1. ENE 325 Electromagnetic Fields and Waves
Lecture 12 Uniform plane waves RS

2. Introduction (2) 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
Knowledge of fields in media and boundary conditions allows useful applications of material properties to microwave components
A uniform plane wave, both electric and magnetic fields lie in the transverse plane, the plane whose normal is the direction of propagation RS

3. Uniform plane wave propagation direction Wavefront 12/3/2018 RS RS
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4. Maxwell’s equations
(1)
(2)
(3)
(4) RS

5. Integral forms of Maxwell’s equationsRS

6. Divergence theorem
Stokes’ theorem RS

7. Maxwell’s equations in phasor form
Assume and ejt time dependence,
(1)
(2)
(3)
(4) RS

8. Fields are assumed to be sinusoidal or harmonic, and time dependence with steady-state conditions
Time dependence form:
Phasor form: RS

9. Fields in dielectric media (1)RS

10. Fields in dielectric media (2)
may be complex then  can be complex and can be expressed as
Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments.
The loss of dielectric material may be considered as an equivalent conductor loss if the material has a conductivity  . Loss tangent is defined as RS

11. Anisotropic dielectrics
The most general linear relation of anisotropic dielectrics can be expressed in the form of a tensor which can be written in matrix form as RS

12. Analogous situations for magnetic media (1)RS

13. Analogous situations for magnetic media (2)
may be complex then  can be complex and can be expressed as
Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments. RS

14. Anisotropic magnetic material
The most general linear relation of anisotropic material can be expressed in the form of a tensor which can be written in matrix form as RS

15. General plane wave equations (1)
Consider medium free of charge
For linear, isotropic, homogeneous, and time-invariant medium, assuming no free magnetic current,
(1)
(2) RS

16. Maxwell’s equations in free space
 = 0, r = 1, r = 1
0 = 4x10-7 Henrys/m
0 = 8.854x10-12 Farad/m
Ampère’s law
Faraday’s law RS

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

18. Helmholtz wave equations
For electric field
For magnetic field RS

19. Plane waves in a general lossy medium
Transformation from time to frequency domain
Therefore RS

20. Plane waves in a general lossy mediumor
where
This  term is called propagation constant or we can write
 = +j
where  = attenuation constant (Np/m)
 = phase constant (rad/m) RS

21. 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 RS

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

23. Wave impedance
For any medium,
For free space RS

24. Propagating fields relationsRS

25. 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

26. Propagation in lossless-charge free media
intrinsic impedance
wavelength 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. Time averaged power density
Time-averaged power density is easily calculated in the phasor form.
For lossless medium
W/m2 RS

29. Uniform plane wave (UPW) power transmission for lossless medium
W/m2
amount of power W RS

30. Reflection and transmission of UPW at normal incidenceRS

31. Reflection coefficient
From
and
We can find the reflection coefficient as RS
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32. Transmission coefficient
and we can find the transmission coefficient as RS

33. Power transmission for 2 perfect dielectrics (1)
1 and 2 are both real positive quantities and 1 = 2 = 0. 
Average incident power densities
W/m2 RS

34. Power transmission for 2 perfect dielectrics (2)
Average reflected power densities
W/m2 RS

35. Power transmission for 2 perfect dielectrics (3)
Average transmitted power densities
W/m2 or RS