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Understanding Plane Waves in Conductors: Skin Depth and Surface Impedance

This document provides a comprehensive overview of plane waves in good conductors, emphasizing the concept of skin depth, particularly in copper. It explores the effects of skin depth on current penetration in both straight and curved conductors and discusses the implications for coaxial cables. Key concepts include surface impedance, surface resistance, and internal inductance, along with examples illustrating high-frequency behavior in solid wires. Understanding these principles is essential for engineers working with high-frequency electronic systems.

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Understanding Plane Waves in Conductors: Skin Depth and Surface Impedance

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  1. ECE 3317 Prof. Ji Chen Spring 2014 Notes 16 Plane Waves in Good Conductors

  2. Good Conductor Requirement: Example: copper Hence so Use

  3. Skin Depth Denote “skin depth” Then we have

  4. Skin Depth (cont.) Hence

  5. Skin Depth (cont.) Example: copper

  6. Skin Depth (cont.) The same penetration principle holds for curved conductors, as long as the radius of curvature is large compared with the skin depth. a a r r The distance z is measured from the boundary of the conductor. c c b b Coax a E H Penetration into inner conductor Penetration into outer conductor

  7. Skin Depth (cont.) a r c a b r c b Coax Regions of strong currents The fields are confined inside the coax if

  8. Surface Impedance x z Equivalent surface current x z

  9. Surface Impedance (cont.) Actual current Surface current model Hence

  10. Surface Impedance (cont.) Define the surface impedance: z

  11. Surface Impedance (cont.) Hence We then have

  12. Surface Impedance (cont.) Define “surface resistance” and “surface reactance” We then have

  13. Skin Depth (cont.) Example: copper

  14. Impedance of Wire - + Find the high-frequency resistance and inductance for a solid wire. V Note: The current mainly flows on the outside surface of the wire!

  15. Impedance of Wire (cont.) Surface-current model: Z=R+ jX= impedance Hence Therefore, we have

  16. Impedance of Wire (cont.) R jX Equivalent circuit:

  17. Impedance of Wire (cont.) Example: copper wire  = 5.8 107 S/m l = 5 cm f = 1.0 GHz Assume:

  18. Impedance of Wire (cont.) Compare with the same wire at DC:  = 5.8 107 S/m l = 5 cm 1.0 GHz DC

  19. Coax We use the surface resistance concept to calculate the resistance per unit length of coax. a r For a length l : c b Resistance per unit length:

  20. Coax (cont.) The skin effect will also contribute to an extra inductance per unit length, called the “internal inductance” per unit length. Internal reactance per unit length: a r c b Internal inductance per unit length: The internal inductance is usually neglected in practice (It is usually small compared with the external inductance, calculated assuming perfect conductors).

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