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Prof. David R. Jackson ECE Dept.

ECE 6341. Spring 2016. Prof. David R. Jackson ECE Dept. Notes 22. Summary of Potentials. In general,. Spherical Resonator. z. PEC (hollow). y. x. TM r modes. Spherical Resonator (cont.). Spherical Resonator (cont.). Hence we choose. Therefore. Denote. Spherical Resonator (cont.).

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Prof. David R. Jackson ECE Dept.

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  1. ECE 6341 Spring 2016 Prof. David R. Jackson ECE Dept. Notes 22

  2. Summary of Potentials In general,

  3. Spherical Resonator z PEC (hollow) y x TMrmodes

  4. Spherical Resonator (cont.)

  5. Spherical Resonator (cont.) Hence we choose Therefore Denote

  6. Spherical Resonator (cont.)

  7. Spherical Resonator (cont.) Then so Note: is independent of For a non-trivial solution: m n

  8. Spherical Resonator (cont.)

  9. Spherical Resonator (cont.) TMmnp mode: Index m: controls oscillations in  Index n: controls oscillations in  Index p: controls oscillations in r

  10. Spherical Resonator (cont.) Note on n=0: If n = 0, we must also have m = 0 (m n). The TM00p mode has a trivialfield. Proof:

  11. Spherical Resonator (cont.) TEr: So choose Denote

  12. Spherical Resonator (cont.)

  13. Spherical Resonator (cont.)

  14. Spherical Resonator (cont.) Then so Note: since Here we are using the notation from the cylindrical chapter.

  15. Spherical Resonator (cont.) Mode ordering (by cutoff frequency) Notation: (m, n, p)

  16. Hollow Conical PEC Horn z Region of interest y x TMr Note:

  17. Conical Horn (cont.) so Hence Note: The integer index m is arbitrary. (This is a transcendental equation for .)

  18. Conical Horn (cont.) Plot of function: (The superscript “C” stands for “cone.”)

  19. Conical Horn (cont.) z y x Note:An upside-down horn would use Alternatively, Region of interest

  20. Conical Horn (cont.) TEr (The superscript “C” stands for “cone.”)

  21. Conical Horn (cont.) Feeding a conical horn from the TE11 mode of circular waveguide This can be approximately modeled as the TE11 mode of the spherical conical horn. Circular waveguide (TE11 mode) z

  22. Wedge z Source Region of interest: x Notes:

  23. Wedge (cont.) TMr: so

  24. Wedge (cont.) TEr: so

  25. Spherical Horn z y x The bottom of the horn is on the z= 0 plane.

  26. Spherical Horn (cont.) TMr: so

  27. Spherical Horn (cont.) (must be found numerically)

  28. Spherical Horn (cont.) Plot of function: (The superscript “H” stands for “horn.”)

  29. Spherical Horn (cont.) z y x TMmp mode:

  30. Spherical Horn (cont.) TEr: so

  31. Spherical Horn (cont.) (must be found numerically)

  32. Spherical Horn (cont.) Plot of function: (The superscript “H” stands for “horn.”)

  33. Spherical Horn (cont.) z y x TEmpmode:

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