Chapter 4 : Linear Wire Antenna

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Ιολανθη Γιάγκος
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1 Chapt 4 : Lina Wi Antnna nfinitsima Dipo Sma Dipo Finit Lngth Dipo Haf-Wavngth Dipo Lina mnts na o on nfinit Pfct Conductos

2 nfinitsima Dipo Lngth << Usd to psnt capacito-pat top-hatoadd antnnas Capacitiv oading to maintain th unifom cunt

3 Radiatd Fid Th cunt on th infinitsima dipo is assumd to b constant, i.., ' ˆ Rca that jkr µ A x, y, J x', y', ' d' 4 C R Sinc th dipo is infinitsima, th foowing appoximations hod: d ' d' x' y' ' R ' constant

4 4 Radiatd Fid jk jk d y x µ µ 4 ˆ ' 4 ˆ,, / / A ; ; A A A A A, ˆ ˆ ˆ A A A A A A A Thus Sinc A has ony componnt, and µ A A ˆ H Th magntic fid bcoms:

5 5 Radiatd Fid jk jk k j H H H 4 4 k jk k j jk jk jk Thfo, Likwis, th ctic fid can b found to b

6 6 Pow Dnsity and Radiation Rsistanc Poynting vcto ˆ ˆ ˆ ˆ ˆ * * * * H H H H W 6 8 k k j W k j W ˆ ˆ ˆ k j d d W d d W W d P S s W Outgoing Pow Componnts of Poynting Vcto

7 7 Pow Dnsity and Radiation Rsistanc ~ ~ k W W m ω ~ ~ m ad W W j P P ω diction A in adia activ pow; tim - avag imaginay ~ ~ tim - avag ctic ngy ~ tim - avag magntic ngy ~ tim - avag adiatd pow pow m m ad W W W W P P ω 8 R Radiation Rsistanc Compx Pow ad R P Ractiv Pow

8 Fa Fid k >> Fo k >>, th fids can b appoximatd as H k j 4 k j 4 H H jk k jk >> TM Wav 8 Z w Ratio of and H: wav impdanc Z w H intinsic impdanc Ω fo f -spac

9 Tim-avag pow dnsity: Wav R H Radiation intnsity: Dictivity k U Wav * k ˆ ˆ 4 Maximum adiation intnsity: Maximum dictivity: 4,, k U max 4 D U max 4 P ad Maximum ffctiv aa: A m D 4 8 9

10 Sma Dipo Lngth /5 < < /

11 Radiatd Fid Th cunt on th sma dipo is assumd to b a tiangua function, i.., ' / ', ˆ / ' ', ˆ ' / / ' ' ' ' 4 ˆ,, jkr jkr d R d R y x µ A wh is a constant. Vcto potntia bcoms: Appoximating R ~ yids th maximum phas o k/ / fo /.

12 Radiatd Fid 8 8 >> k H H k j H k j jk jk jk A y x µ 4 ˆ ˆ,, A Ug R~: Th fa-fid can b givn by which is on-haf of that fo th infinitsima dipo. P R ad Radiation sistanc:

13 Fid Spaation Fo a vy thin dipo, x y, thus ' ' ' ' ' ' ' ' y x y x y y x x R ; wh y x

14 4

15 Fid Spaation 5 Rca aso th Tayo xpansion: f f '' x f f ' x x! which yids th sam sut: Rca that ' Lt x ' / x x x 8, thn L x R ' ' ' ' R ' L L 6

16 Fa Fid 6 By taining ony th fist two tms, i.., R ' Th most significant ngctd tm has th maximum vau ' ' whn max A maximum tota phas o of /8 is accptab, thus ' k 8 Fa-fid appoximation whn D / ' / R fo ampitud tm R ' ˆ ' fo phas tm

17 Radiating Na Fid By taining ony th fist th tms, i.., ' R ' Th most significant ngctd tm is th fouth tm. n od to find its maximum vau, on can diffntiat th fouth tm with spct to, and th sut is st to, i.., yids ' ' [ ] [ ] tan ± 7

18 8 Radiating Na Fid 8 8 ' tan / ' k.6 which ducs to o f th maximum tota phas o is aowd to b /8, Na-fid gion.6 D D

19 Lngth > / Finit Lngth Dipo Cunt on th finit ngth dipo assuming th wi is vy thin ˆ ' ˆ k k ', ', ' / / ' wh is a constant. This distibution assums that th antnna is cnt-fd and th cunt vanishs at th nd points. 9

20 d dh d Radiatd Fid Th ctic and magntic fid componnts in th fa fid fo th infinitsima dipo d a givn by k ' j 4R k ' j 4R d dh dh jkr jkr d' d' Ug th fa-fid appoximation yids d k j ' 4 jk jk' d'

21 Radiatd Fid Th tota ctic fid can b obtaind by summing up contibutions fom a infinitsima dipos, i.., / / jk k / jk' j ' ' d / d mnt facto spac facto tota fid mnt facto spac facto Thus, th ctic fid of th finit ngth dipo can b givn by jk k jk' j k ' / / 4 k ' jk' d' d'

22 Radiatd Fid k k j jk ] [ γ β β γ β α β α γ β α α x x dx x x x Ug Likwis, th magntic fid can b givn by yids k k j H jk

23 Cunt Distibutions and Radiation Pattn

24 4 Radiation Pattn fo.5

25 5 Pow Dnsity and Radiation ntnsity * * 8 ˆ ˆ ˆ ˆ R R k k H av H W Tim-avag pow dnsity: Radiation intnsity: 8 k k W U av

26 6 P ad Ω 4 Radiatd Pow Radiatd pow can b obtaind by UdΩ which can b givn by Pad C n k C 4 wh k[ C i k n k / C U dd k k i k[ S k C i k S i k] C u's constant i d k]

27 Radiatd Pow C i S x i x x x y y dy y dy y x y 5 y dy Co intga Sin intga C i x is atd to C in x by 4 C in x wh C C n x C in x x i x y y dy C i x S i x 4 C in x

28 R Radiation Rsistanc and Radiation sistanc bcoms P ad nput Rsistanc C n k C k k[ S k[ C n k / Ci k C Sinc in Rin R assuming oss-ss nput sistanc can b givn by Rin R k in Fo a dipo of ngth, nput Rsistanc R in in R k i i i k S k] i k] 8

29 9 Dictivity Q F P U D ad max max 4 ] / n [ ] [ n k C k C k C k k S k S k k C k C Q i i i i i Dictivity is givn by and wh k k F

30 Radiation sistanc, input sistanc and dictivity

31 Haf-wavngth Dipo j jk j H jk 8 8 W av Lt /, thn 8 8 W U av Pow dnsity Radiation ntnsity

32 Radiatd Pow P wh Haf-wavngth Dipo ad C Dictivity 4 4 in Radiation Rsistanc nput mpdanc d y y dy C 8 C n C D R U Z in 7 j4.5 i 4.45 max 4 P ad Pad C in 4 in

33 Wi antnnas na o on infinit pfct conducto

34 mag Thoy tan nˆ on PC H tan nˆ H on PMC 4 NOT: Th fids obtaind a vaid ony in th top haf-pan.

35 5 Vtica ctic Dipo 4 k j jk d 4 4 k j k jr jk jk v Dict Componnt Rfctd Componnt

36 Vtica ctic Dipo n fa fid: Paa ay appoximation [ h h ] h [ h h ] h Phas tm Ampitud tm 6

37 Vtica ctic Dipo Tota ctic Fid k j 4 jk [ ] < h numb of obs 7

38 Pow Dnsity Radiation ntnsity Vtica ctic Dipo 4 * W R ˆ av H Maximum Radiation ntnsity Radiatd Pow U P ad / ˆ U max U dd / U d 8

39 9 Vtica ctic Dipo 5 max 4 P U D ad P R ad Dictivity Radiation Rsistanc

40 /4 Monopo Z in Zindipo 7 j j.5 4

41 4 Fids du to y-dictd dipo Vcto Potntia A A A A y µ 4 ˆ jk jk µ Ay 4 jk µ Ay 4 j jωa A ωµε Rca that Fa-fid ctic Fid jωa jωa µ jω 4 µ jω 4 jk jk

42 Fids du to y-dictd dipo ntoduc a nw sphica coodinat systm,ψ,ζ such that ˆ ˆ ψ ζ ψˆ ˆ ψ ζ ˆ ζ ˆ ψ xˆ ψ xˆ ψ ζ yˆ ψ xˆ ψ ζ yˆ ψ This can b obtaind by tting x,y,->,x,y and ψ,ζ->, thn Fa-fid ctic Fid Fa-fid Magntic Fid H ψ ζ µ jω 4 ωµ j 4 jk jk ψ ψ 4

43 4 Hoionta ctic Dipo ψ ψ 4 k j jk d ψ ψ ψ 4 4 k j k jr jk jk h Dict Componnt Rfctd Componnt

44 Hoionta ctic Dipo n fa fid: [ h h ] h [ h h ] h Ampitud tm ψ yˆ ˆ Phas tm 44

45 Hoionta ctic Dipo Tota ctic Fid ψ jk k j [ j ] 4 numb of obs h 45

46 46 Hoionta ctic Dipo 4 ˆ av W U / / d U d d U P ad Pow Dnsity Radiation ntnsity Radiatd Pow P R ad Radiation Rsistanc

47 47 Hoionta ctic Dipo 5 > / 4 / 4 R R D h h R Dictivity Fo sma Maximum Radiation ntnsity > and, /, / max max U

48 Hoionta ctic Dipo 6 wh R Fo sma 8 D

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