Computational Electromagnetics

Week 1:
Introduction. Discussion on the personal projects, on the homework and on the examination. Scope of the lectures. Maxwell-equations, constitutive relations, differential Ohm’s law, understanding and demonstrating the impressed electric field and current density. Continuity of field quantities at the interface of two materials. Macroscopic and microscopic Maxwell-equations. Classification of the material properties: linear and nonlinear; isotropic, anisotropic and bi-anisotropic; time- or spatial dispersion, etc. Mathematical backgrounds: vector fields, reviewing the basics used from vector algebra.

Week 2:
Orthogonal co-ordinate systems (Descartes, Cylindrical, Spherical). Derivatives in general orthogonal co-ordinate systems. Energy balance, electromagnetic power density. Conditions to get unique solution of the Maxwell-equations: proof of the theorem. The radiation condition for unbounded domain. Concept of initial value and boundary value problems. Classification of the practical problems of electrodynamics. Time independent problems: (i) Electrostatics, (ii) Current flow fields, (iii) Magnetostatics, (iv) Magnetic field of steady currents.

Week 3:
Classification of the practical problems of electrodynamics (continuation): Time dependent fields: (v) Quasistatic problems: Magneto-quasistatic (eddy current), Electro-quasistatic and Electromagneto-quasistatic problems, (vi) Electromagnetic waves. The boundary value problem (BVP) of electrostatics. Introduction of the scalar potential. Physical meaning of the boundary conditions. Capacitance, partial capacitance. Proof of the unicity of the solution of the BVP of electrostatics. Short review of selected problems of electrostatics having analytical solutions.

Week 4:
Electrostatic energy of the field due to charged conductors. 2D field problems in general. Solution of an electrostatic problem with PDE Toolbox of Matlab. Introduction to the use of the software: input of the geometry, setting up the BVP, postprocessing the results.

Week 5:
Week formulation of the BVP related to the Laplace-Poisson's equation. Some words about the FEM developed for the solution of problems leading to the BVP of Laplace-Poisson's equation. 1D numerical example for FEM discretization based on the weak formulation. BVP of stationary conduction (current flow) fields. Review of selected problems of stationary conduction having analytical solutions.

Week 6:
Demonstration: solution of two current flow field problems with PDE Toolbox of Matlab. Introduction to the use of the software for problems with cylindrical symmetry, some notes on coupled problem of current flow field and electrostatics.

Week 7:
BVP of magnetostatics. Demonstration: solution of a magnetostatic field problem with PDE Toolbox of Matlab: analysis of Deprez-type multimeter. Introduction to the analysis of configurations containing permanent magnets (ferromagnetic materials with nonlinear magnetization).

Week 8:
Magnetic fields of stationary currents: the governing equations and the BVP based on magnetic vector potential, A. The BVP and its solution for free space, the Biot-Savart law. Review of selected problems of magnetic fields of stationary currents having analytical solutions. Formulation of the BVP in 2D situations: planar 2D problems and configurations with cylindrical symmetry, description of the related BVPs with one component vector potential. Demonstration of the solution of 2D magnetic field problems of stationary currents: analysis of the force of a cylindrical shape magnetic actuator and the inductances of a cylindrical transformer.

Week 9:
Time-dependent electromagnetic fields: introduction, consequences of the time dependence. Steady-state sinusoidal excitation, the complex calculation method: introduction of the method trough a 1D example from vibration theory. Special time dependences: constant excitation, steady-state sinusoidal periodic excitation. The complex calculation method representation of vector fields with complex vectors. Complex power, complex form of the energy balance. Steady-state periodic excitation in linear materials: solution by using the complex calculation method and Fourier-series. Excitation with arbitrary time dependence in linear passive materials: application of Fourier-transform. Consideration of causal excitation (Laplace-transform).

Week 10:
Time dependent electromagnetic fields. Quasistationary fields: Magneto-, electro- and electromagneto-quasistationary fields. Eddy-current (magneto-quasistationary) fields. Analytical expression of eddy-currents in conducting half-space. Review of selected problems of magnetic induction and eddy current fields having analytical solutions.

Week 11:
BVP of the magneto-quasistationary fields (eddy-current fields), the BVP in 2D. Demonstration of the solution of an eddy-current problem in cylindrical symmetry: analysis of induction cooker, discussion on the impedance type boundary condition.

Week 12:
Elektromagnetic waves. Introduction of the topic, typical sets of problems related to EM waves: (i) generation of EM waves, (ii) propagation and scattering of EM waves and (iii) guided waves. Electromagnetic field of a Hertzian dipole. Characterization of antennas. Some typical antenna types. Demonstration of the calculation of the radiated EM waves of a dipole antenna.

Week 13:
Typical boundary conditions related to EM wave problems, the BVP of EM waves, the eigenvalue problems, the guided modes of rectangular waveguides. Propagation and scattering of EM waves: theory and demonsrations. Plane waves. Transmission lines.

Week 14:
Review of the studies of the semester.