A New Direction for Undergraduate Electromagnetics Professor David R. Voltmer Rose-Hulman Institute of Technology 5500 Wabash Avenue Terre Haute, IN 47803 812-877-8289
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ABSTRACT - This paper describes a new approach to undergraduate electromagnetics that responds to changing educational and technological environments. Lumped circuit elements, with which students are already familiar, provide the framework from which the properties of electromagnetic fields naturally evolve. Circuit behavior provides the basis for the introduction, formulation and interpretation of many electromagnetic concepts. This approach emphasizes spatial discretization and numeric formulations in order to exploit the computational power and rich graphical environment of modern computers. Classroom activities include the use of a powerful, PC-based, interactive, electromagnetic computation engine which empowers students to experiment with the geometry and material properties of two-dimensional electromagnetic problems. Course details and software demonstrations are included in the presentation.
INTRODUCTION Recent trends in the design of commercial electrical systems rely more heavily upon electromagnetics. Improved performance described by higher frequencies, faster clock speeds, and increased circuit density have heightened the likelihood of coupling between circuits and of electromagnetic radiation. The expanded use of electronics in emerging areas such as automotive electronics, cellular and personal communications, high-speed computer systems, GPS navigation, high-power switching circuits, and invasive medical electronics all require increased understanding of the underlying principles of electromagnetics. The electromagnetic interaction of devices and systems is an increasingly challenging problem. More than ever before, electrical engineers need a solid grounding in the fundamentals of electromagnetics. Academia has undergone significant changes as well. Accreditation pressures have increased the curricular content for design and other professional skills in spite of trends to reduce the overall credit hours. Student proficiency in mathematics has shifted from analytic to computer-based skills. Moreover, topical coverage in under-
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graduate mathematics courses has changed; the coverage of vector calculus and partial differential equations has been reduced significantly in many undergraduate curricula. These trends have forced many schools to reduce their coverage of traditional topics including electromagnetics. This paper proposes an alternative approach to overcome these pressures in electromagnetics education by maintaining a strong focus on the fundamentals tempered by a modern perspective on calculation methods.
CLASSROOM METHODS AND PROFESSIONAL PRACTICE The vector nature of electromagnetic fields presents a mathematical stumbling block to many students which impedes their understanding of the physical concepts. In the main, present courses and textbooks base the behavior of electromagnetic fields upon differential and integral vector calculus. Because the visualization of three-dimensional vector fields is challenging to many students, introductory electromagnetics courses often become an exercise in which students merely memorize a set of formulas, making minimal connections between the physical phenomena and the mathematics. Only a few students, many of whom have previously mastered the mathematical concepts and operations of vector calculus, are able to gain a sound working knowledge of the subject. Very few textbooks and course syllabi have deviated from this "tried and true," traditional approach--physical concepts follow from vector calculus. Electromagnetics is taught today much as it has been taught for the last several decades. On the other hand several significant changes have occurred in the engineering workplace where electromagnetic fields are used everyday. Firstly, computational power has developed to the point that numerical methods are viable; indeed for most problems they provide the only practical method of solution. Secondly, computer graphics makes visualization of complex electromagnetic fields available in real time. Thirdly, electromagnetic professionals have developed and regularly use simplified scalar models of electromagnetic phenomena. These models retain the essential
features of electromagnetic fields, but enable many of the field’s vector properties to be expressed in scalar form for easier calculation and interpretation. Fourthly, an accompanying set of solution techniques has evolved which exploits the rapidly developing computational power and computer graphics. These practical and useful tools rarely find their way into today's textbooks; when they do, often they take the form of added material to be used at the option of the instructor. These emerging electromagnetics concepts and techniques cannot be crowded into existing courses which are shrinking in number and are populated by students with significantly different skills. However, following a careful evaluation of the present situations in electromagnetics education and practice, we have chosen and propose a novel approach to promote student learning and understanding of electromagnetic fields.
THE APPROACH A revolutionary departure from traditional electromagnetics is necessary to strengthen every student’s understanding of electromagnetics. Implicit in this approach is a change in the presentation of electromagnetics to students; learning is focused on the concepts and tools of today's electromagnetic professionals. Two fundamental principles are synergistically combined in the foundation of this approach--spatial discretization and numeric formulation. Without numeric formulations which can exploit computational power and graphics, the concepts of spatial discretization offer an interesting, but limited, alternative to continuous electromagnetics. Without spatial discretization, numeric methods are limited to formula evaluation and plotting of equations. Spatial Discretization Traditional electromagnetics textbooks are based upon the vector calculus of continuous mathematics. An alternate perspective considers space to be divided into discrete incremental cells each of which can be modeled as a lumped circuit element. This construct allows the laws of electromagnetics to be cast in terms of lumped element circuit theory--resistance, capacitance, inductance, Ohm's law, Kirchoff's voltage and current laws. Basic field concepts are developed within the familiar framework of the terminal behavior of circuit elements. The axioms of this approach are based in circuit theory rather than in physical observations and vector calculus. The vector description of fields naturally follows from the behavior of incremental circuit elements. Spatial variations of fields are manifest in the geometric arrangement of the incremental cells. Macroscopic element values follow directly from the rules for
series and parallel elements of circuits and lead intuitively to the more traditional line and surface integral representation of continuous-variable, vector calculus. Numeric Formulation The elegance and rigor of the continuous-variable vector calculus foundations of electromagnetics have great appeal to many professionals (and most textbook writers) and with good reason--they are a necessity for advanced concepts of electromagnetics. On the other hand, spatial discretization, in addition to its basis in circuit theory, allows the use of computer-oriented (and less intimidating) discrete mathematics. Formulation of electromagnetic laws in numeric form leads beginning students directly to computer-based solutions. Realtime, scalar calculations displayed with color graphics provide a powerful learning environment. Using this process, students can understand the basic principles of electromagnetics before they master differential and integral calculus. This numeric foundation makes students aware of the value of vector calculus; moreover, it enables them to grasp more readily its physical underpinnings. An intuitive sense of electromagnetic field behavior can be acquired by computer experimentation. An added benefit is that numeric formulations enable students to obtain computer solutions to practical problems. Students can solve "real world" problems, not just those which satisfy the artificial boundary geometry of infinite planes, infinite circular cylinders, or spheres.
THE DETAILS Spatial discretization is a powerful method by which the internal electromagnetic details of circuit elements are readily visualized and related to the terminal behavior of lumped circuit elements. Several assumptions are used in this approach in order that the fundamental features of the fields are not hidden by unnecessary, advanced details. In this approach, a student’s first encounter with electromagnetic fields is limited to quasi-static cases so that the structures are small compared to wavelength. This correspond to the conditions of electric circuits. But more importantly, the complexities high-frequency, radiating fields are delayed until students have some experience with electromagnetics. Cartesian coordinates are chosen for most of the course in order to minimize the masking of electromagnetic principles by the complexities of new coordinate systems. Most structures are defined by or can be adequately approximated in Cartesian coordinates so that only a limited number of applications in cylindrical and spherical coordinates are included.
Further simplification is gained by considering mainly two-dimensional structures. One-dimensional structures provide a simplistic view of fields limited to only a single spatial variable and vector direction. Two-dimensional structures are adequate to show the variety of basic properties and spatial variations of potentials and vectors; the added complexity of three-dimensional structures is unnecessary. Equally important, two-dimensional electromagnetic fields quite conveniently take advantage of computer graphics displays. Structures and solution techniques are simplified by assuming that ideal flux-guiding material is present in all devices so that all of the flux is confined to the material, i.e. there is no fringing or leakage flux. The exclusion of fringing flux exactly models the fields of resistors and is a reasonable approximation for most capacitors. However, inductors are limited to closed, magnetic cores with at most a thin air gap. Air core inductors are not covered since the computational complexities associated with leakage flux do not reveal any additional basic principles. Under these conditions, the behavior of all lumped element devices is governed by ∇ 2V = 0, Laplace’s equation, where V represents a scalar electric or magnetic potential. The perpendicular nature of equipotential surfaces and flux lines is exploited to describe local behavior in terms of discretized, incremental elements. In this representation, all incremental element values are calculated by the same general form, ∆X = ∆Ψ /∆V, which can be specialized to give numeric values of incremental elements as ∆G = σ∆a/∆l. ∆C = ε∆a/∆l, and 1/∆R = µ∆a/∆l for conductors, capacitors, and magnetic reluctors, respectively. Figure 1 illustrates the relationship of the flux and potential difference of the incremental elements. The incremental elements are combined by the familiar series and parallel procedures of circuit theory to element values for finite structures.
high-performance computers, today’s students are able to numerically and analytically solve a wide variety of electromagnetic problems. The limited power of the calculator, paper, and pencils are replaced by the powerful tools used by electromagnetics professionals.
THE TOOLS Two new tools have been developed at Rose-Hulman in order to adopt this approach to electromagnetics. Classnotes [1] with the discrete/numeric perspective of this approach have been used as the text in the required undergraduate electromagnetics courses for two years. While these classnotes have provided a discrete/numeric flavor to the electromangetics courses, the full impact of this approach was not felt until the recent availability of the accompanying computer tools. Beginning in Fall 1998, all students at Rose-Hulman have laptop computers with a variety of mathematical software packages including MAPLE, EXCEL, and MATLAB. Students have completed two (2) years of MAPLE-based mathematics and have experience with the other software packages. Classrooms contain network connections including access to the World Wide Web. Computer and video projection systems are available in the classrooms. Design and development of Visual Electromagnetics (VEM) [2], a MATLAB-based application package, is now complete so that students have available an interactive, graphical, twodimensional Poisson equation solver. Students can explore the characteristics of electromagnetic fields that are expressed in analytic form with the calculus features of MAPLE. In addition to providing results in the form of mathematical expressions, MAPLE displays a wide variety of graphical formats.
Figure 1 - Incremental Elements This approach is technologically intensive; analytic tools with a modicum of graphics have been replaced by numeric, highly visual tools. With the ready availability of
Figure 2 - Electrostatic Field Potential Distrib ution.
For example, the equipotentials of the two dimensional potential field, v(x,y) = exp[-(x^2+y^2)], are superimposed upon a plot of the potential v(x,y) in Figure 2. The color of the surface at each point is related to the gradient of the potential at the point; the magenta color indicates that the steepest slope occurs midway down the “potential hill.” In Figure 3 the equipotentials are projected onto the zplane with their color indicative of the potential. The electric field vectors are directed perpendicular to the equipotentials; their color and their length indicate the electric field strength.
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Figure 5 - Potential Distribution within a Corner Resistor: Potential Surface.
Visual EMag (VEM)
Figure 3 - Equipotentials and Electric Field Vectors of Electrostatic Field. Numeric solutions of Laplace’s equation in twodimensions via finite differences are accomplished readily with EXCEL. The active cells of the spreadsheet conform to the shape of the resistor and define the solution region; the equations of each cell are programmed according to the material surrounding the cell and whether it represents an interior, an edge, or a corner node. The numeric potential distribution of a two dimensional resistor is represented by the spreadsheet of Figure 4. The corresponding potential plot is shown in Figure 5. 0.33 0.32 0.29 0.24 0.17 0.09 0
0.35 0.33 0.3 0.25 0.18 0.09 0
0.38 0.37 0.33 0.28 0.2 0.1 0
0.43 0.51 0.61 0.73 0.86 0.42 0.5 0.6 0.72 0.86 0.39 0.46 0.57 0.71 0.85 0.32 0.4 0.51 0.68 0.84 0.24 0.3 0.41 0.65 0.83 0.12 0.15 0.18 0 0 0
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Figure 4 - Potential Distribution within a Corner Resistor: Numeric Values.
Visual EMag (VEM) is a two dimensional, electromagnetic simulator designed and developed as a visualization aid for students in undergraduate electromagnetics. VEM utilizes finite difference techniques in electrostatic and magnetostatic environments. The VEM code, written in MATLAB 5 for Windows 95, provides a user-friendly graphical interface for which is inexpensive and platform independent. VEM consists of a structure window in which the user enters electromagnetic materials and sources via common drawing tools and pop-up menus. The solver button generates the system matrix, solves it, and activates the solutions window in which the results are displayed in a variety of user-selected viewing modes. Though the solution region is finite in extent, its outer boundaries appear open as if the solution region were of infinite extent. A compact simulation of the open boundaries is implemented via the Transparent Grid Termination (TGT) [3] techniques. Several VEM examples are described and shown below. The fields within a microstrip transmission line are shown in Figure 6 The equipotentials and electric field vectors are superimposed upon the microstrip structure. The equipotential strengths are indicated by their color; the electric field intensity is indicated by vector length and color. VEM clearly shows that the electric field intensity is greatest directly under the upper electrode and is perpendicular to the equipotentials and that. fringing fields are present throughout the entire region, but greatly reduced in magnitude. The coordinates and potential at the cursor are displayed in the lower left corner of the solution window.
Figure 6 - Equipotentials and Electric Field Vectors of a Microstrip Line. An alternate view of the solution is shown in Figure 7. The electric flux density is shown by logarithmically scaled white vectors; the background color shows the strength of the electric field intensity. With the wide variety of viewing options, students can select that which they understand best; they can compare the various views to gain a clearer understanding of the electromagnetic phenomena.
Figure 8 - Potential Distribution associated with a Corner Resistor. The full range of VEM features will be demonstrated during the presentation. The latest version of the VEM code is available for free distribution from the principal author.
THE FUTURE There is an air of anticipation about the future of this approach. With the classnotes and computer tools in place and in use, exciting opportunities and some challenging questions present themselves daily. It is too early to report on the success of this approach. An early assessment of this approach will be made this year and reported next year. In the meantime, the details of the course and the software package VEM are available for wider distribution.
REFERENCES [1] Voltmer, D., “Engineering Electromagnetics: Contin uous, Discrete, and Numeric,” EC340 classnotes, RoseHulman Institute of Technology, 1996. [2] Garner, D., “VEM: An Electromagnetics Visualization Aid for Students”, MS Thesis, Rose-Hulman Institute of Technology, May, 1998. Figure 7 - Electric Field Magnitude A final display option is shown in Figure 8 as the plot of the VEM solution to the potential within the corner resistor previously solved with EXCEL. The height and color of the surface are both related to potential. Fields and potentials exterior to the resistor are included here, though they weren’t within the previous solution.
[3] Lebaric, J., “Transparent Grid Termination: A Boundary Simulation Technique for Solving Problems Involving Poisson’s Equation,” unpublished notes.
