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Spring 2011

ECE 6345. Spring 2011. Prof. David R. Jackson ECE Dept. Notes 5. Overview.

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Spring 2011

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  1. ECE 6345 Spring 2011 Prof. David R. Jackson ECE Dept. Notes 5

  2. Overview This set of notes discusses improved models of the probe inductanceof a coaxially-fed patch (accurate for thicker substrates). A parallel-plate waveguide model is initially assumed (at the end of the notes we will also look at the actual finite patch).

  3. Overview (cont.) The following models are investigated: • cosine-current model • gap-source model • frill model Reference: “Comparison of Models for the Probe Inductance for a Parallel Plate Waveguide and a Microstrip Patch,” H. Xu, D. R. Jackson, and J. T. Williams, IEEE Trans. Antennas and Propagation, Vol. 53, pp. 3229-3235, Oct. 2005.

  4. Improved Probe Models Cosine-current model: I (z) We assume a tube of current (as in Notes 4) but with a z variation. Note: The derivative of the current is zero at the top conductor (PEC). Pc= complex power radiated by probe current

  5. Improved Probe Models (cont.) + 1V  - Gap-source model: An ideal gap voltage source of height  is assumed at the bottom of the probe.

  6. b Improved Probe Models (cont.) Frill model: A magnetic frill of radius b is assumed on the mouth of the coax. (TEM mode of coax, assuming 1 V)

  7. Improved Probe Models (cont.) Next, we investigate each of the improved probe models in more detail: • Cosine-current model • Gap-source model • Frill model

  8. Cosine Current Model I (z) Assume that Note: Represent current as:

  9. coax feed Cosine Current Model (cont.) Circuit Model:

  10. Cosine Current Model I (z) Represent the probe current as: This will allow us to find the fields and hence the power radiated by the probe current.

  11. Cosine Current Model (cont.) Using Fourier-series theory: The integral is zero unless m=m. Hence

  12. Cosine Current Model (cont.) or Result: (derivation omitted)

  13. Cosine Current Model (cont.) Note: We have both Ez and E To see this: (Time-Harmonic Fields) so

  14. Cosine Current Model (cont.) For Ez, we represent the field as follows: where

  15. Cosine Current Model (cont.) At (BC 1) so

  16. Cosine Current Model (cont.) Also we have (BC 2) where To solve for E, use

  17. Cosine Current Model (cont.) so Hence we have For the mth Fourier term:

  18. Cosine Current Model (cont.) so that (BC 2) where Hence

  19. Cosine Current Model (cont.) For the mth Fourier term: where

  20. Cosine Current Model (cont.) Hence Using (BC 1) we have

  21. Cosine Current Model (cont.) or or (using the Wronskian identity) Hence

  22. Cosine Current Model (cont.) We now find the complex power radiated by the probe:

  23. Cosine Current Model (cont.) Integrating in z we have Am+ coefficient Hence we have

  24. Circuit Model (cont.) Therefore, Define

  25. Circuit Model (cont.) Also, use We then have The probe reactance is:

  26. Circuit Model (cont.) Thin substrate approximation Keep only the m= 0 term The result is (same as previous result using uniform model)

  27. Circuit Model (cont.) The probe reactance is:

  28. Gap Model + 1V  - Note: It is not clear how best to choose , but this will be re-visited later. Hence From Fourier series analysis:

  29. Gap Model (cont.) + 1V  - where The magnetic field is found from Ez , with the help of the magnetic vector potential Az (the field is TMz): using Setting  = a allows us to solve for the coefficients Am. where

  30. Gap Model (cont.) + 1V  - Final result:

  31. b B z SF A Frill Model 1V frill To find the current I (z) , use reciprocity. Introduce a ring of magnetic current K= 1 in the  direction at z (the testing current “B”).

  32. B z SF A Frill Model (cont.)

  33. Frill Model (cont.) b b The magnetic current ring B may be replaced by a 1V gap source of zero height (by the equivalence principle). Let z 0: The field of the gap source is then calculated as was done in the gap-source model, using  = 0.

  34. b Frill Model (cont.) Final result:

  35. Comparison of Models Next, we show results that compare the various models, especially as the substrate thickness increases.

  36. Comparison of Models Models are compared for changing substrate thickness. r= 2.2 a = 0.635 mm f = 2 GHz Z0 = 50  (b = 2.19 mm)

  37. Comparison of Models (cont.) Models are compared for changing substrate thickness. r= 2.2 a = 0.635 mm f = 2 GHz Z0 = 50  (b = 2.19 mm) Note:

  38. Comparison of Models (cont.) For the gap-source model, the results depend on . r= 2.2 a = 0.635 mm f = 2 GHz Z0 = 50  (b = 2.19 mm)

  39. Comparison of Models (cont.) The gap-source model is compared with the frill model, for varying , for a fixed h. h=20 mm r= 2.2 a = 0.635 mm f = 2 GHz Z0 = 50  (b = 2.19 mm) R X

  40. Comparison of Models (cont.) These results suggest the “1/3” rule: The best  is chosen as This rule applies for a coax feed that has a 50  impedance.

  41. Comparison of Models (cont.) The gap-source model is compared with the frill model, using the optimum gap height (1/3 rule). r= 2.2 a = 0.635 mm f = 2 GHz Z0 = 50  (b = 2.19 mm)

  42. Comparison of Models (cont.) The gap-source model is compared with the frill model, using the optimum gap height (1/3 rule). r= 2.2 a = 0.635 mm f = 2 GHz Z0 = 50  (b = 2.19 mm)

  43. y (x0, y0) W x L Probe in Patch A probe in a patch does not see an infinite parallel-plate waveguide. Exact calculation of probe reactance: Zin may be calculated by HFSS or any other software, or it may be measured. f0= frequency at which Rinis maximum

  44. y PMC (x0, y0) We x Le Probe in Patch (cont.) Cavity Model Using the cavity model, we can derive an expression for the probe reactance: This formula assumes that there is no z variation of the probe current or cavity fields (thin-substrate approximation), but it does accurately account for the actual patch dimensions.

  45. Probe in Patch (cont.) a= probe radius (x0, y0) = probe location This formula assumes that there is no z variation of the probe current or cavity fields (thin-substrate approximation), but it does accurately account for the actual patch dimensions.

  46. Probe in Patch (cont.) Image Theory Image theory can be used to improve the simple parallel-plate waveguide model when the probe gets close to the patch edge. Using image theory, we have an infinite set of “image probes.”

  47. Probe in Patch (cont.) A simple approximate formula is obtained by using two terms: the original probe current in a parallel-plate waveguide and one image. This should be an improvement when the probe is close to an edge. original image

  48. Probe in Patch (cont.) As shown on the next plot, the best overall approximation in obtained by using the following formula: “modified CAD formula”

  49. y (x0, y0) W x L Probe in Patch (cont.) Results show that the simple formula (“modified CAD formula”) works fairly well.

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