NUMERICAL METHODS FOR ELECTROMAGNETIC FIELDS AND CIRCUITS

In the design of devices in the various engineering fields, physical quantities are used which, in general, vary both in space and in time. The complexity of the (differential) equations, which these quantities must satisfy, allows only an approximate solution by means of numerical techniques, in which the physical quantities of interest are discretized both in space and in time. These techniques are so important that you can definitely affirm that all industries (electrical, electronic, mechanical, aerospace, etc.) and research centers of medium-large size are equipped with CAD (Computer Aided Design) tools based on they.

In the engineering studies the student matures sufficient experience in the time discretization of quantities, but, perhaps, not as in the space or space-time discretization of variables.

The aim of the course "Numerical methods for electromagnetic fields and circuits '' is just to study techniques for the space and time discretization of physical quantities. Such techniques have a value that goes well beyond the scope of the Electrical Engineering. However, also in this context there are several applications in which the circuit approach is inappropriate, such as the calculation of antennas and transmission lines for telecommunications (coaxial cables, waveguides), the calculation of the electrical and thermal behavior of the power transistors and the verification of the electromagnetic compatibility of various electrical and electronic devices.

The course will cover mainly the finite element method (FEM), which, devised in the sixties, is now considered as the most powerful numerical method for solving field problems, completely replacing the finite difference method (FDM). The basic idea of the method consists in dividing the spatial domain of interest in a large number of subdomains of simple form, said finite elements (tetrahedra and bricks in 3D, triangles and quadrangles 2D). In each finite element, it is assumed that the field has simple behavior (linear, quadratic, etc.) and that it can be expressed as a function of its values in the finite element vertices (nodes). This spatial discretization process 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 us to obtain a solution whose degree of accuracy increases with the number of finite elements (but with a consequent greater computational cost).

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

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

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

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).

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

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

3. R. Schaumann, M. E. Van Valkenburg: "Design of analog filters". OUP, New York, 2001.

4. Wai-Kai Chen: " Passive, Active and Digital Filters. Taylor&Francis.

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.

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