The paper presents the use of symbolic computation in electromagnetics, where the complexity of the involved mathematics often prevents the students from understanding the underlying physical aspects. The educational strategy we propose is complementary to the numerical approach. It is based on actual symbolic calculation of electromagnetic field, not just on field visualization.
An example of a Maple V worksheet dedicated to the computation of magnetic field with the BiotSavartLaplace method (BSL) is presented. The worksheet contains a theoretical introduction, 8 solved problems (equations, text and plots) and 12 proposed projects.
After going through several solved problems, the students are asked to make changes in the worksheet and to discuss the results. They are finally required to conceive their own worksheets for solving new proposed problems, using the BSL worksheet as a model.
Our experience shows that the symbolic computation has an outstanding educational potential not only in mathematics, circuits and systems, but also in electromagnetics.
The CDROM contains the HTML, PostScript and Maple V version 4 formats of the BSL worksheet.
Many concepts, even basic ones, concerning electromagnetic field theory are difficult to learn because they incorporate advanced mathematics (calculus) and they are not easy to manipulate and understand for many students. Electromagnetics, with application in domains such as electric machines, eddy current heating / nondestructive testing and microwaves, is usually viewed as one of the most difficult and abstract courses in electrical engineering curriculum.
The omnipresence of desktop computers gives students the feeling that a course which is not oriented towards computer use is an obsolete one. Observing these difficulties, more and more teachers aim to use computers to assist teaching of electromagnetics. This trend started already in the early 80's, when development and use of educational software running on PC's began in several areas of electric and computer engineering curricula, including electromagnetics [1].
There are many papers in the literature on how to present electromagnetic field theory to students and how to use computers in that process. A reference paper about the use of computers in teaching undergraduate electromagnetics is that of Hoburg [2]. The author describes motivations and conclusions after twelve years of experience in the development of electromagnetics educational software. Electromagnetics was a prime candidate because it had been traditionally presented through relatively abstract mathematical description and highly simplified models.
The main conclusions and recommendations of Hoburg are:
1. ``Active involvement by the student accomplishes more than the passive observation of results.''
2. ``General purpose tools are, in general, of considerable more value than concept  specific software.''
3. ``Robust and well implemented user interfaces are an essential part of good educational software.''
4. ``What actually works in terms of student understanding'' is the essential aspect in education.
5. ``The value of even well  designed, well  implemented teaching tools is strongly dependent upon how the tools are integrated into the course.''
These ideas guided (consciously or not) many educators trying to elaborate computer  aided teaching strategies. These strategies can be classified in the following three main categories:
1. the computer is used for numerical evaluation and visualization of preprogrammed formulae;
2. the computer is used for numerical analysis of the electromagnetic field, the student being either developer or user of a CAD software;
3. the computer is used for symbolic analysis, actually solving analytically the field equations.
Many papers describing the use of computers in teaching electromagnetics fall into the second category, evaluating the impact of commercial or educational CAD software packages based on FEM/FDM/BEM field analysis [3], [4], [5], [6], [7]. The last one noticed ``the package does not exempt the students to have a regular and durable work''. The educational potential of numerical field analysis for undergraduate students in early, introductory courses proves to be unsatisfactory due to a series of reasons:
· simple packages with educational purpose are able to solve only a very limited class of problems (e.g., electrostatic simple geometric configurations) and leads to the loss of the link to the fundamental equations of electromagnetics;
· the solutions are altered by numerical discretization errors, which are meshdependent;
· the advanced CAD packages are difficult to learn and require focus on ``marginal'' aspects such as: geometry description language, mesh generation, choice of appropriate solvers;
· CAD packages are not appropriate to study neither simple degenerate cases such as filamentary wires, punctual field sources or fluxcarrying surfaces (layers), nor the asymptotic behavior (at great distance from the field sources);
· the dependence of the results on the input data is masked, while this is easily observed from a formula.
However, the CAD packages could represent valuable tools in engineering education, especially for their capabilities in field visualizations and their ability to solve problems with complex geometries. The authors of [8] notice that ``the very complicated analytical solutions, formulas and derivations often lead students to fail the electromagnetic subjects'' and propose a new educational strategy based on the following rules:
· ``Students must be involved in developing the algorithms without becoming bogged down in complex programming for such tasks as matrix inversions and data visualization;''
· ``Students must learn to verify their code using models, which have analytical solutions. This is essential practice for both students code and commercial packages.''
· ``Excessive assessment of simple or trivial exercises is not desirable. Individualized projects following nonassessed, noncompulsory, learning workshops have generated considerable student enthusiasm for the subject.''
The Computational and Visualization Electromagnetics approach (CVEM) proposed in [8] complies to these rules and has as natural consequence the return of students ``to their theoretical textbooks to seek out and understand the analytical solutions and to interpret complex equations and the nature of the field''. CVEM is a projectoriented strategy, in which students develop FEM, MOM or BEM software packages written in the MATLAB programming language.
A completely different approach is based on the use of symbolic instead of numerical analysis in education. The use of generalpurpose symbolic mathematic programs, such as REDUCE (1968), MACSYMA (1970), Maple (1985) or Mathematica (1988), was initially introduced into engineering education in courses dedicated to mathematics [9], or circuits and systems [10].
Paper [11] uses Maple to analyze electric circuits and draws attention to the ``enormous capabilities of symbolic computation in education, because it enables a student to focus on the ideas of the theoretical approach rather than on the calculation difficulties.''
Paper [12] analyzes the use in education of the symbolic analysis programs dedicated to circuits, such as SC, PC  SNAPS, Sspice, SAPWIN, ISAAC, ASAP, SYNAP, SAPFC, SCYMBAL, SCAPP and RANIER.
Article [13] concludes ``the combination of symbolic manipulation and numerical analyses can be used in a complementary way to provide computer  aided instruction in teaching of electronics''. Symbolic manipulation packages like Maple give an additional insight that cannot be obtained by using simulation packages like SPICE.
The following conclusions are valid not only for circuit theory but also for electromagnetics:
One of the first papers concerning the use of symbolic analysis packages in electromagnetics is [14], which proposes an Electromagnetic Notebook package for Mathematica. Each notebook consists of text, ``equations'' and graphics. The ``equations'' are actually Mathematica expressions which evaluate formulas found in typical undergraduate electromagnetics textbooks and do not exploit the symbolic capabilities. The interactivity comes from the possibility to change equations parameters. In addition, much of the graphics can be animated. We consider this approach to be of the first category, as well as [15], which proposes the use of Mathcad for Windows as an effective learning tool in electromagnetics, for manipulation of complex expressions and for the visual display of the results. The author emphasizes that, with the proper use of Mathcad, especially of the ability to enter text, students' organizational and communication skills will also improve.
To conclude the short historic overview, we can say that:
· For electromagnetics education, most of the approaches fall in the first (numerical evaluation and field visualization) and second (numerical analysis of electromagnetic field) categories.
· The truly symbolic approaches are intensively used in electrical engineering education mainly in mathematics, circuits and systems, but not in electromagnetics.
· When generalpurpose symbolic packages are used in electromagnetic education, they are used mainly for visualization and less for finding the analytic solution.
Our approach is different and it is based on the following ideas:
· The generalpurpose symbolic packages (in our case, Maple) are used to do actual symbolic calculation and to analytically solve the electromagnetic field equations.
Although not dedicated to education, the approach of [16], which describes the development of efficient FE analysis codes based on analytical integration procedure, is, in some sense, closer to our point of view. In this paper, FEM matrix element contributions are generated in a symbolic form by a Mathematica notebook that outputs a suitable C or FORTRAN code.
· Of course, full advantage is taken of the evaluation and visualization capabilities, but these represent a marginal aspect of the educational process.
· The approach is projectoriented, in that the students are asked to solve new problems, based on models elaborated by the teaching staff.
· The students' activity is finalized by scientific reports containing text, formulae, graphics and, optionally, automatically generated C or FORTRAN code.
· The advanced /postgraduate students are also required to solve the same problem with numerical methods and to compare the results.
The symbolic approach has several important educational advantages: it highlights the close ties of the fundamental equations to the results of practical problems, has a good potential of improving students' modelling skills, has the capability of treating more complex problems while maintaining the advantages of analytical approaches, and allows highquality visualization of the fields of practical and complex problems.
The rest of the paper presents an example based on this strategy, which was used in the Department of Electrical Engineering at ``Politehnica'' University of Bucharest, Romania.
As an example of how the symbolic analysis can be used in undergraduate electromagnetics, a Maple V [17] worksheet dedicated to the computation of the magnetic field with the BiotSavartLaplace (BSL) method is presented.
The worksheet is conceived in such a way that it can be used both in ``classroom'' (with tutor supervision) and as a standalone tool for learning the BSL method. The provided solved example problems are sequenced in increasing degree of difficulty, so that both undergraduate and more advanced students can benefit from it.
The BSL worksheet comprises three main parts:
· A theoretical part, explaining the background of the method and its algorithm. For students who did not use symbolic computation before, a short part introducing symbolic analysis is also included.
· Solved problems. This part comprises 8 solved problems, with increasing degree of difficulty, meant to provide models for solving magnetics problems with the BSL method.
· Proposed projects. This part comprises 12 unsolved problems, for which students conceive their own worksheets for solving them.
This part starts with a general presentation of the BSL method and gives the main integral BSL formula, the one for the magnetic field. Since the students for which the worksheet was conceived have, in the second year of study, little knowledge about the magnetic field, some comments about the magnetic field lines are made subsequently. The spatial variation of the field magnitude is commented upon.
The theoretical part also comprises a section dedicated to the magnetic vector potential. The BSL formula for the magnetic potential, its spatial variation, as well as the advantages of using the vector potential instead of the field itself, are presented and commented upon.
The form of the BSL integrals for particular (degenerated) current distributions, such as currentcarrying thin sheets or filamentary conductors is also presented.
Finally, the algorithm of the BSL method is given.
Since not all the students for which the worksheet is intended worked with symbolic analysis packages before, the theoretical part also comprises a section on this subject, describing the concept of computeraided symbolic analysis and giving some very simple examples of Maple V syntax.
This part comprises problem formulation, solution hints and full Maple V code for solving 8 stationary magnetic field problems:
1. Currentcarrying segment;
2. System of infinite, currentcarrying filamentary conductors;
3. Interaction force between two parallel conductors;
4. Currentcarrying circular wire;
5. The Helmholtz coil;
6. The short, thin coil;
7. The plane spiral coil;
8. The rectangular currentcarrying bar.
The first four problems (the fourth, if only the field on the axis is required) are easy to solve by hand and are intended as introductory examples of the BSL method and Maple V use. Various Maple V facilities (line, contour and field plots, 3D plots, animation, limits, series developments, gradient, curl and divergence computation, solution of differential equations) are introduced in these problems.
For example, the code below (excerpt of the Problem no. 3) is used for demonstrating the rotating field of a threephase line (Fig. 1). For the clarity of the paper, the Maple instructions and only some of the results are presented; all the results computed by Maple are visible in the worksheet.
> i:=vector([1,0]):j:=vector([0,1]): # unit vectors of the coordinate axes in the problem's plane
> r:= evalm(x*i + y*j): # position of the field point
> r0:= evalm(x0*i + y0*j): # position of the source point
> modR := norm(rr0,2): # distance between source and field points
> kxr:=([(r[2]r0[2]), (r[1]r0[1])]): # the vector product (k x r)
> H0 := evalm(I0 * kxr/(2*Pi*modR)): # field produced by a conductor
> H:=unapply([H0[1],H0[2]], I0, x0, y0): # field of a conductor, function of current and of coordinates
> Htot3 := H(I1,x1,y1) + H(I2,x2,y2) + H(I3,x3,y3); # total field of three conductors

> A:=int(Htot3[1],y): # vector potential
> I1:=sin(t): I2:=sin(t2*Pi/3): I3:=sin(t+2*Pi/3): # Numerical values: harmonic currents
> # ... (instructions for plotting and animation)
Figure 1: Animation 
rotating field lines of a threephase line
Click on the figure to see the animation
The students are asked to interchange the expressions of I2 and I3 and observe the result.
The example below, extracted from problem number 4, shows the difference, at great distance from the source, between the field produced by a circular wire of radius a carrying the current i and an approximate expression thereof, valid at great distances namely, the expression of the field produced by a small permanent magnet of magnetic moment m = i pa^{2}.
The Maple commands and the plots of the relative difference in magnitude of the two fields versus the distance from the circular wire's center are shown in Fig. 2 (the figure shows in red the difference relative to the field in the current point, and in blue the difference relative to the field in the center of the wire). In this way, the notion of asymptotic expression is clarified. By imposing a certain error, the students can evaluate the distance at which the simple asymptotic expression can be used.
> errh:= subs(i=1,a=1,y=0,abs((evalm(Hm)[1]Hsp)/Hsp)); # Error relative to the field in the current point, versus distance

> errhc:=subs(i=1,a=1,y=0,abs((evalm(Hm)[1]Hsp )/subs(i=1,a=1,y=0,x=0,Hsp))); # Error relative to the field in the center, versus distance

> plot([errh,errhc],x=2..10,color=[blue,red]);
Figure 2: Relative
difference between the field of a circular wire and the field of a small magnet
Click to see large version of the picture
> solve(errh=0.01, x);

> solve(errhc=0.01, x);

In conclusion, the field outside the axis can be calculated with the formula valid at great distance, for distances bigger than 3...12 times the circle's radius.
The problems 58 are more complicated ones, for which an analytic expression is difficult to obtain by hand. For example, the last problem requires computation of the magnetic field produced by a massive conducting bar with rectangular crosssection. The expression of the field is complicated, and there would be no way to have the students obtain it in the time allocated for the BSLmethod exercises. Having obtained the required formula using Maple, various postprocessing steps can be undertaken, such as plotting the magnetic field vectors (see figure 3), evaluating the error made when the massive conductor is treated as a filamentary one, at great distances, or automatic generation of C or FORTRAN code lines associated to these expressions.
This part contains 12 proposed problems, requiring one to determine the field produced by:
· A currentcarrying segment of conductor, in an arbitrary point in space
· A conducting currentcarrying triangle, in an arbitrary point in space
· A rectangular filamentary conductor
· An arbitrary polygonal line
· A filamentary conductor whose shape is an arbitrary curve described by a parametric function
· Models for the field at great distance from a circular filamentary conductor
· Magnetic field produced by a system of parallel filaments and sheets
· A thin coil (with a single layer of turns), with rectangular turns
· A thin coil (with a single layer of turns), of arbitrary turn shape
· A thin coil (with a single layer of turns), with spherical core
· A coil with several layers of turns; study of the difference as to the thin coil
· A straight very long bar, of polygonal crosssection
The list is an open one, and other subjects such as: computation of inductivities, magnetic energy, forces or induced voltage can be added.
This part is, we believe, the one with the biggest educational impact. Based on the models provided by the solved problems, each student is invited to write a Maple V worksheet containing the solution of one proposed problem. They are thus obliged to apply ``creative reasoning'' for solving a new problem, to understand the basic BSL integrals, to select the right models from the available ones.
The projects contribute to the development not only of the theoretical knowledge, but also of the students' communication skills, by writing scientific reports. The graduate students are asked to compare the results with those of numerical CAD packages.
The proposed methodology was applied to fifthsemester Electrical Engineering students in an Electromagnetics course.
The instructor feedback included the following observations:
· the efficiency is increased if, in the beginning, the students solve by hand one or two simple problems;
· an introductory class in Maple syntax is useful, otherwise some students may take longer to solve a problem by computer than by hand;
· for undergraduate students, the presence of the instructor is necessary, in order to clarify some theoretical aspects.
The 23 participating students were invited to fill out a questionnaire. The main results on the questionnaire are:
· 78% of the students considered that they better understood the theory taught;
· 78% of the students said they could concentrate better on the interpretation of the results;
· 39% of the students succeeded to solve more problems, 26% about the same number, and 35% a smaller number of problems than by hand;
· 65% of the students thought the Maple V program was relatively easy to use;
· all the students considered that they would benefit if symbolic analysis would be used in other subjects of study.
Based on our previous experience throughout the years with various other techniques (CAEME packages, numerical analysis packages as developers or users of educational and commercial CAD packages, project  oriented symbolic approach but without model worksheet), we can say that the preliminary results obtained with the proposed approach are encouraging.
The paper presented an educational approach for teaching electromagnetics, based on symbolic analysis. In our approach, the computer is used for actually solving analytically the field equations, and not only for mere evaluation or visualization of formulas.
Based on teaching material (Maple V worksheets) prepared by the instructor, the students solve new problems and write scientific reports (containing formulas, text, graphics, and automatically generated C code).
The proposed strategy has as main advantages:
· close link to the fundamental field equations;
· understanding of the physical aspects and of the mathematical ``finesse''s;
· increase of students confidence (``electromagnetics may be difficult, but not impossible to learn'');
· gain of modeling skills, very useful in real, highcomplexity problems;
· better understanding of the modern computers capabilities to manipulate symbols, knowledge or other concepts specific to artificial intelligence;
· improvement of the professional communication skills.
The preliminary results obtained with the proposed approach are encouraging.
This work was supported in part by the Romanian Ministry of Education in the frame of the World Bank Project CNFIS 24354/99 ``Symbolic Analysis in Electrical and Electronic Education''.
The BSL worksheet was developed under Windows in Maple V version 4, Student edition, and translated to L^{A}T_{E}Xusing the ``Export'' facility of Maple.
The translation from L^{A}T_{E}X to HTML was made under the Linux operating system with TtH [18]. The Greek symbols may not appear on some Unix machines: please see [19].
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[18]
``TtH home page ,'' http://hutchinson.belmont.ma.us/tth/.
[19]
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Irina Munteanu*
Numerical Methods Laboratory
Electrical Engineering Department
"Politehnica" University of Bucharest
Spl. Independentei 313
77206 Bucharest
Romania
Phone: +4014106984
Fax: +4014111190
Email: irina@lmn.pub.ro
Daniel Ioan
Numerical Methods Laboratory
Electrical Engineering Department
"Politehnica" University of Bucharest
Spl. Independentei 313
77206 Bucharest
Romania
Phone: +4014106984
Fax: +4014111190
Email: daniel@lmn.pub.ro
Irina Munteanu (S'92M'95) received the M.Sc. and PhD degrees from
``Politehnica'' University of Bucharest, Romania.
She is an Associate Professor at the Electrical Engineering Department of the same university. Currently she teaches courses on numerical methods, electromagnetic waves and professional communication techniques.
Her main academic interests are in computeraided engineering education, and the research interests include electromagnetic field modeling and simulation, reduced order models for distributedparameter systems, absorbing boundary conditions, and highperformance computing.
She authored/coauthored more than 40 papers and 3 textbooks, was nominated for the 1998 Prize of the Romanian Academy  Technical Science Section and received an Alexander von Humboldt Foundation Fellowship in 2000.
Daniel Ioan (M'78) received his MS and PhD degrees in Electrical
Engineering from Polytechnic Institute of Bucharest in 1970 and 1979,
respectively.
Since 1970, he has been actively engaged in teaching and research at ``Politehnica'' University of Bucharest where he is currently Professor of Electrical Engineering. His main professional interest are in computer aided electrical engineering, high performance computing, and computer aided engineering education.
He is founder of Numerical Method
Laboratory and president of Computer Aided Electrical Engineering Research
Center at "Politehnica" University. Initiator of many educational and
research international projects and conferences. Chairman on IEEE Romanian
Section 1997  1998.
File translated
from T_{E}X by T_{T}H,
version 2.62.
On 16 Feb 2000, 00:00.