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Spring 2011

ECE 6345. Spring 2011. Prof. David R. Jackson ECE Dept. Notes 31. Overview. In this set of notes we extend the spectral-domain method to analyze infinite periodic structures. Two typical examples of infinite periodic problems: Scattering from a frequency selective surface (FSS)

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Spring 2011

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  1. ECE 6345 Spring 2011 Prof. David R. Jackson ECE Dept. Notes 31

  2. Overview In this set of notes we extend the spectral-domain method to analyze infinite periodic structures. • Two typical examples of infinite periodic problems: • Scattering from a frequency selective surface (FSS) • Input impedance of a microstrip phased array

  3. FSS Geometry Incident plane wave Reflected plane wave z y a L Metal patch b W x Dielectric layer Transmitted plane wave 3

  4. FSS Geometry (cont.) 4

  5. Phased Array Geometry z Probe y L W a Metal patch b x Dielectric layer Ground plane Probe current mn: 5

  6. Phased Array Geometry (cont.) 6

  7. Periodic SDI Fundamental observation: If the structure is infinite and periodic, and the excitation is periodic except for a phase shift, then all of the currents and radiated fields will also be periodic except for a phase shift. z This is sometimes referred to as “Floquet’s theorem.” y 7 x

  8. Periodic SDI (cont.) From Floquet’s theorem: z layered media y b L W a 8 x

  9. Periodic SDI Let  denote any component of the surface current or the field (at a fixed value of z). 9

  10. Periodic SDI (cont.) where From Fourier-series theory, we know that the periodic function P can be represented as Hence we have 10

  11. Periodic SDI (cont.) Hence, the surface current or field can be expanded in a set of Floquet waves: 11

  12. Periodic SDI (cont.) The surface current on the periodic structure is represented in terms of Floquet waves: To solve for the unknown coefficients, multiply both sides by and integrate over the (0,0) unit cell S0. 12

  13. Periodic SDI (cont.) Use orthogonality: 13

  14. Periodic SDI (cont.) Therefore we have: The current Js00 is the current on the (0,0) patch. Hence, we have 14

  15. Periodic SDI (cont.) Hence the current on the 2D periodic structure can be represented as We now calculate the Fourier transform of the 2D periodic current (this is what we need in the SDI method): 15

  16. Periodic SDI (cont.) Hence Now calculate the field produced by the patch currents: 16

  17. Periodic SDI (cont.) Hence, we have 17

  18. Periodic SDI (cont.) Therefore we have 18

  19. Periodic SDI (cont.) Compare: Single element (non-periodic): Periodic array of phased elements: 19

  20. Periodic SDI (cont.) Conclusion: where The double integral is replaced by a double sum, and a factor is introduced. 20

  21. ky kx Periodic SDI (cont.) Sample points in the (kx, ky) plane 21

  22. z y b L W a x Microstrip Patch Phased Array Example Microstrip Patch Phased Array Find Ex(x,y,0) 22

  23. Phased Array (cont.) Single patch: 23

  24. Phased Array (cont.) 2D phased array of patches: where 24

  25. Phased Array (cont.) The field is of the form where The field is thus represented as a “sum of Floquet waves.” Note: Each Floquet wave repeats from one unit cell to the next, except for a phase shift. 25

  26. ky Scan blindness (-1,0) kx (E-plane scan) Scan Blindness One of the sample points (p,q) lies on the surface-wave circle (shown for (-1, 0)). 26

  27. Scan Blindness (cont.) The scan blindness condition is: The field produced by an impressed set of infinite periodic phased surface-current sources will be infinite. 27

  28. y x Scan Blindness (cont.) Physical interpretation: All of the surface-wave fields excited from the patches add up in phase in the direction of the vector Proof: Start with the surface-wave array factor: N elements 28

  29. y x Scan Blindness (cont.) Hence we have, in this direction, that N elements 29

  30. y x Scan Blindness (cont.) In the direction sw the surface fields from each patch add up in phase. N elements 30

  31. Scan Blindness (cont.) Example y ky kx x 31

  32. Grating Lobes One of the higher-order Floquet waves propagates in space. For a finite-size array, this corresponds to a secondary beam that gets radiated. 32

  33. Grating Lobes (cont.) ky Grating wave (-1,0) kx k0 33

  34. H-plane scan k0sin E-plane scan Pozar Circle Diagram Denote 34

  35. Pozar Circle Diagram (cont.) Grating Lobes or or or or 35

  36. Pozar Circle Diagram (cont.) or or where 36

  37. Grating lobe region + Pozar Circle Diagram (cont.) 37

  38. H-plane scan E-plane scan Pozar Circle Diagram (cont.) 38

  39. Pozar Circle Diagram (cont.) To avoid grating lobes, we require or Hence or 39

  40. Pozar Circle Diagram (cont.) Scan Blindness or or or or 40

  41. Pozar Circle Diagram (cont.) or or where 41

  42. Scan blindness curve Pozar Circle Diagram (cont.) + 42

  43. Pozar Circle Diagram (cont.) To avoid scan blindness, we require or Hence or 43

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