NUMERICAL METHODS FOR ELECTROMAGNETIC FIELDS AND CIRCUITS

Course Description

The course provides the theoretical and methodological tools for the numerical analysis of electromagnetic fields and for the modelling of electromagnetic structures using spatial and spatio-temporal discretisation techniques. Following a review of the fundamentals of classical electromagnetism, the main numerical methods for solving field problems are introduced, with particular emphasis on the Finite Element Method (FEM), the Finite Difference Method (FDM), the Boundary Element Method (BEM) and hybrid methods. The course also covers applications in computational electromagnetics, the use of electromagnetic CAD software and transmission line theory, providing students with the tools necessary to tackle design and simulation problems in the electrical and electronic fields.

Learning Outcomes

By the end of the course, students will be able to:
Knowledge and Understanding
•    Describe the theoretical foundations of classical electromagnetism and the main formulations of field problems.
•    Understand the mathematical principles underlying the spatial and spatio-temporal discretisation of electromagnetic problems.
•    Understand the theoretical foundations of the main numerical methods for solving field problems (FDM, FEM, BEM and hybrid methods).
•    Understand the variational and differential formulations used in the discretisation of electromagnetic problems.

•    Understand transmission line models and the main applications of computational electromagnetics.

By the end of the course, students will be able to:

Knowledge and Understanding

•    Describe the theoretical foundations of classical electromagnetism and the main formulations of field problems.
•    Understand the mathematical principles underlying the spatial and spatio-temporal discretisation of electromagnetic problems.
•    Understand the theoretical foundations of the main numerical methods for solving field problems (FDM, FEM, BEM and hybrid methods).
•    Understand the variational and differential formulations used in the discretisation of electromagnetic problems.

•    Understand transmission line models and the main applications of computational electromagnetics.

Applied Skills

•    Formulate mathematical models of electromagnetic problems using differential and integral equations.
•    Apply the main numerical methods (FDM, FEM, BEM and hybrid methods) to the solution of field problems.
•     Use CAD tools for the simulation and analysis of electromagnetic devices and structures.
•     Analyse transmission lines in the time, frequency and sinusoidal domains.

•     Critically interpret numerical results, assessing their accuracy, convergence, limits of validity and physical significance.

Independent Judgement

•    Select the most appropriate numerical method depending on the problem to be analysed.
•    Critically evaluate the reliability of the numerical solutions obtained.

•    Analyse the causes of any discrepancies between theoretical results and numerical simulations.

Communication Skills

•    Use the terminology of computational electromagnetics correctly.
•    Present mathematical models, numerical procedures and simulation results in written and oral form.

•    Communicate the methodologies, results and limitations of electromagnetic analysis techniques using appropriate technical language.

Learning Skills

•    Develop analytical and problem-solving skills in the modelling of electromagnetic problems.
•    Develop autonomy in learning and in the use of numerical tools, CAD environments and electromagnetic simulation software.
•    Consolidate a study method suitable for tackling advanced topics in computational electromagnetics and computer-aided design.

The course comprises lectures devoted to presenting the theoretical foundations of computational electromagnetics and the main numerical methods for field analysis (Lecture-based Teaching), supplemented by numerical exercises and laboratory activities carried out using electromagnetic CAD software (Interactive Teaching). The exercises enable students to apply the methodologies studied to solving field problems and analysing transmission lines, thereby developing the skills set out in the learning outcomes. Should the course be delivered via blended or distance learning, necessary adjustments may be made to the above arrangements in order to adhere to the programme set out 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.

Contribution of the course to the Goals of the 2030 Agenda for Sustainable Development
The topics covered in the course and the acquired knowledge are directly or indirectly aimed at the development of sustainable technological solutions, as well as contributing to a high quality education, in accordance with Goals 4, 7, 9, 11, 12, 13 of the 2030 Agenda for Sustainable Development.

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.