Fundamentals of Electromagnetics
for Teaching and Learning:
A Two-Week Intensive Course for Faculty in
Electrical-, Electronics-, Communication-, and
Computer- Related Engineering Departments in
Engineering Colleges in India
by
Nannapaneni Narayana Rao
Edward C. Jordan Professor Emeritus
of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign, USA
Distinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, India
Program for Hyderabad Area and Andhra Pradesh Faculty
Sponsored by IEEE Hyderabad Section, IETE Hyderabad
Center, and Vasavi College of Engineering
IETE Conference Hall, Osmania University Campus
Hyderabad, Andhra Pradesh
June 3 – June 11, 2009
Workshop for Master Trainer Faculty Sponsored by
IUCEE (Indo-US Coalition for Engineering Education)
Infosys Campus, Mysore, Karnataka
June 22 – July 3, 2009
4-2
Module 4
Wave Propagation
in Free Space
4.1 Uniform Plane Waves in Time Domain
4.2 Sinusoidally Time-Varying Uniform Plane Waves
4.3 Polarization
4.4 Poynting Vector and Energy Storage
4-3
Instructional Objectives
23. Write the expression for a traveling wave function for a
set of specified characteristics of the wave
24. Obtain the electric and magnetic fields due to an infinite
plane current sheet of an arbitrarily time-varying uniform
current density, at a location away from it as a function of
time, and at an instant of time as a function of distance, in
free space
25. Find the parameters, frequency, wavelength, direction of
propagation of the wave, and the associated magnetic (or
electric) field, for a specified sinusoidal uniform plane
wave electric (or magnetic) field in free space
26. Write expressions for the electric and magnetic fields of a
uniform plane wave propagating away from an infinite
plane sheet of a specified sinusoidal current density, in
free space
4-4
Instructional Objectives (Continued)
27. Obtain the expressions for the fields due to an array of
infinite plane sheets of specified spacings and sinusoidal
current densities, in free space
28. Write the expressions for the fields of a uniform plane
wave in free space, having a specified set of
characteristics, including polarization
29. Express linear polarization and circular polarization as
superpositions of clockwise and counterclockwise circular
polarizations
30. Find the power flow and the electric and magnetic stored
energies associated with electric and magnetic fields
4.1 Uniform Plane Waves
in Time Domain
(EEE, Sec. 3.4; FEME, Secs. 4.1, 4.2, 4.4, 4.5)
4-6
Infinite Plane Current Sheet Source
JS   JS  t  ax
for z  0
Example: JS t   JS 0 cos t ax
4-7
B
×E  
t
D
×H  J +
t
For a current distribution having only an x-component of
current density that varies only with z,
ax
ay
0
0
Ex
Ey
az

B

z
t
Ez
ax
ay
0
0
Hx
Hy
az

D
J+
z
t
Hz
4-8
H y
Ey
Dx

 Jx 
z
t
Bx


z
t
H x Dy

z
t
By
Ex

z
t
Bz
0
t
Dz
0
t
The only relevant equations are:
By
Ex

z
t
Thus, E  Ex  z, t  ax
H y
Dx

 Jx 
z
t
H  H y  z, t  a y
4-9
In the free space on either side of the sheet, Jx = 0
By
H y
Ex

  0
z
t
t
H y
Dx
Ex

 0
z
t
t
2 Ex
  H y 
  0 
Combining, we get

2
z
z  t 
  H y 
  0 

t  z 
Ex 

  0   0

t 
t 
2 Ex
2 Ex
Wave Equation
 0 0 2
2
z
t
4-10
Solution to the Wave Equation
4-11
4-12

 

 A      f  t  z   
 B   g t  z   
    Af   t  z     g   t  z    


Ex  z , t   Af t  z 0 0  Bg t  z 0 0
Ex
z
0 0
0 0
0 0
2 Ex
z 2
0 0
2 Ex
 0 0 2
t
0 0
0 0
0 0
4-13


z 
z 
Ex  z , t   Af  t    Bg  t  




v
v
p
p




1
Where vp 

00
 3  108 m/s = c, velocity of light

f t  z vp represents a traveling wave propagating in the
+z-direction.


g t  z vp represents a traveling wave propagating in the
–z-direction.
4-14
E4.1: Examples of Traveling Waves
f  t  z vp    t  z 5 
2
f
t
t 0
1
5
1
25
1
0
1
1
vp 
 5 m/s
15
2
z
4-15
g  t  z vp   e
 2t  z
2 t  z 2
e
g
1
t
3
2
1
2
t0
1
0
1
1
vp 
 2 m/s
12
2
z
4-16
H y
Ex
From
 
,
0 t
z
H y
1 Ex

t
0 z
1

0 vp



z 
z 
 Af   t    Bg   t   
vp 
vp  



1 
H y  z , t    Af
0 
where 0 


z 
z 
 t     Bg  t  
vp  
vp 


0  0  Intrinsic impedance
 120  377 
4-17
Thus, the general solution is


z 
z 
Ex  z , t   Af  t    Bg  t  


vp 
vp 


1 
H y  z, t    Af
0 


z 
z 
 t     Bg  t  
vp  
vp 


For the particular case of the infinite plane current sheet in the
z = 0 plane, there can only be a () wave for z > 0 and a ()
wave for z < 0. Therefore,
 Af  t  z vp  ax for z  0
E  z, t   
 Bg  t  z vp  ax for z  0
 A
  f  t  z vp  a y for z  0

H  z, t    0
 B g  t  z vp  a y for z  0
 0
4-18
Applying Faraday’s law in integral form
to the rectangular closed path abcda in
the limit that the sides bc and da0,
Lim
bc 0
da 0
 b E d l  d E d l 
c
 a

Lim
bc 0
da 0
 d


B
d
S
 dt abcda

 ab   Ex z 0   dc   Ex z 0
0
Af t   Bgt   say, F t 
4-19
Therefore,

E  z, t   F  t


z
vp

H  z, t    F  t
0 
1

 ax for z

z
vp
0

 a y for z

0
Now, applying Ampere’s circuital law in integral form to the
rectangular closed path efgha in the limit that the sides fg and
he0,
Lim
fg 0
he0
 f H d l  h H d l
g
 e


Lim
fg  0
he  0
d

 efghe J d S  dt

efghe D d S 
4-20
ef H y z 0  hg H y z 0  ef  JS t 
 1

F t    F t   J S t 
0
 0

1
F t  
Thus, the solution is
0
2
J S t 

z 
E  z, t  
JS  t
ax for z 0

2 
vp 
1 
z 
H  z, t    JS  t
a y for z 0

2 
vp 
Uniform plane waves propagating away from the sheet to
either side with velocity vp = c.
0
4-21
In practice, there are no uniform plane waves. However,
many practical situations can be studied based on uniform
plane waves. For example, at large distances from physical
antennas and ground, the waves can be approximated as
uniform plane waves.
4-22
x
z
y

z 
E  z, t  
J S  t   ax
2 
vp 

z 
E  z, t  
J S  t   ax
2 
vp 
1 
z 
H  z, t    JS  t   a y
2 
vp 
1 
z 
H  z, t   JS  t   a y
2 
vp 
0
0
JS  t 
z=0
4-23
E4.2
z<0
z>0
JS  t 
z
z = 0
x
y
z
4-24
a Ex t  for z  300 m
b H y t  for z  450 m
4-25
c Ex  z for t  1s
d H y  z for t  2.5s
4-26
Review Questions
4.1. Outline the procedure for obtaining from the two
Maxwell’s equations the particular differential equations
for the special case of J = Jx(z, t)ax.
4.2. State the wave equation for the case of E = Ex(z, t)ax.
Describe the procedure for its solution.
4.3. What is a uniform plane wave? Why is the study of
uniform plane waves important?
4.4. Discuss by means of an example how a function
f(t – z/vp) represents a traveling wave propagating in the
positive z-direction with velocity vp.
4.5. Discuss by means of an example how a function
g(t + z/vp) represents a traveling wave propagating in the
negative z-direction with velocity vp.
4-27
Review Questions (Continued)
4.6. What is the significance of the intrinsic impedance of
free space? What is its value?
4.7. Summarize the procedure for obtaining the solution for
the electromagnetic field due to the infinite plane sheet of
uniform time-varying current density.
4.8. State and discuss the solution for the electromagnetic
field due to the infinite plane sheet of current density
Js(t) = – Js(t)ax for z = 0.
4-28
Problem S4.1. Writing expressions for traveling wave
functions for specified time and distance variations
4-29
Problem S4.2. Plotting field variations for a specified
infinite plane-sheet current source
4-30
Problem S4.3. Source and more field variations from a
given field variation of a uniform plane wave
4.2 Sinusoidally Time-Varying
Uniform Plane Waves
(EEE, Sec. 3.5; FEME, Secs. 4.1, 4.2, 4.4, 4.5)
4-32
Sinusoidal function of time
4-33
Sinusoidal Traveling Waves
f  t  z vp 
 cos   t  z vp     
 cos t   z    
g  t  z vp 
 cos   t  z vp     
 cos t   z    
where    vp   00
4-34
f  z, t   cos t   z 

t
4
f
t

2
1
t 0
0


2

1
z
4-35
g  z, t   cos t   z 
t

2
t

4
g
1
t 0
z


2
0

1
4-36
For JS  t    JS 0 cos t ax for z  0,
The solution for the electromagnetic field is
E
0 J S 0

cos  t

z vp ax for z >< 0
2
0 J S 0
=
cos t  z  ax for z >< 0
2
JS 0
H
cos  t z vp a y for z >< 0
2
JS 0
=
cos t  z  a y for z >< 0
2

where    vp   00

4-37
Three-dimensional depiction of wave propagation
4-38
Parameters and Properties
1. t
 z     Phase, 

2.   radian frequency =
t
 rate of change of phase with time
for a fixed value of z (movie)

f 
 frequency
2
= number of 2 radians of phase change
per second
4-39
3.   phase constant =

z
= magnitude of rate of change of phase with
distance z for a fixed value of t (still photograph)

4. vp  phase velocity =

 velocity with which a constant phase progresses
along the direction of propagation
follows from d t
 z   0 
4-40
5.  = wavelength =
2

 distance in which the phase changes by 2
for a fixed t
6. Note that
 2 f
vp  
f
 2 
  in m  f in MHz = 300
Ex
Ex
7. 0     
Hy
Hy
= Ratio of the amplitude of E to the amplitude
of H for either wave
4-41
8. E × H (Poynting Vector, P)
ax × a y  az for (+) wave

ax × a y  az for () wave
is in the direction of propagation.
x
x
E
E
z
H
y
P
P
y
H
z
4-42
E4.3
Consider E  37.7 cos  6  108 t  2 z  a y V m.
Then

  6  10 , f 
 3  108 Hz
2
2
  2 ,  
 1m

8
6  108
vp 
 3  108 m s
2
Direction of propagation is –z.
H  0.1 cos  6  108 t  2 z  ax A m
4-43
E4.4 Array of Two Infinite Plane Current Sheets
JS1
z0
JS 2
z 4
J S1   J S 0 cos t ax for z  0
J S 2   J S 0 sin t ax for z   4
For J S 1 ,
0 J S 0
 2 cos t   z  ax for z  0
E1  
0 J S 0 cos t   z  a for z  0
x
 2
4-44
For JS 2 ,
0 J S 0

 


sin t    z    ax for z 

4 
4

 2

E2  
0 J S 0 sin t    z     a for z  

 x

 2
4
4




0 J S 0



 2 sin  t   z  2  ax for z  4




0 J S 0 sin  t   z    a for z  

 x
 2
2
4


0 J S 0
 2 cos t   z  ax for z  4

 0 J S 0 cos t   z  a for z  
x

2
4
4-45
For both sheets,
E = E1  E2
 E1    E2 
for z   4
z 0
z  4


  E1 z 0   E2 z  4 for 0  z   4

 E1 z 0   E2 z  4 for z   4
0 J S 0 cos t   z  ax for z   4

 0 J S 0 sin t sin  z ax for 0  z   4
0
for z  0

No radiation to one side of the array.
“Endfire” radiation pattern.
4-46
Depiction of superposition of the two waves
4-47
Review Questions
4.9. Why is it important to give special consideration for
sinusoidal functions of time and hence sinusoidal
waves?
4.10. Discuss the quantities ω, β, and vp associated with
sinusoidally time-varying uniform plane waves.
4.11. Define wavelength. What is the relationship among
wavelength, frequency, and phase velocity? What is the
wavelength in free space for a frequency of 15 MHz?
4.12. How is the direction of propagation of a uniform plane
wave related to the directions of its fields?
4.13. What is the direction of the magnetic field of a uniform
plane wave having its electric field in the positive zdirection and propagating in the positive x-direction?
4-48
Review Questions (Continued)
4.14. Discuss the principle of antenna array, with the aid of
an example.
4.15. What should be the spacing and the relative phase angle
of the current densities for an array of two infinite,
plane, parallel current sheets of uniform densities, equal
in amplitude, to confine the radiation to the region
between the two sheets?
4-49
Problem S4.4. Finding parameters and the electric field for
a specified sinusoidal uniform plane wave magnetic field
4-50
Problem S4.5. Apparent wavelengths of a uniform plane
wave propagating in an arbitrary direction
4-51
Problem S4.5. Apparent wavelengths of a uniform plane
wave propagating in an arbitrary direction (Continued)
4-52
Problem S4.5. Apparent wavelengths of a uniform plane
wave propagating in an arbitrary direction (Continued)
4-53
Problem S4.6. Ratio of amplitudes of the electric field on
either side of an array of two infinite plane current sheets
4-54
4.3 Polarization
(EEE, Sec. 3.6; FEME, Sec. 1.4, 4.5)
4-55
Sinusoidal function of time
4-56
Polarization is the characteristic which
describes how the position of the tip of the
vector varies with time.
Linear Polarization:
Tip of the vector
describes a line.
Circular Polarization:
Tip of the vector
describes a circle.
4-57
Elliptical Polarization:
Tip of the vector
describes an ellipse.
(i) Linear Polarization
F1  F1 cos (t   ) a x
Magnitude varies
sinusoidally with time
Direction remains
along the x axis
 Linearly polarized in the x direction.
4-58
Linear polarization
4-59
F2  F2 cos (t   ) a y
Magnitude varies
sinusoidally with time
Direction remains
along the y axis
 Linearly polarized in the y direction.
If two (or more) component linearly polarized
vectors are in phase, (or in phase opposition), then
their sum
vector is also linearly polarized.
Ex: F  F1 cos (t   ) a x  F2 cos (t   ) a y
4-60
Sum of two linearly polarized vectors in phase
is a linearly polarized vector
4-61
y
F2

F

F1
x
F2 cos (t   )
–1
 tan
F1 cos (t   )
F2
–1
 tan
F1
 constant
(ii) Circular Polarization
If two component linearly polarized vectors are
(a) equal to amplitude
(b) differ in direction by 90˚
(c) differ in phase by 90˚,
then their sum vector is circularly polarized.
4-62
Circular Polarization
4-63
Example:
F  F1 cos t ax  F1 sin t a y
F 
 F1
cos t    F1 sin t 
2
2
 F1 , constant
  tan
 tan
1
1
F1 sin t
F1 cos t
 tan t   t
y
F2
F

F1
x
4-64
(iii) Elliptical Polarization
In the general case in which either of (i) or (ii) is not
satisfied, then the sum of the two component
linearly polarized vectors is an elliptically polarized
vector.
Example: F  F1 cos t a x  F2 sin t a y
y
F2
F
F1
x
4-65
Example: F  F0 cos t ax  F0 cos t   4 ay
y
F0
F2
F
/4
F1 F0 x
–F0
–F0
4-66
D3.17
F1  F0 cos  2 108t  2 z  ax
F2  F0 cos  2 108t  3 z  a y
F1 and F2 are equal in amplitude (= F0) and differ in
direction by 90˚. The phase difference (say ) depends
on z in the manner –2 z – (–3 z) =  z.
(a) At (3, 4, 0),  =  (0) = 0.
 F1  F2 
is linearly polarized.
(b) At (3, –2, 0.5),  =  (0.5) = 0.5 .
 F1  F2  is circularly polarized.
4-67
(c) At (–2, 1, 1),  =  (1) = .
 F1  F2 
is linearly polarized.
(d) At (–1, –3, 0.2) =  =  (0.2) = 0.2.
 F1  F2 
is elliptically polarized.
4-68
Clockwise and Counterclockwise
Polarizations
In the case of circular and elliptical polarizations for
the field of a propagating wave, one can distinguish
between clockwise (cw) and counterclockwise (ccw)
polarizations. If the field vector in a constant phase
plane rotates with time in the cw sense, as viewed
along the direction of propagation of the wave, it is
said to be cw- or right-circularly (or elliptically)
polarized. If it rotates in the ccw sense, it is said to
be ccw- or left- circularly (or elliptically) polarized.
4-69
For example, consider the circularly polarized electric field
of a wave propagating in the +z-direction, given by
E  E0 cos t   z  ax  E0 sin t   z  ay
Then, considering the time variation of the field vector in
the z = 0 plane, we note that for t  0, E  E0ax , and for
t   2 , E  E0a y .
Since ax × a y  az , the polarization is cw- or right-circular.
If E  E0 cos t   z  ax  E0 sin t   z  a y , then the
polarization is ccw- or left-circular.
4-70
Review Questions
4.16. A sinusoidally time-varying vector is expressed in
terms of its components along the x-, y-, and z- axes.
What is the polarization of each of the components?
4.17. What are the conditions for the sum of two linearly
polarized sinusoidally time-varying vectors to be
circularly polarized?
4.18. What is the polarization for the general case of the sum
of two sinusoidally time-varying linearly polarized
vectors having arbitrary amplitudes, phase angles, and
directions?
4.19. Discuss clockwise and counterclockwise circular and
elliptical polarizations associated with sinusoidally
time-varying uniform plane waves.
4-71
Problem S4.7. Expressing uniform plane wave field in
terms of right- and left- circularly polarized components
4-72
Problem S4.8. Finding the polarization parameters for an
elliptically polarized uniform plane wave field
4-73
4.4 Power Flow
and Energy Storage
(EEE, Sec. 3.7; FEME, Sec. 4.6)
4-74
Consider the quantity E× H . Then, from a vector identity,
  E× H   H   × E  E   × H 
Substituting
B
t
D
D
×H  J 
 J0 
t
t
×E  
where J0 represents source current density, we have
D
B
H
t
t
 1
 1
 E J0   0 E 2     0 H 2     E× H 
t  2
 t  2

 E× H   E J0  E
4-75
Performing volume integration on both sides, and using the
divergence theorem for the last term on the right side, we get

  E J0  dv 
t
V

 1  E 2  dv
2 0 
V

 1
2

 0 H  dv 

t V  2




P dS
S
where we have defined P  E × H , known as the Poynting
vector. The equation is known as the Poynting’s Theorem.
4-76
Poynting’s Theorem


 E J0  dv 
V
Source
power density,
(power per
unit volume),
W/m3
 1
 1
2
2

E
dv


H
dv  P d S




0
0
t  2
t  2


V
V
S


Electric stored
energy density,
J/m3
Magnetic stored
energy density,
J/m3

Power flow
out of S
4-77
Interpretation of Poynting’s Theorem
Poynting’s Theorem says that the power delivered to the
volume V by the current source J0 is accounted for by the sum
of the time rates of increase of the energies stored in the
electric and magnetic fields in the volume, plus another term,
which we must interpret as the power carried by the
electromagnetic field out of the volume V, for conservation of
energy to be satisfied. It then follows that the Poynting vector
P has the meaning of power flow density vector associated
with the electromagnetic field. We note that the units of E x H
are volts per meter times amperes per meter, or watts per
square meter (W/m2) and do indeed represent power flow
density.
4-78
In the case of the infinite plane sheet of current, note that the
electric field adjacent to and on either side of it is directed
opposite to the current density. Hence, some work has to be
done by an external agent (source) for the current to flow, and
 E J0  represents the power density (per unit volume)
associated with this work.


P  E × H  Power flow density W m2 associated
with the electromagnetic field
1
we  0 E 2  Energy density J m3 stored
2
in the electric field
1
wm  0 H 2  Energy density J m3 stored
2
in the magnetic field




4-79
Review Questions
4.20. What is the Poynting vector? What is the physical
interpretation of the Poynting vector over a closed
surface?
4.21. State Poynting’s theorem. How is it derived from
Maxwell’s curl equations?
4.22. Discuss the interpretation of Poynting’s theorem.
4.23. What are the energy densities associated with electric
and
magnetic fields?
4.24. Discuss how fields far from a physical antenna vary
inversely with distance from the antenna.
4-80
Problem S4.9. Finding the Poynting vector and power
radiated for specified radiation fields of an antenna
4-81
Problem S4.10. Finding the electric field and magnetic field
energies stored in a parallel-plate resonator
4-82
Problem S4.10. (Continued)
4-83
Problem S4.11. Finding the work associated with
rearranging a charge distribution
The End