1011149 - NUMERICAL METHODS FOR ELECTROMAGNETIC FIELDS AND CIRCUITS A - Z

In the design of devices in the various engineering sectors, physical quantities are used which, generally, vary both in space and time. The complexity of the (differential) equations that these quantities must satisfy is such as to allow only an approximate resolution with numerical techniques, in which the physical quantities of interest are discretized both spatially and temporally. These techniques are so important that it can certainly be said that there is no industry (electronics, mechanics, aerospace, electromechanics, etc.) or medium-large research center that is not equipped with CAD (Computer Aided Design) tools based on they.

Knowledge and understanding.

During his engineering studies, the student engineer matures sufficient experience in the discretization of temporal quantities, but, perhaps, not so much in the discretization of spatial or space-time quantities.

The purpose of the course of “Numerical methods for electromagnetic fields and circuits” is precisely to study techniques for space-time discretization. These techniques have a value that goes far beyond the field of Electrical and Electronic Engineering, in which, moreover, there are various applications in which the circuit approach is inappropriate, such as the calculation of antennas and transmission lines for telecommunications (coaxial cables, wave guides), the calculation of the electrical and thermal behavior of power transistors and the verification of the electromagnetic compatibility of various electronic and electrical devices.

Applying knowledge and understanding.

The course will mainly deal with the finite element method (FEM) which, conceived in the 1960s, has now completely supplanted the finite difference method and has established itself as the most powerful numerical method for solving field problems. The basic idea of ​​the method consists in dividing the spatial domain in which it is necessary to determine the trend of some quantities of interest into a large number of subdomains of simple form, precisely called finite elements (tetrahedra and parallelepipeds in 3D, or triangles and parallelograms in 2D), in which it is assumed that the quantities have equally simple trends (linear, quadratic, etc.) which can be expressed as a function of the values ​​assumed by the quantity in question in the vertices of the finite element (nodes). This process of spatial discretization leads to transform the system of partial differential equations into an algebraic system whose unknowns are the nodal values. The resolution of this system allows you to obtain a solution whose degree of accuracy increases as the number of finite elements increases (but with a consequent higher computational cost).

Another part of the course is dedicated to presenting the main concepts and results of the theory of transmission lines operating in both transient and sinusoidal conditions, with particular regard to two-conductor lines. In order to clarify the contents of these topics and enable the student to apply them, various transmission line problems are developed during the lessons and some numerical approaches to their analysis are described.

Finally, the course also aims to provide a brief, but not superficial, introduction to the theory of both passive and active electrical filters and their implementation, as well as to the theory of antennas and electromagnetic waves, giving the main definitions and the most important concepts.

Making judgements.

The course also intends to stimulate and increase the ability to exercise the student's critical and judgment skills. In fact, the identification of the most appropriate strategy for solving a specific problem to be faced 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 suitable to solve it. Once the solution is obtained, the student is also asked to verify the correctness of the solution obtained on the basis of the expected result, even if 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 in the course of preparation for the teaching exam, the ability to formulate hypotheses on the expected form of the solution of a problem, albeit having 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 and 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 use. A significant part of the theoretical results of the course are 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 who are 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 field problems having different characteristics involves the student's refinement of the ability to identify the most suitable solution strategy. All this determines an increase in the ability to classify problems and the strengthening of his own and effective method of study, which will certainly be useful in the future.

1. Electromagnetic fields

Maxwell equations in differential and integral forms. Stationary current density field, electric scalar potential, Laplace’s equation, Dirichlet and Neumann boundary conditions. Electro-static field, Poisson’s equation. Magneto-static field, magnetic vector potential. Quasi-stationary electromagnetic field, eddy current and skin effect problems. Dynamic electromagnetic field, Helmholtz’s equation, scattering and radiation problems.

Numerical methods for the computation of electromagnetic fields. The finite-difference method (FDM), FDM post-processing. The finite element method (FEM). Variational formulation for the Poisson equation in 2D. First-order triangular elements, shape functions, local coordinates, standard triangle. Dirichlet and metric matrices of a finite element. Boundary conditions. Building of the global algebraic system and its solution. Evaluation of integral quantities (fluxes, energies, forces) in postprocessing. High-order triangular elements. Integration formulas in the standard triangle. Universal matrices in rational form. Triangular elements with curved sides. Quadrangular elements. Gauss quadrature in triangular and quadrangular domains. Tetrahedral and hexahedral elements. Vector finite elements of the edge type.

Green’s function for the Poisson equation. Green’s formulas. The boundary element method (BEM). Integration of singular function. Hybrid methods: FEM-BEM and FEM-DBCI (Dirichlet Boundary Condition Iteration).

2. Transmission lines

Distributed parameter circuits. Parameters of a two-wire line. Telegrapher’s equations in sinusoidal steady state. Progressive and regressive waves. Propagation constant, attenuation constant, phase constant, characteristic impedance. Reflection coefficient, matched load. Energy balance. Half wave and quarter wave lengths. Lossless lines, Standing Wave Ratio (SWR). Time- and Laplace-domain analyses. Multi-conductor transmission lines.

3. Electrical filters

Characteristic function of an electrical filter. Low-pas, high-pass, band-pass and band-reject filters. Frequency transformations, prototype low-pass filters. Butterworth and Chebyshev approximations. Synthesis of LC ladder filters. First- and second-order active filters. Biquadratic cell. Sallen-Key filters. Negative feedback filters. Design examples.

4. Antenna theory

Main kinds of antennas and their characteristics. Electric and magnetic dipole antennas. Parameters of a transmitting antenna: radiation diagram, radiation resistance, effective length, directivity and gain functions. Parameters of a receiving antenna: effective area, power matching and polarization factors, antenna factor. Half-wave dipole antenna, quarter-wave monopole antenna. Antenna array (phased array).