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Spring 2011

ECE 6345. Spring 2011. Prof. David R. Jackson ECE Dept. Notes 29. Overview. In this set of notes we use the spectral-domain method to find the input impedance of a rectangular patch antenna.

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Spring 2011

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  1. ECE 6345 Spring 2011 Prof. David R. Jackson ECE Dept. Notes 29

  2. Overview In this set of notes we use the spectral-domain method to find the input impedance of a rectangular patch antenna. This method uses the exact spectral-domain Green’s function, so all radiation physics, including surface-wave excitation, is automatically included (no need for an effective permittivity). It does not account for the probe inductance (the way it is formulated here), so the CAD formula for probe inductance is added on at the end.

  3. Spectral Domain Method The probe is viewed as an impressed current. Set S is the patch surface This is the “Electric Field Integral Equation (EFIE)”

  4. Spectral Domain Method (cont.) Let The EFIE is then Pick a “testing” function T(x,y):

  5. Spectral Domain Method (cont.) Galerkin’s Method: (The testing function is the same as the basis function.) Hence The solution for the unknown amplitude coefficient Ax is then

  6. Spectral Domain Method (cont.) The input impedance is calculated as (The probe current is real and equal to 1.0 [A].) The total field comes from the patch and the probe:

  7. Spectral Domain Method (cont.) Hence Define Then we have or

  8. Spectral Domain Method (cont.) We have from reciprocity that Hence or

  9. Spectral Domain Method (cont.) Denote Note: The subscript notation on Zij follows the usual MoM convention. Then we have Using reciprocity again, Note: Zzx is easier to calculate than Zxz.

  10. Spectral Domain Method (cont.) Note: The probe impedance may be approximately calculated by using a CAD formula: This result comes from a probe inside of an infinite parallel-plate waveguide. (Calculating Zprobe exactly from the spectral-domain method would be more difficult.)

  11. Spectral Domain Method (cont.) The next goal is to calculate the reactions Zxx and Zxz in closed form. For the patch-patch reaction we have From previous SDI theory, we have so Hence, integrating over the patch surface, we have

  12. Spectral Domain Method (cont.) Since the Fourier transform of the basis function (cosine function) is an even function of kx and ky, we can write or

  13. Spectral Domain Method (cont.) Converting to polar coordinates, we have Note: The path must extend to infinity.

  14. Spectral Domain Method (cont.) From previous calculations, we have

  15. Spectral Domain Method (cont.) For the patch-probe reaction we have where so To calculate use so that

  16. Spectral Domain Method (cont.) Using spectral-domain theory, we have

  17. Spectral Domain Method (cont.)

  18. Spectral Domain Method (cont.) Hence we have Note:

  19. Spectral Domain Method (cont.) Hence we have Using we then identify that

  20. Spectral Domain Method (cont.) From TL theory, we have the property that (The short circuit at z = -hcauses the current to have a zero derivative there.) Hence

  21. Spectral Domain Method (cont.) For the field due to the patch basis function we then have Note that Recall that

  22. Spectral Domain Method (cont.) Hence we have where

  23. Spectral Domain Method (cont.) The integrand is an even function of ky and an odd function of kx (due to the cosine term). Hence we use the following combinations to reduce the integration to one over the first quadrant: The result is then

  24. Spectral Domain Method (cont.) The final result is then

  25. Spectral Domain Method (cont.) Note on effective loss tangent: The spectral-domain method already accounts for radiation into space and into surface waves. However, in order to account for material losses, we can use It is also possible to account for conductor loss by using a impedance boundary condition on the patch, but using an effective loss tangent is a simpler approach (no need to modify the code – simply increase the loss tangent to account for conductor loss).

  26. Spectral Domain Method (cont.) We now calculate the current function

  27. Spectral Domain Method (cont.) so

  28. Spectral Domain Method (cont.) Also, so Hence

  29. Results D. M. Pozar, "Input impedance and mutual coupling of rectangular microstrip antennas,“ IEEE Trans. Antennas Propagat.,Vol. AP-30. pp. 1191-1196, Nov. 1982. [6] E. H. Newman and P. Tulyathan, “Analysis of microstrip antennas using moment methods,“ IEEE Trans. Antennas Propagat.,Vol. AP-29. pp. 47-53, Jan. 1981. 29

  30. Two Basis Functions Using two basis functions is important for circular polarization. EFIE: 30

  31. Two Basis Functions (cont.) Galerkin testing: Denote 31

  32. Two Basis Functions (cont.) (reciprocity) By symmetry, This follows since (from symmetry) and 32

  33. Two Basis Functions (cont.) Hence, the two testing equations reduce to: Using reciprocity: The solution is: 33

  34. Two Basis Functions (cont.) The input impedance is then Hence, using the previous results for Ax and Ay, we have 34

  35. Two Basis Functions (cont.) Note: To calculate Zyyand Zyz, we can use the formulas for Zxxand Zxz and make a simple set of substitutions: y actual problem rotated coordinates y' W' W Jsx' x' x Jsy L' Zxxand Zxzproblem L In the rotatedproblem the parameters are first labeled with "primes" to avoid confusion with the actual coordinates. 35

  36. Two Basis Functions (cont.) Results: 36

  37. Two Basis Functions (cont.) Results: 37

  38. Two Basis Functions (cont.) where so 38

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