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

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

3. 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

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

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/  = -.
27/02/51

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
27/02/51

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
27/02/51

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

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

10. 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

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

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
27/02/51

13. c) dc resistance
27/02/51

14. 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

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

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

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

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

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

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

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?
27/02/51