ECE Spring Prof. David R. Jackson ECE Dept. Notes 18

Transcription

1 ECE 634 Sprig 06 Prof. David R. Jackso ECE Dept. Notes 8

2 Scatterig by Wedge Lie source y (, φ ) I 0 φ = α Note: We will geeralize to allowig for k z at the ed. φ = π α Assume TM z : A ( φ, ) z Boudary coditios: A = 0 at φ = α,π α z

3 Scatterig by Wedge (cot.) Let Φ φ = h( φ) = Asiυ φ α + Bcosυ φ α ( ) ( ) ( ) (B.C. at φ = α) φ = π -α : (( ) ) siυ π α α = 0 so υ π α = π ( ) υ = υ = π ( π α) 3

4 Scatterig by Wedge (cot.) Az = Bυ ( k)siυ φ α ( ) Bessel fuctio of order υ Note: = 0 υ = 0( trivial solutio) Note: < 0 υ < 0 Jυ ( k) as 0 sice J ( )~ υ υ υ Γ ( υ + ) ( υ egative iteger) 4

5 Scatterig by Wedge (cot.) Hece υ = υ = π ( π α) =,, 3 π υ =, etc. ( π α) 5

6 Scatterig by Wedge (cot.) < ( ) A = a si υ φ α Jυ ( k) z = > υ > 0 For assume to match with the iterior form. > ( ) () A = b si υ φ α Hυ ( k) z = 6

7 Scatterig by Wedge (cot.) where B.C. s = Az H φ = µ E = E z z I Hφ Hφ J ( φ) δφ φ 0 = sz = ( ) (, φ ) I 0 y φ = α φ = π α 7

8 Scatterig by Wedge (cot.) Hece we have = A = A z z Az Az I0 δφ ( φ ) µ = µ 8

9 Scatterig by Wedge (cot.) y First B.C. : A = A z z (, φ ) I 0 φ = α φ = π α aj ( k ) si υ ( φ α) = bh ( k ) si υ ( φ α) υ υ = = ( ) Multiply by si υ ( φ α) m ad itegrate over φ [ α,π α] 9

10 Scatterig by Wedge (cot.) π α 0, m si υ( φ α)si υm( φ α) dφ = α ( π α), m = Note: To evaluate this itegral, use π α α π Recall: υ = ( π α) π = ( φ α) ( π α) ( π α) π si υ( φ α)si υm( φ α) dφ = si ( ) si ( m) d π 0 Hece ( ) aj ( k ) = bh ( k ) m υ m m υ m 0

11 Scatterig by Wedge (cot.) Secod B.C. : ( ( ) si ( ) ) k υ φ α bh υ k aj υ k µ Multiply by = si υ ( φ α) m ( ) ( ) I δφ ( φ ) 0 = ad itegrate over φ [ α,π α] ( ) ( k) (π α) bh m υ ( k ) aj ( ) m m υ k µ I0 = si υm( φ α)

12 Scatterig by Wedge (cot.) Solutio: µ I k π DEN ( α) ( ) ( ) ( ) υ φ α 0 a = si Hυ k ( k ) ( ) ( ) µ 0 J υ ( ) b = siυ( φ α) H υ k k ( π α) DEN Hυ k ( ) where ( ) ( ) ( ) ( ) ( ) ( ) DEN = H k J k H υ υ υ k Jυ k = j πk (Wroskia Idetity)

13 Scatterig by Wedge (cot.) Geeralizatio: To geeralize the solutio for arbitrary etire solutio by ep( jk z) z k z, we simply multiply the ad the make the substitutio k k The solutio is the valid for a lie source of the form: z I( z) = I e jk z 0 3

14 Edge Behavior 0 = ( ) A = a si υ φ α Jυ ( k) z = As, keep term, sice J ( )~ υ υ υ Γ ( υ + ) Hece ( ) υ z A ~ a si υ φ α J ( k ) e z jk z so A z υ υ = π ( π α) 4

15 Edge Behavior (cot.) Therefore we have: E E E z φ k = Az jωµε A jωµε z z υ = A jωµε φ z υ ( k 0) ( k 0) z υ = z z Note: k z = 0 correspods to a uiform lie curret, where there is o charge desity (ad hece o ormal electric field). 5

16 Edge Behavior (cot.) E 0 as 0 z y ( E, E ) -< 0 φ if υ υ < π < ( π α) (, φ ) I 0 φ = α φ = π α Hece ( E, E ) φ if ( π α) π > π α > π α < π Therefore ( E, E ) φ if α < π (cove corer) 6

17 Kife Edge Recall: E, E υ φ υ = ( π α) α = 0 υ = υ = π so E 7

18 Kife Edge (cot.) y Parallel Curret Jsz ( ) + At φ = 0, = : J = H = H sz Az = µ φ a ( ) ( ) υcosυ φ α Jυ k e µ jk z z 8

19 Kife Edge (cot.) so J sz = υ / or J sz / or J sz 9

20 Strip i Free Space y Curret o Strip J sz w From coformal mappig: J sz = I 0 / π w Mawell fuctio 0

21 Kife Edge (cot.) Perpedicular Curret y J ( ) s + φ = J = H At 0, s z Note: To have this compoet, we must use a TE z solutio (e.g., usig a magetic curret source). If we did the TE z solutio, the result would show that Js

22 Microstrip lie y Total Curret Desity o a Strip Note: The curret has both compoets, due to the fact that the mode is ot eactly TEM (due to the substrate). J J = jω s s w jk J = jω z sz s s Logitudial Trasverse The logitudial curret ad the charge desity are eve fuctios, while the trasverse curret is a odd fuctio.

23 Microstrip lie (cot.) y logitudial trasverse w Fourier-Mawell Basis Fuctio Epasio: M m m= 0 jkz z mπ Jsz ( z, ) = e a cos w w ( ) N jkz z w π Js ( z, ) = e b si = w 3

24 Microstrip lie (cot.) y logitudial trasverse w Chebyshev-Mawell Basis Fuctio Epasio: J z e a T ( + δ ) M jk z z m0 sz (, ) = m m w m= 0 w π w N jkz z s (, ) = = w j4 J z e bu w π w 4

25 Meier* Edge Coditio U E < This coditio must be satisfied at all edges. Mathematically, imposig this coditio i the solutio of a problem is ecessary to esure a uique solutio. C. J. Bouwkamp. A ote o sigularities occurrig at sharp edges i electromagetic diffractio theory, Physica (Utrecht), vol., pp Oct., 946. *J. Meier, Dle katebedigug i der theorie du beugug electromagetischer welle a vollkomme leitede ebee schirm, A. Phys., vol. 6, pp -9,

26 Meier Edge Coditio (cot.) Meier coditio: U E < y Let s verify this for the wedge: U E = ε E dv 4 = V 4 ( υ ) ε φ V d d dz V a E E φ υ υ 6

27 Meier Edge Coditio (cot.) We require that a δ ( υ ) d as δ < 0 or or a δ υ υ d < υ a δ < or υ as δ < = 0 This will be satisfied sice υ > 0 Recall: υ = π ( π α) 7