Modern Graduate Electromagnetics Education—A New Perspective
Director, Center for Computational EM and EM Lab.
Department of Electrical and Computer Engineering
University of Illinois
Urbana, IL 61801-2991
Electromagnetics is a branch of applied physics that has evolved over the years. The physics associated with electromagnetics is well understood, but despite the age-old advent of Maxwell theory, electromagnetic engineers are indispensable. The primary reason is the pervasiveness of electromagnetic technology in the modern world, where electromagnetic engineers are needed to design systems related to electromagnetic technology, such as in wireless communications, computer chips, optical networks, etc.
Because of the elusiveness of electromagnetic physics and the complexity of the law that governs it, electromagnetic analysis has always been used in the understanding and design of electromagnetic related systems. As a result, electromagnetic analysis is a continuously evolving science, and an important topic of research up to this day, and recently is an important research topic in computational science.
Over the years, electromagnetic analysis has evolved from the solving for scattering solutions from simple shapes, to approximate and perturbation methods, and then more recently, to the use of numerical and fast methods with the help of digital computers.
Electromagnetic analysis is an active area of research that has attracted the interest of mathematicians, computer scientists and engineers. However, a good understanding of modern electromagnetic analysis requires the melange of a deep insight in electromagnetic physics and ability for mathematical finesse, and knowledge of computational numerical algorithms. Therefore, it is naturally an interdisciplinary field.
Due to the advent of fast algorithms for them, electromagnetic simulations will become an indispensable analysis tool in the arsenal of an electromagnetic engineer. She will be involved with design work where simulation tools will be used repeatedly until she arrives at a satisfactory design. The final test of the pudding will still be in the laboratory where the design is built and tested.
While university research emphasizes computational electromagnetics, we have to be mindful that a student of electromagnetics should understand the underlying physics, and develop the requisite physical and engineering intuition and insight for problem solving. These skills are important both for analysis and design. Therefore, it is still important to educate graduate students on the classical electromagnetic analysis methods. For instance, classical electromagnetic analysis teaches us the concepts of surface waves, creeping waves, lateral waves, Goubaud waves, guided modes, evanescent modes, radiation modes, leaky modes, low-frequency physics, and high-frequency physics that do not emerge from numerical analysis, but such concepts are instrumental in a good physical understanding of many electromagnetic interactions.
In terms of mathematical knowledge, classical analysis methods require students to understand harmonic analysis, complex variables, perturbation and asymptotic methods. However, modern numerical approach to problem solving requires students to understand linear algebra and linear vector spaces. At a more advanced level, students will need to understand functional analysis, algebra, and even topology.
In quantum mechanics, one sees a marriage between functional analysis and physics. The physics in quantum mechanics evolved from wave physics, and for a while, was called wave mechanics. However, in electromagnetics, we also see the use of functional analysis ideas in numerical methods such as Galerkin’s method, method of moments, and finite element method.
As topological concepts found in differential forms become more matured, they will be important for graduate electromagnetic education as well. For instance, the understanding that Stokes’ theorem and Gauss’ theorem are essentially the same concept topologically, and similarly for other vector identities in vector calculus, are beautiful concepts that an electromagnetic graduate student should know.
Electromagnetic theory was fully formulated by James Clerk Maxwell in 1864 in terms of the Maxwell’s equations. Even though it has been around for over a hundred years, scientists and engineers are continuously pursuing new methods to solve these equations. The reason is that Maxwell’s equations govern the law for the manipulation of electricity. Hence, many branches of electrical engineering are directly or indirectly related to the electromagnetic theory. Scientists and engineers solve these equations in order to gain a better understanding of and physical insight into systems related to the use of electromagnetic fields and waves. The solutions of Maxwell’s equations can also be used to predict design and experimental outcomes.
Electromagnetics has persisted as a vibrant field despite it being over a hundred year old is because many electrical engineering technologies depend on it. To name a few, these are: physics based signal processing and imaging, computer chip design and circuits, lasers and optoelectronics, MEMS (micro-electromechanical sensors) and microwave engineering, remote sensing and subsurface sensing and NDE (non-destructive evaluation), EMC/EMI (electromagnetic compatibility/electromagnetic interference) analysis, antenna analysis and design, RCS (radar cross section) analysis and design, ATR (automatic target recognition) and stealth technology, wireless communication and propagation, and biomedical engineering and biotech.
Figure 1. The impact of electromagnetics is far-reaching and affects many different branches of electrical engineering technologies.
For instance, the field of computer chip design has long relied on the use of circuit theory, which is a subset or an approximation of electromagnetic theory when the frequency is very low. As the clock frequency of a computer becomes higher, circuit theory becomes inadequate in describing many of the physical phenomena that occurs within a computer chip. Electromagnetic theory has to be used to correctly describe the physics within a computer chip. An emerging electromagnetic analysis method is computational electromagnetics where the computer is used intensively to analyze electromagnetic problems. The growth of this field has been spurred by the rapid growth of computer speed, and now the further growth and design of faster computers will rely on computational electromagnetics—a symbiotic existence indeed.
Electromagnetics, the study of the solution methods to solving Maxwell’s equations, and the application of such solutions for understanding and engendering new technologies, has a long history of over a hundred years. But the analysis method with Maxwell’s equation is constantly evolving over the years. In the beginning, there was the age of simple shapes: during this period, roughly between the late 19th century to 1950s, solution methods, such as the separation of variables, harmonic analysis, and Fourier transform methods were developed to solve for the scattering solution from simple shapes. We can identify the names of Sommerfeld, Rayleigh, Mie, Debye, Chu, Stratton, Marcuvitz, and Wait for contributions during this era. Many of the solutions are documented in a book by Bowman, Senior and Uslenghi.
Despite the successful closed form solution of many simple geometries, the solutions available were insufficient to analyze many electromagnetic systems. Hence, scientists and engineers started to seek approximation solutions to Maxwell’s equations. This was the age of approximations, roughly between 1950s and 1970s. During this period, asymptotic and perturbation methods were developed to solve Maxwell’s equations. The class of solvable problems for which approximate solutions exist, was greatly enlarged. We can identify names such as Bremmer, Keller, Jones, Kline, Fock, Hansen, Lee, Deschamps, Felsen, and Marcuvitz during this era.
However, the limited range of approximate solutions of Maxwell’s theory still could not meet the demand of many engineering and system designs. As soon as the computer was developed, numerical methods were studied to solve Maxwell’s equations. This was the age of numerical methods (1960s+). Method of moments (MOM), finite difference time domain method (FDTD), and finite element method (FEM) were developed to solve problems alongside with many other numerical methods. In particular, Harrington was noted for popularizing MOM among the electromagnetics community, while it is know as the boundary element method (BEM) in other communities. Yee developed FDTD, for solving Maxwell’s equation. Finite element has been with the structure and mechanics community, and Silvester was an early worker who brought its use into the electromagnetics community. Other names commonly cited in this field are: Wilton, Mittra, and Taflove.
There has always been marriage between electromagnetics and mathematics from the very beginning—a marriage made in heaven perhaps. Actually, quite sophisticated mathematical techniques were used to analyze electromagnetic problems because electromagnetic theory was predated by the theory of fluid and theory of sound. These fields were richly entwined with mathematics with the work of famous mathematicians such as Euler, Lagrange, Stokes, and Gauss. Moreover, many of the mathematics of low-Reynold number flow in fluid theory and scalar wave theory of sound can be transplanted with embellishment to solve electromagnetic problems.
Examples of problems solved during the age of simple shapes are the Mie and Debye scattering by a sphere and Rayleigh scattering by small particles. Rayleigh also solved the circular waveguide problem for electromagnetic waves because he was well versed in the mathematical theory of sound, having written three volumes on the subject while sailing down the Nile River. Sommerfeld solved the half plane problem as far back as 1896 because the advanced mathematical techniques were available then. He also solved the Sommerfeld half space problem in 1949 in order to understand the propagation of radio waves over the lossy half-earth. The problem was solved in terms of, what is now known as, the Sommerfeld integrals, an example of which is as follows:
Figure 2. A dipole over a half space. The problem was first solved by Sommerfeld to understand the propagation of radio waves over the lossy earth.
Evaluating the Sommerfeld integrals was an impossibility during his time, but it is a piece of cake now in the modern era. Subsequently, approximation techniques, such as the stationary phase method, the method of steepest descent, and the saddle point methods were used to derive approximations to the Sommerfeld integrals.
However, even though electromagnetics has been intimately entangled with mathematics, a student of electromagnetics has to be able to read the physics into the mathematical expressions that describe the solutions of Maxwell’s equations. Approximate methods generally help to elucidate the physics of the wave interaction with complex geometry.
The physical insights offered by approximate solutions spurred the age of approximations, roughly between 1950s and 1970s. A large parameter such as frequency is used to derive asymptotic approximations. Moreover, heuristic ideas were used to derive the physical optics approximation, Kirchhoff approximation, and various geometrical optics approximations. These approximations eventually lead to the geometrical theory of diffraction and the uniform asymptotic theory of diffraction. The applications of these approximate methods to scattering by complex structures are usually ansatz based. The ansatz assumes that the scattering solution is of the form:
The leading coefficient and the exponent are found from canonical solutions such as the Sommerfeld half plane problem, or scattering by a sphere or a cylinder, followed by the use of Watson transformation. The use of approximate solution enlarges the class of solvable problems, but the error is usually not controllable. Asymptotic series are semi-convergent series; hence there is not a systematic way to reduce the solution error by including more terms in the ansatz. Moreover, the range of application is limited because the frequency has to be sufficiently high before the ansatz forms a good approximation.
The advent of the transistorized computer in the 1960s almost immediately brought about the birth of numerical methods for electromagnetics. The method of moments (MOM) was popularized among the electromagnetics community by Harrington in the 1960s. The method is integral-equation-based, and is versatile for solving problems with arbitrary geometries. It entails a small number of unknowns since the unknown is the current, but unfortunately, the pertinent matrix equation is dense. The finite-difference time-domain method was proposed by Yee in the 1960s for solving Maxwell’s equations in its partial differential form. The method is extremely simple, and gives rise to a sparse matrix system. Since the field is the unknown to be solved for, the drawback is that it entails a large number of unknowns. Moreover, the field is always propagated from point to point via a numerical grid, hence yielding grid dispersion error, which accentuates with increasing problem size.
However, some of the recent advances in fast computational algorithms will remove the objections to the shortcomings of numerical methods.
Electromagnetics is a branch of applied physics. However, due to the dependency of solution methods on mathematics, both knowledge of physics and mathematics are indispensable in the study of electromagnetics.
We should encourage our students to study modern physics; even though it is not directly relevant to electromagnetics, modern physics embodies the most beautiful of the physical theories that have been developed in this century. If a student can understand the thought processes and abstraction that go on in modern physics, she eventually will become a better thinker and a proficient problem solver. Our goal is to teach a student to think in graduate school. A proverbial saying is that “If you give a man a fish, it lasts him for a day, but if you teach a man how to fish, it lasts him a lifetime.” Moreover, if we can stretch the mind of a student, it does not regain its original dimension.
The long history of electromagnetics has produced much classical knowledge that cannot be ignored by our students. They should have a good understanding of the fundamental solutions that accompany simple shapes. Furthermore, they should understand and should be able to elucidate the physics that arises from the approximate solutions, such as the physics of surface waves, creeping waves, lateral waves, Goubaud waves, evanescent waves (tunneling), guided modes, radiation modes, leaky modes, specular reflection, edge diffraction etc.
A student of electromagnetics should be cognizant of the metamorphosis of the physics over different lengthscales or frequencies. When the wavelength is extremely long, one is in the regime of electrostatics and magnetostatics where the electric field and the magnetic field are decoupled or weakly coupled to each other. This is also the world in which circuit theory lives in. For shorter wavelengths, the coupling between the electric and magnetic field becomes stronger, and we have mid-frequency or high frequency electromagnetics, whose physics is quite different from low frequency electromagnetics. This is also when the wave nature of an electromagnetic field becomes important. Often, the vector nature of electromagnetic field plays an important role in this regime. As the frequency gets higher, then we are in the world of optics and ray optics. In this world, electromagnetic waves can be described by rays, and often be thought of as particles. Equations can be derived to govern only the propagation of the envelop of a pulse. At very high frequencies, the quantization of the energy associated with an electromagnetic field becomes important. A quantum of energy in electromagnetic field is where the Planck constant erg sec. Therefore, to properly understand the interaction of very high frequency electromagnetic field with material, we have to invoke quantum electrodynamics.
Since there has often been an intimate intermarriage between electromagnetics and mathematics, a student of electromagnetics should understand the reason for the finesse, care, and precautions that mathematicians go through in their work. It is to lay a firm foundation for their mathematical work so that others can build upon them. She should understand the fundamental of harmonic analysis and complex variables, which traditionally have been used to analyze classical electromagnetic problems. The fundamental of perturbation and asymptotic methods should be understood in order to appreciate the wide body of knowledge generated by approximate calculations.
The advent of computational electromagnetics however, calls for the use of a different body of mathematical knowledge. A student who is well versed in computational electromagnetics should have an elementary understanding of functional analysis, Hilbert spaces, and operator theory. She should also understand partial differential equation (PDE) theory, the existence, uniqueness, and well-posedness of PDE solutions, and the integral equation theory that follows from PDE. A wonderful aspect of modern computational electromagnetics is the error controllability of the solution—viz., the error can be made increasingly small by devoting increasing computational resources. Hence, the pervasiveness of computational method in the future will require that a student understands the elements of approximation theory and error bounds.
My students, F. Teixeira and E. Forgy have convinced me that often times, a more elegant view of electromagnetic theory can be gotten from a topological viewpoint. Such is the viewpoint of electromagnetic theory from differential forms. Modern computational electromagnetics will inherently deal with complex geometry handling. A more profound understanding of electromagnetic theory through a topological viewpoint may even engender the development of new numerical methods.
Computers are used by all scientists and engineers as a tool. In order to harness the power of this tool effectively, it is important that a student of electromagnetics understands the basics of modern computer languages and computer architectures. A student should understand the basis of object oriented programming paradigm, and how its use will result in better computer code maintenance, reusability, and encapsulation. It is also important for students to learn the basis of parallel computing and large scale computing, and the need to use message passing for distributed processor computing. Many fast algorithms exist in computer science for sorting, searching, matrix manipulations that students should be aware of. They should also be aware of the element of computer architecture, the issue of distributed memory computing versus shared memory computing, the issue of memory latency, cache usage, cache hit and miss. Geometry handling will be indispensable from computational electromagnetics; hence, a student should understand certain aspects of computational geometry.
It is important that one educates independent thinkers rather than just high-class technicians who are automatons good at receiving and executing instructions. After being a professor for so many years, I can roughly classified students into three types:
I. Students who will do A when instructed to do A.
II. Students who will do A+B when instructed to do A.
III. Students who will do C when instructed to do A.
Type I students are what we call high-class technicians. If a professor has many of them, they will help expand the capabilities of a professor. However, it is the Type II student who will truly empower and amplify the capabilities of a professor, and provide work beyond the call of duty. Just imagine how much more productive our workforce will be if all our students are of Type II. Type III students are an interesting class. They may be either extremely creative or unproductive. They may also be students who like to take on a completely different path and break new grounds. So while professors may not be comfortable with them, they should be allowed to thrive if there is inkling that they are doing some good. It is important that we allow people to go outside a beaten path and explore new grounds.
We should always seek to bring the best people into our field and stimulate their creativity. Good people will always bring new ideas, forge new frontiers, and create new areas to work on. We should also cultivate independent thinking in our students. Students should be encouraged to challenge the thinking of a professor. If we force our students to blindly follow the instructions of a professor, it will no doubt stifle the creativity of a student. There is an old Chinese adage:
“If one believes completely in the teacher, it’s better not to have teachers; if one believes completely in books, it’s better not to have books.”
There should be a constant dialogue between professors and students. These dialogues provide a window into the mind of the student, and provide a venue to link minds. Modern technology and scientific problems are getting increasing sophisticated that many of them cannot be solved by just a single mind. Teamwork and linking of minds are important for the solutions to many of our advanced technological and scientific problems.
It is well known that our right brain is responsible for creative and artistic expressions, and that our left brain is for logical thinking. Students should not precipitate an early burnout by being one-dimensional and working only with their left-brain. They should be encouraged to pursue the appreciation of the arts, music and sports to maintain a healthy mind. A student with a healthy mind will eventually be more creative than a mentally exhausted one.
When educating students, we have to be aware that students often time, play a very different role in the academe as oppose to the industry. In the university, we are pushing the frontiers of knowledge, and our prerogatives are quite different. In the industry, a student is often involved with the development or improvement of a technology. Therefore, when educating students, we should be mindful that students would not work on the kind of problems they work on while they were graduate students.
In the academe, since the current hot area of research is electromagnetic analysis using computational electromagnetics, many universities are doing software research related to computational electromagnetics. A student typically studies and develops algorithms and methodologies for computational electromagnetics. Some may even apply these methodologies to some applications. Often times, these students are heavily involved with computer programming.
In another scenario, a student may learn how to build a component of a larger system. She may be using existing CAD tools to design a component for her research goals.
However, when a student graduates, she will most likely be a system or component design engineer, who helps develop the next generation electromagnetics technology. The work is very different from her graduate schoolwork. Therefore, it is imperative that we teach our students to be thinkers rather than high-class technicians. It is instrumental that the students understand the physics of electromagnetics in graduate schools. Understanding the physics deeply means the understanding of the mechanism behind how things work. Therefore, an EM graduate student has a tall calling, but an exciting one involving a lifetime of learning.
Since electromagnetic analysis has been used as a prediction tool important for many branches of electrical engineering, electromagnetics will always remain important in electrical engineering technologies. However, the long and rich history of electromagnetics offers us a challenge on how we should educate our graduate students in this rapidly changing world. The total amount of knowledge cannot be taught to all our students within the short span of their graduate education. The important knowledge changes with changing times. Therefore, it is extremely important that we educate our students with the fundamental knowledge, so that she can read further independently if it needs be. Learning everything relevant to electromagnetics technology will entail a lifetime of learning. Also, it is important that we educate our students to become thinkers rather than one who regurgitates our knowledge mechanically. Furthermore, if we can educate a student who can empower an organization, the student will undoubtedly be more valuable to the society.
Since the role of a student changes from the university to the industry, it is important that students develop physical insight into their work as graduate students, so that they will become better system and design engineers in the industry. They will also become better problem solvers with better physical insight.
It is important that we bring the best and brightest, and the most creative people to work in our field. These new people will always create new frontier problems to work on. Consequently, there is no shortage of problems to work on and the field of electromagnetics will be constantly rejuvenated.