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Lecture 12 Plane Waves in Conductor, Poynting Theorem, and Power Transmission

ENE 325 Electromagnetic Fields and Waves. Lecture 12 Plane Waves in Conductor, Poynting Theorem, and Power Transmission. Review (1). Wave equations Time-Harmonics equations where. Review (2). where This  term is called propagation constant or we can write  = +j

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Lecture 12 Plane Waves in Conductor, Poynting Theorem, and Power Transmission

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  1. ENE 325Electromagnetic Fields and Waves Lecture 12 Plane Waves in Conductor, Poynting Theorem, and Power Transmission

  2. Review (1) • Wave equations • Time-Harmonics equations where

  3. Review (2) where This  term is called propagation constant or we can write  = +j where  = attenuation constant (Np/m)  = phase constant (rad/m)

  4. Review (3) • The instantaneous forms of the solutions • The phasor forms of the solutions reflected wave incident wave

  5. Attenuation constant  • Attenuation constant determines the penetration of the wave into a medium • Attenuation constant are different for different applications • The penetration depth or skin depth,  is the distance z that causes to reduce to z = -1  z = -1/  = -.

  6. Good conductor • At high operation frequency, skin depth decreases. • A magnetic material is not suitable for signal carrier. • A high conductivity material has low skin depth.

  7. Currents in conductor • To understand a concept of sheet resistance from Rsheet() sheet resistance At high frequency, it will be adapted to skin effect resistance

  8. Currents in conductor Therefore the current that flows through the slab at t   is

  9. Currents in conductor From Jxor current density decreases as the slab gets thicker.

  10. Currents in conductor For distance L in x-direction Ris called skin resistance Rskinis called skin-effect resistance For finite thickness,

  11. Currents in conductor Current is confined within a skin depth of the coaxial cable.

  12. Ex1 A steel pipe is constructed of a material for which r = 180 and  = 4106 S/m. The two radii are 5 and 7 mm, and the length is 75 m. If the total current I(t) carried by the pipe is 8cost A, where  = 1200rad/s, find: • skin depth • skin resistance

  13. c) dc resistance

  14. The Poynting theorem and power transmission Poynting theorem Total power leaving the surface Joule’s law for instantaneous power dissipated per volume (dissi- pated by heat) Rate of change of energy stored In the fields Instantaneous poynting vector

  15. Example of Poynting theorem in DC case Rate of change of energy stored In the fields = 0

  16. Example of Poynting theorem in DC case From By using Ohm’s law,

  17. Example of Poynting theorem in DC case Verify with From Ampère’s circuital law,

  18. Example of Poynting theorem in DC case Total power W

  19. Uniform plane wave (UPW) power transmission • Time-averaged power density W/m2 amount of power for lossless case, W

  20. Uniform plane wave (UPW) power transmission for lossy medium, we can write intrinsic impedance for lossy medium

  21. Uniform plane wave (UPW) power transmission from W/m2 Question: Have you ever wondered why aluminum foil is not allowed in the microwave oven?

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