IMST:

Institute of Mobile and Satellite Communication Techniques

Carl-Friedrich-Gauss-Str. 2

Germany

D-47475 Kamp-Lintfort

Tel. #49-2842/981 214

Fax #49-2842/981 299

Abstract

A multigrid solver was developed to determine the static potential
distribution of 2-dimensional microwave quasi TEM-guides. The boundary
conditions of these rectangular structures can be Dirichlet or Neumann
walls, which are electrical or magnetical walls in the case of
waveguides. An arbitrary number of dielectric layers or blocks can be
defined within the rectangular domain. There is also no restriction in
the number and dimensions of the metal conductors. The idea of the new
multigrid solver is to overcome the drawback of a bad overall
convergence
of traditional smoothers like the Gauss-Seidel or Jacobi method. In /1/
the
SOR method is already used to characterize about 14 coplanar elements
for
the RF circuit design in a frequency range from about 100 MHz to 90 GHz.
However, the simulation time increases drastically with the complexity
of
3-dimensional structures, which is inconvenient for design tools. An
other
aim is to achieve more flexibility in finding and defining new
structures
for high frequency circuits. Future fields of applications are
multilayer
boards for communication devices (e.g. frontends).
The problem now is, to find a discretisation from a very coarse to a
fine
level. The basic idea is, to place grid lines at each edge point of the
structure. The next finer grid will be generated by dividing the
distance
of two coarser grid points in half, to get a new grid line. The
advantage
is, that the structure is correctly described in all levels, while the
disadvantages are a strong non-equidistant grid and a concentration of
gridlines, which are not necessarily on the physical right place. The
solution is a minimum distance of 2 grid points in x- and y-direction.
This makes the grid more and more homogeneous, while the level
increases.

The implementation of the present algorithm in C++ starts with the code
suggested in /2/, while some basic ideas where taken from /3/. The
multigrid program is able to determine the static potential of an above
described structure with a lot of variations of the solver. The smoother
is the Gauss-Seidel method with the enhancement of successive
overrelaxation
or underrealxation. Four options for the ordering of the grid point are
available. This can be combined with a multigrid or full-multigrid
formulation with a free number of pre and post smoothing steps and
cycles.
Since the restriction operator has a strong influence on the convergence
(especially on non-equidistant rectangular grids), injection, 5-point-
and
9-point-restriction can be chosen. Results from the program as well as
some convergence considerations will be given on the convergence. In
addition
a demonstration of the software may be possible on a LINUX-PC.

The result from the multigrid simulation is the static potential
distribution in the
non-metallization areas. The boundary conditions are 1V on the signal
line and 0V on the
ground electrodes and all other lines. The capacitance between the
signal line and ground will be
calculated by an integration of the electrical flux around this line.
The result C^Ò in [F/m] is the
first element of the equivalent circuit. In the case of N lines C^Ò will
be a matrix C^Ò with NxN
capacity coefficients. From the same geometry but filled with air
instead of dielectric blocks or
layers we determine C0^Ò in exactly the same way. With the assumption,
that the structure is a
TEM-waveguide, which means that no z-component of the electrical field
in the propagation
direction of the wave exists, a definite relationship between the
electrical and magnetical fields is
given. Then, the inductivity L^Ò will be calculated from the capacitance
C0^Ò. The last element of
the equivalent circuit describes the losses of the line. They can be
divided into DC-losses and a
frequency dependent part, which is depending on the skin effect, where
the AC-current flow is
concentrated in the surface of the conductor. To calculate R^Ò(freq), the
electrical field
components within the structure are needed. Ex and Ey are the
derivations of the potential in x-
and y-direction (grad-operator). The resulting equivalent circuit, which
consists of a serial C'
and L' and a parallel R' can be taken as an interface to any RF- or
digital circuit design tool.

/1/ P. Pogatzki, R. Kulke, T. Sporkmann, D. Köther, R. Tempel,
I. Wolff: "A Comprehensive Evaluation of Quasi-Static 3D-FD
Calculations for more than 14 CPW Structures - Lines, Discontinuities
and Lumped Elements", IEEE MTT-S, Volume 2, pp. 1289-1292, San Diego,
May 1994

/2/ W. Hackbusch: "Multi-Grid Methods and Applications", Springer
Verlag New York Tokyo, 1985

/3/ M. Schuster: "Lösung dreidimensionaler elektrostatischer
Randwertprobleme mit dem
Multi-Grid-Verfahren", Diplomarbeit an der Uni-Duisburg, Fachgebiet
Allgemeine und
Theoretische Elektrotechnik, März 1993