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Analysis of Spherical and Conical Resonators in Electromagnetics

Explore the properties and modes of spherical and conical resonators in electromagnetics, with detailed notes and numerical methods. Understand the oscillations and field patterns governed by different mode indices, providing insights into waveguide feeding and mode approximations.

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Analysis of Spherical and Conical Resonators in Electromagnetics

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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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