1. ENE 325 Electromagnetic Fields and Waves
Lecture 11 Uniform Plane Waves

2. Introduction From Maxwell’s equations, if the electric field
From Maxwell’s equations, if the electric field
is changing with time, then the magnetic field
varies spatially in a direction normal to its orientation direction
A uniform plane wave, both electric and magnetic fields lie in the transverse plane, the plane whose normal is the direction of propagation
Both fields are of constant magnitude in the transverse plane, such a wave is sometimes called a transverse electromagnetic (TEM) wave.

3. Maxwell’s equations
(1)
(2)
(3)
(4)

4. Maxwell’s equations in free space
 = 0, r = 1, r = 1
Ampère’s law
Faraday’s law

5. General wave equations
Consider medium free of charge where
For linear, isotropic, homogeneous, and time-invariant medium,
(1)
(2)

6. General wave equations
Take curl of (2), we yield
From
then
For charge free medium

7. Helmholtz wave equation
For electric field
For magnetic field

8. Time-harmonic wave equations
Transformation from time to frequency domain
Therefore

9. 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)

10. Solutions of Helmholtz equations
Assuming the electric field is in x-direction and the wave is propagating in z- direction
The instantaneous form of the solutions
Consider only the forward-propagating wave, we have
Use Maxwell’s equation, we get

11. Solutions of Helmholtz equations in phasor form
Showing the forward-propagating fields without time-harmonic terms.
Conversion between instantaneous and phasor form
Instantaneous field = Re(ejtphasor field)

12. Intrinsic impedance
For any medium,
For free space

13. Propagating fields relation
where represents a direction of propagation.

14. 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),

15. Propagation in lossless-charge free media
intrinsic impedance
Wavelength

16. 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

17. c) propagation velocity
d) intrinsic impedance
e) amplitude of the magnetic field intensity

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

19. Propagation in dielectrics
Define
from
and

20. Propagation in dielectrics
We can derive
and

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

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

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

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

25. 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

26. 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 

27. c) phase constant 
d) intrinsic impedance

28. 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

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