The course provides an introduction to analytical and numerical methods for the analysis and design of low-frequency electromagnetic systems relevant to electrical engineering. The first part of the course is devoted to analytical modelling of electromagnetic field problems under low-frequency assumptions. Starting from vector analysis and Maxwell’s equations, the course addresses electrostatic fields, steady-current fields, magnetostatic fields, magneto-quasi-static fields and transmission lines. Particular attention is devoted to the computation of engineering quantities such as capacitance, resistance, conductance, self and mutual inductance, partial inductance, magnetic forces, Joule losses, eddy-current losses and distributed line parameters. The second part of the course introduces numerical techniques for electromagnetic field computation. After discussing basic concepts of numerical approximation, quadrature, numerical differentiation and time integration, the course presents the finite difference method and the finite element method for low-frequency electromagnetic problems. The numerical part emphasizes the formulation of field problems, the implementation of simple computational procedures, the validation of results against analytical reference solutions and the critical interpretation of numerical simulations. Classroom exercises and computer-based activities are integrated throughout the course. Analytical exercises are used to consolidate the theoretical concepts, while numerical mini-projects allow students to develop and discuss computational models for selected electromagnetic problems using Matlab, Python or other suitable tools.

At the end of the course, students are expected to have acquired both theoretical and practical skills for the analysis and numerical modelling of low-frequency electromagnetic problems relevant to electrical engineering. In particular, students will be able to: 1. Understand and formulate low-frequency electromagnetic field problems Students will be able to identify the appropriate electromagnetic approximation — electrostatic, steady-current, magnetostatic or magneto-quasi-static — starting from Maxwell’s equations and from the physical characteristics of the problem. 2. Apply analytical methods to canonical electromagnetic configurations Students will be able to solve simplified electromagnetic field problems using analytical or semi-analytical techniques, with particular reference to electric potential, magnetic vector potential, field energy and interface conditions. 3. Compute engineering quantities from field solutions Students will be able to compute capacitance, resistance, conductance, self and mutual inductance, partial inductance, magnetic forces, Joule losses and eddy-current losses in configurations of interest for electrical engineering. 4. Relate field models to circuit and transmission-line models Students will be able to extract lumped and distributed parameters from electromagnetic field solutions and use them in circuit-level or transmission-line descriptions. 5. Understand the foundations of numerical field computation Students will understand the basic principles of numerical approximation, numerical integration, finite differences and the finite element method, including the role of discretization, boundary conditions, convergence and validation. 6. Develop simple numerical codes for electromagnetic problems Students will be able to implement basic numerical procedures for field-related computations using Matlab, Python or other suitable tools. These procedures may include quadrature, time integration, finite-difference solvers and simple finite-element formulations. 7. Use numerical tools critically Students will be able to interpret the results of numerical simulations, compare them with analytical reference solutions when available, identify modelling assumptions and assess the reliability and limitations of the computed results. 8. Communicate modelling choices and results Students will be able to explain the formulation of an electromagnetic problem, justify the adopted analytical or numerical method, describe the structure of a computational implementation and discuss the physical meaning of the results.

Students are expected to have a basic knowledge of: • electric circuit theory, including DC circuits, sinusoidal steady-state analysis, transients, phasors, power and three-phase systems; • basic electromagnetism, including electric and magnetic fields, Maxwell’s equations in elementary form, electric potential, magnetic flux and electromagnetic induction; • differential and integral calculus for functions of one and several variables; • basic linear algebra, including vectors, matrices and systems of linear equations; • elementary ordinary differential equations; • basic programming skills in Matlab, Python or an equivalent computational environment. Previous experience with circuit simulation tools, such as SPICE or Simulink, is useful but not mandatory.

Part I — Analytical low-frequency electromagnetics 1. Vector analysis and electromagnetic field quantities Scalar and vector fields. Gradient, divergence, curl and Laplacian. Flux and circulation. Integral theorems. Electromagnetic field quantities and material properties: electric field, electric flux density, current density, magnetic flux density, magnetic field intensity, permittivity, conductivity and permeability. 2. Maxwell’s equations and low-frequency approximations Maxwell’s equations in differential and integral form. Constitutive laws. Classification of electromagnetic problems: electrostatic, steady-current, magnetostatic, magneto-quasi-static and full time-varying regimes. Validity of low-frequency approximations in electrical engineering applications. 3. Electrostatic fields and capacitance computation Electric potential formulation. Laplace and Poisson equations. Conductors and dielectrics. Interface conditions at conductor/dielectric and dielectric/dielectric interfaces. Electric field energy. Capacitance and capacitance matrices. Analytical solutions for canonical geometries: parallel plates, coaxial systems, cylindrical configurations, multi-conductor systems and method of images. Applications to insulation systems, electrodes, cables and line capacitance. 4. Steady-current fields and resistance computation Local Ohm’s law and charge conservation. Scalar potential formulation in conducting media. Interface conditions between conductive media. Joule losses. Analogy between electrostatic and steady-current fields. Resistance and conductance matrices. Applications to grounding systems, current spreading, contact resistance and leakage-current problems. Analogies between electrostatic, steady-current and thermal conduction problems are discussed to highlight common mathematical structures. 5. Magnetostatic fields, inductance and magnetic forces Ampčre’s law, Biot–Savart law and magnetic vector potential. Magnetic material interface conditions. Magnetic flux, flux linkage, self and mutual inductance. Partial self and mutual inductance of conductors and current paths. Inductance matrices. Magnetic energy and co-energy. Magnetic forces from energy methods. Magnetic circuits as engineering approximations. Applications to coils, busbars, magnetic cores, actuators, cables and multi-conductor systems. 6. Magneto-quasi-static fields Faraday’s law and induced electric fields. Magnetic diffusion in conducting media. Eddy currents. Skin effect and proximity effect. Penetration depth. Losses in conducting and magnetic materials. Interface conditions in conductor/air and magnetic/conductive configurations. Magnetic shielding and induction heating may be discussed as application examples in connection with numerical modelling. 7. Transmission lines from electromagnetic field parameters Distributed parameters of electrical lines: resistance, inductance, conductance and capacitance per unit length. Extraction of line parameters from electrostatic, steady-current and magnetostatic field solutions, including capacitance and inductance matrices. Telegrapher’s equations. DC transients and sinusoidal steady-state operation of lines. Engineering applications to cables, busbars and multi-conductor systems. Circuit-based simulation of transmission lines using tools such as SPICE or Simulink may be used to connect field-derived parameters with system-level behaviour. Part II — Numerical methods for electromagnetic field computation 8. Numerical approximation and error concepts Approximation, discretization, truncation error, round-off error and convergence. Modelling error versus numerical error. Use of analytical solutions as benchmarks for numerical models. 9. Numerical integration and quadrature Newton–Cotes formulas. Gaussian quadrature in one dimension. Extension to multidimensional integrals. Error estimation. Applications to electromagnetic field integrals, fluxes, energy quantities and mutual inductance calculations. 10. Numerical differentiation and discrete field operators Forward, backward and central differences. Higher-order schemes. First and second derivatives. Boundary nodes. Partial and mixed derivatives. Discrete gradient, divergence, curl and Laplacian operators. 11. Time integration of first-order differential systems Initial value problems written in the general form M\,dx/dt + Kx(t) = b(t). Explicit and implicit Euler methods. Stability and accuracy. Theta method and Crank–Nicolson scheme. Predictor–corrector methods. Basic adaptive step-size concepts. The formulation is first introduced using simple circuit examples and then connected to transient field and finite-element formulations. 12. Finite difference method for field problems Finite-difference discretization of Laplace and Poisson equations. Structured grids. Algebraic system assembly. Implementation of Dirichlet, Neumann and Robin boundary conditions. Equivalent network interpretation. Post-processing of potentials, fields, fluxes and energies. Limitations for irregular geometries and non-homogeneous materials. 13. Fundamentals of the finite element method Motivation for FEM. Meshes, elements, nodes, connectivity and element quality. Shape functions in 1D and 2D. Field interpolation. Weighted residuals and weak formulation. Galerkin method. Element matrices, assembly and treatment of boundary conditions. 14. FEM for low-frequency electromagnetic problems Finite-element formulations for electrostatic, magnetostatic and magneto-quasi-static problems. Scalar potential formulation for electrostatic problems. Magnetic vector potential formulation for 2D and axisymmetric magnetostatic problems. Capacitance and inductance extraction from FEM solutions. Coils, imposed current density, permanent magnets and magnetic materials. Magnetic energy, Joule losses, eddy-current losses and force computation. Frequency-domain and time-domain formulations for eddy-current problems. Validation against analytical solutions. Magnetic shielding and induction-heating configurations may be considered as application examples. 15. Brief overview of 3D FEM modelling Differences between 2D, axisymmetric and 3D models. Computational cost and mesh requirements. Nodal versus edge-based formulations at a conceptual level. Examples of 3D electromagnetic applications. Limitations of simplified 2D models. 16. Optional numerical solver topics Basic notions on sparse linear systems, direct and iterative solvers, convergence criteria and nonlinear iterations. Fixed-point and Newton-type approaches for nonlinear magnetic materials. This topic may be adapted depending on available time and on the needs emerging from the computer-based activities.

The course combines analytical modelling, numerical methods and computer-based activities. Students are expected to participate actively in classroom exercises and computational sessions, since these activities are closely connected to the final assessment. The course makes use of Matlab, Python or equivalent computational tools. Students may also use symbolic computation software, circuit simulation environments and other appropriate tools to support their work. The use of generative AI systems is permitted as a support for coding, checking, documentation and study activities, provided that students remain fully responsible for the submitted material and are able to explain, justify and critically assess their work. The computer-based activities require students to bring or have access to a personal computer suitable for running the selected computational tools. Further organizational details, including deadlines for numerical mini-projects and specific instructions for the oral examination, will be provided during the course.

The course consists of lectures, classroom exercises and computer-based activities. Lectures introduce the theoretical foundations and the modelling approaches required to analyse low-frequency electromagnetic problems. The course is divided into two main parts. The first part is devoted to analytical methods for electrostatic, steady-current, magnetostatic, magneto-quasi-static and transmission-line problems. The second part introduces numerical techniques for electromagnetic field computation, with emphasis on numerical integration, finite differences and the finite element method. Classroom exercises are distributed throughout the first part of the course and are organized by topic. They include both theoretical and application-oriented problems, with the aim of consolidating the analytical methods presented during the lectures. Typical activities include the computation of capacitances, resistances, inductances, magnetic fields, eddy-current effects and transmission-line parameters in canonical and engineering-relevant configurations. In selected cases, calculations may be carried out with the support of Matlab, Python or other suitable computational tools, at the student’s discretion. In the second part of the course, the exercises are mainly computer-based. Students develop numerical codes and small computational projects related to the topics discussed in class. These activities include, for example, the numerical integration of electromagnetic quantities, the time integration of first-order differential systems, the finite-difference solution of potential problems, and the finite-element modelling of electrostatic, magnetostatic or magneto-quasi-static configurations. The computer-based activities are intended to help students understand not only how numerical tools are used, but also how they work, what assumptions they rely on, and how their results should be interpreted and validated. Particular attention is devoted to comparison with analytical reference solutions, parameter studies, field post-processing and the extraction of engineering quantities such as capacitance, resistance, inductance, forces and losses. Students may use any appropriate tool for developing their computational work, including programming environments, symbolic manipulation software, circuit simulation tools, technical documentation and generative AI systems. These tools are considered aids for modelling, coding, checking and documenting the work. They do not replace the student’s responsibility for understanding the formulation, verifying the correctness of the implementation, interpreting the results and critically assessing the validity of the final product. The numerical codes and mini-projects developed during the second part of the course will contribute to the final assessment, as specified in the “Assessment and grading criteria” section.

Main references • D. K. Cheng, Field and Wave Electromagnetics, Addison-Wesley. • J. D. Jackson, Classical Electrodynamics, Wiley. Numerical methods and FEM references • M. N. O. Sadiku, Numerical Techniques in Electromagnetics, CRC Press. • P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, Cambridge University Press. My suggestion

Dispense; Esercitazioni di laboratorio; Materiale multimediale ; Strumenti di simulazione;

Modalita di esame: Prova scritta (in aula); Prova orale obbligatoria;

Exam: Written test; Compulsory oral exam;

... The final exam consists of two parts: a written examination and an oral examination. The written examination assesses the student’s ability to apply analytical methods to low-frequency electromagnetic problems. It has an indicative duration of 1.5 hours and consists of two exercises related to Part I of the course. The exercises are similar in style and level of difficulty to those discussed during the classroom practice sessions and may cover electrostatic fields, steady-current fields, magnetostatic fields, magneto-quasi-static fields and transmission-line parameters. During the written examination, students are allowed to use their own notes. No electronic devices are allowed, except for a scientific calculator. The written examination is graded out of 16 points overall. The oral examination assesses the student’s understanding of the numerical methods introduced in Part II of the course. During the oral examination, students discuss the numerical codes and mini-projects developed during the computer-based activities. Students must bring their own computer in order to show, run and explain the codes developed during the course. The oral discussion focuses on the formulation of the problem, the numerical method adopted, the structure of the code, the assumptions made, the validation of the results, and the interpretation of the computed electromagnetic quantities. The assessment does not only consider whether the code produces a numerical result, but also whether the student can explain the formulation, justify the numerical choices and critically discuss the validity of the results. Students are allowed to use suitable computational and documentation tools during the development of their numerical projects, including programming environments, symbolic computation software and generative AI systems. However, students remain fully responsible for the submitted and discussed material. They must be able to explain all relevant parts of their codes, justify the adopted modelling choices, identify possible limitations and demonstrate a critical understanding of the results. The oral examination is graded out of 16 points. The final score is obtained as the sum of the written-examination score and the oral-examination score, for a maximum of 32 points. Each part has a minimum passing threshold of 8 points. The minimum final score required to pass the exam is 18 points. Final scores higher than 30 are recorded as “30 cum laude”. The assessment is designed to verify that students are able to formulate and solve analytical electromagnetic problems, compute relevant engineering quantities such as capacitance, resistance, inductance, forces and losses, implement basic numerical methods for field computation, critically interpret numerical results, and compare numerical solutions with analytical reference cases where available.

Gli studenti e le studentesse con disabilita o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unita Special Needs, al fine di permettere al/la docente la declinazione piu idonea in riferimento alla specifica tipologia di esame.