The subject of electromagnetics is still a core subject of the undergraduate electrical
engineering
(EE) curriculum; however, at most of the universities in United States, the time
allotted to teach it is cut into half (one 3-credit course instead of two). The present graduates
with BS degree in EE being rushed through the same curriculum content in a shorter time
often miss the concepts and depend on a lot of formulas which they use as a recipe for some
calculations based on an example worked out in the book. Some of them are fortunate to
take a follow-up special elective course in microwaves or RF design or antennas or fiber
optics, and so on, thus partly reinforcing one application area. Readily available commercial
software allows them to do routine calculations and design without having a conceptual
understanding of the expected solution. The commercial software is so user-friendly that
we usually get a beautiful colored visualization of the solution, even if it is a wrong simulation
of the physical problem. After getting one or two mild reprimands from the boss in a
new employment after graduation, the new graduate realizes the need to have a fairly good
idea of what is the appropriate model to be simulated and what qualitative result is to be
expected. Though the software is very useful, it is not a substitute for a conceptual understanding
of the steps involved in solving the problem. Fortunately, for him, there is probably
a university which offers graduate courses and there is an instructor or professor who
understands that these bright students recruited by some of the top companies are not less
smart than the employees recruited by the company, say a decade or two ago. On the other
hand, they are very knowledgeable and comfortable using the computers and online
resources. They are willing to challenge themselves to learn quickly to think in terms of
concepts and analysis rather than routine calculations; however, they would like to learn
these through examples that connect them to a technological application. Also, they find it
interesting if they find that the technique they learnt in one technological area by an indepth
study of that particular area can be applied to another technological area having the
same basis of engineering science, in this case electromagnetics. Such graduate students,
even if they enjoy the electromagnetics per se, cannot afford to take more than one or two
graduate courses in electromagnetics before specializing in one of the technological areas
for which electromagnetics is a base.
In a discipline as classical as electromagnetic theory, there are many excellent textbooks.
Many of us who teach and do research in electromagnetics had the benefit of these
graduate-
level courses based on classical textbooks, which are precommercial electromagnetic
software. Those who are motivated to continue this classical mode of learning and
doing research in electromagnetics will continue to be inspired by the thorough mathematical
treatment of all aspects learning them from these classical graduate-level textbooks
over a period of 2 or 3 years.
I believe that in teaching electromagnetics to EE students as opposed to the physics students,
we can make some subtle changes in the presentation of the material. The first
change is to exploit the strong circuit background of the EE students and treat transmission
lines as distributed circuits.
Given below are some thoughts on the motivation, reasoning, and general themes in
developing the material in this book presented in Parts I through V.
1. Transmission lines as distributed circuits are a logical extension of the lumped
parameter circuit theory. For electrical engineers, scalar waves on the transmission
lines with voltage and current as the dependent variables somehow seem to
be less abstract and give the basic framework (clutch) in which electrical engineers
can think. Transmission line analogies even if they are not physical (artificial)
seem to help electrical engineers to grasp more abstract concepts.
2. I have taken the liberty of defining a simple electromagnetic medium as one where
ε, μ, and σ are all scalar constants. This is to correspond to the gross parameter
description of the circuits as capacitance C, inductance L, and conductance G, or
resistance R. It also roughly corresponds to the problems we usually solve in the
undergraduate course. Some purists may object to this definition. They may like
to think of a free-space medium as the only simple medium. They are willing to
extend the definition of a simple medium to an ideal isotropic dielectric. Anything
beyond isotropic dielectric is a complex media.
3. I have taken a utilitarian view in distinguishing the simple medium problem
from the complex medium problem. The four Maxwell’s equations are the same
for both, and the electromagnetic properties of the materials are introduced
through constitutive relations. Specification of the boundaries and the sources
completes the specification of the problem. Many practical problems can involve
complex media as well as complicated boundaries. However, from the pedagogical
view point, one can classify the problems as (a) involving a simple medium
with complicated boundaries or (b) a complex medium bounded by simple boundaries.
For example, a simple boundary may be a planar surface, allowing Cartesian
coordinate descriptions.
4. Part I of this book deals with electromagnetics of bounded simple media. After
introducing the equations in the time domain, the time-harmonic equations, wave
propagation solutions, and their applications are obtained for one-dimensional,
two-dimensional, and then three-dimensional problems. In one-dimensional
problems, planar boundaries and then the cylindrical boundary problems and
applications are considered. Starting from the first principles, the process of
obtaining the one-dimensional model (for the z-component of the vector potential,
Az ) for the ideal problem of an infinitely long conducting filament along the z-axis
excited by a harmonic current is explained. Then considering the symmetries
involved in the problem, it is shown that Az is at the most a function of the cylindrical
radial coordinate ρ. This is a simple example of building a model appropriate
to the objectives of the investigation rather than getting bogged down with
unnecessary details, which could increase the complexity of the problem. As an
example of increasing the complexity, one could solve the same problem by considering
a differential length of the filament as a Hertzian dipole and do the integration
with infinite limits for the infinitely long filament.
5. The ordinary differential equation of the above-mentioned problem is shown to
have a singularity at the origin and is shown to have two independent solutions,
one of them having a singularity at the origin. After mentioning that the solution
to such equations can be obtained by power series, the series solution is given and
designated as the Bessel function of the first kind of zero order. Bessel functions
are thus introduced, compared with trigonometric functions, and their applications
are illustrated.
6. The rectangular and cylindrical waveguides are used as examples of two-dimensional
problems. After defining the waveguide problem, the well-known
separation of variable–product solution technique of solving partial differential
(PD) equations is illustrated. It is shown that the technique converts the PD equations
to the ordinary differential equations with constraints on the separation constants.
In the discussions of special functions, the emphasis is on developing an
interest for these functions, facilitating their use in obtaining the eigenvalues, and
eigenvectors of the ordinary differential equations. Use of fractional Bessel functions
is illustrated through sector waveguides. In these examples, the technique of
choosing the appropriate functions from a template of admissible functions for the
problem based on given and implied (based on the physics of the problem) boundary
conditions are illustrated.
7. Chapter 4 deals with a rectangular cavity as an example of a three-dimensional
problem. The well-known approximation technique of obtaining the fields (eigenvectors)
and resonant frequencies (eigenvalues), assuming the boundaries are perfect
conductors, and then calculating the losses based on the surface current-flow
on the walls of the cavity is illustrated. Homework problems are given to test whether
the students are able to write by inspection, the solution for a cylindrical cavity.
8. The waveguide and cavity problems of Chapters 4 and 5 are essentially based on the
solution of a scalar Helmholtz equation for the potential (Ez for the TM problems
and Hz for the TE problems). It became possible to do such decomposition because
for these problems we could identify a longitudinal direction and a transverse plane.
In the spherical geometry, we do not have such easily identifiable scalar potentials.
In principle, the more general vector Helmholtz equation for the fields has to be
solved. The mathematics thus becomes more involved. We can relate to the previous
techniques by first considering the solution F of the scalar Helmholtz equation in
spherical coordinators and then relating it to the TMr and TEr modes through the
defined vectors M and N. In a one-semester course, one can omit this chapter since
it can distract a student from a simpler conceptual understanding aimed so far.
9. Chapter 6 approximates the scalar Helmholtz equation to the Laplace equation
for low-frequency (quasistatic) or static applications. A quick review of the onedimensional
problems, the technique of using the template of the admissible functions
in the three main coordinate systems, and the expansion of an arbitrary
function in terms of the orthonormal functions which were the modes (eigenvectors)
of the solutions in Chapters 3 and 4 are illustrated. A large number of homework
problems are given to illustrate the application to the electromagnetic
problems. Miscellaneous topics on waves, particularly Section 7.2, is written at a
comparatively intuitive and comfortable level suitable for an undergraduate EE
student. Section 7.3 is particularly interesting for those who would like to extend
their strength in circuits and networks to high-frequency engineering. Sections 7.5
through 7.7 are usually studied in greater depth as separate courses and are
included here as an introduction to these topics. This concludes Part I of this book,
dealing with the electromagnetics of simple bounded media.
10. Part II of this book deals with electromagnetics of complex media. At least one of
the electromagnetic parameters is not a scalar constant. Chapter 8 develops the
constitutive relations for various complex materials, including superconductors,
mostly using classical simple models for the microscopic interactions.
11. Effects of temporal dispersion, spatial dispersion, nonhomogeneity, and anisotropy
on wave propagation are investigated taking cold plasma, warm plasma, magnetoplasma,
anisotropic crystals as examples. A special case of time-varying medium
problems, that is, a moving medium, is discussed in Chapter 14. Several techniques
of electromagnetic analysis are considered in some depth. Electromagnetic modeling
and experimental simulations of plasmas, chiral materials, and left-handed
materials are discussed in Chapter 9, under the heading of Artificial Electromagnet
Materials.
12. In Part II, the dominant effect of each kind of complexity is brought out. The goal
of this part is to bring the system approach of relating the kind-of-complexityresultant
dominant effect as an input–output description of a system element. A
combination of the system elements through interconnection of the system elements
in an approach of synthesis can bring a desired output. Section 10.10 mentions
one example: the combination of two undesirable dominant effects of (a)
dispersion in broadening the pulse and (b) the nonlinearity in steepening the
pulse into a desirable overall effect of preserving the pulse shape in the propagation
of a soliton in the dispersive nonlinear medium.
13. The purpose of Part III, Electromagnetic Computation, is to bring out the basis of
the engines of various commercial electromagnetic software, widely used in the
industry. Algorithms of finite differences, moment method, finite-element method,
and finite-difference time-domain method are developed and illustrated. Handcomputed
simple examples and MATLAB®-coded simple examples with only a
few elements are used to explain the concepts behind the algorithms. The coding
is also kept very simple translating the equations of the algorithm as directly as
possible. A few case studies of practical examples from transmission lines, waveguides,
and electrostatic problems are given so that the student is able to develop
the code and solve the problems. The students are encouraged to run the same
problems on the commercial software to verify their result and get a feel for the
algorithm. Of course, some of the Commercial software have a lot more postprocessing
capabilities and more efficient and accurate engines, and the purpose of
this part was not to discourage the student from using these commercial softwares
but to use them with greater confidence and satisfaction.
The three parts have enough material to serve as a textbook for two senior-level/firstyear-
graduate-level courses, each of three semester credits. At the University of
Massachusetts Lowell, the material in various versions was used for such a purpose during
the past 24 years (for the courses 16.507: Electromagnetic Waves and Materials, 16.532:
Computational Electromagnetics). In each of these classes, about two-thirds of the students
were from industries based on electromagnetic technologies.
Part IV consists of appendices for various chapters. Some of them contain the details of
a derivation or explanation that is not central to the concept and likely to distract the reader
from the main point being made and hence relegated to the appendix for completeness. On
the other hand, some of the appendices contain advanced topics or newer topics of interest
to a subset of the students. It gives the instructor a choice of advanced topics he can
include as examples of topics of current research interest to the electromagnetic community.
A third category of appendices are a basic exposure to an electromagnetic topic.
Advanced discussion of the topic is not pursued but it is pointed out that it can proceed on
lines very similar to the one in the chapter. For example, Chapter 13 deals with “Optical
Waves in Anisotropic Crystals.” The analysis is based on a constitutive relation relating D
with E through permittivity tensor. Appendix 13A formulates the permeability tensor for
the complex medium of a ferrite in the presence of a background static magnetic field.
Part V is an important pedagogic tool containing homework problems, 15-minute quizzes,
and take-home examinations. The author used them in the following fashion. After
the lecture, some problems are assigned as homework, in the next class, the homework is
briefly discussed mostly to tell the importance of the problem in terms of a technological
application, modeling tip, and the solution outline is provided. A quiz of 15-minute duration
is administered periodically (every third or fourth 50-minute lecture class to check
whether the central concepts in the homework are learnt). Midway through the semester
and at the end of the course, take-home or open-book examination is given where more
substantial problems are set. The feedback from the students was always positive with the
comment that the questions in Part V was the most effective way they learnt the deeper
implication of the material in the other parts of the book.
The solutions to the questions in Part V will be provided to the instructor through the
downloadable online component of this book. This book is so structured that a course
outline can be picked from the table of contents to serve the needs of courses of three to
six semester credits with different starting points on different aspects of electromagnetics.
Examples of such courses-outline will be included in the online component of this book
for the benefit of the instructor. These are (UML stands for University of Massachusetts
Lowell):
Course Outline A: one-semester 3-credit senior-elective-first-year graduate course
with a prerequisite of one-semester 3-credit core undergraduate course in
electromagnetics;
Course Outline B: (UML 16.507 Electromagnetic Waves and Materials) one-semester
3-credit senior-elective-first-year graduate course with a prerequisite of twosemester
3-credit each core undergraduate course in electromagnetics (UML
16.360, UML 16.461);
Course Outline C: (UML 16.532 Computational Electromagnetics) one-semester
3-credit senior-elective-first-year graduate course with a prerequisite of twosemester
3-credit each core undergraduate course in electromagnetics (UML
16.360, UML 16.461);
Course Outline D: (UML 16.607 Electromagnetics of Complex Media) one-semester
3-credit second-year advanced graduate course in electromagnetics with a prerequisite
of first-year graduate course in Electromagnetics 16.507. This course
includes an additional project/extra material.
Though the book contains more material than can be reasonably covered in a one-semester
3-credit course, I contend that this book is useful even for students who take only
one graduate course in electromagnetics, since any of the course outlined above sets the
tone and the rest of the material can be understood in a self-study as and when needed by
the student. I believe a graduate-level book should also serve as a starting point for some
of the current and active research areas as well as spark an interest in such areas.
This book is a companion to my research monograph (K10882) Electromagnetics of
Time-
Varying Complex Media: Frequency and Polarization Transformer, Second Edition, which was
published by CRC Press (Taylor & Francis Group) in April 2010. The connection between
the publications is established through the common theme of a few chapters in the two
books. Chapters 10, 12, 19, and Appendix 10E of this book are the modified versions of
Chapters 2, 6, 11, and Overview of the book K10882, respectively.
This book aims to strike a balance between theory, intuitive approximate solutions, and
the use of commercial software and interpretation of the software solutions, of electromagnetic
problems.
