8. Synopsis
**I.
Introduction, review of previous studies**

Week 1

Review of
the mathematics background. Maxwell-equations. Electromagnetic field
quantities, forces. Constitutive relations, spatial- and time
dispersion in materials. The meaning of imposed electric field.
*Macroscopic and microscopic Maxwell-equations* and their
relations. Electromagnetic field quantities at the boundary of two
different materials.

Week 2

*Energy balance,
electromagnetic power density*. Concept of initial value and
boundary value problems. *Special time dependences* in linear
materials: steady-state periodic excitation, arbitrary shape
excitation in passive materials, causal excitations. *Complex
**power**,* complex form of the energy balance.

**II. Boundary
value problems in electrodynamics**

Week 3

Conditions to get
*unique solution of the Maxwell-equations*, the radiation
condition. *Boundary value problems (BVP) of electrodynamics*.
Scalar problems leading to the Laplace-Poisson equation: (i)
electrostatics, (ii) magnetostatics, (iii) current flow problems. Boundary conditions leading to the unique solution of the
Laplace-Poisson equation, the physical representation of this
boundary conditions.

Week 4

Further BVPs of
electrodynamics: (iv) magnetic field due to stationary currents
defined by vector potentials or by reduced magnetic scalar potential,
(v) eddy current fields (quasi-stationary fields), (vi)
electromagnetic waves. Electromagnetic field representation of
n-poles of Kirchhoff type networks.

Week 5

Demonstration (4
hours): (i) solution of electrostatic problems with finite element
method (FEM) software. Definition of the BVP, derivation of the
design parameters from the numerical solution of the BVP. Usage of
the applied FEM software. Some further practical examples solved by
FEM: magnetic field due to stationary currents, eddy current field
and electromagnetic waves.

**III. Numerical
solution of boundary value problems, basics of the analysis softwares
used in the electrical engineering practice**

Week 6

Review of methods
used for the numerical solution of BVPs (global/local approximations,
integral/differential formulations, etc.). Application of* FEM*
for the solution of BVPs. Residuum theory, derivation of the
discretized equations for Poisson-type problems. Approximating
function used for FEM.

Week 7

*Green's functions*
for scalar BVPs. Some 1D Green's functions. Green's functions of the
scalar Poisson- and wave equations in free space. Dyadic Green's
functions. Dyadic Green's functions related to the vectorial Poisson-
and wave equations in free space. *Method of integral equations*
used for the solution of the BVPs of electrodynamics.

Week 8

Demonstration (2
hours): examples for the solution of BVPs of electrodynamics. *Finite
difference time domain (FDTD) method*. Discretization of the
differential operator, the Yee algorithm for 1 and 3 dimensional
cases. Demonstration (1 hours): analysis of multilayer
anti-reflection coating for various excitations with FDTD method.

**IV. Classical
electromagnetic field analysis problems in electrical engineering **

Week 9

*Time-dependent
problems in lossy transmission lines*, application of
Fourier-transform method. Time-dependent problems in ideal
transmission lines, use of the Laplace-transform method,
understanding the graphical solution. Demonstration (1 hour):
numerical code for the analysis of lossy transmission lines using the
Fourier-transform method. *Inverse and optimization methods* in
electrodynamics.

Week 10

Demonstration (2
hours): example, (i) inverse and optimization problem of
electromagnetic nondestructive testing, (ii) solution of an
optimization problem. *High power engineering *applications,
eddy currents in electrical machines.

Week
11

*Electromagnetic
wave problems. Plane waves*:
reflection of plane waves with arbitrary incidence, total reflection,
representation of arbitrary electromagnetic field as superposition of
plane waves. *Waveguides*:
eigenvalue problems, definition of modes in waveguides with arbitrary
cross-sections, guided modes of waveguides with rectangular cross
section. Open waveguides: microstrip waveguides, dielectric
waveguides. *Hertz-dipole*:
near- and far fields, radiation
pattern, input impedance, directivity, gain. *Patch
antennas*.

Week
12

Demonstration
(2
hours): analysis of high
frequency devices using HFSS FEM software.

**V.
Selected topics from novel applications**

*Electromagnetic
waves in periodic structures, *investigation
of some meta-materials. Homogenization.

Week
13

Demonstration
(2
hours): reflection of
electromagnetic waves on periodic structures. Coupled mode theory,
basics of *wireless power transfer*.

Week
14

Maxwell-equations
in moving media: relativistic Maxwell-equations and its
approximations. Example: scattering from moving objects.