NUMERICAL METHODS FOR ELECTROMAGNETIC FIELDS AND CIRCUITS

      In the design of structures and devices inherent in the various fields of Engineering, models formulated in terms of physical quantities that typically possess both a spatial and a temporal coordinate dependence are extensively employed nowadays. The complexity of the links that exist between these quantities, expressed mathematically in the form of systems of functional equations, frequently differential, integral or integral-differential in nature, is such that only an approximate resolution is possible, obtained by resorting to numerical methods whose use requires discretization of the said quantities with respect to both types of coordinates.

This approach to design and the consequent implementation techniques have become so efficient and well-established that one can, without fear of contradiction, state that there is no industry (electrical, electronic, mechanical, aerospace, etc.) or medium-to-large research center that has not equipped itself with a special section specializing in the development and use of CAD (Computer Aided Design) tools for the purpose of implementing such design procedures.

Knowledge and understanding.

During the Engineering studies, the student acquires some knowledge of methods suitable for the temporal discretization of quantities that intervene in the physical-mathematical modeling of structures or processes of interest in the disciplines inherent to such studies, but much more rarely, than those used in the spatial or spatial-temporal discretization of said quantities.

One of the primary objectives of the “Numerical Methods for Electromagnetic Fields” course is precisely to acquire knowledge and develop the ability to understand the main spatial and/or temporal discretization methods, presenting their theoretical bases. methodologies and illustrating their concrete application within that vast and continually expanding research sector called Computational Electromagnetism. These numerical methods and the algorithms that implement their actual use have a scope that goes well beyond the sectors of Electrical and Electronic Engineering, in which, moreover, there are numerous application contexts in which the circuit approach, which requires at most the temporal discretization of the network variables, appears to be completely inadequate and must therefore be replaced by the field one. Common contexts of this nature are, for example, the analysis and design of structures whose operation is based on the free propagation of the electromagnetic field (typically antennas) and those which instead use guided propagation (typically transmission lines, metallic and dielectric waveguides, optical fibers, etc.). Other engineering fields which are gradually assuming increasing importance and for which the use of these methods is indispensable, are that of the analysis of the joint electrical and thermal behavior of power devices and that of the design of electrical systems and equipment. and electronic, in compliance with international standards resulting from electromagnetic compatibility studies, both industrial and environmental, as well as from those studies aimed at ensuring the quality of electrical quantities.

The other primary objective of the course, closely connected with the first and which consequently reflects on the contents of the course itself, is to provide a brief review of the fundamental laws of electromagnetism, aimed at the knowledge and understanding of some, but significant, their applications in the field of Electrical Engineering. To this end, after having recalled and commented on the formulation of electromagnetism both in global terms (i.e. integral) and in local terms (i.e. differential), we will discuss some specific applications concerning the stationary regime of the electric field, the magnetic field and the current field concerning respectively the calculation of capacitance, inductances and electrical resistances, as well as the calculation of mechanical stresses between conductors to subsequently move on to those inherent to the magnetic quasi-stationary regime, far more relevant than the electrical quasi-stationary one, represented by the study of phenomena of induced currents and the skin effect. This part of the course will conclude with brief notes on the behavior of electromagnetic fields in non-stationary conditions, regarding in particular their propagation by plane waves in material media and their classification in relation to the polarization state.

This part of the course will be followed by that concerning the exposition of a significant part of the theory of uniform transmission lines, operating both in the transient regime and in the sinusoidal regime, illustrating both the general and specific aspects of these operating conditions, all motivated and exemplified through discussion and resolution of various exercises and problems.

Applying knowledge and understanding.

Of the numerical methods, the course will mainly deal with the Finite Element Method (FEM) which, developed in the sixties and having now never completely supplanted the Finite Difference Method, or FDM , has established itself as the most powerful numerical method for solving field problems formulated in differential terms. The basic idea of the method consists in dividing the problem definition domain into a large number of subdomains of simple shape, called finite elements (typically tetrahedra and parallelepipeds in 3D, triangles and parallelograms in 2D), at within which it is assumed that the quantities of interest have trends that can be expressed through simple mathematical expressions (linear, quadratic, etc.), whose coefficients depend on the values that the quantity in question has at specific points of the finite element, called nodes. Furthermore, integral methods are of particular importance, the most important of which is the Boundary Element Method (BEM) and hybrid methods, such as FEM-BEM and numerous others, particularly suitable for solving problems defined in domains of unlimited extension, extremely frequent in electromagnetism. This spatial discretization process leads to transforming the system of functional equations (differential and/or integral) that arises from the mathematical formulation of the problem, into a system of algebraic equations whose unknowns are the values of the quantities representing the fields introduced in the discretization process, such as, for example, nodal values. The resolution of this system then allows us to obtain, using appropriate algorithms, a solution whose degree of accuracy essentially depends on the number of finite elements used and on the order of these, with an obvious consequent computational cost increasing with the value of said parameters. of discretization.

Finally, a praying objective of the course is to exemplify its contents, carrying out an appropriate number of exercises based on the use of specific calculation codes, aimed both at the resolution of some types of field problems and at the analysis of lines of transmission. In this way, the student is provided with the opportunity to strengthen and bring to an application level the knowledge and understanding previously acquired theoretically.

Making judgements.

The course also intends to stimulate and increase the student's ability to exercise critical and judgmental skills. In fact, the identification of the most appropriate strategy for solving a specific problem to be addressed with numerical techniques, in relation to its nature and the quantities to be calculated, requires the student to carry out a careful examination of the problem and a reflection on the knowledge already acquired to solve it. Once the solution has been obtained, the student is also asked to verify the correctness of the expected result, albeit approximate. A further source of acquiring independent judgment is the ability to provide an explanation for possible initially unexpected results, which further contributes to improving the understanding of the calculation method used and to developing during the preparation for the teaching exam, the ability to formulate hypotheses on the expected form of the solution to a problem, even if we have non-exhaustive information on it.

Communication skills.

One of the outcomes that the course aims to achieve is learning the correct use of both terminology and mathematical tools, as well as the physical knowledge, learned in the preparatory courses, necessary for solving specific field problems. During the lessons, particular attention was obviously dedicated to the units of measurement of electrical quantities and their correct use. A significant part of the theoretical results of the course are then demonstrated, further contributing to increasing the understanding of the results themselves and their implications, as well as their appropriate and flexible use in problem solving. This stimulates and advances the student's communication skills, enabling him to communicate clearly and without uncertainty both with subjects cultured in the discipline and with subjects who are not, providing both categories with valid arguments.

Learning skills.

The study activity required by the course, traditionally and equally divided between the acquisition of concepts and theoretical results and the progressive increase in the ability to solve specific problems, leads to an improvement in the student's ability to reflect and learn. Specifically, the analysis of problems of electromagnetic fields and transmission lines, having different characteristics, requires the student to refine his ability to identify the most suitable solution strategy. All this determines an increase in the ability to classify problems and the strengthening of one's own effective study method, which will certainly be useful in the future.

The knowledge to be acquired during the course is the content of the frontal lessons carried out in the classroom by the teachers and - in order to facilitate personal study - the topics are listed in detail in the course program, with explicit references to the parts in which they are covered in the main texts recommended.

The examples carried out by the teachers in the classroom which follow the theoretical explanation of a new topic and the learning activity carried out independently by the student, represent the means by which he learns to apply the theoretical topics covered in class. The student is also invited to delve deeper into the topics covered, using materials other than those proposed, especially as regards the personal study phase, thus developing the ability to apply the knowledge acquired to contexts different from those presented during the course. If the teaching is taught in mixed or remote mode, the necessary variations may be introduced with respect to what was previously declared, in order to respect the planned program and reported in the syllabus.

The prerequisite is having attended and passed a basic course in Electrical Engineering.

Class attendance is not compulsory.

1. Introductory notions.

- Scientific notation, order of magnitude, significant figures. Electrical quantities and their systems of units; fundamental concepts and main rules of use of SI.
- Scalar fields and vector fields. Differential operators and integral operators. Simple-connected and multiple-connected domains. Generalized gradient, divergence and rotor theorem. Conservative fields and solenoidal fields; scalar potentials and vector potentials; Helmholtz theorem. Orthogonal curvilinear coordinate systems. Lemmas and Green's formulas. Harmonic functions and their main properties. Boundary value problems for the Poisson equation; representation theorem; types of potentials and their properties. Green's function for Dirichlet-type and Neumann-type boundary value problems.

2. Computational electromagnetism.

- Numerical methods for the calculation of electromagnetic fields.
- The Finite Difference Method (FDM).
- The Finite Element Method (FEM). Domain discretization; linear scalar triangular finite elements, shape functions, local coordinates, standard simplex. Variational formulation of the scalar Poisson equation. Dirichlet matrix and metric matrix of a finite element. Dirichlet-type, Neumann-type and Robin-type boundary conditions. Evaluation of integral quantities (flows, energies, forces). Higher order triangular elements. Quadrangular and hexahedral elements.
- The Boundary Element Method (BEM). Integration of singular functions.
- Hybrid Methods (HMs). The FEM-BEM and FEM-DBCI (Dirichlet Boundary Condition Iteration) methods.
- Edge-type Vector Finite Elements.
- FDM and FEM methods in the time domain.
- Newton-Raphson and fixed-point methods for the solution of non-linear cases.

3. Electromagnetism.

- Electric charge and its properties.
- Stationary electric field. Global properties and local properties of the electric field in vacuum. Electric scalar potential. Electric field and electric potential of a charge distribution. Electric field in conductors. Electrostatic induction. Capacity coefficients and potential coefficients, proper and partial, of a system of conductors; electrostatic screens. Equivalent network of capacitors. Capacitor. Capacity calculation examples. Electric field energy. Mechanical stresses on charges and conductors. Polarization of a dielectric, polarization intensity vector. Electric field in matter; electric displacement vector. Form and name of constitutive equations. Notes on dissipative phenomena in dielectrics: loss angle, corona effect; dielectric strength and discharge phenomena in dielectrics.
- Stationary current field. Conduction, convection, advection, diffusion electric current; electromotive force generators. Current density vector. Global properties and local properties of the current field. Form and name of constitutive equations; Ohm's law for specific quantities, electromotive field. Calculation of the steady-state current field in linear media. Joule's law for specific quantities and energy balance. Conductances and resistances, proper and partial, of a system of conductors; equivalent network of resistors. Examples of resistance calculations.
- Stationary magnetic field. Global properties and local properties of the magnetic field in vacuum. Poisson vector equation; Biot-Savart law. Magnetic scalar potential. Magnetic field energy. Inductance matrix of a conductor system. Internal inductance and external inductance of a conductor. Examples of inductance calculations. Magnetic field in matter; magnetization intensity vector. Form and name of constitutive equations. Diamagnetic, paramagnetic and ferromagnetic media. First magnetization curve; normal, incremental and differential permeability; hysteresis loop, permanent magnets. Magnetic circuits. Flow tube. Hopkinson's law, reluctance, loss figure, equivalent electrical circuit. Mechanical stresses on the conductors.
- Quasi-stationary magnetic field. Equation of the diffusion of the magnetic field in conductors; skin and proximity effect, depth of penetration; conducting half-space, conducting plate, conductor of circular section. Induced currents.
- Electromagnetic field. Global properties, for fixed and mobile domains, of the electromagnetic field. Faraday-Neumann-Lenz law; dynamic and motivational induced electromotive force. Ampere-Maxwell law; displacement current. Local properties of the electromagnetic field; Maxwell's equations, interface conditions and regularity conditions at infinity. Electromagnetic potentials; auxiliary (gauge) conditions. Wave equations of potentials. Energy and mechanical stresses of the electromagnetic field; vector and Poynting theorem. Uniform electromagnetic waves; propagation of plane waves and spherical waves in material media. Sinusoidal electromagnetic field. Helmholtz equation; solution, boundary conditions and radiation condition. Monochromatic plane waves; dispersion relation, types of polarization. Vector and complex Poynting theorem.
- Deduction of the circuit model of an electrical system. Kirchhoff's laws.

- Elements of electromechanical energy conversion.

4. Transmission lines.

- Transmission Line Model. Free propagation and guided propagation; distributed parameter systems and propagation modes. Assumptions, deductions and validity limits of the Transmission Line model. Two-conductor lines, uniform lines, non-distorting lines, ideal lines. Calculation of the primary parameters of a coaxial line.
- Analysis in the time domain. System of first order equations, second order equations; initial conditions and boundary conditions. Energy balance of a line. Study of non-distorting lines and ideal lines; group velocity and phase velocity; form, properties and physical interpretation of the solutions. Elementary case studies of ideal lines with bipolar terminations. Notes on multi-conductor lines.
- Analysis in the pulsation domain. Line equations in the pulsation domain; propagation parameter and characteristic impedance. Form, properties and physical interpretation of solutions. Impedance, admittance, reflection coefficient, transmission coefficient. Representations of immittance, transmission, diffusion. Study of non-distorting lines of finite length. Adapted line. Elementary case studies of ideal lines with resistive terminations.
- Analysis of lines in sinusoidal regime. Deduction of the equations and properties of the lines in the sinusoidal regime from the analogues valid in the pulsation domain. Study of non-distorting lines; voltage trend and current trend, VSWR, lines at λ/2, lines at λ/4. Elementary case studies of ideal lines with bipolar terminations. Energy balance. Smith diagram and use of it; fitting a line using stubs.

REFERENCE TEXTS

 

1. P. P. Silvester, R. L. Ferrari: “Finite elements for electrical engineers”, 3rd edition, Cambridge University Press, 2003.

2. Jian-Ming Jin: “The Finite Element Method in Electromagnetics”, 3rd Edition, Wiley-IEEE Press, 2014

3. S. Alfonzetti: " Lecture Notes of the Course on Numerical Methods".

4. A. Laudani: “ Lecture Notes of the Course on Numerical Methods”.

5. Clayton R. Paul: " Analysis of Multiconductor Lines". Wiley.

6. G. Miano, A. Maffucci: " Transmission Lines and Lumped Circuits". Academic Press.

7. G. Franceschetti: " Electromagnetics: theory, techniques and engineering paradigms". Plenum Press, New York, 1997.

8. S. Ramo, J. R. Whinnery, T. Van Duzer: "Fields and waves in communication electronics", 3rd edition, 1977.

 

OTHER EDUCATIONAL MATERIAL It is available at the following address:

https://studium.unict.it/dokeos/2024/

The examination consists of a single oral test, during which the candidate is asked some questions the topics covered by the two lecturers and in which an optional end-of-course paper prepared by the student is discussed. The learning assessment may also be carried out also by telematic means, should the conditions require it.

Examples of typical questions are available on the Studium platform and the course website.