BASIC ANTENNA PARAMETERS AND WIRE ANTENNAS

Transcription

1 Naval Postgraduat School Distanc Larning Antnnas & Propagation LCTUR NOTS VOLUM II BASIC ANTNNA PARAMTRS AND WIR ANTNNAS by Profssor David Jnn λ kˆ H PROPAGATION DIRCTION ANTNNA (vr 1.3)

2 Antnnas: Introductory Commnts Classification of antnnas by siz: Lt l b th antnna dimnsion: 1. lctrically small, l << λ : primarily usd at low frquncis whr th wavlngth is long. rsonant antnnas, l λ / : most fficint; xampls ar slots, dipols, patchs 3. lctrically larg, l >> λ : can b composd of many individual rsonant antnnas; good for radar applications (high gain, narrow bam, low sidlobs) Classification of antnnas by typ: 1. rflctors. lnss 3. arrays Othr dsignations: wir antnnas, aprtur antnnas, broadband antnnas 1

3 Radiation Intgrals (1) Considr a prfct lctric conductor (PC) with an lctric surfac currnt flowing on S. In th cas whr th conductor is part of an antnna (a dipol), th currnt may b causd by an applid voltag, or by an incidnt fild from anothr sourc (a rflctor). Th obsrvation point is dnotd by P and is givn in trms of unprimd coordinat variabls. Quantitis associatd with sourc points ar dsignatd by prims. W can us any coordinat systm that is convnint for th particular problm at hand. x O r z S r PC WITH SURFAC CURRNT R r r J s ( x, y, z ) OBSRVATION POINT P( x, y, z) or P( r,, ) y ( x x ) + ( y y ) + ( z z ) R R Th mdium is almost always fr spac ( µ o, εo), but w continu to us ( µ, ε ) to covr mor gnral problms. If th currnts ar known, thn th fild du to th currnts can b dtrmind by intgration ovr th surfac.

4 Radiation Intgrals () Th vctor wav quation for th lctric fild can b obtaind by taking th curl of Maxwll s first quation: k jωµ J s A solution for in trms of th magntic vctor potntial A(r ) is givn by ( A( r) ) ( r ) jωa( r) + (1) jωµε whr (r µ J jkr ) is a shorthand notation for (x,y,z) and A r s ( ) ds 4π R S W ar particularly intrstd in th z cas wr th obsrvation point is in th far zon of th antnna ( P ). As P rcds to infinity, th vctors r and R r bcom paralll. r rˆ OBSRVATION POINT R r rˆ ( r rˆ ) x r y 3

5 Radiation Intgrals (3) In th xprssion for A(r ) w us th approximation 1/ R 1/ r in th dnominator and rˆ R rˆ r rˆ r rˆ in th xponnt. quation () bcoms [ ( )] A( r ) µ 4πr S [ jkrˆ r rˆ J ( r rˆ ) ] µ ds jkr J jk ( r rˆ ) s s d s 4πr Whn this is insrtd into quation (1), th dl oprations on th scond trm lad to 1/ r 3 and 1/ r trms, which can b nglctd in comparison to th jωa trm, which dpnds only on 1 / r. Thrfor, in th far fild, ( r ) jωµ 4πr jkr S J s jk ( r rˆ ) ds xplicitly rmoving th r componnt givs, ( r ) jkη 4π r jkr S (discard th S r componnt) (3) [ J r ( J )] jk ( r rˆ ˆ ) rˆ ds Th radial componnt of currnt dos not contribut to th fild in th far zon. s s 4

6 Radiation Intgrals (4) Notic that th filds hav a sphrical wav bhavior in th far zon: ~. Th r sphrical componnts of th fild can b found by th appropriat dot products with. Mor gnral forms of th radiation intgrals that includ magntic surfac currnts ( J ms) ar: jkη jkr Jms ˆ jkr r r J s ˆ ˆ (,, ) + ds 4π r S η jkη jkr Jms ˆ jkr r r J s ˆ ˆ (,, ) ds 4π r η S Th radiation intgrals apply to an unboundd mdium. For antnna problms th following procss is usd: 1. find th currnt on th antnna surfac, S,. rmov th antnna matrials and assum that th currnts ar suspndd in th unboundd mdium, and 3. apply th radiation intgrals. jkr 5

7 Hrtzian Dipol (1) Prhaps th simplst application of th radiation intgral is th calculation of th filds of an infinitsimally short dipol (also calld a Hrtzian dipol). Not that th critrion for short mans much lss than a wavlngth, which is not ncssarily physically short. x l z z a Js( ρ,, z ) J y ρ a o zˆ For a thin dipol (radius, a << λ ) th surfac currnt distribution is indpndnt of. Th currnt crossing a ring around th antnna is I J s π a A/m For a thin short dipol ( l << λ ) w assum that th currnt is constant and flows along th cntr of th wir; it is a filamnt of zro diamtr. Th two-dimnsional intgral ovr S bcoms a on-dimnsional intgral ovr th lngth, J ds πa I d l S s L 6

8 Hrtzian Dipol () Using r zˆ z and r ˆ xˆsin cos + yˆ sin sin + zˆ cos givs r rˆ z cos radiation intgral for th lctric fild bcoms l l jkη jkr jk( r rˆ ) jkηi zˆ jkr ( r,, ) I zdz ˆ jkz cos dz 4πr 4πr 0 Howvr, bcaus l is vry short, k z 0 and 1. Thrfor, dl jkz cos l jk I z jkr jk I z r η ˆ r η l ˆ (,, ) 4π 0 4 π r lading to th sphrical fild componnts z jkr ( 1) dz 0. Th ˆ ˆ jkηil 4πr jkηilsin jkr 4 πr ˆ 0 zˆ jkr x l z y SHORT CURRNT FILAMNT 7

9 Hrtzian Dipol (3) Not that th lctric fild has only a 1/r dpndnc. Th absnc of highr ordr trms is du to th fact that th dipol is infinitsimal, and thrfor r ff 0. Th fild is a sphrical wav and hnc th TM rlationship can b usd to find th magntic fild intnsity kˆ ˆ rˆ ˆ jki jkr H ˆ lsin η η 4πr Th tim-avragd Poynting vctor is W av 1 R k I { ˆ * 1 * η l sin H } R{ H } rˆ rˆ Th powr flow is outward from th sourc, as xpctd for a sphrical wav. Th avrag powr flowing through th surfac of a sphr of radius r surrounding th sourc is π π π π ηk I l ηk I l P rad Wav nˆ ds sin ˆ ˆ sin r r r d d W 3π 1π 0 0 r 0 0 8π / 3 3π r 8

10 Solid Angls and Stradians Plan angls: s R, if s R thn 1 radian R ARC LNGTH s Solid angls: Ω A / R, if A R, thn Ω 1 stradian Ω A / R R SURFAC ARA A 9

11 Dirctivity and Gain (1) Th radiation intnsity is dfind as dp rad U(, ) r rˆ Wav r W dω and has units of Watts/stradian (W/sr). Th dirctivity function or dirctiv gain is dfind as powr radiatd pr unit solid angl dprad / dω r Wav D(, ) 4π avrag powr radiatd pr unit solid angl P /(4π ) P For th Hrtzian dipol, D(, ) 4π r W rad av 4π ηk ηk I 3π I l 1π Th dirctivity is th maximum valu of th dirctiv gain D o P r l sin r av rad Dmax (, ) D( max, max ) 3 3 sin rad 10

12 Dipol Polar Radiation Plots Half of th radiation pattrn of th dipol is plottd blow for a fixd valu of. Th halfpowr bamwidth (HPBW) is th angular width btwn th half powr points (1/ blow th maximum on th voltag plot, or 3dB blow th maximum on th dcibl plot). FILD (VOLTAG) PLOT DCIBL PLOT Th half powr bamwidth of th Hrtzian dipol, B : ( ) norm ~ sin sin HP HP B HP 11

13 Dipol Radiation Pattrn Radiation pattrn of a Hrtzian dipol alignd with th z axis. D n is th normalizd dirctivity. Th dirctivity valu is proportional to th distanc from th cntr. 1

14 Dirctivity and Gain () Anothr formula for dirctiv gain is whr Ω A is th bam solid angl Ω A 4π D(, ) Ω π π 0 0 A norm norm (, ) (, ) sin d d and norm(, ) is th normalizd magnitud of th lctric fild pattrn (i.., th normalizd radiation pattrn) ( r,, ) norm(, ) ( r,, ) Not that both th numrator and dnominator hav th sam 1/r dpndnc, and hnc th ratio is indpndnt of r. This approach is oftn mor convnint bcaus most of our calculations will b conductd dirctly with th lctric fild. Normalization rmovs all of th cumbrsom constants. max 13

15 Dirctivity and Gain (3) As an illustration, w r-comput th dirctivity of a Hrtzian dipol. Noting that th maximum magnitud of th lctric fild is occurs whn π /, th normalizd lctric fild intnsity is simply norm (, ) sin Th bam solid angl is Ω A π π 0 0 π π and from th dfinition of dirctivity, norm 4 / 3 (, ) 3 8π sin d 3 0 sin d d 4π 4π D(, ) norm (, ) sin Ω 8π /3 which agrs with th prvious rsult. A 3 sin 14

16 xampl Find th dirctivity of an antnna whos far-lctric fild is givn by ( r,, ) r r jkr jkr cos, cos, Th maximum lctric fild occurs whn cos 1 max 10/ r. Th normalizd lctric fild intnsity is cos, 0 90 (, ) norm 0.1 cos, which givs a bam solid angl of Ω A π π 0 / 0 cos sin d d π π 0 π / cos sin d d π 3 (1.1) and a dirctivity of D db. o 15

17 Bam Solid Angl and Radiatd Powr In th far fild th radiatd powr is P π π 1 π π * 1 rad R H rˆ ds r sin d d Frad η / η Frad From th dfinition of bam solid angl ηp rad Ω A π π max norm r (, ) π π 0 0 quat th xprssions for Frad sin d r sin d F rad d d P rad Ω A max η r F rad Ω A max r 16

18 Gain vs. Dirctivity (1) Dirctivity is dfind with rspct to th radiatd powr, P rad. This could b lss than th powr into th antnna if th antnna has losss. Th gain is rfrncd to th powr into th antnna, P in. I ANTNNA P inc Prf P in R l Ra Dfin th following: P inc powr incidnt on th antnna trminals P rf powr rflctd at th antnna input P powr into th antnna in P loss powr loss in th antnna (dissipatd in rsistor Rl, Ploss I Rl) P rad powr radiatd (dlivrd to rsistor Ra, Prad I Ra, Ra is th radiation rsistanc) Th antnna fficincy,, is P P whr 0 1. rad in

19 Gain vs. Dirctivity () Gain is dfind as dp / / (, ) rad dω dp 4π rad dω G P /(4π ) P / in rad D(, ) Most oftn th us of th trm gain rfrs to th maximum valu of G (, ). xampl: Th antnna input rsistanc is 50 ohms, of which 40 ohms is radiation rsistanc and 10 ohms is ohmic loss. Th input currnt is 0.1 A and th dirctivity of th antnna is. 1 1 in in loss I R 0.1 (10) l 1 1 rad I R 0.1 (40) 0. a Th input powr is P I R.1 (50) 0. 5 W Th powr dissipatd in th antnna is P W Th powr radiatd into spac is P W Prad 0. If th dirctivity is D o thn th gain is G D D () 1. 6 Pin

20 Azimuth/lvation Coordinat Systm Radars frquntly us th azimuth/lvation coordinat systm: (Az,l) or (α,γ ) or (, a ). Th antnna is locatd at th origin of th coordinat systm; th arth's surfac lis in th x-y plan. Azimuth is gnrally masurd clockwis from a rfrnc (lik a compass) but th sphrical systm azimuth angl is masurd countrclockwis from th x axis. Thrfor α 360 and γ 90 dgrs. ZNITH z CONSTANT LVATION P α γ r y x HORIZON 19

21 Approximat Dirctivity Formula (1) Assum th antnna radiation pattrn is a pncil bam on th horizon. Th pattrn is constant insid of th lvation and azimuth half powr bamwidths (, ) rspctivly: z 0 a y a x 0 π / 0

22 1 Naval Postgraduat School Antnnas & Propagation Distanc Larning Approximat Dirctivity Formula () Approximat antnna pattrn ( ) ( ) + ls 0, / / and / / / / ˆ, ), ( a a jkr o r π π Th bam solid angl is ( ) ( ) [ ] [ ] a a a A d d a a π π Ω + ) / ( / / sin / sin sin 1 This lads to an approximation for th dirctivity of a A D o 4π 4π Ω. Not that th angls ar in radians. This formula is oftn usd to stimat th dirctivity of an omnidirctional antnna with ngligibl sidlobs.

23 Thin Wir Antnnas (1) Thin wir antnnas satisfy th condition a << λ. If th lngth of th wir (l) is an intgr multipl of a half wavlngth, w can mak an ducatd guss at th currnt basd on an opn circuitd two-wir transmission lin FD POINTS z λ / 4 λ / I(z) OPN CIRCUIT For othr multipls of a half wavlngth th currnt distribution has th following faturs l λ / l λ FD POINT LOCATD AT MAXIMUM CURRNT GOS TO ZRO AT ND l 3λ /

24 Thin Wir Antnnas () On a half-wav dipol th currnt can b approximatd by Using this currnt in th radiation intgral I( z) I cos( kz) for λ / 4 < z < λ / 4 jkη ( r,, ) 4πr jkηi 4πr From a tabl of intgrals w find that cos( Bz ) Az o λ / 4 jkr zˆ Io λ / 4 λ / 4 o jkr dz Az zˆ λ / 4 cos( kz ) cos( kz ) 0 jkz cos jkz cos ± 1 dz dz [ Acos( Bz ) + B sin( Bz )] A + B whr A jk cos and B k, so that A + B k cos + k k sin. Th componnt rquirs th dot product z ˆ ˆ sin. 3

25 Thin Wir Antnnas (3) valuating th limits givs π cos cos jπ cos / jπ cos / jkr [ ( 1) sin ] o k 4 jkηi πr k sin Th magntic fild intnsity in th far fild is H kˆ η ˆ H ji πr o jkr jηi πr o π cos cos sin ˆ jkr π cos cos sin Th dirctivity is computd from th bam solid angl, which rquirs th normalizd lctric fild intnsity norm max cos ( π ) cos sin 4

26 Thin Wir Antnnas (4) Ω A π π 0 cos Th dirctiv gain is 4π D Ω A ( π cos / ) cos ( π cos / ) sin norm Th radiatd powr is P rad Ω A (, ) max η r sin d π π d sin 0 Intgrat numrically 4π cos (π )(1.18) sin π (1.18) η η (πr) Io (π )(1.18) ( π ) ( ) π cos cos cos r I whr R a is th radiation rsistanc of th dipol. Th radiatd powr can b viwd as th powr dlivrd to rsistor that rprsnts fr spac. For th half-wav dipol th radiation rsistanc is P R a rad ()(36.57) I ohms o o sin 1 I o R a 5

27 Numrical Intgration (1) Th rctangular rul is a simpl way of valuating an intgral numrically. Th ara undr th curv of f (x) is approximatd by a sum of rctangular aras of width and hight f ( x n ), whr x n + ( n 1) + a all of th rctangls ar of qual width b a is th cntr of th intrval nth intrval. Thrfor, if f ( x) dx f ( x ) b a Clarly th approximation can b mad as clos to th xact valu as dsird by rducing th width of th triangls as ncssary. Howvr, to kp computation tim to a minimum, only th smallst numbr of rctangls that provids a convrgd solution should b usd. n 3 f ( x) 1 4 N a x1 x x N b x 6

28 Numrical Intgration () xampl: Matlab programs to intgrat π 0 cos ( π cos /) d sin Sampl Matlab cod for th rctangular rul % intgrat dipol pattrn using th rctangular rul clar radpi/180; % avoid 0 by changing th limits slightly a.001; bpi-.001; N5 dlta(b-a)/n; sum0; for n1:n thtadlta/+(n-1)*dlta; sumsum+cos(pi*cos(thta)/)^/sin(thta); nd Isum*dlta Convrgnc: N5, 1.175; N10, 1.187; N50, Sampl Matlab cod using th quad8 function % intgrat to find half wav dipol solid angl clar Iquad8('cint',0.0001,pi-.0001,.00001); disp(['cint intgral, I: ',numstr(i)]) function Pcint(T) % function to b intgratd P(cos(pi*cos(T)/).^)./sin(T); 7

29 Thin Wirs of Arbitrary Lngth For a thin-wir antnna of lngth l along th z axis, th lctric fild intnsity is xampl: kl kl cos cos cos jηi jkr πr sin l 1.5λ (lft: voltag plot; right: dcibl plot)

30 Fding and Tuning Wir Antnnas (1) Whn an antnna trminats a transmission lin, as shown blow, th antnna impdanc ( Z a ) should b matchd to th transmission lin impdanc ( Z o) to maximiz th powr dlivrd to th antnna Zo Za Za Zo Γ and Z + Z a o VSWR Γ Γ Th antnna s input impdanc is gnrally a complx quantity, Z a ( Ra + Rl ) + jxa. Th approach for matching th antnna and incrasing its fficincy is 1. minimiz th ohmic loss, R l 0. tun out th ractanc by adjusting th antnna gomtry or adding lumpd lmnts, X a 0 (rsonanc occurs whn Z a is ral) 3. match th radiation rsistanc to th charactristic impdanc of th lin by adjusting th antnna paramtrs or using a transformr sction, R Z a o 9

31 Fding and Tuning Wir Antnnas () xampl: A half-wav dipol is fd by a 50 ohm lin Za Zo Γ Za + Zo Γ VSWR Γ Th loss du to rflction at th antnna trminals is τ 1 Γ ( ) log τ db which is statd as db of rflction loss (th ngativ sign is implid by using th word loss ). 30

32 Fding and Tuning Wir Antnnas (3) Th antnna impdanc is affctd by 1. lngth. thicknss 3. shap 4. fd point (location and mthod of fding) 5. nd loading Although all of ths paramtrs affct both th ral and imaginary parts of Z a, thy ar gnrally usd to rmov th ractiv part. Th rmaining ral part can b matchd using a transformr sction. Anothr problm is ncountrd whn matching a balancd radiating structur lik a dipol to an unbalancd transmission lin structur lik a coax. UNBALANCD FD BALANCD FD V g + COAX λ / + V g TWO-WIR LIN λ / DIPOL DIPOL 31

33 Fding and Tuning Wir Antnnas (4) If th two structurs ar not balancd, a rturn currnt can flow on th outsid of th coaxial cabl. Ths currnts will radiat and modify th pattrn of th antnna. Th unbalancd currnts can b liminatd using a balun (balancd-to-unbalancd transformr) UNBALANCD FD BALANCD FD Va V b I(z) b.... a a b V a V b Baluns frquntly incorporat choks, which ar circuits dsignd to chok off currnt by prsnting an opn circuit to currnt wavs. 3

34 Fding and Tuning Wir Antnnas (5) An xampl of a balun mploying a chok is th slv or bazooka balun I I 1 λ / 4 Z o I I 1 HIGH IMPDANC (OPN CIRCUIT) PRVNTS CURRNT FROM FLOWING ON OUTSID Z o SHORT CIRCUIT Th chok prvnts currnt from flowing on th xtrior of th coax. All currnt is confind to th insid of surfacs of th coax, and thrfor th currnt flow in th two dirctions is qual (balancd) and dos not radiat. Th intgrity of a short circuit is asir to control than that of an opn circuit, thus short circuits ar usd whnvr possibl. Originally balun rfrrd xclusivly to ths typs of wir fding circuits, but th trm has volvd to rfr to any fd point matching circuit. 33

35 Calculation of Antnna Impdanc (1) Th antnna impdanc must b matchd to that of th fd lin. Th impdanc of an antnna can b masurd or computd. Usually masurmnts ar mor tim consuming (and thrfor xpnsiv) rlativ to computr simulations. Howvr, for a simulation to accuratly includ th ffct of all of th antnna s gomtrical and lctrical paramtrs on Z a, a fairly complicatd analytical modl must b usd. Th rsulting quations must b solvd numrically in most cass. On popular tchniqu is th mthod of momnts (MM) solution of an intgral quation (I) for th currnt. z 1. I (z ) is th unknown currnt distribution on th l/ wir. Find th z componnt of th lctric fild in a trms of I (z ) from th radiation intgral + 3. Apply th boundary condition V g g b / b / l / z (ρ a) 0, b / z l / g, b / z in ordr to obtain an intgral quation for I (z ) 34

36 Calculation of Antnna Impdanc () On spcial form of th intgral quation for thin wirs is Pocklington s quation k + z l / I( z ) d z 4πR l / jkr 0, b/ z l / jωε g, b / z whr R a + ( z z ). This is calld an intgral quation bcaus th unknown quantity I (z ) appars in th intgrand. 4. Solv th intgral quation using th mthod of momnts (MM). First approximat th currnt by a sris with unknown xpansion cofficints { } I n N I( z ) I n Φ n ( z ) n1 Th basis functions or xpansion functions { } n Φ ar known and slctd to suit th particular problm. W would lik to us as fw basis functions as possibl for computational fficincy, yt nough must b usd to insur convrgnc. 35

37 Calculation of Antnna Impdanc (3) xampl: a stp approximation to th currnt using a sris of pulss. ach sgmnt is calld a subdomain. Problm: thr will b discontinuitis btwn stps. I( z ) CURRNT I Φ STP APPROXIMATION I 1 Φ 1 l z 1 z 0 z N IN ΦN A bttr basis function is th ovrlapping picwis sinusoid I( z ) CURRNT IΦ l PICWIS SINUSOID z I1Φ1 l z 1 z 0 z N INΦN l z 36

38 Calculation of Antnna Impdanc (4) A picwis sinusoid xtnds ovr two sgmnts (ach of lngth ) and has a maximum at th point btwn th two sgmnt z z n 1, z n 1 z z n z n z n 1 z n ( zn 1) zn Φn( z ) z n + ( zn + 1) z Φ n ( z ) z n+1 z, z n+1 z n z n z 0, lswhr z n+1 Φn( z ) ntir domain functions ar also possibl. ach ntir domain basis function xtnds ovr th ntir wir. xampls ar sinusoids. l Φ3 Φ 1 Φ l z 37

39 Calculation of Antnna Impdanc (5) Solving th intgral quation: (1) insrt th sris back into th intgral quation k + l / N jkr z I n Φ n ( z ) d z 0, b / z l / n1 4πR jωε g, b/ z l / Not that th drivativ is with rspct to z (not z ) and thrfor th diffrntial oprats only on R. For convninc w dfin nw functions f and g: N n 1 I n l jkr z z / (, ) k n z dz + Φ l z ( ) R z z / 4π (, ) f ( Φ n ) f n ( z) 0, b/ z l / j g b z ωε, / Onc Φn is dfind, th intgral can b valuatd numrically. Th rsult will still b a function of z, hnc th notation f n (z). () Choos a st of N tsting (or wighting) functions { Χ m }. Multiply both sids of th quation by ach tsting function and intgrat ovr th domain of ach function m to obtain N quations of th form m N n n 1 g Χ ( z) I f ( z) dz Χ ( z) g( z) dz, m 1,,..., N m n m m 38

40 Calculation of Antnna Impdanc (6) Intrchang th summation and intgration oprations and dfin nw impdanc and voltag quantitis N In Χ m( z) fmn( z) dz Χm( z) g( z) dz n 1 m m Z V mn m This can b cast into th form of a matrix quation and solvd using standard matrix mthods N In Z mn Vm [ Z] [ I] [ V] [ I] [ Z] 1 [ V] [ ] n1 Z is a squar impdanc matrix that dpnds only on th gomtry and matrial charactristics of th dipol. Physically, it is a masur of th intraction btwn th currnts on sgmnts m and n. [ V ] is th xcitation vctor. It dpnds on th fild in th gap and th chosn basis functions. [ I ] is th unknown currnt cofficint vctor. Aftr [ I ] has bn dtrmind, th rsulting currnt sris can b insrtd in th radiation intgral, and th far filds computd by intgration of th currnt. 39

41 Slf-Impdanc of a Wir Antnna Th mthod of momnts currnt allows calculation of th slf impdanc of th antnna by taking th ratio Z slf V g / I o RSISTANC RACTANC 40

42 Th Fourir Sris Analog to MM Th mthod of momnts is a gnral solution mthod that is widly usd in all of nginring. A Fourir sris approximation to a priodic tim function has th sam solution procss as th MM solution for currnt. Lt f (t) b th tim wavform a f ( t) o + [ an cos( ωnt) + bn sin( ωnt) ] T T n1 For simplicity, assum that thr is no DC componnt and that only cosins ar ncssary to rprsnt f (t) (tru if th wavform has th right symmtry charactristics) f ( t) a n cos( ωnt) T n 1 Th constants ar obtaind by multiplying ach sid by th tsting function cos(ω m t) and intgrating ovr a priod T / T / 0, m n f ( t)cos( ωmt) dt an cos( ωnt) cos( ωmt) dt T / T T / n 1 an, m n This is analogous to MM whn f ( t) I ( z ), an In, Φ n cos( ω n t), and Χ m cos( ω m t). (Sinc f (t) is not in an intgral quation, a scond variabl t is not rquird.) Th slction of th tsting functions to b th complx conjugats of th xpansion functions is rfrrd to as Galrkin s mthod. 41

43 Rciprocity (1) Whn two antnnas ar in clos proximity to ach othr, thr is a strong intraction btwn thm. Th radiation from on affcts th currnt distribution of th othr, which in turn modifis th currnt distribution of th first on. ANTNNA 1 (SOURC) I 1 ANTNNA (RCIVR) I V 1 + g V 1 Z 1 Z Considr two situations (dpictd on th following pag) whr th gomtrical rlationship btwn two antnnas dos not chang. 1. A voltag is applid to antnna 1 and th currnt inducd at th trminals of antnna is masurd.. Th situation is rvrsd: a voltag is applid to antnna and th currnt inducd at th trminals of antnna 1 is masurd. 4

44 Rciprocity () Cas 1: V 1 + ANTNNA 1 NRGY FLOW I ANTNNA CURRNT MTR Cas : ANTNNA 1 + V I 1 NRGY FLOW CURRNT MTR ANTNNA 43

45 Mutual Impdanc (1) Dfin th mutual impdanc or transfr impdanc as Z 1 V I 1 and Z 1 V 1 I Th first indx on Z rfrs to th rciving antnna (obsrvr) and th scond indx to th sourc antnna. Rciprocity Thorm: If th antnnas and mdium ar linar, passiv and isotropic, thn th rspons of a systm to a sourc is unchangd if th sourc and obsrvr (masurr) ar intrchangd. With rgard to mutual impdanc: This implis that Z 1 Z 1 Th rciving and transmitting pattrns of an antnna ar th sam if it is constructd of linar, passiv, and isotropic matrials and dvics. In gnral, th input impdanc of antnna numbr n in th prsnc of othr antnnas is obtaind by intgrating th total fild in th gap (gap width b n ) Z n V n I n 1 I n n b n dl 44

46 Mutual Impdanc () n is th total fild in th gap of antnna n (du to its own voltag plus th incidnt filds from all othr antnnas). V n n In 45

47 Mutual Impdanc (3) If thr ar a total of N antnnas n n1 + n + + Thrfor, Dfin Th impdanc bcoms Z n 1 I Z n 1 I n V N nm b n b n V N m1 nm nm d ln nn dl n N nm n m 1 m 1 m 1 Vn N Vnm I n Znm For xampl, th impdanc of dipol n1 is writtn xplicitly as Z 1 Z 11 + Z Z 1N Whn m n th impdanc is th slf impdanc. This is approximatly th impdanc that w hav alrady computd for an isolatd dipol using th mthod of momnts. Z nm 46

48 Mutual Impdanc (4) Th mthod of momnts can also b usd to comput th mutual coupling btwn antnnas. Th mutual impdanc is obtaind from th dfinition Z mn V m I n Im 0 mutual impdanc at port m du to a currnt in port n, with port m opn ciruitd By rciprocity this is th sam as Z mn Z nm V n. Say that w hav two dipols, on I m In 0 distant (n) and on nar (m). To dtrmin Zmn a voltag can b applid to th distant dipol and th opn-circuitd currnt computd on th nar dipol using th mthod of momnts. Th ratio of th distant dipol s voltag to th currnt inducd on th nar on givs th mutual impdanc btwn th two dipols. Plots of mutual impdanc ar shown on th following pags: 1. high mutual impdanc implis strong coupling btwn th antnnas. mutual impdanc dcrass with incrasing sparation btwn antnnas 3. mutual impdanc for wir antnnas placd nd to nd is not as strong as whn thy ar placd paralll 47

49 Mutual Impdanc of Paralll Dipols λ / d R or X (ohms) X 1 R Sparation, d (wavlngths) 48

50 Mutual Impdanc of Colinar Dipols s λ / R or X (ohms) X 1 R Sparation, s (wavlngths) 49

51 Mutual-Impdanc xampl xampl: Assum that a half wav dipol has bn tund so that it is rsonant (Xa 0). W found that a rsonant half wav dipol fd by a 50 ohm transmission lin has a VSWR of 1.46 (a rflction cofficint of ). If a scond half wav dipol is placd paralll to th first on and 0.65 wavlngth away, what is th input impdanc of th first dipol? From th plots, th mutual impdanc btwn two dipols spacd 0.65 wavlngth is Z 1 R 1 + jx j7.7ω Noting that Z 1 Z1 th total input impdanc is Th rflction cofficint is Γ Z in Z o Z in + Z o Z in Z 11 + Z ( 4.98 j7.7) 48.0 j7.7 Ω (48.0 j7.7) 50 (48.0 j7.7) + 50 j j which corrsponds to a VSWR of In this cas th prsnc of th scond dipol has improvd th match at th input trminals of th first antnna. 50

52 Broadband Antnnas (1) Th rquird frquncy bandwidth incrass with th rat of information transfr (i.., high data rats rquir wid frquncy bandwidths). Dsigning an fficint widband antnna is difficult. Th most fficint antnnas ar dsignd to oprat at a rsonant frquncy, which is inhrntly narrow band. Two approachs to oprating ovr wid frquncy bands: 1. Split th ntir band into sub-bands and us a sparat rsonant antnna in ach band Advantag: Th individual antnnas ar asy to dsign (potntially inxpnsiv) Disadvantag: Many antnnas ar rquird (may tak a lot of spac, wight, tc.) BROADBAND INPUT SIGNAL FRQUNCY MULTIPLXR 1 N f 1 f f N TOTAL SYSTM BANDWIDTH τ BANDWIDTH OF AN INDIVIDUAL ANTNNA f 1 f f N f 51

53 Broadband Antnnas (). Us a singl antnna that oprats ovr th ntir frquncy band Advantag: Lss aprtur ara rquird by a singl antnna Disadvantag: A widband antnna is mor difficult to dsign than a narrowband antnna Broadbanding of antnnas can b accomplishd by: 1. intrlacing narrowband lmnts having non-ovrlapping sub-bands (stppd band approach) xampl: multi-fd point dipol TANK CIRCUIT USING LUMPD LMNTS d min d max OPN CIRCUIT AT HIGH FRQUNCIS SHORT CIRCUIT AT LOW FRQUNCIS 5

54 Broadband Antnnas (3). dsign lmnts that hav smooth gomtrical transitions and avoid abrupt discontinuitis xampl: biconical antnna dmin spiral antnna d max FD POINT d max d min WIRS 53

55 Circular Spiral in Low Obsrvabl Fixtur 54

56 Broadband Antnnas (4) Anothr xampl of a broadband antnna is th log-priodic array. It can b classifid as a singl lmnt with a gradual gomtric transitions or as discrt lmnts that ar rsonant in sub-bands. All of th lmnts of th log priodic antnna ar fd. DIRCTION OF MAXIMUM RADIATION d min dmax Th rang of frquncis ovr which ana antnna oprats is dtrmind approximatly by th maximum and minimum antnna dimnsions d min λ H and d max λ L 55

57 Yagi-Uda Antnna A Yagi-Uda (or simply Yagi) is usd at high frquncis (HF) to obtain a dirctional azimuth pattrn. Thy ar frquntly mployd as TV/FM antnnas. A Yagi consists of a fd lmnt and at last two parasitic (non-xcitd) lmnts. Th shortr lmnts in th front ar dirctors. Th longr lmnt in th back is a rflctor. Th convntional dsign has only on rflctor, but may hav up to 10 dirctors. POLAR PLOT OF RADIATION PATTRN RFLCTOR FD LMNT DIRCTORS BACK LOB MAXIMUM DIRCTIVITY MAIN LOB MAXIMUM DIRCTIVITY Th front-to-back ratio is th ratio of th maximum dirctivity in th forward dirction to that in th back dirction: D max / D back 56

58 Ground Plans and Imags (1) In som cass th mthod of imags allows construction of an quivalnt problm that is asir to solv than th original problm. Whn a sourc is locatd ovr a PC ground plan, th ground plan can b rmovd and th ffcts of th ground plan on th filds outsid of th mdium accountd for by an imag locatd blow th surfac. h ORIGINAL PROBLM I dl RGION 1 RGION I dl PC ORIGINAL SOURCS IMAG SOURCS QUIVALNT PROBLM GROUND PLAN RMOVD Th quivalnt problm holds only for computing th filds in rgion 1. It is xact for an infinit PC ground plan, but is oftn usd for finit, imprfctly conducting ground plans (such as th arth s surfac). I dl I dl h h I dl I dl 57

59 Ground Plans and Imags () Th quivalnt problm satisfis Maxwll s quations and th sam boundary conditions as th original problm. Th uniqunss thorm of lctromagntics assurs us that th solution to th quivalnt problm is th sam as that for th original problm. Boundary conditions at th surfac of a PC: th tangntial componnt of th lctric fild is zro. ORIGINAL PROBLM h PC Idz 0 h h QUIVALNT PROBLM Idz A similar rsult can b shown if th currnt lmnt is orintd horizontal to th ground plan and th imag is rvrsd from th sourc. (A rvrsal of th imag currnt dirction implis a ngativ sign in th imag s fild rlativ to th sourc fild.) Idz 1 1 TANGNTIAL COMPONNTS CANCL 58

60 Ground Plans and Imags (3) Half of a symmtric conducting structur can b rmovd if an infinit PC is placd on th symmtry plan. This is th basis of a quartr-wav monopol antnna. Vg + I λ / 4 I λ / 4 + V g b b z 0 PC ORIGINAL DIPOL MONOPOL Th radiation pattrn is th sam for th monopol as it is for th half wav dipol abov th plan z 0 Th fild in th monopol gap is twic th fild in th gap of th dipol Sinc th voltag is th sam but th gap is half of th dipol s gap R a R monopol a ohms dipol 59

61 Crossd Dipols (1) Crossd dipols (also known as a turnstil) consists of two orthogonal dipols xcitd 90 dgrs out of phas. I x I o I y I o jπ / ji o I y y I x x z Th radiation intgral givs two trms jkη 4πr If k l << 1 thn l / l / o jkr jx k sin cos jy k sin sin Io cos cos + cos sin dx j Io dy l / / ˆ l xˆ ˆ yˆ l/ jx k sin cos dx l l/ jkηiol 4π o and similarly for th y intgral. Thrfor, jkr cos r jkr ( cos + j sin ) o cos ( cos + j sin ) r 60

62 Crossd Dipols () A similar rsult is obtaind for jkηiol 4π o jkr r jkr ( sin j cos) o ( sin j cos ) r Considr th componnts of th wav propagating toward an obsrvr on th z axis 0 : o, j o, or o jkr r ( ) which is a circularly polarizd wav. If th obsrvr is not on th z axis, th projctd lngths of th two dipols ar not qual, and thrfor th wav is lliptically polarizd. Th axial ratio (AR) is a masur of th wav s llipticity at th spcifid,: For th crossd dipols AR AR max min, xˆ + jyˆ 1 AR 1 cos ( cos +sin ) 1 cos 61

63 Crossd Dipols (3) Th rotating linar pattrn is shown. A linar rciv antnna rotats lik a propllr blad as it masurs th far fild at rang r. Th nvlop of th oscillations at any particular angl givs th axial ratio at that angl. For xampl, at 50 dgrs th AR is about 1/ db NORMALIZD FILD ROTATING LINAR POLARIZATION PATTRN ANGL, (DGRS) 6

64 Crossd Dipols (4) xampls of rotating linar pattrns on crossd dipols that ar not qual in lngth VOLTAG PLOT DCIBL PLOT 63

65 Polarization Loss (1) For linar antnnas an ffctiv hight ( h ) can b dfind h Voc inc h V oc inc Th opn circuit voltag is a maximum whn th antnna is alignd with th incidnt lctric fild vctor. Th ffctiv hight of an arbitrary antnna can b dtrmind by casting its far fild in th following form of thr factors jkr ( r,, ) [ o] [ h(, )] r Th ffctiv hight accounts for th incidnt lctric fild projctd onto th antnna lmnt. Th polarization loss factor (PLF) btwn th antnna and incidnt fild is inc h PLF, p inc h 64

66 Polarization Loss () xampl: Th Hrtzian dipol s far fild is ( r,, ) jηki 4π o jkr r [ sinˆ ] l h (, ) If w hav a scond dipol that is rotatd by an angl δ in a plan paralll to th plan containing th first dipol, w can calculat th PLF as follows. First, V oc inc h inc z ˆ h z ˆ inc h z ˆ z ˆ inc h cosδ z y inc z δ h z y Voc x TRANSMIT RCIV 65

67 66 Naval Postgraduat School Antnnas & Propagation Distanc Larning Polarization Loss (3) Th PLF is δ δ inc inc inc inc cos cos h h h h p Whn th dipols ar paralll, p1, and thr is no loss du to polarization mismatch. Howvr, whn th dipols ar at right angls, p0 and thr is a complt loss of signal. A mor gnral cas occurs whn th incidnt fild has both and componnts inc i ˆ + i ˆ ( ) ˆ ˆ ˆ ˆ i i i i h h p + + xampl: Th ffctiv hight of a RHCP antnna which radiats in th +z dirction is givn by th vctor ( ) ˆ ˆ j h h o. A LHCP fild is incidnt on this antnna (i.., th incidnt wav propagats in th z dirction): ( ) jkz o j ˆ ˆ inc

68 67 Naval Postgraduat School Antnnas & Propagation Distanc Larning Polarization Loss (4) Th PLF is ( ) ( ) 0 ˆ ˆ ˆ ˆ o o jkz o o h j j h p If a RHCP wav is incidnt on th sam antnna, again propagating along th z axis in th ngativ dirction, ( ) jkz o j ˆ ˆ inc +. Now th PLF is ( ) ( ) 1 ˆ ˆ ˆ ˆ + o o jkz o o h j j h p Finally, if a linarly polarizd plan wav is incidnt on th antnna, jkz ˆ o inc ( ) 1/ ˆ ˆ ˆ o o jkz o o h j h p If a linarly polarizd antnna is usd to rciv a circularly polarizd wav (or th rvrs situation), thr is a 3 db loss in signal.

69 Antnna Polarization Loss TRANSMIT ANTNNA RCIV ANTNNA Summary of polarization losss for polarization mismatchd antnnas RHCP 0 db > 5 db RHCP 3 db LHCP 3 db 0 db V V 3 db > 5 db H 68

70 Aircraft Blad Antnna Blad antnnas ar usd for tlmtry and communications. Thy hav narly hmisphrical covrag, allowing th aircraft to manuvr without a complt loss of signal. Th rsulting polarization is oftn calld slant bcaus it contains both horizontal and vrtical componnts. Comparison of masurd and calculatd Blad antnna pattrns for a blad installd on a cylindr λ / 4 THIN WIR h (4.01 cm) 3. 0 (7.6 cm) Mthod of momnts modl: Location of sourc xcitation 69

71 Blad Antnnas Installd on an Aircraft Th top antnna (A) provids covrag in th uppr hmisphr, whil th bottom antnna (B) covrs th lowr hmisphr. Th two antnnas can b duplxd (switchd) or thir signals combind using a couplr. Mthod of momnts patch modl: lvation pattrn: (signals combind with 3 db couplr) y Pit ch Yaw z x Ro ll A B 70

72 Small Loop Antnna (1) By symmtry, w xpct that th fild of a small wir loop locatd in th z 0 plan will dpnd only on th angl off of th wir axis,. Bcaus of th azimuthal symmtry cylindrical coordinats ar rquird to solv this problm. If th wir is vry thin (a filamnt) and has a constant currnt I oˆ flowing in it, th radiation intgral is x z r aρ ˆ jkη ( r,, ) 4πr r R y π jkr 0 I jk( r rˆ ) ˆ o a d dl Using th transformation tabls rˆ xˆ sin cos + yˆ sin sin + zˆ cos r aρˆ a rˆ r a sin sin ( xˆ cos + yˆ sin ) ( cos cos + sin ) W also nd ˆ in trms of th cartsian unit vctors (ˆ is not a constant that can b movd outsid of th intgral) ˆ xˆsin + yˆ cos 71

73 Small Loop Antnna () Th radiation intgral bcoms π jkηioa jkr ( ) ( + jkasin cos cos + sin sin ( r,, ) xˆ sin yˆ cos ) d 4πr 0 For a small loop ka << 1 and th xponntial can b rprsntd by th first two trms of a sin ( cos sin + sin cos ) Taylor s sris to gt jka 1+ jkasin ( cos cos + sin sin ) Insrting th approximation in th intgral π π ( xˆ sin + yˆ cos )[ 1+ jkasin ( cos cos + sin sin )] d 0 π Sinc sin d cos d 0 th 1 in th squar brackts can b droppd. Th 0 0 rmaining trms in th intgrand involv th following factors: sin sin sin cos cos intgrats to zro bcaus sin cos is an odd function of cos cos sin sin cos intgrats to zro bcaus sin cos is an odd function of 7

74 73 Naval Postgraduat School Antnnas & Propagation Distanc Larning Small Loop Antnna (3) Th only two trms that do not intgrat to zro ar of th form π π π 0 0 cos sin d d Thrfor, ( ) ( ) jkr o jkr o jkr o r a I k y x r a I k d y d x r jka I jka r + + η η π η π π sin 4 ˆ ˆ cos ˆ sin ˆ sin 4 cos cos ˆ sin ˆsin 4 sin ),, ( 0 0 Th radiation pattrn of th small loop is th sam as that of a short dipol alignd with th loop axis. Th radiatd powr is rad a I k P o π η and th radiation rsistanc rad 30 6 ) (10 ) / ( 6 λ π π π λ π ηπ a a a k I P R o a

75 Hlix Antnna (1) A hlix is dscribd by th following paramtrs: z D diamtr A axial lngth C circumfrnc ( π D) S cntr-to-cntr spacing btwn turns L lngth of on turn N numbr of turns α tan 1 ( S / πd) pitch angl S D d C πd α S L GROUND PLAN Convntional hlics ar constructd with air or low-dilctric cors. A hlix is capabl of oprating in svral diffrnt radiation mods and polarizations, dpnding on th combination of paramtr valus and th frquncy. 74

76 Hlix Antnna () Radiation mods of a hlix: NORMAL MOD AXIAL MOD In both cass th radiation is circularly polarizd: SMALL (LOOKS LIK A LOOP) LARG C > λ Normal mod: Axial mod: AR AR Sλ D n + 1 n In th axial mod, th bamwidth dcrass with incrasing hlix lngth, NS for BWFN 1 < α < π ( C / λ ) NS / λ 75