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Summary of Modes and Potentials in Spherical and Conical Resonators

This document presents a comprehensive summary of electromagnetic potentials and modes in spherical and conical resonators, with a focus on the Transverse Magnetic (TM) and Transverse Electric (TE) modes. Key topics include the characteristics of the TM modes designated by indices (m, n, p), the implications of varying these parameters, and the significance of the combined mode structures. Additionally, the document discusses the relationship between different resonator shapes and their mathematical descriptions, highlighting the importance of numerical solutions in certain cases.

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Summary of Modes and Potentials in Spherical and Conical Resonators

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  1. ECE 6341 Spring 2014 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.) unpvalues

  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 trivial field

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

  12. Spherical Resonator (cont.)

  13. Spherical Resonator (cont.) unpvalues

  14. Spherical Resonator (cont.) Then so Note: since 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 Note: The integer index m is arbitrary. Hence (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 smoothly 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

  23. Wedge (cont.) TMr: so

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

  25. Spherical Horn (cont.) TMr: so

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

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

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

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