1. EEE 498/598 Overview of Electrical Engineering
Lecture 10:
Uniform Plane Wave Solutions to Maxwell’s Equations 1

2. Lecture 10 Objectives
To study uniform plane wave solutions to Maxwell’s equations:
In the time domain for a lossless medium.
In the frequency domain for a lossy medium. 2

3. Overview of Waves
A wave is a pattern of values in space that appear to move as time evolves.
A wave is a solution to a wave equation.
Examples of waves include water waves, sound waves, seismic waves, and voltage and current waves on transmission lines. 3

4. Overview of Waves (Cont’d)
Wave phenomena result from an exchange between two different forms of energy such that the time rate of change in one form leads to a spatial change in the other.
Waves possess
no mass
energy
momentum
velocity 4

5. Time-Domain Maxwell’s Equations in Differential Form5

6. Time-Domain Maxwell’s Equations in Differential Form for a Simple Medium6

7. Time-Domain Maxwell’s Equations in Differential Form for a Simple, Source-Free, and Lossless Medium7

8. Obviously, there must be a source for the field somewhere.
Time-Domain Maxwell’s Equations in Differential Form for a Simple, Source-Free, and Lossless Medium
Obviously, there must be a source for the field somewhere.
However, we are looking at the properties of waves in a region far from the source. 8

9. Derivation of Wave Equations for Electromagnetic Waves in a Simple, Source-Free, Lossless Medium9

10. The wave equations are not independent.
Wave Equations for Electromagnetic Waves in a Simple, Source-Free, Lossless Medium
The wave equations are not independent.
Usually we solve the electric field wave equation and determine H from E using Faraday’s law. 10

11. Uniform Plane Wave Solutions in the Time Domain
A uniform plane wave is an electromagnetic wave in which the electric and magnetic fields and the direction of propagation are mutually orthogonal, and their amplitudes and phases are constant over planes perpendicular to the direction of propagation.
Let us examine a possible plane wave solution given by 11

12. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
The wave equation for this field simplifies to
The general solution to this wave equation is 12

13. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
The functions p1(z-vpt) and p2 (z+vpt) represent uniform waves propagating in the +z and -z directions respectively.
Once the electric field has been determined from the wave equation, the magnetic field must follow from Maxwell’s equations. 13

14. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
The velocity of propagation is determined solely by the medium:
The functions p1 and p2 are determined by the source and the other boundary conditions. 14

15. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
Here we must have
where 15

16. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
h is the intrinsic impedance of the medium given by
Like the velocity of propagation, the intrinsic impedance is independent of the source and is determined only by the properties of the medium. 16

17. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
In free space (vacuum): 17

18. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
Strictly speaking, uniform plane waves can be produced only by sources of infinite extent.
However, point sources create spherical waves. Locally, a spherical wave looks like a plane wave.
Thus, an understanding of plane waves is very important in the study of electromagnetics. 18

19. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
Assuming that the source is sinusoidal. We have 19

20. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
The electric and magnetic fields are given by 20

21. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
The argument of the cosine function is the called the instantaneous phase of the field: 21

22. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
The speed with which a constant value of instantaneous phase travels is called the phase velocity. For a lossless medium, it is equal to and denoted by the same symbol as the velocity of propagation. 22

23. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
The distance along the direction of propagation over which the instantaneous phase changes by 2p radians for a fixed value of time is the wavelength. 23

24. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
The wavelength is also the distance between every other zero crossing of the sinusoid.
Function vs. position at a fixed time l 24

25. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
Relationship between wavelength and frequency in free space:
Relationship between wavelength and frequency in a material medium: 25

26. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
b is the phase constant and is given by
rad/m 26

27. Uniform Plane Wave Solutions in the Time Domain (Cont’d)
In free space (vacuum):
free space wavenumber (rad/m) 27

28. Time-Harmonic Analysis
Sinusoidal steady-state (or time-harmonic) analysis is very useful in electrical engineering because an arbitrary waveform can be represented by a superposition of sinusoids of different frequencies using Fourier analysis.
If the waveform is periodic, it can be represented using a Fourier series.
If the waveform is not periodic, it can be represented using a Fourier transform. 28

29. Time-Harmonic Maxwell’s Equations in Differential Form for a Simple, Source-Free, Possibly Lossy Medium 29

30. Derivation of Helmholtz Equations for Electromagnetic Waves in a Simple, Source-Free, Possibly Lossy Medium 30

31. The Helmholtz equations are not independent.
Helmholtz Equations for Electromagnetic Waves in a Simple, Source-Free, Possibly Lossy Medium
The Helmholtz equations are not independent.
Usually we solve the electric field equation and determine H from E using Faraday’s law. 31

32. Uniform Plane Wave Solutions in the Frequency Domain
Assuming a plane wave solution of the form
The Helmholtz equation simplifies to 32

33. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
The propagation constant is a complex number that can be written as
attenuation constant (Np/m)
phase constant (rad/m)
(m-1) 33

34. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
a is the attenuation constant and has units of nepers per meter (Np/m).
b is the phase constant and has units of radians per meter (rad/m).
Note that in general for a lossy medium 34

35. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
The general solution to this wave equation is
wave traveling in the -z-direction
wave traveling in the +z-direction 35

36. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Converting the phasor representation of E back into the time domain, we have
We have assumed that C1 and C2 are real. 36

37. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
The corresponding magnetic field for the uniform plane wave is obtained using Faraday’s law: 37

38. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Evaluating H we have 38

39. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
We note that the intrinsic impedance h is a complex number for lossy media. 39

40. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Converting the phasor representation of H back into the time domain, we have 40

41. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
We note that in a lossy medium, the electric field and the magnetic field are no longer in phase.
The magnetic field lags the electric field by an angle of fh. 41

42. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Note that we have
These form a right-handed coordinate system
Uniform plane waves are a type of transverse electromagnetic (TEM) wave. 42

43. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Relationships between the phasor representations of electric and magnetic fields in uniform plane waves:
unit vector in direction of propagation 43

44. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Example:
Consider 44

45. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Snapshot of Ex+(z,t) at wt = 0 45

46. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Properties of the wave determined by the source:
amplitude
phase
frequency 46

47. Uniform Plane Wave Solutions in the Frequency Domain (Cont’d)
Properties of the wave determined by the medium are:
velocity of propagation (vp)
intrinsic impedance (h)
propagation constant constant (g=a+jb)
wavelength (l)
also depend on frequency 47

48. Dispersion
For a signal (such as a pulse) comprising a band of frequencies, different frequency components propagate with different velocities causing distortion of the signal. This phenomenon is called dispersion.
input signal
output signal 48

49. Plane Wave Propagation in Lossy Media
Assume a wave propagating in the +z-direction:
We consider two special cases:
Low-loss dielectric.
Good (but not perfect) conductor. 49

50. Plane Waves in a Low-Loss Dielectric
A lossy dielectric exhibits loss due to molecular forces that the electric field has to overcome in polarizing the material.
We shall assume that 50

51. Plane Waves in a Low-Loss Dielectric (Cont’d)
Assume that the material is a low-loss dielectric, i.e, the loss tangent of the material is small: 51

52. Plane Waves in a Low-Loss Dielectric (Cont’d)
Assuming that the loss tangent is small, approximate expressions for a and b can be developed.
wavenumber 52

53. Plane Waves in a Low-Loss Dielectric (Cont’d)
The phase velocity is given by 53

54. Plane Waves in a Low-Loss Dielectric (Cont’d)
The intrinsic impedance is given by 54

55. Plane Waves in a Low-Loss Dielectric (Cont’d)
In most low-loss dielectrics, er is more or less independent of frequency. Hence, dispersion can usually be neglected.
The approximate expression for a is used to accurately compute the loss per unit length. 55

56. Plane Waves in a Good Conductor
In a perfect conductor, the electromagnetic field must vanish.
In a good conductor, the electromagnetic field experiences significant attenuation as it propagates.
The properties of a good conductor are determined primarily by its conductivity. 56

57. Plane Waves in a Good Conductor
For a good conductor,
Hence, 57

58. Plane Waves in a Good Conductor (Cont’d)58

59. Plane Waves in a Good Conductor (Cont’d)
The phase velocity is given by 59

60. Plane Waves in a Good Conductor (Cont’d)
The intrinsic impedance is given by 60

61. Plane Waves in a Good Conductor (Cont’d)
The skin depth of material is the depth to which a uniform plane wave can penetrate before it is attenuated by a factor of 1/e.
We have 61

62. Plane Waves in a Good Conductor (Cont’d)
For a good conductor, we have 62

63. Wave Equations for Time-Harmonic Fields in Simple Medium63