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Far-Field Analysis Using Reciprocity for Rectangular Patch Antennas

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This set of notes explores the application of the reciprocity theorem to calculate the far-field radiation of a source within layered dielectric structures. Focusing on the dominant mode of a rectangular patch antenna, we employ the electric current model assuming infinite substrate conditions. Key aspects include deriving the far-field equations for a horizontal electric dipole and the implications of the patch's current distribution. This material is essential for understanding antenna radiation behavior and the utilization of reciprocity in electromagnetic theory.

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Far-Field Analysis Using Reciprocity for Rectangular Patch Antennas

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  1. ECE 6345 Spring 2011 Prof. David R. Jackson ECE Dept. Notes 7

  2. Overview In this set of notes we use reciprocity to calculate the far field of a source in a layered structure, and use this to calculate the far-field for the dominant mode of a rectangular patch using the electric-current model, assuming an infinite substrate • Review of reciprocity to calculate the far field. • Far field of horizontal electric dipole in the x direction (hex) on top of a grounded the substrate. • Far field of dominant mode of rectangular patch, using the electric current model and infinite substrate.

  3. Far-Field Reciprocity theorem: Consider an electric current source:

  4. Far-Field (cont.) Let

  5. Far-Field (cont.) Hence ipw = “incident plane wave” Assume Hence FF = “far field”

  6. Far-Field (cont.) Consider a dipole in free space: at

  7. Far-Field (cont.) In general, At O : where At (0, 0, 0):

  8. Far-Field (cont.) For an arbitrary observation point (x, y, z) we have where

  9. Far-Field (cont.) X X Hence Similarly, if

  10. Far-Field (cont.) body Now consider an object near the radiating current: If the “body” is an infinite layered dielectric structure, the scattered field can be calculated exactly.

  11. Far-Field (cont.) body For magnetic current source: where

  12. Electric Dipole infinite substrate Far field of HED “hex” = unit-amplitude horizontal electric dipole in the x direction. rpw = “reflected plane wave”

  13. Electric Dipole (cont.) For For

  14. Magnetic Dipole X The source is now a unit-amplitude magnetic dipole in the y direction.

  15. Magnetic Dipole (cont.)

  16. Transverse Equivalent Network (TEN)

  17. TEN (cont.) where and or

  18. TEN (cont.) From Snell’s law, Define

  19. TEN (cont.) Then Also,

  20. TEN (cont.) For a wave traveling in the z = - z direction, we use: This is the situation for our incident wave.

  21. Electric Dipole

  22. Electric Dipole: TEN so

  23. Electric Dipole: TEN (cont.) Hence Hence

  24. Electric Dipole: TEN (cont.) or

  25. Electric Dipole: TEN (cont.) Note that After simplifying, we obtain the following results. Similarly,

  26. Electric Dipole: Final Results Hence we have the following final results

  27. Electric Dipole: Final Results (cont.) y-directed electric dipole We don’t need this for modeling the (1,0) mode, however.

  28. Far Field of Patch Current

  29. Far Field of Patch Current (cont.) Hence hex = horizontal electric dipole in the x direction. Assume

  30. Far Field of Patch Current (cont.) For this patch current we have the following Fourier transform:

  31. Summary Assumption:

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