical media, IEEE Trans. Antennas Prpag., Vl. 57, N. 12, , Dec Harringtn, R. F., Time-Harmnic Electrmagnetic Fields, IEEE Press, Jhn Wiley & Sns, Inc., Pzar, D. M. Micrwave Engineering, 3rd Editin, Jhn Wiley & Sns, Inc., CST Reference Manual, Cmputer Simulatin Technlgy, Darmstadt, Germany, Hansen, R. C., Gemetrical Thery f Diffractin, IEEE Press, New Yrk, 1981.