International Journal of Scientific & Engineering Research, Volume 7, Issue 9, September ISSN

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1 Intnational Jounal of cintific & Engining Rsach, Volum 7, Issu 9, ptmb Analysis and Dsign of Pocklingotn s Equation fo any Abitay ufac fo Radiation Pavn Kuma Malik [1], Haish Pathasathy [], M P Tipathi [3] Abstact Elctomagntic fild adiation mchanism fo any abitay conducting sufac is mathmatically computd by solving th pocklington s quation, and which is gnaliz that can b usd to any typ of antnna in fild thoy. Tangntial vcto on any abitay sufac is dfind with th hlp of sufac quation. Using Lontz gaug condition, scala potntial is dfind in tms of vcto potntial and scattd lctic fild is calculatd on th abitay sufac. Mathmatical psntation of E and H fild fo paabolic flcto is also divd. Indx Tms Elctomagntic scatting, Intgal quations, Numical solutions, Pocklington s quation, paabolic flcto. T I. INTRODUCTION IJER I = εω + s' HE intgal quation mthod, with a Momnt Mthod numical solution, is usd fist to find th slf- and diving-point impdancs of any antnna, and mutual impdanc of wi typ of antnnas. This mthod casts th solution fo th inducd cunt in th fom of an intgal wh th unknown inducd cunt dnsity is pat of th intgand. Numical tchniqus, such as th Momnt Mthod, can thn b usd to solv th cunt dnsity. In paticula two classical intgal quations fo lina lmnts, Pocklington s and Hall n s Intgal quations, is usd. Hall n s quation is usually stictd to th us of a dltagap voltag souc modl at th fd of a wi antnna. Pocklington s quation, howv, is mo gnal and it is adaptabl to many typs of fd soucs (though altation of its xcitation function o xcitation matix), including a magntic fill. Th staight, thin, cnt-divn wi is oftn usd as a tansmitting antnna. In thotical studis, th basic unknown is th cunt along th antnna, which satisfis a ondimnsional, fist-kind, Fdholm-lik intgal quation usually fd to as Hallén s quation, o a cosponding intgodiffntial quation calld Pocklington s quation. Pocklington s intgo-diffntial quation is a stapl of thinwi antnna analysis, and appas in most antnna txt books [1]-[3], as wll as foming th basis of antnna simulation cods such as th Numical Elctomagntic Cod (NEC) []. In 1897 Pocklington dducd its quation fo staight stuctus, and in 1965 Mi, usd a huistically pocdu to dfin it fo bnt wis [4]; fo an abitay shapd wi, it is possibl to dduc th quation using a fomal way, stating fom Maxwll quations [6] gtting: k ' E Is( s ') k s. s ' ds ' ss' 4 π ' -- (1) I Wh E is th tangntial incidnt lctic fild. Considing th thin-wi appoximation and skin ffct, is possibl to xpss th lctic fild as a lina intgation ov th ach lngth s. Th gnal Pocklington quation (1) can b usd fo any possibl thin wi gomty. Th wi s gomty is xpssd by th dot poduct ss. ', wh ss () a th unit tangntial vcto fo th wi s axis and s'. s' th sam fo th paalll cuv psnting th cunt filamnt. dx() s dy() s dz() s ss () = i+ + k ds ds ds dx '( s ') dy '( s ') dz '( s ') s'( s') = i+ + k ds ' ds ' ds ' () Th gomty is also xpssd by th diffnc btwn th vctos ' as: M. Pavn K. Malik is doing Ph D fom Mwa Univsity, Raasthan, INDIA (-mail: malikbailly@gmail.com). Pof. Haish Pathasathy is with th Elctonics and Communication dpatmnt, NIT, Nw Dlhi, INDIA (-mail: haishp@nsit.ac.in) Pof. M P Tipathi is with Elctonics and Communication dpatmnt, M.A.I.T, Ghaziabad, INDIA (-mail: munishpashadtipathi@diffmail.com) IJER 016

2 Intnational Jounal of cintific & Engining Rsach, Volum 7, Issu 9, ptmb R= R = ' = [ x( s) x'( s')] + y( s) y'( s')] + z( s) z'( s')] --(3) Dfind all fom quations, th wok is ducd to find th vctos psnting th paalll and axis cuvs fo th considd wi and solv it by Mthod of Momnts [7] II. THEORY AND FORMULATION Gnalization of a pocklington s quation fo any abitay sufac. Lt th sufac quation b givn by uv (, ) wh u and v a f paamt. Dfin th following tangntial vcto on th sufac [8]-[10]. (, ) δ δ u uv = And v ( uv, ) = δu δ v Th sufac lmnt is givn by: d( u, v) = u vdudv Th sufac cunt dnsity can b wittn as: Js( uv, ) = Ju( uv, ) u( uv, ) + Jv( uv, ) v( uv, ) It follows that th vcto potntial poducd by this sufac cunt dnsity is givn by [11] kr( u, v) µ A ( ) = Js ( uv, ). duv (, ) 4 π Ruv (, ) H, Ruv (, ) = uv (, ) Wh, Ruv (, ) = uv (, ) = nuv (, ) kr d 1 kr And (,,) R k Guv = = ( + ) dr R R It follows that Φ= ( Js, n)( uvguvduv, ) (,, ) (, ) ωε o that, E = Φ ω A= ( Js, n) Guvduv (,, ) (, ) 4 πωε ωµ JsuvHuvduv (, ) (,, ) (, ) kr( u, v) Wh, Huv (,,) = Ruv (, ) Th lctic fild, w dnot by Es () th subscipt s standing fo cattd [13]. W can wit in componnt fom as: m= 3 = E ψ (, u, v) J ( u, v) dudv sk km m m= 1 k = 1,,3... Wh ψ km(, uv,) = ωµ Gk ( uvn,, ) m( uv, ) Huv (,, ) δkm ωε Th tangntial componnts of th scattd lctic fild on th givn sufac (along th u and v diction) a [14] 3 3 (, E ) = x E,(, E ) = x E u s k, u sk v s k, v sk k= 1 k= 1 This can b usd to gnaliz th pocklington s quation. IJER III. APPLICATION FOR PARABOLIC REFLECTOR It has bn shown by gomtical optics that if a bam of paalll ays is incidnt upon a flcto whos gomtical shap is a paabola, th adiation will convg (focus) at a spot which is known as th focal point. A paabola is dfind as th locus of a point th atio of whos distanc fom a point W calculat th scala potntial fom this making us of th P and fom a lin is qual to unity. Th point P is calld th Lontz gaug condition [1] focus. Equation of a paabola in th x y plan can b Φ= ( ) diva xpssd as y = 4ax. Focus of such a paabola is givn ωµε by P= ( a,0). Th symmtical point on th paabolic Now diva = µ sufac is known as th vtx. Rays that mg in a paalll ( (, ), (, )) (,, ) (, ) Jsuv Ruv Guvduv Fomation a usually said to b collimatd. A paaboloid is a sufac obtaind by otating a paabola about th nomal to its apx. Equation of paabola can b givn as: - x + y = 4az = Wh, y= distanc at y axis, x= distanc at x axis, a = distanc of focus point fom axis As, y = 4ax = 4az And = x + y o that x + y 4az = 0 Paabolic cylinds hav widly bn usd as high-gain aptus fd by lin soucs. Th analysis of a paabolic cylind (singl cuvd) flcto is simila, but considably simpl than that of a paaboloidal (doubl cuvd) flcto. Th pincipal chaactistics of aptu amplitud, phas, and IJER 016

3 Intnational Jounal of cintific & Engining Rsach, Volum 7, Issu 9, ptmb polaization fo a paabolic cylind, as contastd to thos of a paaboloid, a as follows: 1. Th amplitud tap, du to vaiations in distanc fom th fd to th sufac of th paabolic flcto, is popotional to 1 1/ in a cylind compad to in a paaboloid.. Th focal gion, on which incidnt plan wavs convg, is a lin souc fo a cylind and a point souc fo a paaboloid. 3. Whn th filds of th fd a linaly polaizd and which a paalll to th axis of th cylind, no coss-polaizd componnts a poducd by th paabolic cylind. That is not th cas fo a paaboloid. δ δ d(, ) = x d. d δ δ O, d(, ) = d. d Using coss poduct, i.. aa und th infinitsimal cuv of, x y z i.. cos sin a sin cos 0 ( )cos( ) x ( )sin( ) = y+ z a a = ( )cos( ) + ( )sin( ) + (1) a a IJER = 1+ a Fig 1. Aft otating a paabola about th nomal to its apx. Thfo, d(, ) = 1 + /4a d d IV. CALCULATION OF INFINITEIMAL AREA Fom tigonomty, w hav x = cos y = sin And, z = 4a Th quation of th paaboloid can b xpssd in paamtic fom as (, ).cos( ).sin( ) = x+ y+ ( ) z 4a Tangnt vcto on this sufac lativ to th paamtic coodinat (, ) will b δ cos( ) sin( ) = = x+ y+ ( ) z δ a And 1 δ sin( ) x cos( ) = = + y δ Aa lmnt (diffntial) on th sufac is givn by o that, V. CALCULATION OF CURRENT DENITY It is wll awa that, lctic and magntic psntation in tms of filds gnatd by an lctic cunt souc is J and a magntic cunt souc is M. Th pocdu quis that th auxiliay lctic and magntic vcto potntial functions A and F gnatd, spctivly, by J and M a found fist. In tun, th cosponding lctic and magntic filds a thn dtmind (E A, H A du to A and E F,H F du to F). Th total filds a thn obtaind by th supposition of th individual filds du to A and F (J and M). As w know that sufac cunt dnsity is givn by: J = aj + aj + aj x x y y z z ufac cunt dnsity fo paaboloid can b xpssd in paamtic fom as. J(, ) = J + J + J Z Z Wh, JZ Z = 0 Thfo, IJER 016

4 Intnational Jounal of cintific & Engining Rsach, Volum 7, Issu 9, ptmb J (, ) = VII. CALCULATION OF VECTOR POTENTIAL J (cos( ) x J ( sin( ) x cos( ) y) sin( ) It is a vy common pactic in th analysis pocdu to y+ ( ) z ) intoduc auxiliay functions, known as vcto potntials, a Combining thm in x, y and z coodinats, which will aid in th solution of th poblms. Th most J (, ) = common vcto potntial function a th A (magntic vcto potntial) and F (lctic vcto potntial). To calculat th E [ J cos( ) J sin( )] x [ J sin( ) J cos( )] + + y+ [ Jand ] z H fild, dictly fom th lctic and magntic cunt a dnsity (J and M) qui high dg of intgation [15], which intoduc th xta computational tim. Thfo, calculation of E and H filds a don though vcto potntial VI. CALCULATION OF DITANCE R A and F, though J and M. Fist intgation is don to k calculat th vcto potntial A and F, and thn diffntiation Fild adiatd by a point souc vais as, to obtain E and H filds [16]-[18]. Wh, k is phas facto and 1 is th amplitud facto Fo fa fild calculation R 'cosψ = Fo phas tms (ψ is th angl btwn th two vctos and ' as shown) R=, Fo amplitud, and 'cos ψ =. ' IJER ' ' = 'cos( ') x+ 'sin( ') y+ ( ) z Fig. Calculation of E and H fild using J and M fild. a ( ' = xx ' + yy ' + zz As p th lctomagntic fild thoy, w know that ') magntic vcto potntial can b givn as: Fom quation kr µ x sinθcos cosθcos sin A = J s ds R y = sinθsin cosθsin sinθ k + k 'cosψ θ µ A = J s ds z cosθ sinθ 0 = x sinθcos+ ysinθsin+ z cosθ. ' = ' 'sin( θ) cos( ) cos( ') + 'sin( θ)sin( )sin( ') + cosθ a '. ' = 'cos ψ = 'sin( θ)cos( ') + cosθ a R = ' ' R= 'sin( θ)cos( ') + cosθ a This can b wittn as, R= 'cosψ Wh, ' 'cos ψ = 'sin( θ)cos( ') + cosθ a k + k 'cosψ µ A= Js 1 /4ad d + [ cos( ) sin( )] J J x+ k µ A [ J sin( ) J cos( )] = y + + [ J ] z a + k 'cosψ Multiply by 1 + /4add [ J cos( ) J sin( )] x + k µ A= [ J sin( ) + J cos( )] y + [ J ] z a IJER 016

5 Intnational Jounal of cintific & Engining Rsach, Volum 7, Issu 9, ptmb Multiply by ' + k ( 'sin( θ)cos( ') + cos θ) a ad d 1 + /4 With th hlp of Maxwll quations in homognous, lina and isotopic mdium: Χ E = ωµ H M Χ H = ωε E+ J. B = 0 and. D = 0 Wh all vaiabls a having thi taditional psntation. Aft som mathmatical computation (Fist by putting, M = 0, J 0 And thn J = 0, M 0 W gt. IJER E = ω A (. A) ωµε And H = ωf (. F) ωµε VIII. CONCLUION Th adiation chaactistics of any abitay sufac hav bn invstigatd by applying th mthod of momnt and Pocklington s quation, and adiatd fild quation has bn obtaind. Th intgal quation fo a paabolic flcto is divd; som poptis of intgal quation a psntd and utilizd to duc th computation of intgal quation to som spas matix notation. Th mthod is computationally stiking, and accuatly fomulation is dmonstatd though illustativ xampl. REFERENCE [1] Kaus J D, Elmnts of Elctomagntic McGaw-Hill, Nw Dlhi 3 d dition, 010. [] Lonad L. Tsai, Chal. mith, "Momnt Mthods in Elctomagntic fo und gaduats", IEEE Tansactions on Education, vol. -1, no. 1, Fbuay 1978 [3] Antnna thoy, analysis and synthsis, Advancd signal analysis and its applications to mathmatical physics, Haish Pathasathy, pag (I K Intnational publication) 009. [4] J.J.H. Wang, PhD "Gnalizd momnt mthods in lctomagntic", IEEE pocdings, vol.137, pt. H, no. Apil [5] C A. Balanis, Antnna Thoy : Analysis and dsign, John Wily & sons, Nw Dlhi, 008. [6] Uma-shanka, K. Application of intgal quation and mthod of momnts fo lctically vy lag scatts using spatial dcomposition tchniqu, IEEE Antnnas and Popagation ocity Intnational ymposium, 7-11 May 1990, pp Vol 1. [7] Elctomagntism and lativity thoy, Application of advanc signal analysis, Haish Pathasathy, pag (I K Intnational publication) 008. [8] Fuscon, Fundamntal of Antnna Thoy and Tchniqu Pason Education, Nw Dlhi 011 [9] J. D. Lilly, Application of th Momnt Mthod to Antnna Analysis, MEE Thsis, Dpatmnt of Elctical Engining, Wst Viginia Univsity, [10] J. D. Lilly and C. A. Balanis, Cunt Distibutions, Input Impdancs, and Radiation Pattns of Wi Antnnas, Noth Amican Radio cinc Mting of URI, Univsitis Laval, Qubc, Canada, Jun 6, [11] R. F. Haington, Fild Computation by Momnt Mthods, Macmillan, Nw Yok, [1] J. H. Richmond, Digital Comput olutions of th Rigoous Equations fo catting Poblms, Poc. IEEE, Vol. 53, pp , August [13] L. L. Tsai, Momnt Mthods in Elctomagntic fo Undgaduats, IEEE Tans. Educ., Vol. E 1, No. 1, pp. 14, Fbuay [14] R. Mitta (Ed.), Comput Tchniqus fo Elctomagntic, Pgamon, Nw Yok, [15] J. Moo and R. Piz, Momnt Mthods in Elctomagntic, John Wily and ons, Nw Yok, [16] ignal Analysis, Advancd signal analysis and its applications to mathmatical physics, Haish Pathasathy, pag (I K Intnational publication) 009 [17] Elctomagntism, Advancd signal analysis and its applications to mathmatical physics, Haish Pathasathy, pag (I K Intnational publication) 009 [18] K. Y, Numical solution of initial bounday valu poblms involving Maxwll s quations in isotopic mdia, IEEE Tans. Antnnas Popagation, vol. AP-14, no. 3, pp , May [19] G. J. Buk and A. J. Poggio, Numical lctomagntic cod (NEC) Mthod of momnts, Naval Ocan ystms Cnt, Tch.Doc. 116, Jan [0] D. Foigo, P. Gianola, R. cotti & R. Vallai, Masumnts and numical valuation of th lctic fild in th na-zon of adio bas station antnnas, in Poc. IEEE Int. ymp. Antnnas and Pop. ocity, vol. 3, Ma. 001, pp ACKNOWLEDGMENT Autho would lik to thanks Pof. Haish Pathsathy, Dpt. of Elctonics and Communication, N..I.T, Nw Dlhi fo his suppot. I fl gatful fo th suppot of Pof. M P Tipathi, Pof. NIT, Nw Dlhi, fo witing and publishing pap. Th autho will also lik to xpss sinc appciation and gatitud to dpatmnt of Elctonics and Communication, Ra Kuma Gol Institut of Tchnology, Ghaziabad UP India, which has povidd tmndous assistanc thoughout th wok. Fist Autho M. Pavn Kuma Malik A pofssional qualifid and xpincd TECHNICAL and CREATIVE pson with xtnsiv knowldg and skills in Embddd ystm and Antnna. H is B Tch and M Tch fom Elctonics and Communication systm sp. H is having paps in intnational fd ounals. Having mo than 10 paps in intnational and national confnc. E mail: malikbailly@gmail.com IJER 016

6 Intnational Jounal of cintific & Engining Rsach, Volum 7, Issu 9, ptmb cond Autho Pof. Haish Pathasathy Pof. (D.) HARIH PARTHAARATHY is woking in dpatmnt of Elctonics and Communication, Nta i ubhash Institut of Tchnology, Nw Dlhi, INDIA. H is B Tch fom IIT Kanpu, Ph D fom IIT Dlhi and Post-doctoal fllowship fom IIA Bangalo. His mao aa of intst is, DP, statistically signal pocssing, Antnna. H is having mo than 10 paps in intnational fd ounals and IEEE tansaction. Having mo than 0 paps in intnational confnc and has wittn mo than 6 tchnical and sound books also. -mail: haishp@nsit.ac.in Thid Autho Pof. M P Tipathi Pof. (D.) M P Tipathi is woking in dpatmnt of Elctonics and Communication, Mahaaa Agsn Collg of Engining & Tchnology, Ghaziabad, UP INDIA. H is B Tch, M Tch and Ph D fom put govnmnt collg of India. His mao aa of intst is ignal thoy, DP, and Fild thoy. H is having mo than 08 paps in intnational fd ounals. Having mo than 0 paps in National and Intnational confnc H has wittn 4 tchnical and sound books also. -mail: munishpashadtipathi@diffmail.com IJER IJER 016