VIII. Further Aspects of Edge Diffraction

Transcription

1 VIII. Futhe Aspects f Edge Diffactin Othe Diffactin Cefficients Oblique Incidence Spheical Wave Diffactin by an Edge Path Gain Diffactin by Tw Edges Numeical Examples Septembe 3 3 by H.L. Betni

2 Othe Diffactin Cefficients Felsen s Rigus Slutin f Absbing Sceen ( Γ = ) D = + πk θ π θ Cnducting Sceen D = ΓEH, + πk φ φ φ+ φ cs cs Reflected plane RSB wave φ φ θ ISB Use : Γ = f E paallel t edge Γ E H = f H paallel t edge Incident plane wave Septembe 3 3 by H.L. Betni

3 Cmpaisn f Diffactin Cefficients 5 4 θ RSB ISB D π k 3 D 4 D 3 Edge D D RSB θ ISB angle, θ 9 wedge D : Kichhff -Huygens D : Felsen D 3 : Cnduct f TE plaizatin D 4 : 9 cnducting wedge f TM plaizatin Septembe 3 3 by H.L. Betni 3

4 Diffactin f Oblique Incidence π/ Ψ y Diffacted ays lie n a cne whse angle is the same as that between incident ay and edge. z π/ Ψ All waves have wave numbe k sinψ alng edge k csψ in (x,y) plane x Replace k f nmal incidence by k csψ E D = E e jπ / 4 jk cs ψ jk sin ψz e e D θ Septembe 3 3 by H.L. Betni 4 csψ whee is the pependicula distance t the edge

5 Diffactin f an Incident Spheical Wave (f paths that ae nealy pependicula t the edge) diple θ < θ da da In the hizntal plane, ays spead as if they came fm a pint behind the edge. Nea the edge E D ( λ<< << ) diffacted field is nealy a cylindical wave π = ZI e jk jk j / 4 e e f ( θ) D( θ) Field incident n the edge Diffacted cylindical wave Septembe 3 3 by H.L. Betni 5

6 Tp and Side Views f the Diffacted Rays Diple Diple W() W() L() L() Tp View Side View + Since W() = W and L () = L Then + da W L W L da + = () () = = Septembe 3 3 by H.L. Betni 6

7 Diffacted Field Amplitude Must Cnseve Pwe in a Ray Tube diple θ θ< da da Field amplitude pptinal t da / da da da = whee + T cnseve pwe alng a ay tube, E () da= E da E () = E Accunting f phase change alng the ay E E D D D D D D jk da da e ZIf j / 4 jk ( ) ( + ) π e e θ D θ + ()= = Septembe 3 3 by H.L. Betni 7 da da

8 Path Gain f Diffacted Field F istpic antennas D θ λ PR = ED Ae = ZI η η ( + ) 4π But fm the desciptin f antenna adiatin P T 4π = ZI η Theefe PG P R = = P λ 4π T D ( θ) ( + ) Septembe 3 3 by H.L. Betni 8

9 UTD Diffactin f Pependicula Incidence f Rays Fm a Pint Suce Diffactin cefficient whee D GO ( θ) and D( θ) = D ( θ) F( S) plane wave diffactin cefficient fm gemetic ptics js FS tansitin funcitn FS = j Se exp( ju) du F spheical waves fm a pint suce GO S k = sin θ + S Septembe 3 3 by H.L. Betni 9

10 Example f Path Gain f Diffacted Field Using Felsen diffactin cefficient m m 7.3 m θ = 3 f = 9 MHz, λ =/3 m, k =6π m - m π θ = = π adians 8 3 / 6 ( θ)= D + =. 6 πk θ π θ λ Since = = PG P R P 4π D( θ) ( + ) = 3 / (. 6) PG = 46. 4π 4 PLdB = lgpg = db T 9 F antennas 4 m appat in fee space PL db = db Septembe 3 3 by H.L. Betni

11 Diffactin f Pint Suce Rays Incident Oblique t the Edge z 9 ο ψ θ da da diple 9 ο ψ csψ da da csψ = + Septembe 3 3 by H.L. Betni

12 Diffactin f Pint Suce Rays Incident Oblique t the Edge - cnt. Nea the edge λ<< << diffacted field is nealy a cylindical wave E D ψ π θ = ZI e jk j( k cs ) j / 4 e e D f ( θ) csψ Field incident n the edge Diffacted cylindical wave T cnseve pwe alng a ay tube: ED() = ED Accunting f phase change alng the ay E ZIf D ()= ( θ ) e e jk + jπ / 4 csψ + D θ csψ da da Septembe 3 3 by H.L. Betni

13 Path Gain f Paths Oblique t the Edge F istpic antennas Since the pwe adiated by the tansmitte is the path gain is PR = ED Ae = ZI η η D θ ( + ) (cs ψ) PG P D R = = PT λ θ (cs ψ) 4π ( + ) Nte that the angel θ, ( φ', φ), used t cmpte D ae thse seen in the pjectin f suce and eceive pints int the plane pepedicula t the edge. P T λ 4π = ( 4π η) ZI Septembe 3 3 by H.L. Betni 3

14 UTD Diffactin f Oblique Incidence f Rays Fm a Pint Suce Diffactin cefficient whee D GO ( θ) and D( θ) = D ( θ) F( S) plane wave diffactin cefficient fm gemetic ptics js FS tansitin funcitn FS = j Se exp( ju) du F blique incidence f spheical waves fm a pint suce S k = sin θ + GO cs ψ S Septembe 3 3 by H.L. Betni 4

15 Example f Diffactin n Oblique Paths Cdless telephnes ve a bick wall-pespective view -7 φ y x = 4 + = 43. = ( 5 ) = z w z w Tx Lcated at (-7,-.5,) ' = = tan = ' zw = ( 5 zw) Slving gives z = = 9 - ψ = z = 9. w w Septembe 3 3 by H.L. Betni 5 = 9 - ψ 5 Rx Lcated at (4,-,5) z

16 Example f Diffactin n Oblique Paths Cdless telephnes ve a bick wall-end view y -7 φ 4 x ' Tx φ Rx (-7,-.5) (4,-) Band λ S F(S) 45 MHz / MHz / GHz /8 8.4 φ' = tan ( 75. )= 36. ad φ = π tan ( 4 )= ad θ = π ( φ φ') =. 455 ad ψ = 9 Tansitin functin : tan ( ' z )= S= kcs ψ sin ( θ ) = + λ Fs () = πs f( S π)+ jg( S π) u f( u) u+ 3. 4u gu u u u w [ ] 3 Septembe 3 3 by H.L. Betni 6

17 Diffactin n Oblique Paths - cnt. Cdless telephnes ve a bick wall Path Gain PG = λ 4π ( θ) F( S) cs ψ ( + ) =. 43 λ DGO( θ) F( S) Using the Felsen diffactin cefficient Thus D GO ( θ) DGO = + πk θ π θ PG =. λ F( S) λ π = Band 45 MHz 9 MHz.4 GHz λ DGO( θ) F( S) 4π cs ( )(. 9)( 8. 76) = + =. 6λ. 455 π Septembe 3 3 by H.L. Betni 7 λ /3 /3 /8 S F(S).799 PG.87x x -8.37x -9 L db

18 Diffactin by Successive, Paallel Edges --Tp and Side Views-- Diple W() W() L() L() Tp View Side View + + Since W() = W and L () = L + Then da W L W L da = () () = = + + Septembe 3 3 by H.L. Betni 8

19 Diffactin f Vetical Diple Fields by Successive, Paallel Edges θ θ θ Assume the secnd edge is nt nea the shadw bunday f the fist edge. jk jπ /4 E E e e in D θ ()= e jk E () cylindical wave nea edge da/ da jk( ) j / + π 4 e e jk jπ / = ZIf ( θ) D( θ) e e D( θ) ( + ) = ZIf ( θ ) e π / e D ( θ ) + + D ( θ) jk + + j + ( + + ) 4 Septembe 3 3 by H.L. Betni 9

20 Path Gain f Diffactin at Paallel Edges Assuming istpic antennas PG = λ 4π D( θ) D( θ) ( + + ) Example : D( θ )= D( θ)= PG = / (. ) 4π 6 PL = lgpg = 7dB F tw antennas m apat in fee space PL db db = 75. 8dB =. 3 m m θ = 3 m θ = 3 m f = 9 MHz λ =/3 m k =6π m m 6 m 7.3 m Septembe 3 3 by H.L. Betni

21 Walk Abut Tansmissin Ove a Building θ = tan - (/5) =.7 ad m. m m m f = 45 MHz λ =/3 m k =3π m - 5 m m 5 m Using Felsen diffactin cefficient D( θ)= + πk θ π θ = + π π + = λ D θ D θ PG = = (. 998) = 6. > 6. 4π + + 4π (. ) ( )( 4. 4) 4 Septembe 3 3 by H.L. Betni

22 Diffactin f Diple Fields by Successive Pependicula Edges θ θ θ jπ / 4 jk E E e e in D θ ()= + e jk ( ) cylindical wave nea edge da/ da jπ / 4 jk( + ) e e Ein = ZIf ( θ) D( θ) ( + ) E ()= ZIf ( θ ) jπ / jk + + e e ( + )( D θ D θ + ) Septembe 3 3 by H.L. Betni

23 Path Gain f Pependicula Edges PG = λ 4π D( θ ) D( θ) ( + )( + ) f = 9 MHz λ =/3 m k =6π m - m -3 m m 6 m 3 m m D( θ ) = D( θ) =. 6 PG PL db = / (. ) 4π 8 8 = lg PG = 7. db = 89. Septembe 3 3 by H.L. Betni 3 3