1. ENE 428 Microwave Engineering
Lecture 2 Uniform plane waves RS

2. Propagation in lossless-charge free media
Attenuation constant  = 0, conductivity  = 0
Propagation constant
Propagation velocity
for free space up = 3108 m/s (speed of light)
for non-magnetic lossless dielectric (r = 1), RS

3. Propagation in lossless-charge free media
intrinsic impedance
wavelength RS

4. Ex1 A GHz uniform plane wave is propagating in polyethelene (r = 2.26). If the amplitude of the electric field intensity is 500 V/m and the material is assumed to be lossless, find
a) phase constant
b) wavelength in the polyethelene RS

5. c) propagation velocity
d) Intrinsic impedance
e) Amplitude of the magnetic field intensity RS

6. Propagation in dielectrics
Cause
finite conductivity
polarization loss ( = ’-j” )
Assume homogeneous and isotropic medium RS

7. Propagation in dielectrics
Define
From
and RS

8. Propagation in dielectrics
We can derive
and RS

9. Loss tangent
A standard measure of lossiness, used to classify a material as a good dielectric or a good conductor RS

10. Low loss material or a good dielectric (tan « 1)
If or < 0.1 , consider the material ‘low loss’ , then
and RS

11. Low loss material or a good dielectric (tan « 1)
propagation velocity
wavelength RS

12. High loss material or a good conductor (tan » 1)
In this case or > 10, we can approximate
therefore
and RS

13. High loss material or a good conductor (tan » 1)
depth of penetration or skin depth,  is a distance where the field decreases to e-1 or times of the initial field
propagation velocity
wavelength RS

14. Ex2 Given a nonmagnetic material having r = 3. 2 and  = 1
Ex2 Given a nonmagnetic material having r = 3.2 and  = 1.510-4 S/m, at f = 3 MHz, find
a) loss tangent 
b) attenuation constant  RS

15. d) intrinsic impedance
c) phase constant 
d) intrinsic impedance RS

16. Ex3 Calculate the followings for the wave with the frequency f = 60 Hz propagating in a copper with the conductivity,  = 5.8107 S/m:
a) wavelength
b) propagation velocity RS

17. c) compare these answers with the same wave propagating in a free space

18. 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/  =  RS

19. 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 RS

20. 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 RS

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

22. Currents in conductor From
Jx or current density decreases as the slab gets thicker RS

23. Currents in conductor For distance L in x-direction
R is called skin resistance
Rskin is called skin-effect resistance
For finite thickness, RS

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

25. Ex 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:
The skin depth
The skin resistance RS

26. c) The dc resistance RS

27. 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 RS

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

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

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

31. Example of Poynting theorem in DC case
Total power W RS

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

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