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EEE 498/598
Overview of Electrical
Engineering
Lecture 10:
Uniform Plane Wave Solutions to
Maxwell’s Equations
1
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
Lecture 10
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
Lecture 10
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
Lecture 10
Time-Domain Maxwell’s
Equations in Differential Form
Kc  Ki
B
  E  K 
t
D
 H  J 
t
  D  qev
  B  qmv
Jc  Ji
5
Lecture 10
Time-Domain Maxwell’s Equations in
Differential Form for a Simple Medium
D E
B  H
J c   E Kc  m H
H
  E   m H  K i  
t
E
 H   E  J i 
t
6
E 
qev
H 
qmv


Lecture 10
Time-Domain Maxwell’s Equations in Differential Form for a
Simple, Source-Free, and Lossless Medium
J i  K i  0 qev  qmv  0    m  0
H
  E  
t
E
 H  
t
E  0
H  0
7
Lecture 10
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
Lecture 10
Derivation of Wave Equations for Electromagnetic Waves
in a Simple, Source-Free, Lossless Medium
0
    E    E    E
2
   H 
2 E
 
   2
t
t
0
    H    H    H
2
   E 
 H

   2
t
t
2
9
Lecture 10
Wave Equations for Electromagnetic Waves
in a Simple, Source-Free, Lossless Medium
 E
 E   2  0
t

2
2

 H
 H   2  0
t
2
2
10
The wave equations are
not independent.
Usually we solve the
electric field wave
equation and determine
H from E using
Faraday’s law.
Lecture 10
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
E  aˆ x Ex z, t 
11
Lecture 10
Uniform Plane Wave Solutions
in the Time Domain (Cont’d)

The wave equation for this field simplifies to
 2 Ex
 2 Ex
 
0
2
2
z
t

The general solution to this wave equation is
Ex  z, t   p1 z  v pt   p2 z  v pt 
12
Lecture 10
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
Lecture 10
Uniform Plane Wave Solutions in
the Time Domain (Cont’d)

The velocity of propagation is determined
solely by the medium:
vp 

1

The functions p1 and p2 are determined by
the source and the other boundary
conditions.
14
Lecture 10
Uniform Plane Wave Solutions in
the Time Domain (Cont’d)

Here we must have
H  aˆ y H y z, t 
where
H y  z, t  
1

p z  v t   p z  v t 
1
p
15
2
p
Lecture 10
Uniform Plane Wave Solutions in
the Time Domain (Cont’d)

 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
Lecture 10
Uniform Plane Wave Solutions
in the Time Domain (Cont’d)

In free space (vacuum):
v p  c  3 10 m/s
8
  120  377
17
Lecture 10
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
Lecture 10
Uniform Plane Wave Solutions
in the Time Domain (Cont’d)

Assuming that the source is sinusoidal. We
have


p1 z  v p t   C1 cos z  v p t   C1 cost  z 
v

p




p2 z  v p t   C2 cos z  v p t   C2 cost  z 
v

 p



vp
19
Lecture 10
Uniform Plane Wave Solutions
in the Time Domain (Cont’d)

The electric and magnetic fields are given
by
E x  z, t   C1 cost  z   C2 cost  z 
H y z, t  
1

C1 cost  z   C2 cost  z 
20
Lecture 10
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:
 z, t   t  z
21
Lecture 10
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.
t   0
t   z   0  z 

dz 
1
vp 
 
dt 

Lecture 10
22
Uniform Plane Wave Solutions in
the Time Domain (Cont’d)

The distance along the direction of
propagation over which the instantaneous
phase changes by 2 radians for a fixed
value of time is the wavelength.
  2   
23
2

Lecture 10
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
1
0.8
0.6

0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
2
4
24
6
8
10
12
14
16
18
Lecture 10
20
Uniform Plane Wave Solutions in
the Time Domain (Cont’d)

Relationship between wavelength and
frequency in free space:
c

f

Relationship between wavelength and
frequency in a material medium:

vp
f
25
Lecture 10
Uniform Plane Wave Solutions in
the Time Domain (Cont’d)

 is the phase constant and is given by
    

vp
rad/m
26
Lecture 10
Uniform Plane Wave Solutions in
the Time Domain (Cont’d)

In free space (vacuum):
   0 0 

c
 k0 
2
0
free space wavenumber
(rad/m)
27
Lecture 10
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
Lecture 10
Time-Harmonic Maxwell’s Equations in Differential Form for
a Simple, Source-Free, Possibly Lossy Medium
  E   j H
  H  j E
E  0
H  0

     j   j

m
     j   j

29
Lecture 10
Derivation of Helmholtz Equations for Electromagnetic
Waves in a Simple, Source-Free, Possibly Lossy Medium
0
    E    E    E
2
  j  H   2  E
0
    H    H    H

2
2
 j  E   2  H

30
2
Lecture 10
Helmholtz Equations for Electromagnetic Waves
in a Simple, Source-Free, Possibly Lossy Medium

 E  E  0
2
2

 H  H  0
2
2
31
The Helmholtz
equations are not
independent.
Usually we solve the
electric field equation
and determine H from
E using Faraday’s law.
Lecture 10
Uniform Plane Wave Solutions in the
Frequency Domain

Assuming a plane wave solution of the form
E  aˆ x Ex z 

The Helmholtz equation simplifies to
2
d Ex
2
  Ex  0
2
dz
32
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)

The propagation constant is a complex
number that can be written as
      j     j
2
(m-1)
attenuation
constant
(Np/m)
33
phase constant
(rad/m)
Lecture 10
Uniform Plane Wave Solutions in the
Frequency Domain (Cont’d)
  is the attenuation constant and has
units of nepers per meter (Np/m).
  is the phase constant and has units of
radians per meter (rad/m).
 Note that in general for a lossy medium
   
34
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)

The general solution to this wave equation is
Ex  z   C1e
 C1e
 z
 C2 e
 z  j z
e
 z
z
 C2 e e
j z
Ex  z 

Ex  z 

• wave traveling in
the -z-direction
• wave traveling in
the +z-direction
35
Lecture 10
Uniform Plane Wave Solutions in the
Frequency Domain (Cont’d)
 Converting the phasor representation of E
back into the time domain, we have

Ex  z, t   Re Ex  z e
 C1e
 z
jt

cost  z   C2e cost  z 
z
• We have assumed that C1 and C2 are real.
36
Lecture 10
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:
 E
  E  j H  H 
j
37
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)

Evaluating H we have
H y z  


Ce

1
1

E

1

x
38
 z
 C2 e
z

z   Ex z 

Lecture 10
Uniform Plane Wave Solutions in the
Frequency Domain (Cont’d)
 We note that the intrinsic impedance  is a
complex number for lossy media.
 e
39
j
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)

Converting the phasor representation of H
back into the time domain, we have

H y  z , t   Re H y z e

C1


jt

e  z cost  z   
C2

e z cost  z   
40
Lecture 10
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 .

41
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)

Note that we have

E  H  aˆ z

These form a righthanded coordinate
system
Uniform plane
waves are a type of
transverse
electromagnetic
(TEM) wave.
âE
â z
âH
42
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)

Relationships between the phasor
representations of electric and magnetic
fields in uniform plane waves:
H
1
aˆ p  E
unit vector in
direction of
propagation

E   aˆ p  H
43
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)

Example:
9
f  1 10 Hz  0  0.300 m
  2.5 0


  0

  0.01 S/m 

Consider
Ex  z , t   e

α  1.191 Np/m
  33.16 rad/m
 z
cost   z 
44
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)
Snapshot of Ex+(z,t) at t = 0
1
0.8
e
0.6
 z
0.4
x
E+ (z,t)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.5
1
1.5
2
2.5
z/0
45
3
3.5
4
4.5
5
Lecture 10
Uniform Plane Wave Solutions in the
Frequency Domain (Cont’d)

Properties of the wave determined by the
source:
 amplitude
 phase
 frequency
46
Lecture 10
Uniform Plane Wave Solutions in
the Frequency Domain (Cont’d)

Properties of the wave determined by
the medium are:
• also depend on
frequency
 velocity
of propagation (vp)
 intrinsic impedance ()
 propagation constant constant (=j)
 wavelength ()

vp
f
47

2

Lecture 10
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.
25
input signal
20
15
10
output signal
5
0
-5
0
100
200
300
48
400
500
600
Lecture 10
Plane Wave Propagation in Lossy
Media

Assume a wave propagating in the +zdirection:
Ex z , t   Ex 0 e


 z
cost  z 
We consider two special cases:
 Low-loss
dielectric.
 Good (but not perfect) conductor.
49
Lecture 10
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

r 
  


0
     j    1  j 
 

  1  j tan     r  0 1  j tan  

50
Lecture 10
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:
 
tan    1

51
Lecture 10
Plane Waves in a Low-Loss
Dielectric (Cont’d)

Assuming that the loss tangent is small, approximate
expressions for  and  can be developed.
  j   j  0 1  j tan  
1  x 1/ 2  1  x
2
tan  

 j  0   1  j
    j
2 

wavenumber
    0    r k 0  k
tan  k tan 
    0 

2
2
52
Lecture 10
Plane Waves in a Low-Loss
Dielectric (Cont’d)

The phase velocity is given by
 
c
vp   
 k
r
53
Lecture 10
Plane Waves in a Low-Loss
Dielectric (Cont’d)

The intrinsic impedance is given by



0

r

1 / 2
1  j tan  


tan 
j
tan   0

2
1

j

e


2
r


1  x 
1 / 2
x
 1
2
1  x   e x
54
Lecture 10
Plane Waves in a Low-Loss
Dielectric (Cont’d)
 In most low-loss dielectrics, r is more or
less independent of frequency. Hence,
dispersion can usually be neglected.
 The approximate expression for  is used
to accurately compute the loss per unit
length.
55
Lecture 10
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
Lecture 10
Plane Waves in a Good
Conductor

For a good conductor,

 1
 

Hence,

 j

57
Lecture 10
Plane Waves in a Good Conductor
(Cont’d)
 
  j   j    j  
 
 1  j 


j

2

2

2
58
Lecture 10
Plane Waves in a Good Conductor
(Cont’d)

The phase velocity is given by

2
vp  
 c


59
Lecture 10
Plane Waves in a Good Conductor
(Cont’d)

The intrinsic impedance is given by


j






j

1  j 
 j 45


e

2 
60
Lecture 10
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

e

1  
61
1

Lecture 10
Plane Waves in a Good Conductor
(Cont’d)

For a good conductor, we have

1

2


62
Lecture 10
Wave Equations for Time-Harmonic
Fields in Simple Medium
 1

 Ki 
2
     E   k0  r E       j 0 J i
 r

 r 
1

 Ji 
2
     E   k0  r E       j 0 K i
 r

 r 
k0    0  0
63
Lecture 10