ENE 325 Electromagnetic Fields and Waves Lecture 12 Plane Waves in Conductor, Poynting Theorem, and Power Transmission 27/02/51

Review (1) Wave equations Time-Harmonics equations where 27/02/51

Review (2) where This  term is called propagation constant or we can write  = +j where  = attenuation constant (Np/m)  = phase constant (rad/m) 27/02/51

Review (3) The instantaneous forms of the solutions The phasor forms of the solutions incident wave reflected wave 27/02/51

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/  = -. 27/02/51

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 27/02/51

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 27/02/51

Currents in conductor Therefore the current that flows through the slab at t   is 27/02/51

Currents in conductor From Jx or current density decreases as the slab gets thicker 27/02/51

Currents in conductor For distance L in x-direction R is called skin resistance Rskin is called skin-effect resistance For finite thickness, 27/02/51

Currents in conductor Current is confined within a skin depth of the coaxial cable. 27/02/51

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 27/02/51

c) dc resistance 27/02/51

The Poynting theorem and power transmission 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 27/02/51

Example of Poynting theorem in DC case Rate of change of energy stored In the fields = 0 27/02/51

Example of Poynting theorem in DC case From By using Ohm’s law, 27/02/51

Example of Poynting theorem in DC case Verify with From Ampère’s circuital law, 27/02/51

Example of Poynting theorem in DC case Total power W 27/02/51

Uniform plane wave (UPW) power transmission Time-averaged power density W/m2 amount of power for lossless case, W 27/02/51

Uniform plane wave (UPW) power transmission for lossy medium, we can write intrinsic impedance for lossy medium 27/02/51

Uniform plane wave (UPW) power transmission from W/m2 Question: Have you ever wondered why aluminum foil is not allowed in the microwave oven? 27/02/51