1. Notes 14 ECE 6340 Intermediate EM Waves Fall 2016
Prof. David R. Jackson
Dept. of ECE
Notes 14

2. Plane Wave: Lossless Mediaz x y E
Assume
Then
Lossless region: or
Solution:

3. Plane Wave: Lossless Media (cont.)
The H field is found from: so

4. Plane Wave: Lossless Media (cont.)or
But
(real number)
Hence
TEMz

5. Plane Wave: Lossless Media (cont.)so
Lossless:  Is real.
There is no reactive power.

6. Plane Wave: Lossy Media

7. Plane Wave: Lossy Media (cont.)
The “depth of penetration” dp is defined.
Hence

8. Plane Wave: Lossy Media (cont.)

9. Plane Wave in a Good Conductor

10. Plane Wave in Good Conductor (cont.)
Assume
The we have
Hence
Therefore

11. Plane Wave in Good Conductor (cont.)
Denote
“skin depth”
Then we have

12. Surface Impedance x z
Equivalent surface current
Actual
Model z x

13. Surface Impedance (cont.)
Actual current through y
Surface current model
Hence

14. Surface Impedance (cont.)
Define
Integrating, we have

15. Surface Impedance (cont.)
Hence
Hence,

16. Surface Impedance (cont.)
Define “surface resistance” and “surface reactance” so

17. Impedance of a Bulk Conductorz x
Electrodes + -  y
Assume no x or y variation
Electrodes are attached at the outer (top) surface.

18. Impedance of a Bulk Conductor (cont.)jX

19. DC Equivalent Model DC problem Hence
The thickness in the DC problem is chosen as the skin depth in the high-frequency bulk conductor problem.
Hence

20. DC Equivalent Model (cont.)
High-frequency problem (R)
DC current problem (RDC)

21. Example
Find the high-frequency resistance and inductance for a solid wire. + -
Ring electrode
The current is uniform along the boundary ( direction), and therefore we can treat this as a “rolled-up” version of a bulk conductor (w = 2 a).

22. Example (cont.)
We can also get the same result by using the equivalent DC model.
Equivalent DC Model:

23. Example (cont.)
High-frequency equivalent circuit: R jX

24. Find the high-frequency resistance and inductance for a hollow tube.
Example
Find the high-frequency resistance and inductance for a hollow tube.
The electrodes are attached from the outside. + -
Ring electrode

25. Find the high-frequency resistance and inductance for a hollow tube.
Example
Find the high-frequency resistance and inductance for a hollow tube.
The electrodes are attached from the inside. + -
Ring electrode

26. Power Dissipation x S z = time-average power dissipated / m2 on S
Fields are evaluated on this plane.
= time-average power dissipated / m2 on S

27. Power Dissipation (cont.)
Use
where
Note:

28. Power Dissipation (cont.)
We then have
In general,

29. Power Dissipation (cont.)
For a good conductor,
(exactly true for a PEC)
Hence, using this approximation, we have

30. Power Dissipation (cont.)
For the reactive power absorbed by the conducting surface (VARS/m2) we have:
Hence
VARS / m2 = Watts / m2