ECE Spring Prof. David R. Jackson ECE Dept. Notes 20

Transcription

1 ECE 6345 Spring 5 Prof. David R. Jacson ECE Dep. Noes

2 Overview In his se of noes we apply he SDI mehod o invesigae he fields produced by a pach curren. We calculae he field due o a recangular pach on op of a subsrae. We examine he pole and branch poin singulariies in he complex plane. We examine he pah of inegraion in he complex plane.

3 y Pach Fields W x J sx π x xˆ cos L L Find E x (x,y,) z J sx ε r L h x 3

4 From Noes 9 we have ( π ) Pach Fields (con.) + + jx ( x + y y ) E G z J e d d (,, ) (, ) x xx x y sx x y x y where ~ G xx x V i ( z) y V TE i ( z) For he pach curren, we have: L cos x π W J sx ( x, y ) LW sinc y π L x 4

5 Pach Fields (con.) From he TEN we have V T i T () () Zin () T Yin () Y jy co h ( ) T T z T denoes or TE z / / z h T Z V / T T Z ( ) + - x y Amp ( z or TE z ) z Y Y Y Y TE TE ωε z ωε z z ωµ z ωµ 5

6 Pach Fields (con.) Define he denominaor erm as T T D ( ) Y () in so ha D ( ) Y jy co h ( ) z D ( ) Y jy co h ( ) TE TE TE z 6

7 Pach Fields (con.) We hen have ~ G xx + x y TE D D The final form of he elecric field a he inerface is hen E ( xy,,) + y x ( ) + π D D jx ( x + yy) J e d d x TE (, ) sx x y x y 7

8 Polar Coordinaes Use he following change of variables: y d d x y d dφ ( is also ofen called ρ ) φ x + π + ( ) jx ( x + y y ) jx ( x + y y ) e d d e d dφ ( ) x y π / + 4 cos cos ( ) ( x) ( ) yddφ x y cosφ sinφ y x Advanage: The poles and branch poins are locaed a a fixed posiion in he complex plane. z / / z 8

9 Hence Polar Coordinaes (con.) + y x jx ( x + y y ) E ( x, y, ) J (, TE ) e d d ( π ) + D D x sx x y x y π / ( ) ( ) Ex xy,, J, cos sin sx φ φ φ TE π + D ( ) ( ) D This is in he following general form: cos ( x) cos( ) yddφ x y π / (, φ) E F d dφ x 9

10 Poles Poles occur when eiher of he following condiions are saisfied: D ( ) ( ) D TE ( ) ( TE ) p p z : D ( ) Y jy co h z

11 + V ( ) V ( ) Y + I ( ) I ( ) Y Hence Y Y Poles (con.) This coincides wih he well-nown Transverse Resonance Equaion (TRE) for deermining he characerisic equaion of a guided mode. (e.g, SW mode) V I + ( ) V ( ) + ( ) I( ) Y Y (Kirchhoff s laws) so ha ( ) jy co h Y z SW: β z β z / / Z z Z I( z ) + V( z) -

12 Comparison: Poles (con.) Poles in plane ( ) Y jy co h z TRE (surface-wave mode) ( ) Y jy co h z Y Y ωε z ωε z Y Y ωε z ωε z z / p z / p β z β z / / (A similar comparison holds for he TE case)

13 Poles (con.) Hence, we have he conclusion ha p TE p β β TE Tha is, he poles are locaed a he wavenumbers of he guided modes (he surface-wave modes). Noe: In mos pracical cases, here is only a surface-wave mode. 3

14 Poles (con.) The complex plane hus has poles on he real axis a he wavenumbers of he surface waves. Im + x y β β Re 4

15 Pah of Inegraion The pah avoids he poles by going above hem. Lossy case Im β C Re Lossy case Im This pah can be used for numerical compuaion. h R L R β C Re 5

16 Pah of Inegraion (con.) The pah avoids he poles by going above hem. Lossless case Im h R L R C Re h. 5 R L. R (ypical choices) Pracical noe: If h R is oo small, we are oo close o he pole. If h R is oo large, here is oo much round-off error due o exponenial growh in he sin and cos funcions. 6

17 Branch Poins To explain why we have branch poins, consider he funcion: wih ( ) co ( ) D Y jy h z ωε ωε j co z z ( h) z z z ( ) ( ) / / Noe: There are no branch cus for z (he funcion D is an even funcion of z ). If z z D ( ) changes (We need branch cus for z.) 7

18 Branch Poins (con.) z ( ) / ( ) ( ) / / + ( ) ( ) / / j + ( ) / ( ) ( ) j / Noe: The represenaion of he square roo of as j is arbirary here. Im ( ) Re 8

19 Branch Poins (con.) ( ) / ( ) ( ) j z / jφ jφ ( ) j e e / / Branch cus are necessary o preven he angles from changing by π : Noe: The shape of he branch cus is arbirary, bu verical cus are shown here. Im φ φ Re Noe: The branch cus should no cross he real axis when here is loss in he air (he inegrand mus be coninuous). 9

20 Branch Poins (con.) We obain he correc signs for z if we choose he following branches: ( ) ( ) π /< Arg < 3 π / ( ) 3 π /< Arg < π / z j( + ) Im z jφ jφ ( ) j e e z ( ) / / + j( + ) z The wave is hen eiher decaying or ougoing in he air region when we are on he real axis. Re

21 Branch Poins (con.) The wavenumber z is hen uniquely defined everywhere in he complex plane: z jφ jφ ( ) j e e / / Example φ.9π φ 5 π /4 Im φ π /4 φ.π π /< φ < 3 π / 3 π / < φ < π / φ Arg ( ) ( ) ( ) φ Arg Re φ.π φ 3 π /4 φ π /4 φ.π

22 Riemann Surface The Riemann surface is a pair of complex planes, conneced by ramps (where he branch cus used o be). The angles (and hence he funcion) change coninuously over he surface. All possible values of he funcion are found on he surface.

23 Riemann Surface (con.) Riemann surface for z / Top shee π < φ < π Boom shee y r φ z j re φ x π < φ < 3π Top view MATLAB : π < φ π Noe: A horizonal branch cu has been arbirarily chosen. A ramp now exiss where he branch cu used o be. 3

24 3D view Riemann Surface (con.) Riemann surface for z / x Top y B D Boom D B y B D D Side view B Top view x 4

25 Riemann Surface The Riemann surface can be consruced for he wavenumber funcion. / ( ) ( ) ( ) / / ( ) j z Example: Im We go couner-clocwise around he branch poin a saring on he op shee on he real axis, and end up bac where we sared. Top shee o 3 o 45 Re φ π /4 φ π /6 Boom shee φ π /4+ π φ π /6 5

26 Sommerfeld Branch Cus Sommerfeld branch cus are a convenien choice for heoreical purposes (discussed more in ECE 634): Im ( ) on branch cu z Im z ( ) / Im ( ) < z (everywhere in complex plane) Re Assume : z j( + ) 6

27 Sommerfeld Branch Cus (con.) Im z ( ) / Im ( ) < z (everywhere in complex plane) Re z is real here, for a lossless air. < < Pracical noe: If we give he air a small amoun of loss, we can simply chec o mae sure ha Im( z ) <. Noe: The branch poins move off of he axes for a lossy air. 7

28 Sommerfeld Branch Cus (con.) The Riemann surface wih Sommerfeld branch cus: Im Im ( ) < z Im ( ) > z (op shee) (boom shee) Re Noe: Surface wave poles mus lie on he op shee, and leay-wave poles mus lie on he boom shee. 8

29 Pah of Inegraion where Im π / ( φ ) Ex I dφ ( φ) (, ) φ I F d Top shee Im ( ) < z h R L R β C Re 9