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Prof. David R. Jackson ECE Dept.

ECE 6341 . Spring 2014. Prof. David R. Jackson ECE Dept. Notes 18. Scattering by Wedge. y . Line source. x . Note: We will generalize to allowing for k z at the end. Assume TM z :. Boundary conditions:. Scattering by Wedge (cont.). Let. (B.C. at  =  ).  = 2  -  :. so.

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Prof. David R. Jackson ECE Dept.

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  1. ECE 6341 Spring 2014 Prof. David R. Jackson ECE Dept. Notes 18

  2. Scattering by Wedge y Line source x Note: We will generalize to allowing for kz at the end. Assume TMz: Boundary conditions:

  3. Scattering by Wedge (cont.) Let (B.C. at =) =2 -: so

  4. Scattering by Wedge (cont.) Bessel function of order n Note: Note: since

  5. Scattering by Wedge (cont.) Hence

  6. Scattering by Wedge (cont.) For assume to match with the interior form.

  7. Scattering by Wedge (cont.) y x B.C.’s

  8. Scattering by Wedge (cont.) Hence we have

  9. Scattering by Wedge (cont.) First B.C.: Multiply by and integrate over

  10. Scattering by Wedge (cont.) Note: To evaluate this integral, use Hence

  11. Scattering by Wedge (cont.) Second B.C.: Multiply by and integrate over

  12. Scattering by Wedge (cont.) Solution: where (Wronskian Identity)

  13. Scattering by Wedge (cont.) Generalization: To generalize the solution for arbitrary , we simply multiply the entire solution by and then make the substitution The solution is then valid for a line source of the form:

  14. Edge Behavior As , keep term, since Hence so

  15. Edge Behavior (cont.) Therefore we have: Note: kz= 0 corresponds to a uniform line current, where there is no charge density (and hence no normal electric field).

  16. Edge Behavior (cont.) as Hence if (convex corner)

  17. Knife Edge Recall:

  18. Knife Edge (cont.) y Parallel Current x At

  19. Knife Edge (cont.) so or or

  20. Strip in Free Space y Current on Strip x w “Maxwell function”

  21. Knife Edge (cont.) Perpendicular Current y x At Note: To have this component, we must use a TEz solution (e.g., using a magnetic current source). If we did the TEz solution, the result would show that

  22. Microstrip line y Longitudinal Transverse x w Total Current Density on Strip Note: The current has both components, due to the fact that the mode is not exactly TEM (due to the substrate). The longitudinal current and the charge density are even functions, while the transversecurrent is an oddfunction.

  23. Microstrip line (cont.) y longitudinal transverse x w Fourier-Maxwell Basis Function Expansion:

  24. Microstrip line (cont.) y longitudinal transverse x w Chebyshev-Maxwell Basis Function Expansion:

  25. Meixner* Edge Condition This condition must be satisfied at all edges. Mathematically, imposing this condition in the solution of a problem is necessary to ensure a unique solution. C. J. Bouwkamp. “A note on singularities occurring at sharp edges in electromagnetic diffraction theory,” Physica(Utrecht), vol. 12, pp. 467-474. Oct., 1946. *J. Meixner, “Dle kantenbedingung in der theorie du beugung electromagnetischer wellen an vollkommen leitenden ebenen schirm,” Ann. Phys., vol. 6, pp 1-9, 1949.

  26. Meixner Edge Condition (cont.) V a Meixner condition: Let’s verify this for the wedge: y x

  27. Meixner Edge Condition (cont.) We require that or or or This will be satisfied since

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