Finite Elements in Electromagnetics 4. Wave problems Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24Graz, Austria email: biro@igte.tu-graz.ac.at

Overview Maxwell‘s equations Resonators Filters Wave propagation in free space

Maxwell‘s equations Time harmonic case

Resonators GE: GH:

Resonators, H-formulation A problem without excitation: Eigenvalue problem

Resonators, operator equation for H

Resonators, finite element Galerkin equations for H Generalized algebraic eigenvalue problem

Filters GH: GE: GH:

Filters, E-formulation A problem with excitation: Driven problem

Filters, operator equation for E

Filters, finite element Galerkin equations for E conditioning of [A] strongly depends on frequency

Filters, A,V-formulation

Filters, operator equation for A,V

Filters, finite element Galerkin equations for A,V (1)

Filters, finite element Galerkin equations for A,V (2) i = 1, 2, ..., n(n)

Filters, finite element Galerkin equations for A,V (3) [A] is singular and its conditioning depends less on frequency R. Dyczij-Edlinger and O. Biro, "A joint vector and scalar potential formulation for driven high frequency problems using hybrid edge and nodal finite elements," IEEE Transactions on Microwave Theory and Techniques, vol. 44, pp. 15-23, January 1996.

Wave propagation in free space Finite element method needs closed domain Modeling of infinite space necessary Perfectly matched layers (PMLs) PML

PMLs Nonphysical material properties z x y No reflection on the interface between air and PML