Radiation Conductance and Pattern of Array Antenna on a Non-Confocal Dielectric-Coated Elliptic Cylinder

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1 Radiatin Cnductanc and Pattn f Aay Antnna n a Nn-Cnfca Dictic-Catd Eiptic Cyind A. HELALY* and A. SEBAK Ectica and Cput Engining Dpt Cncdia Univsity Mnta, QC, H3G M8, CANADA * Engining Mathatics and Physics Dpt, Cai Univsity, Giza, EGYPT aahay@yah.c abd@c.cncdia.ca Abstact- Th fuatin f th adiatin pattn and cnductanc f axia sts aay antnna n a dicticcatd iptic cyind is psntd. Th cating is assud t b nn-cnfca. Th anaytica sutin, givn h, is basd n th ign functin tchniu and th additin th f Mathiu functins. Th xcitd aptus a assud t gnat a TM paizd wav. Accdingy, th btaind sis sutin is tuncatd t gnat nuica suts. Sap f cacuatd aziutha adiatin pattns and adiatin cnductanc a psntd f diffnt antnna and cating paats. Th iptic cyind has n xta dg f fd cpad t a cicua cyind t cnt th adiatin pattn and cnductanc. Th cputd suts shw th fxibiity f th antnna t cnt th shap and dictin f its adiatin pattn by changing th funcy, th xcitatin, th cating thicknss f th cyind, and th cnstitutiv paats f th cating. Ky-Wds: Axia st antnnas, nn-cnfca cating, anaytica thds.. Intductin Aays f antnnas untd n vtica stuctus a usfu f th badcasting f wavs and f bas statins [-3. Many such gtica stuctus hav bn anayticay tatd. Th axia and cicufntia sts n cicua and iptica css-sctina cnducting cyinds w tatd in [4-6. Th dictic-catd cicuas iptica css-sctina cnducting cyinds w cnsidd in [7-3. Th iptic gty is bcing ppua sinc it aws a btt cnt f th paizatin chaactistics and faciitats th dsign by changing bth ccnticity and fca ngth t tun th paats f intst. In additin, th nn-unif cating ffs anth dg f dsign fd. As, iptic cyindica cavitis xcit ppagating ds cpad t cicua cavitis [4. F th anaysis f such antnnas, th hgnus Hhtz uatin in th iptic cdinats is pyd. Sving fid pbs f stuctus with iptica gtis uis th cputatin f Mathiu and difid Mathiu functins [5. Ths a th ign sutins f th wav uatin in iptica cdinats. Th anaytica dvpnts f th sutin f fa fid pattns f th uti-sttd iptic cyind with cncntic cating hav bn psntd in [2. Th studis psntd in this pap a fcusd n th anaysis and dsign f a finit aay f axia sts n a nn-cnfca dictic-catd cnducting iptic cyind. Th wk is an xtnsin f th fuatin f a sing st antnna n an iptic cyind [. On f th tivatins f this wk is t study th ffct f a dictic ay n th pfanc f antnnas untd n a spac shutt. W assud that th aptu fid i.., th xcitatin vtags, is a spcifid functin n ach st. Th fids in th dictic cating and xti gins a xpandd in ts f Mathiu and difid Mathiu functins with unknwn xpansin cfficints. Ths unknwn xpansin cfficints a dtind by nfcing apppiat bunday cnditins at th intfac btwn th dictic cating and f spac and at th sttd cyind. In d t gnat nuica suts and f a pactica pint f viw, th sis sutin ust b tuncatd in a suitab fashin t btain a finit ISSN: Issu, Vu 7, Nvb 28

2 siz atix. Th d f such a atix dpnds n th ctica siz and pptis f th cating gin. 2. Pb Fuatin Th cnfiguatin f th axia aay antnna untd n iptic cyind is shwn in Fig.. Th antnna adiats thugh a nn-cnfca dictic cating (gin II) t f spac (gin I). Tw systs f cdinats a usd. Th gba cdinats at th cnt f th ut sufac f th dictic cating a idntifid by (x, y). Th ca cdinats at th cnt f th sttd cnducting cyind a (x c, y c ). Th si aj and si in axs f th cnducting cyind a dntd by a c and b c. Th cspnding paats f th cating gin a a and b. Th fca ngth f th dictic ut sufac is 2F and that f th cnducting cyind is dntd by 2F c. Th aj axis f th css sctin f th cnducting cyind is incind by an ang β with spct t th x axis and its cnt is catd at (d, ψ) with spct t th gba cdinats. Th iptica cdinats (u, ν, z) a pyd thughut with x F csh (u) cs (ν) and y F sinh (u) sin (ν) wh F is th si fca ngth f th cspnding iptica css sctin. i th st Δ i ξ 2 y y y c d ψ β x c x gba cdinats (u, ν), th ctic fid cpnnt is givn by E ( I ) z + B B R R, ξ ) S, ξ ) S, η), η) () wh ξ csh (u), η cs (ν) and c kf, k 2π / λ and λ is th f spac wavngth. S, S a, spctivy, th vn and dd angua Mathiu functins f d, R, R a th vn and dd difid Mathiu functins f futh kind. Th cspnding agntic fid cpnnts a givn by wh j E 2ωμ h ξ Hη z j E 2ωμ h ξ Hξ z h F 2 ξ 2 2 ξ η (2a) (2b) Th ctic fid cpnnt in gin II ust vanish n th cnducting pat f th c cyind xcpt at th sts. In gin II and using th ca cdinats, w ay xpss it in th f Rgin II (ε,µ ) ξ Rgin I Fig. : Cnfiguatin f axia sttd-iptic cyind aay antnna catd with a nn-cnfca dictic. E ( II ) z () R (, ) C c3 ξc S (, ) c3 ηc + D R ξc) () C ( 3, ) R c ξc + S ( 3, ) c ηc + DR ξc) This pap cnsids th TM paizatin in which th nn z fid cpnnts a E z, H ρ and H φ. It is assud that th fid distibutin n th sts dpnds n th angua vaiab v. In gin I and using th (3) wh ξ c csh (u c ), η c cs (ν c) and c 3 kfc with ( ) k k ε. R i, (i) R a th vn and dd ISSN: Issu, Vu 7, Nvb 28

3 difid Mathiu functins f th th kind. Th cspnding agntic fid cpnnts a givn by but using th ca cdinats. In () and (3) B, C and D a unknwn xpansin cfficints t b dtin by nfcing pp bunday cnditins n th antnna and th intfac btwn th dictic and f spac. 2. Bunday Cnditins Th th sts f unknwn xpansin cfficints a dtind by appying bunday cnditins n th tw sufacs: th c cnducting cyind and th intfac btwn th cating atia and f spac. 2.. C (antnna) bunday On th cnducting sufac ξ ξ 2, th tangntia ctic fid cpnnt ust vanish xcpt n th sts. Using th ca cdinats, th bunday cnditin ay b xpssd as: and dd unknwn xpansin cfficints. F th vn functins th suting subsyst f uatin is givn by, n,, 2,., () N n 3)[ Cn Rn ξ2) + Dn Rn, ξ2) F( η ) S ) d F 3) sts c n ηc ηc F th dd functins cas and f n, 2, 3, () N n 3)[ CnRn ξ2) + DnRn, ξ2) F( η ) S ) d F 3) sts c n ηc ηc (7) (8) N n, N n a naizd cnstants and givn by [2; ( II ) F( ηc ) Ez n th sts thwis N 2π 2 3 ) [ S η) dv, η csv (9) n n Wh N F( η c ) F i ( ηc ) (5) i is th aptu fid distibutin n th sts. On th i th st, it is assud that th xcitatin vtag is vaying as jδ i F η ) V cs( π ( v v ) / 2Δ ), (6) i ( c i c i i j i V i δ is th appid vtag, v i is th angua cdinat f its cnt, Δ i is its width and η c cs (ν c ) in th (x c, y c ) ca cdinats. Substituting (3) and (5) in, utipying th suts by c, η ) and S n ( 3 c appying th thgnaity f Mathiu functins [6, th ts invving vn functins dcup cpty f ths f dd functins. Thus, n btains tw subsysts f ina uatins f th vn And F, F a th sts xcitatin vct and givn by; F n N 3 ) th F i st i( η ) S n ) dv () i η Cating f spac bunday On th ut sufac f th dictic cating, i.., ξ ξ, th cntinuity f bth E and H fid tangntia cpnnts ust b nfcd. Th fid cpnnts insid th cating gin can b xpssd in ts f th gba cdinat syst using th tansitin th f Mathiu functins [7, s th Appndix. Again using th thgna pptis f Mathiu functins, n btains f th tangntia E z fid cpnnt: ISSN: Issu, Vu 7, Nvb 28

4 Mn c) B R, ξ) () Nn 3) Rn ξ) [ WEnC + WEnC + Nn 3) Rn ξ) () [ WEnD + WEnD μ Mn, c)[ BR ', ξ) μ () Nn ) Rn ', ξ) [ WOnC + WOnC + Nn ) Rn ', ξ) [ WOnC + WOnC And f th tangntia H v fid cpnnt, w btain: μ Mn c) B R ', ξ) μ () Nn 3) Rn ' ξ) [ WEnC + WEnC + Nn 3) Rn ' ξ) [ WEnD + WEnD Wh c kf, th pi dnts divativ with spct t ξ and M n, c ) S, η) S, η) dv, (5) i j 2π n i Euatins (7)-(8) and ( 4) cnstitut a syst f ina uatins f unknwn xpansin cfficints f th fids: nay: B, B, C, C, D and D. Expssins f th tansfatin cfficints WE and WO a givn in th appndix. n j n Mn c)[ BR, ξ) () Nn 3) Rn ξ) [ WOnC + WOnC + Nn 3) Rn ξ) (3) [ WOnD + WOnD 3 Radiatd Pw Dnsity Epying th asypttic xpansin f R and R in uatin (), th fa adiatd fid, which is f paticua intst, is givn by; E z jkρ j /( kρ) j [ B S + B S, η), η) (6) ISSN: Issu, Vu 7, Nvb 28

5 Th ti-avag pw dnsity adiatd by th antnna is thn givn by: P( ρ, φ) 2π kρ j [ B (, cs ) S c φ + B S, csφ) Th avag pw dnsity is dfind as P av ( ρ) 2π 2π P( ρ, φ)dφ Th antnna dictivity is thn givn by P( ρ, φ) D ( ρ, φ) ( φ) P av F a sing st antnna, w dfin th adiatin cnductanc p unit ngth as G Pav( φ) πρ V 2 2 Wh V is th st vtag. 4 Nuica Rsuts 2 (7) (8) (9) F iustatin pupss, w cnsid th cas f tw sts ach with width Δ2. F a cnsidd cass, th ctica and gtica paats a givn in ach figu captin. Fig. 2 shws th adiatin pattn f a sing st cas and whn th tw sts a n th in axis f th cnducting cyind. Th pattn is siia t that f a pai f dips. Th ffct f catin f th sts pai n th pattn is shwn in Fig.3. F this cas th ut bunday f th cating dictic is f cicua shap. Fig. 4 psnts th ffct f th xcitatin phas n th adiatin pattn. It is sn that th ain ba dictin is std 9 whn th phas is switchd f th in-phas status t ut f phas cnditin. Figu 5 shws th adiatin pattns f an axiay sttd iptic cyind with n cating but with inphas xcitatin. Th ffct f th catin f th tw sts n th adiatin pattns f an axiay sttd iptic cyind is shwn in Fig. 6. In Fig. 7, w cpa th ffct f th c shap n th adiatin pattns f an axiay sttd cicua and iptica n bth catd with th sa atia. Figu 8 shws th adiatin pattns f an axiay sttd iptic cyind catd with a dictic f tw typs f xcitatin: inphas and ut f phas. Finay, w invstigat th ffct f diffnt dsign paats n th adiatin cnductanc f s sing st antnna. In th fwing tw figus Δ/λ psnts th ctica thicknss f th cating. Figu 9 shws that th adiatin cnductanc dpnds n th cating atia and its thicknss. Th ffct f th st catin is dnstatd in Fig. wh th tw cass cnsidd a whn th st is catd at th nd f th aj and in axs. It is vy ca f Figs. 9 and that th adiatin cnductanc dpnds stngy n th cating paats: dictic typ, thicknss f th cating and its shap. 5 Cncusin An anaytic sutin is givn f th adiatin by an aay f axia sts untd n a cnducting iptic cyind catd with a nn-cnfca dictic. Rsuts a psntd f bth th adiatin pattn and adiatin cnductanc f sva gtica and atia paats. Th cputd suts shw th fxibiity f th antnna t cnt th shap and dictin f its adiatin pattn. ISSN: Issu, Vu 7, Nvb 28

6 Tw sts Sing st Fig 2: Radiatin pattn f an axiay sttd iptic cyind (a c.5λ, a c /b c 2) catd with a dictic (ε 4, a.4λ, a /b2, d, ψ, β9 ). Th tw ut f phas sts with ua apitud a catd at th nds f th in axis. Fig. 3: Radiatin pattn f an axiay sttd iptic cyind (a c.25λ, a c /b c 2) catd with a cicua dictic (ε 4, ab.3λ, d, ψ, β). Th tw sts a fd with ut f phas ua apitud vtags: ffct f th sts catins. ISSN: Issu, Vu 7, Nvb 28

7 Fig. 4: Radiatin pattns f an axiay sttd iptic cyind (a c.25λ, a c /b c 2) catd with a dictic (ε 4, a.3λ, a /b2, d, ψ, β). Th tw sts a catd at th nds f th aj axis: in-phas and ut f phas xcitatin cpaisn. Fig. 5 Radiatin pattns f an axiay sttd iptic cyind (a c.5λ, a c /b c 2) with n cating. Th tw sts a catd at th nds f th aj axis with in-phas xcitatin. ISSN: Issu, Vu 7, Nvb 28

8 Fig. 6: Radiatin pattns f an axiay sttd iptic cyind (a c.5λ, a c /b c 2) ) catd with a dictic (ε 4, a.6λ, a /b2, d, ψ). Th tw in-phas sts a catd at th nds f th aj axis (sid in) and at th nds f in axis (dashd in). Fig. 7: Radiatin pattns f an axiay sttd cicua (a c.25λ, a c /b c : d dashd in) and iptica (a c.25λ, a c /b c 2: sid bu in) cyind catd with a dictic (ε 4, a.6λ, a /b2, d, ψ9 ). Th tw ut f phas sts a catd at th nds f th in axis aj axis (sid in) and at th nds f in axis (dashd in). ISSN: Issu, Vu 7, Nvb 28

9 Fig. 8: Radiatin pattns f an axiay sttd iptic cyind (a c.25λ, a c /b c 2) catd with a dictic (ε 4, a.6λ, a /b2, d, ψ, β). Th tw sts a catd at th nds f th aj axis: in-phas (sid bu in) and ut f phas ( dashd d in) xcitatin cpaisn. - - ε 2 -x- ε ε G (S) Fig. 9: Effct f cating atia and thicknss n th adiatin cnductanc (a c.25λ, a/ba c /b c 2, aa c +Δ, d, ψ, β). Th st is catd at th nd f th aj axis. ISSN: Issu, Vu 7, Nvb 28

10 G (S) Fig. : Effct f st catin n th adiatin cnductanc (a c.5λ, a c /b c 2, a/ba c /b c 2, aa c +Δ, d, ψ, β). Th st is catd at th nd f th aj axis (xx) and in axis (). Appndix Cnsid tw iptic cdinat systs: ( ξ,η, z ) and ( ξ,η, z ). Outsid th gin which is bundd by th cic C ( cnt at O, adius d c ) and utsid th cic C (cnt at O, adius siaj axis f th th cyind), th fids dscibd in th th cyind cdinats can b xpssd in ts f th th cyind cdinats. Saak [7 has shwn that th fwing atin wi hd within th abv ntind gin: ( i) R, ξ ) S, η ) Wh WE WO X ip π ( j) N ' ' i+ p ( j) D i ) D p ) ) i p π ( j) N J X (A2) ' ' i+ p ( j) D i ) D p ) ) i p cs i kd ) ψ + ( ) J sin p i p+ i( c (A3) ip Y ip cs + kd ) ψ sin (A4) WE R ( i), ξ ) S, η ) ( i) + R, ξ ) S, η ) (A) WO Y ip J sin kd ) ψ ( ) J cs i p i ( c p+ i ( c + ψ iψ + pψ ψ iψ pψ sin + kd ) ψ cs (A5), ISSN: Issu, Vu 7, Nvb 28

11 and d c is th distanc btwn th cnt ins f th th cyind and th th cyind, ψ is th ang btwn th x axs f th th and th th cyinds asud f th th cyind's x axis and J p (u) is th Bss functin f d p and agunt u. Th pi v th suatin sign ' indicats that th su is v ny vn dd vaus f i ( p ) dpnding n whth () is vn dd. Th cfficints n i D and D a th i Fui sis cfficints f th Mathiu functins [5. Rfncs: [ Z. Zahais, On th Dsign f Mbi Bas Statin Antnna Aays by Cbining an Itativ Ptubatin Pcss and th Othgna Mthd, WSEAS Tans. n Cunicatins, Vu 6, Issu, pp. -8, 27. [2 E. Haiti, L. Aha, and A. Sbak, Cput Aidd Dsign f U-Shapd Rctangua Patch Micstip Antnna f Bas Statin Antnnas f 9 MHz Syst, WSEAS Tans n Cunicatins, V. 5, pp , 26. [3 Z. Zahais, D. Kapitaki, A. Papastgiu, A. Hatzigaidas, P. Lazaidis, M. Spass, Optia Dsign f a Lina Antnna Aay using Patic Swa Optiizatin, WSEAS Tans. n Cunicatins, Vu 5, Issu 2, pp , 26. [4 S. Siv and W. K. Saunds, Th adiatin f a tansvs ctangua st in a cicua cyind, J. app. Phys., V. 2, 95, pp [5 L. I. Baiin, Th adiatin fid pducd by a st in a ag cicua cyind, IRE Tans, V. 3, 955, pp [6 J. Y. Wng, Radiatin pattns f sttd iptic cyind antnnas, IRE Tans.,V. 3, 955, pp [7 R. A. Hud, Radiatin pattn f a dictic-catd axiay sttd cyind, Can. J. Phys., V.34, 956, pp [8 J. R. Wait, and W. Mintka, Sttd-cyind antnna with a dictic cating, J. Rs. Nat. Bu. Stand, V. 58, 957, pp [9 W. F. Cssw, G. C. Wstick, and C.M. Knp, Cputatins f th aptu adittanc f an axia st n a dictic catd cyind, IEEE Tans., V. 2, 972, pp [ J. H. Richnd, Axia st antnna n dictic-catd iptic cyind, IEEE Tans. Ant. Ppag., V. 37, 989, pp [ H. Raghb, A. Sbak, and L. Shafai, Radiatin by axia sts n a dictic-catd nn-cnfca cnducting iptic cyind, IEE Pc. Micw. Ant. Ppag., V. 43, 996, pp [2 M. Hussin, and A. Haid, Radiatin chaactistics f N axiay sttd antnna n a ssy dictic-catd iptic cyind, Can. J. Phys., V. 82, 24, pp [3 M. Hussin, and A. Haid, Exact adiatin f sttd cicua iptica antnna catd by a cncntic isfactiv taatias, Int. J. App. Ecta. Mch., V. 26, 27, pp. -. [4 A. Tadjai, A. R. Sbak, and T. A. Dnidni, Rsnanc Funcis and Fa Fid Pattns f Eiptica Dictic Rsnat Antnna: Anaytica Appach, Pgss In Ectagntics Rsach, PIER 64, 26, pp [5 F. A. Ahagan, A cpt thd f th cputatins f Mathiu chaactistic nubs f intg ds, SIAM Rv. 38, 2, 996, pp [6 P.M. Ms and H. Fshbach, "Mthds f thtica Physics", Vs. I and II, McGaw-Hi, Nw Yk, 953. [7 K. Saak, "A nt n additin ths f Mathiu functins," MZ. Math. Phys., v., 959, pp ISSN: Issu, Vu 7, Nvb 28

GUC (Dr. Hany Hammad) 11/2/2016

GUC (Dr. Hany Hammad) 11/2/2016 GUC (D. Han Hammad) //6 ctu # 7 Magntic Vct Ptntial. Radiatin fm an lmnta Dipl. Dictivit. Radiatin Rsistanc. Th ng Dipl Th half wavlngth Dipl Dictivit. Radiatin Rsistanc. Tavling wav antnna. Th lp antnna.