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

Plane Wave: Lossless Media z x y E Assume Then Lossless region: or Solution:

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

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

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

Plane Wave: Lossy Media

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

Plane Wave: Lossy Media (cont.)

Plane Wave in a Good Conductor

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

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

Surface Impedance x z Equivalent surface current Actual Model z x

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

Surface Impedance (cont.) Define Integrating, we have

Surface Impedance (cont.) Hence Hence,

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

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

Impedance of a Bulk Conductor (cont.) jX

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

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

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).

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

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

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

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

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

Power Dissipation (cont.) Use where Note:

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

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

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