A closed form analytical solution to the radiation problem from a short dipole antenna above flat ground using spectral domain approach

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1 A closd fom analytcal soluton to th adaton poblm fom a shot dpol antnna abov flat gound usng spctal doman appoach S. Sautbov*, *uasan Natonal Unvsty,5, Munatpasov St., Astana, Kazashtan sautb@mal.u P. Fangos**, Ch. Chstas** and K. oannd** **Dvson of nfomaton ansmsson Systms and Matals chnology, School of lctcal and Comput ngnng, Natonal chncal Unvsty of Athns, oon Polytchnou St.9, 57 7 Zogafou, Athns, Gc pfangos@cntal.ntua.g Abstact n ths pap w consd th poblm of adaton fom a vtcal shot (tzan dpol abov flat gound wth losss, whch psnts th wll nown n th ltatu Sommfld adaton poblm. W nd up wth a closd fom analytcal soluton to th abov poblm fo th cvd lctc and magntc fld vctos abov th gound n th fa fld aa. h mthod of soluton s fomulatd n th spctal doman, and by nvs th dmnsonal Fou tansfomaton and subsqunt applcaton of th Statonay Phas Mthod (SPM th fnal solutons n th physcal spac a dvd. o ou nowldg, th abov closd fom solutons a novl n th ltatu fo th Sommfld adaton poblm. Fnally, th physcal ntptaton fo th cvd flds fomula dvd ths pap a povdd. nd ms Sommfld adaton poblm, Spctal doman soluton, Statonay Phas Mthod.. NRODUCON h so - calld Sommfld adaton poblm s a wll nown poblm n th aa of popagaton of lctomagntc (M wavs abov flat lossy gound fo obvous applcatons n th aa of wlss tlcommuncatons [,]. h classcal Sommfld soluton to ths poblm s povdd n th physcal spac by usng th so- calld tz potntals and t dos not nd up wth closd fom analytcal solutons. K. A. Noton [] concntatd n subsqunt yas mo n th ngnng applcaton of th abov poblm wth obvous applcaton to wlss tlcommuncatons, and povdd appomat solutons to th abov poblm, whch a psntd by ath long algbac pssons fo ngnng us, n whch th so calld attnuaton coffcnt fo th popagatng sufac wav plays an mpotant ol. n ths pap th authos ta advantag of pvous sach wo of thm fo th M adaton poblm n f spac [4] by usng th spctal doman appoach. Futhmo, n Rf. [5] th authos povdd th fundamntal fomulaton fo th poblm consdd h, that s th soluton n spctal doman fo th adaton fom a dpol momnt at a spcfc angula fquncy (ω n sotopc mda wth a flat nfnt ntfac. At that pap, th authos nd up wth ntgal psntatons fo th cvd lctc and magntc flds abov o blow th ntfac [Ln of St ( plus flctd fld tansmttd flds, spctvly], wh th ntgaton tas plac ov th adal spctal coodnat. hn, n th psnt pap th authos concntat to th soluton of th classcal Sommfld adaton poblm dscbd abov, wh th adaton of a vtcal dpol momnt at angula fquncy ω tas plac abov flat lossy gound [ths s quvalnt to th adaton of a vtcal small (tzan dpol abov th flat lossy gound, as t wll b pland by fomula n th man tt]. By usng th Statonay Phas Mthod (SPM mthod, [6], [7] ntgaton ov th adal spctal coodnat s pfomd and novl, to ou nowldg, closd fom analytcal solutons fo th cvd lctc and magntc flds n th fa fld zon (wh SPM mthod s applcabl a dvd. Fnally, physcal ntptaton of ths novl closd fom analytcal pssons a povdd.. GOMRY OF RADAON PROBLM h gomty of th poblm s gvn n Fg.. a tzan (small dpol wth dpol momnt p dctd to postv as, at alttud abov th nfnt, flat and lossy gound, adats tm hamonc lctomagntc (M wavs at angula fquncy ωπf [p(-ωt tm dpndnc s assumd n ths pap]. th latv compl pmttvty of th gound (mdum s / +, wh σ/ω 9 σ/f, σ bng th gound

2 conductvty, f th fquncy of adaton and.54 - F/m s th absolut pmttvty n vacuum o a. hn th wavnumbs of popagaton of M wavs n a and lossy gound, spctvly, a gvn by th followng / c ω ω μ ω μ μ ω μ ω c ω μ ω μ μ / + ( h Mawll quatons fo th tm hamonc M flds consdd abov a gvn by ot ωμ μ ot + ω Η j ( wh j s cunt dnsty (souc of M flds consdd h. p p ˆ (, μ NGRAL RPRSNAON FOR RCVD LCRC AND MAGNC FLDS A. M flds n tms of spctal doman cunt dnsts Followng [4]-[5], th M fld n physcal spac s dvd fom cunt dnsty J % n spctal doman and Gn s functon ψ%, also n th spctal doman, though nvs th dmnsonal (D Fou tansfomaton as followng : wh opato and F ψ % ( J% (4 { % % % } ψ μ < > ω F F J,J (5 s th nvs D Fou ansfom (F ( ( ψ % (6 s th D Gn s functon n spctal doman and cylndcal coodnats. Futhmo, by notng that, fo th poblm consdd h, cunt dnsty J % [ J % (,,] has only componnt, and that wavvcto (,, dos not possss azmuthal componnt, by pfomng th coss poduct and nvs F opaton of q. (4 w obtan : π ( J %( p( ψ ( π % d dd Smlaly, by pfomng th nn poduct and nvs F opaton of q. (5 w obtan : (7 (, μ Fg.. Gomty of th adaton poblm consdd n ths pap, namly adaton fom a vtcal tzan dpol at poston (,, abov nfnt, flat and lossy gound, as an ntfac stuatd at plan (Sommfld poblm []- [].. FORMULAON OF SOMMRFLD RADAON PROBLM N SPCRAL DOMAN : π π ω Ε ( μ ( J %( ψ% p( dd d ( wh (,, + (9 s th wavvcto of popagaton, and (,, s th

3 pont of obsvaton (s Fg., all n cylndcal coodnats. Futhmo, by tang nto account q. (9, q. ( fo th cvd lctc fld can also b wttn n th fom π π ω Ε ( ( μ ( J % ( ψ% p( dd d Futhmo, n od to ntgat pssons (7 and wth spct to azmuthal vaabl, w ta nto account that + cos( β ( Τhn, by usng th followng dntts fo Bssl functons R ( J % ( ψ % ( p( d d πω (5 R ( (( μ % ( % ( p( J ψ d d (6 ( J % ( ψ % ( p( d d (7 π π p ( cos d J ( J( ( d d Τ ( (( μ πω % ( % ( p( J ψ d d ( wh J s th Bssl functon of fst nd and zo od and s th anl functon of fst nd and zo od, w obtan ( J %( ψ % ( p( d d ( (( μ πω %( % ( p( J ψ d d B. Fomulaton of th bounday valu poblm ( (4 Fo th poblm consdd n ths wo (Fg., abov, w now us qs. ( and (4 abov, to wt th appopat pssons fo th flctd (R and tansmttd ( M fld, as followng : wh and a gvn by qs. and ( abov, ψ (9 ψ ( J [ J (,,], J [ J (,,] a th Fou componnts of sufac cunt dnsty. Futhmo, th ln-of-sght ( M fld of th tzan dpol (n th fa fld, as t wll b pland n Scton, blow s gvn by [6,7] ω p p( (, θ μ snθ 4π wh sphcal coodnats (,θ a gvn n tms of cylndcal coodnats (, [s Fg., abov] by ( + (

4 π θ tan ( ( and (, θ ζ cosθ ζ snθ (4 + ω J ( ( d ( wh ( wh s gvn by qs. - ( abov. ( hn, th total M fld n th gons > and < (s Fg. s gvn by ( ( ( + (, < R ( + (, < (, > R (, > (5 (6 Futhmo, by pfomng th ntgatons of pssons (5 ( ov, by usng th sdu thoy [], w obtan th followng ntgal pssons fo th M flds : n th upp half spac (> : ( ( J ( ( ( ω ( d + + ω J ( whl fo th low half spac (< : ( ( J ( d d (7 ( C. Applcaton of th Bounday Condtons (BC - Soluton fo th unown cunt dnsts at th ntfac n spctal doman W now apply th BC that at th ntfac ( th tangntal componnts of lctc fld and magntc fld must b contnuous, namly R + ( R wh + (4 ωp ( ( d (5 ωp ( d J ( π R ( d R J ( (6 (7 ( d ω ( J ( π ( d (9 ( J ( ( ω ( d + ( J ( d (9 ( ω J ( hn fom qs. ( and (4 w fnd : ( d (4 4

5 ωp + J ( ( d J ( ( d ( ωp + J ( J ( ( d (4 ( d (4 hfo, fom qs. (4 and (4 w obtan th followng systm of algbac quatons : ωp ωp + J ( + J ( J (, J (. (4 h soluton of systm of quatons (4 a th unown Fou componnts of sufac cunt dnsts, as followng : J% ( ω p ( + % (44 J ( ω p + p ( ( π ( + ( + ( p ( + π ( + d d + + ( n th low half-spac (tansmttd flds, < : ωp ( 4π + d (46 ( ( (47 p ( 4π + d ( ( ( (4. LCROMAGNC (M FLDS RFLCD FROM NFN, FLA AND SY GROUND ΙΝ ΤΗΕ FAR FLD RGON : ANALYCAL CD FORM XPRSSONS OBAND ROUG APPLCAON OF SAONARY PAS MOD (SPM. D. pssons fo th flctd and tansmttd M flds n ntgal psntatons Substtutng pssons of qs. (44 fo th unown cunt dnsts (at th ntfac, n spctal doman n qs. (7_ - (, w obtan th flctd and tansmttd M flds n ntgal psntatons, as followng : n th hgh half-spac ( >: ( + d fld plus flctd fld, ωp ( π ( + ( (45 n od to calculat th M fld abov lossy gound (.. fo >, w wt qs. (45 (46 n th followng fom : p p ˆ ˆ > Ι Ι π ω p Η Η Ι ˆ > wh ο πο ( ( + o Ηο d + (49 (5 (5 5

6 ( ( + Η ( o d ( + ο df ( f ( (59 d and (5 whch fnally ylds th followng psson fo th statonay pont (only on statonay pont sts : ( ( ( + o Ηο d + (5 s ( (6 Futhmo, n od to calculat ntgal (n an almost dntcal mann ntgals and wll b calculatd, usng SPM mthod [6], []-[], lt us assum lag agumnt appomaton fo th anl functons of qs. (5 (5, namly lt us assum that >> (54 fo whch cas functon ( ( bcoms a hghly oscllatng functon of. hn, snc Statonay Phas Mthod (SPM s to b appld, w just plac ( n q. (5 by ts asymptotc lag agumnt appomaton : + ( (55 π hn ntgal of q. (5 tas th followng fom : π ( + o + d + (56 Moov, n od to apply SPM mthod, w dfn adal dstanc (s Fg. as lag paamt, and w also dfn : - Phas functon : ( + o f ( + (57 - Ampltud functon : Not h that fo th a - lossy gound poblm consdd h s s al and postv, and s <. Also, w can asly s that lm lm s ( + o s (6 Futhmo, accodng to SPM mthod ([6], [] [], w also hav to calculat th scond dvatv of th phas functon, whch n ou cas s calculatd, fom q. (57, as ( + o f ( s ( s Not h that f ( s always ngatv, that s : (6 sgn f ( s (6 whch laton s ndd n th applcaton of SPM mthod. hn, by actually applyng SPM mthod ([6], [] [], fom q. (56 w fnd : π f ( sgn ( s s 4 f π F( s p( π / 4 f ( π o f ( s ( s F f ( s s (64 (65 hn, by usng pssons (57 (5 and (6, w fnally nd up wth th pssons : s s s s ( + s s s + s ( + (66 F ( + (5 5 s s s s ( + s s s + s ( + Nt, accodng to SPM mthod ([6], [] [] th statonay pont s calculatd fom th laton : (67 6

7 s s s s ( + s s s + s ( + (6 wh (69 s s s s (7 hn ou fnal closd-fom analytcal soluton conssts of qs. (49-(5 and (66-(7, wh s gvn by q. (6. s ntstng sults of ths pap.. As [and onc ( + s pt fnt] t can b asly found fom qs. (66 (7 that s, s, s, th factons appang n qs. (66 (6 ta (- valu at th lmt, and ntgals, and, as wll as th adatd flds of qs. (49 and (5, ta th lmtng valu of zo, as pctd.. Fo const. and ( + [sufac wav bhavo], smlaly wth cas abov w fnd that s, s, th factons appang n qs. (66 6 ta (- valu at th lmt amnd h, and : s ( + (7 V. PYSCAL NRPRAON OF DRVD CD FORM ANALYCAL XPRSSONS FOR RFLCD M FLDS s ( + (74 Rgadng th physcal ntptaton of th dvd solutons fo th cvd M fld abov th nfnt, flat and lossy gound (whch a novl, to ou nowldg, qs. (49 - (5, (6 and (66 (7, w ma th followng mas :. Fom qs. (6-(6 and (66 (7, w can asly alz that as o as ( +,.. vy fa away fom th adatng dpol o vy na th gound (.. sufac wav th M wavs popagat only paalll to th gound wth wavnumb. n th gnal cas, always n th fa fld gon amnd n Scton, namly fo >>λ (7 as t s asly sn fom th phas factos of qs. (66 (6 and p(-ωt tm dpndnc assumd thoughout ths pap, th M flds popagat wth wavnumb n dcton paalll to th gound, q. (6 ( s <, and wth wavnumb s n upwads dcton [s q. (69]. hn, by tang nto account th hozontal and vtcal wavnumbs of popagaton mntond just abov, th total wavnumb of popagaton quals to, as dvd though th us of q. (69 : s + s (7 whch, of cous, had to b pctd. h abov sults,.. th valus of wavnumbs of popagaton of M wavs n th hozontal and vtcal dctons psnt on of th s s ( + (75 hn, as a concluson, n th cas ( + amnd h [sufac wav bhavo], t can b asly alzd fom qs. (49 and (5 that (vtcal polazaton of lctc fld and (azmuthal componnt of magntc fld domnat n ths cas. 4. n th cas ( + >> [.. adatng dpol and obsvaton pont wll abov th gound, n whch cas th spac wav domnats], t can b asly shown that th factons n qs. (66 (6 ta, n ths lmt,th valu of th wll nown Fsnl flcton coffcnt (n -n / (n +n, wh n psnts th factv nd, n. 5. Fnally, gadng fquncy dpndnc of flctd M flds, not that fom qs. (66 (6 ntgals and vay n popoton to ω, whl ntgal vas n popoton to ω. Τhn, fnally, fom qs. (4 and (49 (5 all M flds (ln of sght and flctd M flds vay n popoton to angula fquncy ω [ω πf, wh f s th fquncy of adaton], wh th tzan dpol stngth (h p ω (n magntud has bn consdd h as gvn (constant. V. CONCLUSONS FUUR RSARC n ths pap w hav dvd analytcal closd-fom solutons fo th cvd lctomagntc (M fld fo th 7

8 poblm of adaton of vtcal tzan (small dpol antnna abov nfnt, flat and lossy gound. o ou nowldg ths pssons a novl n th ltatu, and thy hav bn dvd h fom a fomulaton n th spctal doman [4,5]. Futhmo, vy ntstng mas gadng th physcal ntptaton of th analytcal pssons mntond abov a psntd n ths pap, ncludng wavnumbs of popagaton (n hozontal and vtcal dctons, sufac wav bhavo and fomula fo th Fsnl flcton coffcnt n th poblm amnd h, as wll as n th lmtng cas of spac wavs (wh th usual psson fo th Fsnl flcton coffcnt s obtand. Rlatd sach n th na futu by ou sach goup wll nclud : compason wth M fld valus to b obtand usng K.A. Noton s appomat solutons [], dvaton of cospondng M fld psson fo th tansmttd M fld (gon <, soluton of th cospondng poblm fo hozontal adatng tzan dpol abov flat and lossy gound, popagaton n sotopc and ansotopc cystals wth ntfac (at tc. RFRNCS [] A. N. Sommfld, Popagaton of Wavs n Wlss lgaphy, Ann. Phys.,, pp , Mach 99; and, pp. 5 5, Dcmb 96. [] K. A. Noton, h Popagaton of Rado Wavs Ovw th Sufac of th ath, Pocdngs of th R, 4, pp. 67 7, 96; and 5, pp. 6, 97. []. K. Saa t. al., lctomagntc Maco Modlng of Popagaton n Mobl Wlss Communcaton : hoy and pmnt, Antnnas and Popagaton Magazn, Vol. 54, No. 6, pp. 7 4, Dc.. [4] S. Sautbov, h Gnalzd Solutons of a Systm of Mawll s quatons fo th Unaal Ansotopc Mda, Chapt n boo lctomagntc Wavs Popagaton n Compl Matt, dtd by A. A. Ksh, Coata, pp. 4, Jun. [5] S. Sautbov, R. Kasmhanova and P. Fangos, Modfd soluton of Sommfld s poblm, Communcatons, lctomagntcs and Mdcal Applcatons (CMA ntnatonal Confnc, Natonal chncal Unvsty of Athns (NUA, Athns, Gc, 7-9//, pp. 5. [6] C. A. Balans, Antnna hoy : Analyss and Dsgn, Appnd V : Mthod of Statonay Phas, pp. 9 97, J. Wly and Sons nc., Nw Yo, 997. [7] J. Fos, ntoducton to Antnna hoy and Popagaton of lctomagntc Wavs, Boo n G, Natonal chncal Unvsty of Athns (NUA, Athns, Gc, 9. []. Moshovts, Asymptotc mthods and ogh Fquncy chnqus fo th Calculaton of lctomagntc Scattng by Usng th Modfd Statonay Phas Mthod, Doctoal Dsstaton, n G, Natonal chncal Unvsty of Athns (NUA, Athns, Gc, Dcmb. [9] Ch. Moschovts, K. Kaaatslos,. Papls,. Anastassu,. Ouanos, A. zouls and P. Fangos, gh Fquncy Analytcal Modl fo Scattng of lctomagntc Wavs fom a Pfct lctc Conducto Plat usng an nhancd Statonay Phas Mthod Appomaton, ans. Antnnas and Popagaton, Vol. 5, No., pp., Januay. [] Ch. G. Moschovts,.. Anastassu, and P. V. Fangos, Scattng of lctomagntc wavs fom a ctangula plat usng an tndd Statonay Phas Mthod basd on Fsnl functons (SPM-F, Pogss n lctomagntcs Rsach (PR, Vol. PR 7, pp. 6-99, August. [] G. Afn, Mathmatcal Mthods fo Physsts, d dton, pp. 4 44, Acadmc Pss nc., Olando, Floda, USA, 95.