Aperture Radiation: Huygen s Equation
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- Ξενοκράτης Θεοδωρίδης
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1 perture Radiatin: Hugen s quatin = Radiating patch, assume unifrm plane wave: H, /H = 377Ω perture in X-Y plane ˆ ˆ α r(, R θ ϕ ϕ z Superpsitin f cntributins frm radiating patches H J surface current S ff radiated b integral f current elements Hugen s superpsitin integral j j ( π r, ( eff ( θφ,,r ( 1+ csθ( αˆ (, e dd R phase lag (, R1
2 Hugen s quatin: Gemetric apprimatins eff j R j ( ( ( ( ( π r θ φ + θ α (,,,R 1 cs ˆ, e - case perture in - plane r( R 0 sin ϕ ϕ z phase lag dd (, Fr ϕ, ϕ << 1: r (, R sinϕ sinϕ R ϕ ϕ Thus π -j R j e R ( θ, φ,r ( 1+ csθ αˆ (, K e π + j ( ϕ + ϕ dd R
3 Hugen s quatin: Gemetric apprimatins Thus π -j R j e R ( θ, φ,r ( 1+ csθ αˆ (, K e π + j ( ϕ + ϕ dd ( ( 1 ϕ, ϕ vm K (, ˆ ˆ K ( ( 1, vm ( ϕ, ϕ e π + j ( ϕ + ϕ dd ( ϕ + ϕ π j π e dϕdϕ R3
4 Hugen s quatin: Gemetric apprimatins ( ( 1 ϕ, ϕ vm K (, ˆ ( ( 1 ϕ, ϕ vm K (, ˆ ˆ K ( ( 1, vm ( ϕ, ϕ ( ( 1, vm ( ϕ, ϕ Let ˆ K, =, 1+ csθ e ( ϕ + ϕ π + j e dd π j ( ϕ + ϕ ϕ ϕ π e d d + jπ ( ϕ + ϕ ( ϕ + ϕ jπ π e dϕdϕ d d This is a Furier transfrm pair [ jπft + jπft ( ( ( ( ] Recall X f = t e dt ; t = X f e df R4
5 Furier Transfrm Relatins Thus perture (pulse signal (, R ( 1 Vm ~ ( [ 1] ϕ, ϕ Vm at R ( [ -1] ( [ -1] τ, τ Vm ~ ϕ, ϕ Vm S ( ϕ, ϕ = ( ϕ, ϕ η at R [ -] W m R5
6 irectivit (θ, φ f an perture ntenna Let P = radiatin intensit and P = ttal pwer radiated (W TR ( θ, φ ( ϕ ϕ 1, << ( 1+ csθ η ( R 1 η (, e π j (, P P ( θ, φ,f,r TR ( ϕ + ϕ d d 4πR d d 4πR [W m [W m - - ] ] = π ( 1+ csθ (, e π j ( ϕ + ϕ (, d d d d R6
7 irective Gain (θ,φ f an perture ntenna = π Bunds n (ϕ,ϕ, (ϕ,ϕ Recall Schwartz Inequalit ( 1+ csθ ( ( f d g f g d d Therefre: [] ( j ( e d d d d 1 d d 4π ( ϕ, ϕ = (, e π j d d d d ( ϕ + ϕ (, d d d d 4π (m is phsical area f aperture R7
8 irective Gain (θ, φ f an perture ntenna 4π ( ϕ, ϕ = d d d d 4π ( But 4π e ϕ, ϕ = e ( ϕ, ϕ (effective area η η R where radiatin efficienc η R 1. 0 R efine aperture efficienc η η e η ( ma R 0.65 in practice; = 1 fr unifrm illuminatin Therefre e = η η R R8
9 Unifrml Illuminated Circular perture ntennas φ r ϕ θ φ z = π ( 1+ csθ 0 3 db π j 1 e P = 0 ( ϕ + ϕ r dr dφ' r dr dφ' perture crdinates = r, φ Surce crdinates = ϕ, ϕ fr θ << db 8.8 db side lbes π sin θ ( π ( 1 π θφ = (f,, 1+ cs θ Λ sin θ Lambda functin ( π = = 4π at θ = 0 where Λ 1 ( q = J1(q q Bessel functin f first kind T1
10 Nn-Unifrml Illuminated Circular pertures ( ( r ssume r = 1 (r P = 0 P = 1 P = P G(θ 0 / θ P θ B1/ θ NULL #1 First Side - Lbe η Mre tpical / 1. / 17.6 db / 1.63 / 4.6 db /.93 / 30.4 db 0.56 T
11 Sidelbes and Backlbes f perture ntennas Spillver Main lbe Feed iffractin Backlbes Sidelbes Backlbes Reflectr T3
12 Waveguide Hrn Feeds Pramidal Hrn φ ( φ 0 Null at ϕ = / G(ϕ ( minant Waveguide Mde T 10 G(ϕ High Sidelbes ( 17.6 db Lwer Sidelbes ( 5 db T4
13 Scalar Feed perture ( Yields ver lw sidelbes Side view /4 minimizes return ech /4 pen circuit at wall /4 grves cut int wall T5
14 amples f Parablic Reflectr ntennas Fcus N aperture blckage Circularl Smmetric Parablic Reflectr Off-is Parablid Lateral scan via phased arra line feed Clindrical Parabla T6
15 Spherical Reflectr ntennas Fcus at R/ R/ Center f curvature Fcal plane Variable-pitch linear phased arra, psitined at line fcus Feed Supprt Beam Line Feed Illuminated Prtin e.g. riceb (1000 =, 600 is illuminated T7
16 Tridal Parablic Reflectr ntenna irectin f View Spinning Feeds is f Revlutin z r(z ω Spin ntenna Feeds Tridal Reflectr Feed Surface (N perture Blckage z Tridal Reflectr Tridal Reflectr dvantage: man rapidl scanning spinning feeds T8
17 Multifeed rras θ Parabla C B Fcus G B (θ G (θ z G C (θ Fcal length = f Fr f = 0.5, n 3-5 beams (sa ~1 db gain lss with useable G and sidelbes η e.g. ( f in - directin if f = 7, n ( Can d much better with gd lens sstems U1
18 Scalar Feed perture ( Yields ver lw sidelbes Side view /4 minimizes return ech /4 pen circuit at wall /4 grves cut int wall U
19 Multiple-Hrn Feeds Parablic Reflectr Thus Fcal Plane ( djacent Feeds B Crss-ver Pint belw 3 db, Far-Field B Furier Transfrm Crss-ver 6 10 db fr Lw Sidelbes Scalar Feeds U3
20 Multiple-Hrn Feeds Patterns Three-rra Slutin B C C B C B C B Feeds Pr Cverage Feed is assembl f ecited adjacent feeds U4
21 Near-Field ntenna Cupling Near-field f aperture >> near field f Hertzian diple (r << /π r Unifrm Phase Frnt r r z Spherical Phase Frnt "Far field" r ~ > U5
22 Near-Field ntenna Cupling Cnsider near-field link: Sa: unifrml illuminated apertures /3 P T1 (vm -1 P T PT 1 = η watts r << PT P T P 1 r = =, P r =? η η PT P r 1 Claim Pr = (reciprcit i.e. = = 1 3 P 3 T1 = 3 Pr1 PT U6
23 Near-Field ntenna Cupling, Mde Orthgnalit ( ( received Illuminate nl half = η [ W] Half gets in ( η Half reflected, rthgnal t dminant (cupled mde Onl half the pwer is accepted here! Waves are nt a sum f independent bullets; the have phase, mdal structure (classic wave/particle issue. U7
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