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

1
Fundamentals of Electromagneticsfor Teaching and
LearningA Two-Week Intensive Course for Faculty
inElectrical-, 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

2
Program for Hyderabad Area and Andhra Pradesh
FacultySponsored by IEEE Hyderabad Section, IETE
Hyderabad Center, and Vasavi College of
EngineeringIETE Conference Hall, Osmania
University CampusHyderabad, Andhra PradeshJune
3 June 11, 2009 Workshop for Master Trainer
Faculty Sponsored byIUCEE (Indo-US Coalition for
Engineering Education)Infosys Campus, Mysore,
KarnatakaJune 22 July 3, 2009
3

• Module 6
• Statics, Quasistatics, and Transmission Lines
• 6.1 Gradient and electric potential
• 6.2 Poissons and Laplaces equations
• 6.3 Static fields and circuit elements
• 6.4 Low-frequency behavior via quasistatics
• 6.5 Condition for the validity of the quasistatic
  approximation
• 6.6 The distributed circuit concept and the
  transmission-line

4
Instructional Objectives

• 42. Understand the geometrical significance of
  the gradient
• operation
• 43. Find the static electric potential due to a
  specified charge
• distribution by applying superposition in
  conjunction
• with the potential due to a point charge,
  and further find
• the electric field from the potential
• 44. Obtain the solution for the potential between
  two
• conductors held at specified potentials,
  for one-
• dimensional cases (and the region between
  which is filled
• with a dielectric of uniform or nonuniform
  permittivity,
• or with multiple dielectrics) by using the
  Laplaces
• equation in one dimension, and further find
  the
• capacitance per unit area (Cartesian) or
  per unit length
• (cylindrical) or capacitance (spherical) of
  the
• arrangement

5
Instructional Objectives (Continued)

• 45. Perform static field analysis of arrangements
  consisting
• of two parallel plane conductors for
  electrostatic,
• magnetostatic, and electromagnetostatic
  fields
• 46. Perform quasistatic field analysis of
  arrangements
• consisting of two parallel plane conductors
  for
• electroquastatic and magnetoquasistatic
  fields
• 47. Understand the condition for the validity of
  the quasistatic
• approximation and the input behavior of a
  physical
• structure for frequencies beyond the
  quasistatic
• approximation
• 48. Understand the development of the
  transmission-line
• (distributed equivalent circuit) from the
  field solutions
• for a given physical structure and obtain
  the transmission-
• line parameters for a line of arbitrary
  cross section by
• using the field mapping technique

6
6.1 Gradient and Electric Potential (EEE, Secs.
5.1, 5.2 FEME, Sec. 6.1)
7
Gradient and the Potential Functions
8
6-7
B can be expressed as the curl of a vector.
Thus
A is known as the magnetic vector potential.
Then
9
F is known as the electric scalar potential.
is the gradient of F.
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11
6-10
Basic definition of
For a constant ? surface, d? 0. Therefore
is normal to the surface.
12
6-11
Thus, the magnitude of at any point P is
the rate of increase of ? normal to the surface,
which is the maximum rate of increase ? at that
point. Thus
Useful for finding unit normal vector to the
surface.
13
6-12
D5.1 Finding unit normal vectors to the surface

at several points
14
6-13
6-13
15
6-14
(1)
(2)
(3)
(4)
(4)
(1)
(3)
16
6-15
(2)
Potential function equations
17
Laplacian of scalar
Laplacian of vector
In Cartesian coordinates,
18
6-17
For static fields,
But,
also known as the potential difference between A
and B, for the static case.
19
Given the charge distribution, find V using
superposition. Then find E using the above.
For a point charge at the origin,
since
agrees with the previously known result.
20
Thus for a point charge at an arbitrary location P
P
R
Q
P5.9
21
Considering the element of length dz? at (0, 0,
z?), we have
Using
22
6-21
23
Magnetic vector potential due to a current element
P
R
Analogous to
24
Review Questions

• 6.1. What is the divergence of the curl of a
  vector?
• 6.2. What is the expansion for the gradient of a
  scalar in
• Cartesian coordinates? When can a vector
  be expressed
• as the gradient of a scalar?
• 6.3. Discuss the basic definition of the gradient
  of a scalar.
• 6.4. Discuss the application of the gradient
  concept for the
• determination of unit vector normal to a
  surface.
• 6.5. Define electric potential. What is its
  relationship to the
• electric field intensity?
• 6.6. Distinguish between voltage as applied to
  time-varying
• fields and potential difference.
• 6.7. What is the electric potential due to a
  point charge?
• Discuss the determination of electric
  potential due to a
• charge distribution.

25
Review Questions (Continued)

• 6.8. What is the Laplacian of a scalar? What is
  the expansion
• for the Laplacian of a scalar in Cartesian
  coordinates?
• 6.9. What is the magnetic vector potential? How
  is it related
• to the magnetic flux density?

26
Problem S6.1. Finding the gradient of a
two-dimensional function and associated discussion
27
Problem S6.2. Finding the angle between two
plane surfaces, by using the gradient concept
28
Problem S6.3. Finding the image charge(s) for a
point charge in the presence of a conductor
29
Problem S6.3. Finding the image charge(s) for a
point charge in the presence of a conductor
(Continued)
30
6.2 Poissons and Laplaces Equations (EEE, Sec.
5.3 FEME, Sec. 6.2)
31
Poissons Equation
For static electric field,
Then from
If e is uniform,
Poissons equation
32
If e is nonuniform, then using
Thus
Assuming uniform e, we have
For the one-dimensional case of V(x),
33
D5.7
Anode, x d V V0
Vacuum Diode
Cathode, x 0 V 0
(a)
34
(b)
35
(c)
36
6-35
Laplaces Equation
If r 0, Poissons equation becomes
Let us consider uniform e first.
E6.1. Parallel-plate capacitor
x d, V V0
x 0, V 0
37
Neglecting fringing of field at edges,
General solution
38
Boundary conditions
Particular solution
39
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area of plates
For nonuniform e,
For
41
E6.2
x d, V V0
x 0, V 0
42
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43
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44
Review Questions

• 6.10. State Poissons equation for the electric
  potential. How
• is it derived?
• 6.11. Outline the solution of the Poissons
  equation for the
• potential in a region of known charge
  density varying in
• one dimension.
• 6.12. State Laplaces equation for the electric
  potential. In
• what regions is it valid?
• 6.13. Outline the solution of Laplaces equation
  in one
• dimension by considering a
  parallel-plate arrangement.
• 6.14. Outline the steps in the determination of
  the capacitance
• of a parallel-plate capacitor.

45
Problem S6.4. Solution of Poissons equation for
a space charge distribution in Cartesian
coordinates
46
Problem S6.5. Finding the capacitance of a
spherical capacitor with a dielectric of
nonuniform permittivity
47
6.3 Static Fields and Circuit Elements(EEE, Sec.
5.4 FEME, Sec. 6.3)
48
Classification of Fields
6-47
Static Fields ( No time variation )

Static electric, or electrostatic fields Static
magnetic, or magnetostatic fields Electromagnetost
atic fields
Dynamic Fields (Time-varying) Quasistatic Fields
(Dynamic fields that can be analyzed as though
the fields are static) Electroquasistatic
fields Magnetoquasistatic fields
49
Static Fields
6-48
6-48
For static fields, , and the
equations reduce to


50
6-49
Solution for Potential and Field
Solution for charge distribution
Solution for point charge
Electric field due to point charge
51
Laplaces Equation and One-Dimensional Solution
6-50
For ????????Poissions equation reduces to
Laplaces equation
52
Example of Parallel-Plate ArrangementCapacitance
6-51
53
6-52
Electrostatic Analysis of Parallel-Plate
Arrangement
Capacitance of the arrangement, F
54
Magnetostatic Fields
6-53


ò
ò
H
d
l

J
d
S
S
C

B
d
S

0
ò
S
Poissons equation for magnetic vector potential
55
6-54
Solution for Vector Potential and Field
Solution for current distribution
Solution for current element
Magnetic field due to current element
?2A 0
For current-free region
56
Example of Parallel-Plate ArrangementInductance
6-55
57
6-56
Magnetostatic Analysis of Parallel-Plate
Arrangement
58
Magnetostatic Analysis of Parallel-Plate
Arrangement (Continued)
Inductance of the arrangement, H
59
Electromagnetostatic Fields

E
d
l

0
ò
C



H
d
l

ò
J
d
S
ò

s
E
d
S
ò
c
C
S
S

D
d
S

0
ò
S

B
d
S

0
ò
S
60
Example of Parallel-Plate Arrangement
6-59
61
6-60
Electromagnetostatic Analysis of Parallel-Plate
Arrangement
62
Electromagnetostatic Analysis of Parallel-Plate
Arrangement (Continued)
Conductance, S
Resistance, ohms
63
6-62
Electromagnetostatic Analysis of Parallel-Plate
Arrangement (Continued)
0
1


z
æ
ö

ò
è
ø
I
l



z
l
c
Internal Inductance
64

Electromagnetostatic Analysis of Parallel-Plate
Arrangement (Continued)
Alternatively, from energy considerations,
Equivalent Circuit
65
Review Questions

• 6.15. Discuss the classification of fields as
  static, dynamic,
• and quasistatic fields.
• 6.16. State Maxwells equations for static fields
  in (a) integral
• form, and (b) differential form.
• 6.17. Outline the steps involved in the
  electrostatic field
• analysis of a parallel-plate structure
  and the
• determination of its capacitance.
• 6.18. Outline the steps involved in the
  magnetostatic field
• analysis of a parallel-plate structure
  and the
• determination of its inductance.
• 6.19. Outline the steps involved in the
  electromagnetostatic
• field analysis of a parallel-plate
  structure and the
• determination of its circuit equivalent.
• 6.20. Explain the term, internal inductance.

66
Problem S6.6. Finding the internal inductance
per unit length of a cylindrical conductor
arrangement
67
6.4 Low Frequency Behaviorvia Quasistatics
(EEE, Sec. 5.5 FEME, Sec. 6.4)
68
Quasistatic Fields
For quasistatic fields, certain features can be
analyzed as though the fields were static. In
terms of behavior in the frequency domain, they
are low-frequency extensions of static fields
present in a physical structure, when the
frequency of the source driving the structure is
zero, or low-frequency approximations of
time-varying fields in the structure that are
complete solutions to Maxwells equations. Here,
we use the approach of low-frequency extensions
of static fields. Thus, for a given structure,
we begin with a time- varying field having the
same spatial characteristics as that of the
static field solution for the structure and
obtain field solutions containing terms up to
and including the first power (which is
the lowest power) in w for their amplitudes.
69
Electroquasistatic Fields
J
H
S
I
(
t
)
1








g
y
z
x

0
E
????


0
x
x

d








z
z

0
z


l
70
6-69
Electroquasistatic Analysis of Parallel-Plate
Arrangement
71
Electroquasistatic Analysis of Parallel-Plate
Arrangement (Continued)


I
(
t
)

w
H
g
y
1

-
z
l
e
w
l
æ
ö

-
w
V
sin
w
t
è
ø
0
d
dV
(
t
)
g

C
d
t
where
72
Electroquasistatic Analysis of Parallel-Plate
Arrangement (Continued)


P

wd
E
H
in
y
x
0
1
z

0
e
wl
æ
ö
2

-
w
V
sin
w
t
cos
w
t
è
ø
0
d
d
1
2
æ
ö

CV
è
ø
g
dt
2
73
Magnetoquasistatic Fields
????
74
6-73
Magnetoquasistatic Analysis of Parallel-Plate
Arrangement
75
Magnetoquasistatic Analysis of Parallel-Plate
Arrangement (Continued)


V
(
t
)

d
E
g
x
1
z

-
l
m
dl
æ
ö

-
w
I
sin
w
t
0
è
ø
w
dI
(
t
)
g

L
dt
where
76
Magnetoquasistatic Analysis of Parallel-Plate
Arrangement (Continued)


P

wd
E
H
in
x
1
y
0

-
z
l
m
d
l
æ
ö
2

-
w
I
sin
w
t
cos
w
t
0
è
ø
w
d
1
æ
ö
2

LI
è
ø
g
dt
2
77
Quasistatic Fields in a Conductor
??
????
78
6-77
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor
79
6-78
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor (Continued)
80
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor (Continued)
V
w
m
s
V
(
)
2
2
0
0
E

cos
w
t
-
z
-
l
sin
w
t
x
d
2
d
81
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor (Continued)


I

w
H
g
y

-
z
l
2
3
æ
ö
m
s
wl
s
wl
e
wl


j
w
-
j
w
V
ç

g
d
d
3
d
è
ø
2
I
æ
ö
s
wl
m
s
l
e
wl
g
Y


j
w

1
-
j
w
ç

in
d
d
3
è
ø
V
g
e
wl
1

j
w

(
)
d
m
s
l
2
d
1

j
w
3
s
wl
82
6-81
Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor (Continued)
Equivalent Circuit
83
Review Questions

• 6.21. What is meant by the quasistatic extension
  of the static
• field in a physical structure?
• 6.22. Outline the steps involved in the
  electroquasistatic field
• analysis of a parallel-plate structure
  and the
• determination of its input behavior.
  Compare the input
• behavior with the electrostatic case.
• 6.23. Outline the steps involved in the
  magnetoquasistatic
• field analysis of a parallel-plate
  structure and the
• determination of its input behavior.
  Compare the input
• behavior with the magnetostatic case.
• 6.24. Outline the steps involved in the
  quasistatic field
• analysis of a parallel-plate structure
  with a conducting
• slab between the plates and the
  determination of its
• input behavior. Compare the input
  behavior with the
• electromagnetostatic case.

84
Problem S6.7. Frequency behavior of a capacitor
beyond the quasistatic approximation
85
Problem S6.7. Frequency behavior of a capacitor
beyond the quasistatic approximation (Continued)
86
6.5 Condition for the validity ofthe quasistatic
approximation (EEE, Sec. 5.5 FEME, Secs. 6.5,
7.1)
87
We have seen that quasistatic field analysis of a
physical structure provides information
concerning the low-frequency input behavior of
the structure. As the frequency is increased
beyond that for which the quasistatic
approximation is valid, terms in the infinite
series solutions for the fields beyond the
first-order terms need to be included. While one
can obtain equivalent circuits for frequencies
beyond the range of validity of the quasistatic
approximation by evaluating the higher order
terms, we shall here obtain the exact solution
by resorting to simultaneous solution of
Maxwells equations to find the condition for
the validity of the quasistatic approximation,
and further investigate the behavior for
frequencies beyond the quasistatic approximation.
We shall do this by considering the
parallel-plate structure, and obtaining the
wave solutions, which will then lead us to the
distributed circuit concept and the
transmission-line.
88
Wave Equation
6-87

For the one-dimensional case of
One-dimensional wave equation
89
Solution to the One-Dimensional Wave Equation
6-88
Traveling wave propagating in the z direction
90
Solution to the One-Dimensional Wave Equation
6-89
Traveling wave propagating in the z direction
91
6-90
General Solution in Phasor Form

f
f

j
j
A

Ae
,
B

Be
,
Phase constant
,
Phase velocity
,
Intrinsic impedance
92
6-91
Example of Parallel-Plate Structure Open-Circuited
at the Far End
?s
????
H

0
at
z

0
ü
y
ï
B.C.
V
ý
g
E

at
z


??l
ï
x
d
þ
93
Standing Wave Patterns (Complete Standing Waves)
94
Complete Standing Waves
Complete standing waves are characterized by pure
half-sinusoidal variations for the amplitudes of
the fields. For values of z at which the
electric field amplitude is a maximum, the
magnetic field amplitude is zero, and for values
of z at which the electric field amplitude is
zero, the magnetic field amplitude is a maximum.
The fields are also out of phase in time, such
that at any value of z, the magnetic field and
the electric field differ in phase by t p /
2w.
95
Input Admittance
j
w
V


g
I

w
H

tan
b
l
g
y
h
d

-
z
l
3
5
é
ù
(
b
l
)
2
(
b
l
)
w
Y

j
b
l



L
ê
ú
in
h
d
3
15
ë
û

For ?l ltlt 1,
96
Condition for the Validity of the Quasistatic
Approximation
The condition bl ltlt 1 dictates the range of
validity for the quasistatic approximation for
the input behavior of the structure. In terms of
the frequency f of the source, this condition
means that f ltlt vp/2p?l, or in terms of the
period T 1/f, it means that T gtgt 2p?(l/vp).
Thus, quasistatic fields are low-frequency
approximations of time-varying fields that are
complete solutions to Maxwells equations, which
represent wave propagation phenomena and can be
approximated to the quasistatic character only
when the times of interest are much greater than
the propagation time, l/vp, corresponding to the
length of the structure. In terms of space
variations of the fields at a fixed time, the
wavelength l ( 2p/?b ), which is the distance
between two consecutive points along the
direction of propagation between which the phase
difference is 2p, must be such that l ltlt l /2p
thus, the physical length of the structure must
be a small fraction of the wavelength.
97
For frequencies slightly beyond the approximation
?l ltlt1,
6-96
98
In general,
Y
i
n
c
a
p
a
c
i
t
i
v
e
3
v
5
v
v
p
p
p
4
l
4
l
4
l
f
0
v
v
3
v
p
p
p
2
l
l
2
l
Y
i
n
i
n
d
u
c
t
i
v
e
99
Example of Parallel-Plate Structure Short-Circuite
d at the Far End
?s
?????
100
Standing Wave Patterns (Complete Standing Waves)
101
Input Impedance
3
5
é
ù
h
d
(
b
l
)
2
(
b
l
)
Z

j
b
l



L
ê
ú
in
w
3
15
ë
û

For ?l ltlt 1,
102
For frequencies slightly beyond the approximation
?l ltlt1,
103
In general,
f
104
Review Questions

• 6.25. Outline the steps in the solution for the
  electromagnetic
• field in a parallel-plate structure
  open-circuited at the
• far end.
• 6.26. What are complete standing waves? Discuss
  their
• characteristics.
• 6.27. What is the input admittance of a a
  parallel-plate
• structure open-circuited at the far end?
  Discuss its
• variation with frequency.
• 6.28. State and discuss the condition for the
  validity of the
• quasistatic approximation.
• 6.29. Outline the steps in the solution for the
  electromagnetic
• field in a parallel-plate structure
  short-circuited at the
• far end.
• 6.30. What is the input impedance of a a
  parallel-plate
• structure short-circuited at the far
  end? Discuss its
• variation with frequency.

105
Problem S6.8. Frequency behavior of a
parallel-plate structure from input impedance
considerations
106
Problem S6.8. Frequency behavior of a
parallel-plate structure from input impedance
considerations (Continued)
107
6.6 The Distributed Circuit Concept and the
Transmission Line (EEE, Secs. 6.1, 11.5 FEME,
Secs. 6.5, 6.6)
108
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134
Review Questions

• 6.31. Discuss the phenomenon taking place in a
  parallel-plate
• structure at any arbitrary frequency.
• 6.32. How is the voltage between the two
  conductors in a
• given cross-sectional plane of a
  parallel-plate
• transmission line related to the
  electric field in that
• plane?
• 6.33. How is the current flowing on the plates
  across a
• given cross-sectional plane of a
  parallel-plate
• transmission line related to the
  magnetic field in that
• plane?
• 6.35. Discuss transverse electromagnetic waves.
• 6.36. What are transmission-line equations? How
  are they
• derived from Maxwells equations?

135
Review Questions

• 6.37. Discuss the concept of the distributed
  equivalent circuit.
• How is it obtained from the
  transmission-line
• equations?
• 6.38. Discuss the solutions for the
  transmission-line equations
• for the voltage and current along a
  line.
• 6.39. Explain the characteristic impedance of a
• transmission line.
• 6.40. Discuss the relationship between the
  transmission-line
• parameters.
• 6.41. What are the transmission-line parameters
  for a parallel-
• plate line?
• 6.42. Describe the curvilinear squares technique
  of finding
• the line parameters for a line with an
  arbitrary cross
• section.

136
Problem S6.9. Transmission-line equations and
power flow from the geometry of a coaxial cable
137
Problem S6.10. Application of the curvilinear
squares technique for an eccentric coaxial cable
138
The End