Title: Fundamentals of Electromagnetics for Teaching and Learning: A TwoWeek Intensive Course for Faculty in Electrical, Electronics, Communication, and Computer Related Engineering Departments in Engineering Colleges in India
1Fundamentals of Electromagneticsfor Teaching and
LearningA TwoWeek 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 UrbanaChampaign, USA
 Distinguished Amrita Professor of Engineering
 Amrita Vishwa Vidyapeetham, India
2Program 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 (IndoUS 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 Lowfrequency behavior via quasistatics
 6.5 Condition for the validity of the quasistatic
approximation  6.6 The distributed circuit concept and the
transmissionline
4Instructional 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
5Instructional 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
transmissionline  (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
66.1 Gradient and Electric Potential (EEE, Secs.
5.1, 5.2 FEME, Sec. 6.1)
7Gradient and the Potential Functions
867
B can be expressed as the curl of a vector.
Thus
A is known as the magnetic vector potential.
Then
9F is known as the electric scalar potential.
is the gradient of F.
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11610
Basic definition of
For a constant ? surface, d? 0. Therefore
is normal to the surface.
12611
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.
13612
D5.1 Finding unit normal vectors to the surface
at several points
14613
613
15614
(1)
(2)
(3)
(4)
(4)
(1)
(3)
16615
(2)
Potential function equations
17Laplacian of scalar
Laplacian of vector
In Cartesian coordinates,
18617
For static fields,
But,
also known as the potential difference between A
and B, for the static case.
19Given 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.
20Thus for a point charge at an arbitrary location P
P
R
Q
P5.9
21Considering the element of length dz? at (0, 0,
z?), we have
Using
22621
23Magnetic vector potential due to a current element
P
R
Analogous to
24Review 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
timevarying  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.
25Review 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?
26Problem S6.1. Finding the gradient of a
twodimensional function and associated discussion
27Problem S6.2. Finding the angle between two
plane surfaces, by using the gradient concept
28Problem S6.3. Finding the image charge(s) for a
point charge in the presence of a conductor
29Problem S6.3. Finding the image charge(s) for a
point charge in the presence of a conductor
(Continued)
306.2 Poissons and Laplaces Equations (EEE, Sec.
5.3 FEME, Sec. 6.2)
31Poissons Equation
For static electric field,
Then from
If e is uniform,
Poissons equation
32If e is nonuniform, then using
Thus
Assuming uniform e, we have
For the onedimensional case of V(x),
33D5.7
Anode, x d V V0
Vacuum Diode
Cathode, x 0 V 0
(a)
34(b)
35(c)
36635
Laplaces Equation
If r 0, Poissons equation becomes
Let us consider uniform e first.
E6.1. Parallelplate capacitor
x d, V V0
x 0, V 0
37Neglecting fringing of field at edges,
General solution
38Boundary conditions
Particular solution
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40area of plates
For nonuniform e,
For
41E6.2
x d, V V0
x 0, V 0
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44Review 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
parallelplate arrangement.  6.14. Outline the steps in the determination of
the capacitance  of a parallelplate capacitor.
45Problem S6.4. Solution of Poissons equation for
a space charge distribution in Cartesian
coordinates
46Problem S6.5. Finding the capacitance of a
spherical capacitor with a dielectric of
nonuniform permittivity
476.3 Static Fields and Circuit Elements(EEE, Sec.
5.4 FEME, Sec. 6.3)
48Classification of Fields
647
Static Fields ( No time variation )
Static electric, or electrostatic fields Static
magnetic, or magnetostatic fields Electromagnetost
atic fields
Dynamic Fields (Timevarying) Quasistatic Fields
(Dynamic fields that can be analyzed as though
the fields are static) Electroquasistatic
fields Magnetoquasistatic fields
49Static Fields
648
648
For static fields, , and the
equations reduce to
50649
Solution for Potential and Field
Solution for charge distribution
Solution for point charge
Electric field due to point charge
51Laplaces Equation and OneDimensional Solution
650
For ????????Poissions equation reduces to
Laplaces equation
52Example of ParallelPlate ArrangementCapacitance
651
53652
Electrostatic Analysis of ParallelPlate
Arrangement
Capacitance of the arrangement, F
54Magnetostatic Fields
653
ò
ò
H
d
l
J
d
S
S
C
B
d
S
0
ò
S
Poissons equation for magnetic vector potential
55654
Solution for Vector Potential and Field
Solution for current distribution
Solution for current element
Magnetic field due to current element
?2A 0
For currentfree region
56Example of ParallelPlate ArrangementInductance
655
57656
Magnetostatic Analysis of ParallelPlate
Arrangement
58Magnetostatic Analysis of ParallelPlate
Arrangement (Continued)
Inductance of the arrangement, H
59Electromagnetostatic 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
60Example of ParallelPlate Arrangement
659
61660
Electromagnetostatic Analysis of ParallelPlate
Arrangement
62Electromagnetostatic Analysis of ParallelPlate
Arrangement (Continued)
Conductance, S
Resistance, ohms
63662
Electromagnetostatic Analysis of ParallelPlate
Arrangement (Continued)
0
1
z
æ
ö
ò
è
ø
I
l
z
l
c
Internal Inductance
64 Electromagnetostatic Analysis of ParallelPlate
Arrangement (Continued)
Alternatively, from energy considerations,
Equivalent Circuit
65Review 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 parallelplate structure
and the  determination of its capacitance.
 6.18. Outline the steps involved in the
magnetostatic field  analysis of a parallelplate structure
and the  determination of its inductance.
 6.19. Outline the steps involved in the
electromagnetostatic  field analysis of a parallelplate
structure and the  determination of its circuit equivalent.
 6.20. Explain the term, internal inductance.
66Problem S6.6. Finding the internal inductance
per unit length of a cylindrical conductor
arrangement
676.4 Low Frequency Behaviorvia Quasistatics
(EEE, Sec. 5.5 FEME, Sec. 6.4)
68Quasistatic 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 lowfrequency extensions of static fields
present in a physical structure, when the
frequency of the source driving the structure is
zero, or lowfrequency approximations of
timevarying fields in the structure that are
complete solutions to Maxwells equations. Here,
we use the approach of lowfrequency 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.
69Electroquasistatic Fields
J
H
S
I
(
t
)
1
g
y
z
x
0
E
????
0
x
x
d
z
z
0
z
l
70669
Electroquasistatic Analysis of ParallelPlate
Arrangement
71Electroquasistatic Analysis of ParallelPlate
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
72Electroquasistatic Analysis of ParallelPlate
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
73Magnetoquasistatic Fields
????
74673
Magnetoquasistatic Analysis of ParallelPlate
Arrangement
75Magnetoquasistatic Analysis of ParallelPlate
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
76Magnetoquasistatic Analysis of ParallelPlate
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
77Quasistatic Fields in a Conductor
??
????
78677
Quasistatic Analysis of ParallelPlate
Arrangement with Conductor
79678
Quasistatic Analysis of ParallelPlate
Arrangement with Conductor (Continued)
80Quasistatic Analysis of ParallelPlate
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
81Quasistatic Analysis of ParallelPlate
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
82681
Quasistatic Analysis of ParallelPlate
Arrangement with Conductor (Continued)
Equivalent Circuit
83Review 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 parallelplate 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 parallelplate
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 parallelplate structure
with a conducting  slab between the plates and the
determination of its  input behavior. Compare the input
behavior with the  electromagnetostatic case.
84Problem S6.7. Frequency behavior of a capacitor
beyond the quasistatic approximation
85Problem S6.7. Frequency behavior of a capacitor
beyond the quasistatic approximation (Continued)
866.5 Condition for the validity ofthe quasistatic
approximation (EEE, Sec. 5.5 FEME, Secs. 6.5,
7.1)
87We have seen that quasistatic field analysis of a
physical structure provides information
concerning the lowfrequency 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
firstorder 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
parallelplate structure, and obtaining the
wave solutions, which will then lead us to the
distributed circuit concept and the
transmissionline.
88Wave Equation
687
For the onedimensional case of
Onedimensional wave equation
89Solution to the OneDimensional Wave Equation
688
Traveling wave propagating in the z direction
90Solution to the OneDimensional Wave Equation
689
Traveling wave propagating in the z direction
91690
General Solution in Phasor Form
f
f
j
j
A
Ae
,
B
Be
,
Phase constant
,
Phase velocity
,
Intrinsic impedance
92691
Example of ParallelPlate Structure OpenCircuited
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
93Standing Wave Patterns (Complete Standing Waves)
94Complete Standing Waves
Complete standing waves are characterized by pure
halfsinusoidal 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.
95Input 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,
96Condition 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 lowfrequency
approximations of timevarying 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.
97For frequencies slightly beyond the approximation
?l ltlt1,
696
98In 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
99Example of ParallelPlate Structure ShortCircuite
d at the Far End
?s
?????
100Standing Wave Patterns (Complete Standing Waves)
101Input 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,
102For frequencies slightly beyond the approximation
?l ltlt1,
103In general,
f
104Review Questions
 6.25. Outline the steps in the solution for the
electromagnetic  field in a parallelplate structure
opencircuited at the  far end.
 6.26. What are complete standing waves? Discuss
their  characteristics.
 6.27. What is the input admittance of a a
parallelplate  structure opencircuited 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 parallelplate structure
shortcircuited at the  far end.
 6.30. What is the input impedance of a a
parallelplate  structure shortcircuited at the far
end? Discuss its  variation with frequency.
105Problem S6.8. Frequency behavior of a
parallelplate structure from input impedance
considerations
106Problem S6.8. Frequency behavior of a
parallelplate structure from input impedance
considerations (Continued)
1076.6 The Distributed Circuit Concept and the
Transmission Line (EEE, Secs. 6.1, 11.5 FEME,
Secs. 6.5, 6.6)
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134Review Questions
 6.31. Discuss the phenomenon taking place in a
parallelplate  structure at any arbitrary frequency.
 6.32. How is the voltage between the two
conductors in a  given crosssectional plane of a
parallelplate  transmission line related to the
electric field in that  plane?
 6.33. How is the current flowing on the plates
across a  given crosssectional plane of a
parallelplate  transmission line related to the
magnetic field in that  plane?
 6.35. Discuss transverse electromagnetic waves.
 6.36. What are transmissionline equations? How
are they  derived from Maxwells equations?

135Review Questions
 6.37. Discuss the concept of the distributed
equivalent circuit.  How is it obtained from the
transmissionline  equations?
 6.38. Discuss the solutions for the
transmissionline 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
transmissionline  parameters.
 6.41. What are the transmissionline 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.

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