1. ECE 6341
Spring 2016
Prof. David R. Jackson
ECE Dept.
Notes 42

2. Efficiency of Patch
Patch
Lossless substrate and metal

3. Total Power The total complex power radiated by the patch is:
Use Parseval’s Theorem:

4. Total Power (cont.)
Hence
From SDI analysis:
where

5. Total Power (cont.) We then have Using symmetry, we have
Polar coordinates:

6. Total Power (cont.)
Note that
Hence
Note: is analytic but is not. This last form for Pc is preferable!

7. Total Power (cont.) For real kt we have (proof on next page): TM0 pole
Branch point
For real kt we have (proof on next page):
Hence, we can neglect the region
kt > k0, except possibly for the pole.

8. Total Power (cont.) Proof of complex property:
Consider the following term (De is similar):
always imaginary

9. Space and Surface-Wave Powers
The region gives the power
radiated into space.
The residue contribution gives the power
launched into the TM0 surface wave.

10. Space-Wave Power The region gives the power radiated into space. Note:
The power radiated into space can also be found by integrating the far-field Poynting vector.

11. Surface-Wave Power The pole contribution gives the surface-wave power:
This form is not very convenient, as it involves an integration around a pole.

12. Surface-Wave Power (cont.)
Calculation of surface-wave power
From the Cauchy residue theorem, we have:

13. Surface-Wave Power (cont.)
Residue calculation:
The residue of the spectral-domain Green’s function at the TM pole is:
Note:
The derivative can be calculated in closed form, but the result is omitted here.
where

14. Surface-Wave Power (cont.)
Since the transform of the current is real (assuming that ktp is real), we have
Since the residue is pure imaginary, we then have

15. Summary of Powers

16. Total Power (Alternative)
The total power can also be calculated directly: hR
Note: hb = 0.05 k0 is a good choice for numerical purposes.

17. Surface-Wave Power: Alternative Method
The surface-wave power may then be calculated from:
This avoids calculating any residues. hR
Total
Space

18. Summary hR
Total
Space

19. Lossy Patch hR Lossy Dielectric
For a lossy dielectric, the path must extend to infinity, since the integrand is now complex everywhere along the real axis. hR
Total
Space

20. Lossy Patch Lossy Metal
For lossy metal, the power dissipation in the conductors can be accounted for using the surface resistance. This conductive loss is added to the total power. or or

21. Results
Results: Conductor and dielectric losses are neglected.
2.2
10.8
r = 2.2 or 10.8
W/L = 1.5

22. Results
Results: Accounting for all losses (including conductor and dielectric loss).
r = 2.2 or 10.8
W/L = 1.5