1. ENE 429 Antenna and Transmission lines Theory
Lecture 2 Uniform plane waves

2. Review
Wave equations
Time-Harmonics equations
where

3. Time-harmonic wave equationsor
where
This  term is called propagation constant or we can write
 = +j
where  = attenuation constant (Np/m)
 = phase constant (rad/m)

4. Transverse ElectroMagnetic wave (TEM)

5. Solutions of Helmholtz equations
The instantaneous forms of the solutions
The phasor forms of the solutions
incident wave
reflected wave

6. 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/  = 

7. 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

8. 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

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

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

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

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

13. 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:
The skin depth
The skin resistance

14. c) The dc resistance

15. 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

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

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

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

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

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

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

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

23. Polarization
UPW is characterized by its propagation direction and frequency.
Its attenuation and phase are determined by medium’s parameters.
Polarization determines the orientation of the electric field in a fixed spatial plane orthogonal to the direction of the propagation.

24. Linear polarization Consider in free space,
At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave”

25. Linear polarization
A pair of linearly polarized wave also produces linear polarization
At z = 0 plane
At t = 0, both linearly polarized waves
Have their maximum values

26. More generalized linear polarization
More generalized of two linearly poloraized waves,
Linear polarization occurs when two linearly polarized waves are
in phase
out of phase

27. Elliptically polarized wave
Super position of two linearly polarized waves that
If x = 0 and y = 45, we have

28. Circularly polarized wave
occurs when Exo and Eyo are equal and
Right hand circularly polarized (RHCP) wave
Left hand circularly polarized (LHCP) wave

29. Circularly polarized wave
Phasor forms:
for RHCP,
for LHCP,
from
Note: There are also RHEP and LHEP

30. Ex2 Given ,determine the polarization of this wave

31. Ex3 The electric field of a uniform plane wave in free space is given by , determine
The magnetic field intensity

32. c)
d) Describe the polarization of the wave