1. Radiation from an infinitesimal dipole (current element).

Transcription

1 LECTURE 3: Radiation fom Infinitsimal (Elmntay) Soucs (Radiation fom an infinitsimal dipol. Duality in Maxwll s quations. Radiation fom an infinitsimal loop. Radiation zons.). Radiation fom an infinitsimal dipol (cunt lmnt). Dfinition: Th infinitsimal dipol is a dipol whos lngth dl is much small than th wavlngth λ of th xcitd wav, i.. dl λ (dl < λ /50). Th infinitsimal dipol is quivalnt to a cunt lmnt Idl dq,wh Idl = dl. dt dl Q Idl + Q z A cunt lmnt is bst illustatd by a vy shot (compad to λ) pic of infinitsimally thin wi with cunt I. Sinc th cunt lmnt is vy shot, th cunt is assumd constant along dl. Th idal cunt lmnt is pactically unalizabl, but a vy good appoximation of it is th shot top-hat antnna. To aliz a unifom cunt distibution along th wi, capacitiv plats a usd to povid nough chag stoag at th nd of th wi, so that cunt is not zo th. z I I l/ l/.. Magntic vcto potntial du to cunt lmnt adiation. Th magntic vcto potntial (VP) A du to a lina souc is (s qn..55, Lctu ): AP ( ) = µ 0IQ ( ) dl 4 Q (3.) π L Sinc I is constant along l, A= µ 0I l zˆ (3.)

2 Equation (3.) givs th fild du to an lctic cunt lmnt (infinitsimal dipol) xpssd via th magntic VP A. Th fild adiatd by any complx antnna in lina mdium can b psntd as a supposition of th filds du to th cunt lmnts on th antnna sufac. Th A vcto will now b psntd with its sphical componnts. In antnna thoy, th pfd coodinat systm is th sphical on. This is mostly bcaus th fa fild adiation is of most significant intst, i.. th fild is analyzd so vy fa fom th souc, that it is assumd to popagat only adially away fom th souc. Th tansfomation fom ctangula to sphical coodinats is givn by: A sincos sinsin cos Ax A cos cos cossin sin A = y (3.3) A sin cos 0 Az Applying tansfomation (3.3) to th A vcto in (3.) poducs: A = Az cos = µ 0Il cos A = Az sin = µ 0I l sin (3.4) A = 0 z A A A y x

3 Not that: ) A dos not dpnd on (which is du to th cylindical symmty of th dipol); ) th dpndnc on, whichis( / ), is spaabl fom th dpndnc on... Fild vctos du to cunt lmnt adiation. Lt us now find th fild vctos H and E. a) H = A µ Th cul opato is xpssd in sphical coodinats to obtain: A H ( A ) = ˆ µ R Th magntic fild H has only a -componnt. Finally, H = jβ ( Il) sin + H = H = 0 jβ (3.5) (3.6) (3.7) b) j E = H jω A A jωε = ωµε Explicitly, in sphical coodinats on finds: ( Il) cos E = η π + jβ ( Il) β sin E = jη + jβ ( β ) E = 0 (3.8) (3.9) Not: ) Equations (3.7) and (3.9) show that th EM fild gnatd by th cunt lmnt is ath complicatd unlik th VP A. Th advantag of using th VP instad of th fild vctos is obvious vn in this simplst xampl. 3

4 ) Th fild vctos contain tms, which dpnd on th distanc fom th souc as (/), (/ )and(/ 3 ), and, thfo, som of thm can b nglctd at lag distancs fom th dipol. 3) Th longitudinal ( ˆ ) componnts of th fild vctos dcas fast as th fild popagats away fom th souc (as / and / 3 only). 4) Th non-zo tansvs fild componnts, E and H, a othogonal to ach oth, and thy hav tms, which dpnd on th distanc as /. Ths tms diff by a facto of η. Thy psnt th so-calld fa fild. Th concpt of fa fild will b -visitd lat, whn th adiation zons a dfind..3. Pow dnsity and ovall adiatd pow of th infinitsimal dipol. Th complx vcto of Poynting P dscibs th complx pow dnsity flux. It is calculatd as P= ( E H * ) = ( E ˆ ˆ) ( * ˆ) ( * ˆ * ˆ + E H = EH EH ) (3.0) Substituting (3.7) and (3.9) into (3.0) yilds: η Il sin P = j 3 8 λ ( β ) (3.) ( Il ) cossin P = jηβ 3 + 6π ( β ) Th ovall pow Π will b calculatd ov a sph, and, thfo, only th adial componnt of th vcto of Poynting P will hav contibution: Π= P ds = Pˆ ˆ + P ˆ sindd S S ( ) (3.) j 3 π Il Π= η (3.3) 3 λ ( β ) Th adiatd pow is qual to th al pat of th complx pow (th timavag of th total pow flow, s Lctu, Sction ). Thfo, th adiatd pow of an infinitsimal lctic dipol is: π Il Π ad = η (3.4) 3 λ 4

5 H, it is appopiat to intoduc th concpt of adiation sistanc R,which can dscib th pow loss du to adiation in an quivalnt cicuit of th antnna: Π id π l Π= RI R =, R = η I 3 λ Th adiation sistanc will b discussd lat in much mo dtail.. Duality in Maxwll s quations. Duality in Elctomagntics mans that th EM fild is dscibd by two sts of quantitis, which cospond to ach oth in such a mann that substituting th quantitis fom on st with th spctiv quantitis fom th oth st in any givn quation poducs a valid quation (th dual of th givn on). W shall dduc ths dual sts by simpl compaison of Maxwll s quations dscibing two dual filds: th fild of lctic soucs, and th fild of magntic soucs. As a wod of caution, duality xists vn if th a no soucs psnt in th gion of intst. Tabl. is just an asy intuitiv way to illustat duality and to dfin th dual sts of EM quantitis. TABLE.. DUALITY IN ELECTROMAGNETIC EQUATIONS Elctic soucs ( J 0, M = 0) Magntic soucs( J = 0, M 0) E = jωµ H H = jωε E + J H = jωε E E = jωµ H M D = ρ B = ρ m B = 0 D = 0 J = jωρ A + β A = µ J M = jωρ m F + β F = ε M R R A= µ J dv F = V R εm dv H = A V R E = F µ ε j j E = jωa A H = jωf F ωµε ωµε 5

6 TABLE.. DUAL QUANTITIES IN ELECTROMAGNETICS givn dual E H H J E Additional: M M A J η / η F F / η η A β β ε µ µ ε 3. Radiation fom an infinitsimal magntic dipol (lctic cunt loop). 3.. Th vcto potntial and th fild vctos of a magntic dipol (magntic cunt lmnt) Im l. Using th duality thom, th fild of a magntic dipol is adily found by simpl substitution of th dual quantitis in quations (3.4), (3.7) and (3.9) accoding to Tabl.. W shall dnot th magntic cunt, which is th dual of th lctic cunt I,byI m (masud in Volts). F = Fz cos = ε0 ( Iml) cos F = Fzsin = ε0( Im l) sin (3.5) F = 0 E E H H H jβ ( Iml) sin = + jβ = E = 0 ( I l) m cos = η π + jβ ( I l) j β m sin = η + jβ β = 0 (3.6) (3.7) 6

7 3.. Equivalnc btwn a magntic dipol (magntic cunt lmnt) and an lctic cunt loop. Fist, w shall pov th quivalnc of th filds xcitd by th following soucs E = jωµ H H = j E + J E = jωµ H + M (3.9) ωε (3.) ωε H = j E jωµ J M E ωµεe = jωµ J E ωµεe = M (3.8) (3.0) (3.) If th bounday conditions (BCs) fo E of poblm (3.0) a th sam as th BCs fo E in poblm (3.), and th xcitations of both filds fulfill jωµ J = M, (3.3) thn both filds will b idntical, i.. E E and H H. I m =0 [C] S[ C ] I I m l I m =0 A [L] [L] I m =0 Consid a loop [L] of lctic cunt I. Equation (3.3) can b wittn in intgal fom as: jωµ Jds = S [ C ] C Mdc (3.4) Th intgal at th lft-hand sid is th lctic cunt I. M is assumd nonzo and constant only at th sction ( l ), which is nomal to th loop s plan and passs though th loop s cnt. Thn, jωµ I = M l (3.5) 7

8 Im Th magntic cunt cosponding to th loop [L] will b obtaind by multiplying th magntic cunt dnsity M by th aa of th loop A [L],which yilds jωµ IA[L] = Im l (3.6) It was shown that a small loop of lctic cunt I and of aa A [L] cats EM fild quivalnt to that of a small magntic dipol (magntic cunt lmnt) ( Im l), such that (3.6) holds. H, it was assumd that th lctic cunt is constant along th loop, which is tu only fo vy small loops ( a < 0.0λ, wh a is th loop s adius). If th loop is lag, thn th fild xpssions povidd blow will b inaccuat, and oth solutions should b usd VP and fild vctos of an infinitsimal loop antnna. Th xpssions blow a divd simply by insting (3.6) into (3.6)- (3.7). ( IA) sin E = ηβ + jβ (3.7) ( IA) cos H = jβ π + jβ (3.8) ( IA) sin H = β + jβ β (3.9) E = E = H = 0 (3.30) Th fa-fild tms (which hav / dpndnc on th distanc fom th souc) show th sam bhaviou as in th cas of an infinitsimal dipol antnna: th lctic fild E is othogonal to th magntic fild H and diffs just by a facto of η ; th longitudinal ˆ componnts hav no fa-fild tms. Th dpndnc of th Poynting vcto and th complx pow on th distanc is th sam as in th cas of an infinitsimal lctic dipol. Th adiatd pow can b found to b: 4 Π ( ) ad = ηβ IA (3.3) π 8

9 4. Radiation zons basic concpts. Th spac suounding th antnna is dividd into th gions accoding to th pdominant fild bhaviou. Th boundais btwn th gions a not distinct and th fild bhaviou changs vy gadually as ths boundais a cossd. In this cous, w shall b mostly concnd with th fa-fild chaactistics of th antnnas. 4.. Ractiv na-fild gion This is th gion immdiatly suounding th antnna, wh th activ fild pdominats. Fo most antnnas, it is assumd that this gion is a sph with th antnna at its cnt, and with a adius of D, (3.3) λ wh D is th lagst dimnsion of th antnna, and λ is th wavlngth of th adiatd fild. Th abov xpssion will b divd in Sction 5. It must b notd that this limit is most appopiat fo wi and wavguid aptu antnnas, whil it is not valid fo lctically lag flcto antnnas. At this point, w shall discuss th gnal fild bhaviou making us of ou knowldg of th infinitsimal dipol fild. Whn (3.3) is tu, it is also tu fo most antnnas that β. Thn, th most significant tms in th fild xpssions (3.7) and (3.9) will b ( Il ) H sin ( Il ) E jη sin 3 β, β (3.33) ( Il ) E jη cos 3 β H = H = E = 0 This appoximatd fild is puly activ ( H and E a in phas quadatu). j Actually, th β can b nglctd, and aft som simpl mathmatical tansfomations it can b claly shown that: ) th H componnt is xactly th magntostatic fild of a cunt filamnt ( ) Il ;)th E and E componnts a xactly th lctostatic fild of a dipol. 9

10 That th fild is puly activ at points, which a vy clos to th infinitsimal dipol, is obvious fom th quation (3.3) dscibing th total complx pow. Its imaginay pat is: π Il Im{ Π } = η 3 3 λ β Im{ Π} ( ) will obviously dominat ov th adiatd pow (3.34) π Il R{ Π } = η =Πad, (3.35) 3 λ whn 0,sincΠad dos not dpnd on at all. In th na-fild gion, th adial activ pow flow dnsity P has pdominant magntic chaact (ngativ imaginay numb) and dcass as (/ 5 ): na η Il sin P = j (3.36) λ β Th P has th sam od of dpndnc on but has pdominant lctic chaact: na ( Il ) cossin P = jηβ 3 + (3.37) 6π ( β ) Rmind: Accoding to Poynting s thom * * * i i ( H M + E J ) = ( E H ) + σ E + jω µ H ε E 4 4 o ps = pad + ploss + jω ( wm w ) wh: * i i ps = ( H M + E J ) is th supplid complx pow dnsity, W/m 3 ; * pad = ( E H ) = P is th complx pow dnsity nting o laving th point, W/m 3 ; ploss = σ E is th loss pow dnsity (al only), W/m 3 ; wm = µ H is th tim-avag magntic ngy dnsity, J/m 3 ; 4 0

11 w = ε E 4 is th tim-avag lctic ngy dnsity, J/m Radiating na-fild (Fsnl) gion This is an intmdiat gion btwn th activ na-fild gion and th fa-fild gion, wh th adiation fild pdominats but th angula fild distibution is still dpndnt on th distanc fom th antnna. In this gion, β. Fo most antnnas, it is assumd that th Fsnl gion is nclosd btwn two sphical sufacs: D D (3.38) λ λ H, D is th lagst dimnsion of th antnna. This gion is calld Fsnl gion bcaus its fild xpssions duc to Fsnl intgals. Th filds of an infinitsimal dipol in th Fsnl gion a obtaind by nglcting th high-od (/) n -tms: jβ ( I l) H sin ( Il ) E η cos π, β (3.39) β ( Il ) E jη sin H = H = E = 0 Th adial componnt E is still not ngligibl, but th tansvs componnts ( E and H ) a dominant Fa-fild (Faunhof) gion Only tms / a considd, whn β. Th angula fild distibution dos not dpnd on th distanc fom th souc any mo, i.. th fa-fild pattn is alady wll stablishd. Th fild is a tansvs EM wav. Fo most antnnas, th fa-fild gion is dfind as: D (3.40) λ

12 Th fa-fild of th infinitsimal dipol is obtaind as: jβ ( I l) H sin β ( Il ) E jη sin, β (3.4) E 0 H = H = E = 0 Impotant fatus of th fa fild: ) no adial componnts; ) th angula fild distibution is indpndnt of ; 3) E H ; E 4) η = Zi = ; H * 5) ( ) E P= E H = ˆ= η H ˆ. (3.4) η 5. Rgion spaation and accuacy of adiation intgal appoximations. In most pactical cass, th closd fom solution of th adiation intgal (th VP intgal) is impossibl. Fo th valuation of th fa filds o th filds in th Fsnl gion, standad appoximations a applid, fom which th boundais of ths gions a divd. Consid th VP intgal fo a lina cunt souc: µ I( l' ) R A = dl ', (3.43) L R wh R= ( x x' ) + ( y y' ) + ( z z' ). Th obsvation point is at Pxyz, (,, ) and th souc point is locatd at Qx ( ', y', z '), which is along th intgation contou L. So fa, w hav analyzd just th infinitsimal dipol, whos cunt is constant along L. In pactical antnnas, this is aly tu, and th solution of (3.43) can b vy complicatd dpnding on th function I ( l' ). Bsids, bcaus of th infinitsimal siz of th souc, th distanc btwn th intgation point and th obsvation point R was considd constant and qual

13 to th distanc fom th cnt of th dipol R= = x + y + z. Th clos th obsvation point to a finit-siz antnna, th lss accuat this assumption is. R Th intgal knl will b dividd into two factos: th amplitud R j R facto ( / R ), and th phas facto β. Th amplitud facto is not vy snsitiv to os in R. In both, th Fsnl and th Faunhof gions, th appoximation (3.44) R is accptabl. Th abov appoximation is unaccptabl in th phas tm. To kp th phas tm o low nough, th maximum o in ( β R) must b kpt blow π /8=.5. Nglct th antnna dimnsions along th x and th y-axs (infinitsimally thin wi). Thn, ( ) x' = y' = 0 R= x + y + z z' (3.45) ( ' ') ( ' ' cos ) R x y z z zz z z = = + (3.46) Using th binomial xpansion : / / 3/ R ( ) + ( ) ( z ' z 'cos ) + ( ) ( z ' z 'cos) + z' z' cos R z'cos + + O (3.47) O dnots tms of th od (/ ) and high. Simplifying futh, on aivs at: R z z O = 'cos + ' sin + (3.48) n n n nn ( ) n nn ( )( n ) n 3 3 a+ b = a + na b+ a b + a b +! 3! ( ) 3

14 (a) fa-fild appoximation Only th fist two tms in th xpansion (3.48) a takn into account: R z'cos (3.49) z ' R P (,, ) dz Q(') z x D/ 0 = ' y (a) Finit-siz dipol z R P (,, ) ' = x D/ dz Q( z') 0 = ' z'cos y (b) Finit-siz dipol - fa-fild appoximations Th most significant o tm in R is z ' = sin, z ' which has its maximum at = π /, max =. Th consqunc is a β, which has to b kpt blow π /8: maximum o in ( R) 4

15 z ' max π β 8 z ' max is actually half th lagst dimnsion of th antnna, z max ' = D/.Now,it is asy to find th smallst distanc fom th antnna cnt,atwhichth phas o is accptabl: D (3.50) λ This sult is idntical with th fa-zon limit dfind in (3.40). (b) adiating na-fild (Fsnl gion) appoximation This gion is adjacnt to th Faunhof gion, so its upp bounday is spcifid by: D (3.5) λ Whn th obsvation point blongs to this gion, on should tak on mo tm in th xpansion of R as givn by (3.48) to duc sufficintly th phas o. Th appoximation this tim is: R z'cos + z' sin (3.5) Th most significant o tm is: 3 z ' = cossin (3.53) Th angls o must b found, at which has its xtma. 3 z' = sin ( sin + cos ) = 0 (3.54) Th oots of (3.54) a: () o = 0 min (),(3) o = actan ( ± ) ± 54.7 max (3.55) Following a pocdu simila to cas (a), on obtains: 3 π z ' () () βmax = cos sin o o λ 3 π D π βmax = 3 λ 8 5

16 3 3 D D λ = λ (3.56) Equation (3.56) stats th low bounday of th Fsnl gion (fo wi antnnas) and is idntical to th lft-hand sid of (3.38). 6