A Multigrid-Algorithm Solves the Laplace-Equation in 2D-Electrostatic Structures for the Microwave Circuit Design

R. Kulke

Institute of Mobile and Satellite Communication Techniques
Carl-Friedrich-Gauss-Str. 2
D-47475 Kamp-Lintfort
Tel. #49-2842/981 214
Fax #49-2842/981 299


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^ in [F/m] is the first element of the equivalent circuit. In the case of N lines C^ will be a matrix C^ with NxN capacity coefficients. From the same geometry but filled with air instead of dielectric blocks or layers we determine C0^ 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^ will be calculated from the capacitance C0^. 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^(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. Kther, 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: "Lsung dreidimensionaler elektrostatischer Randwertprobleme mit dem Multi-Grid-Verfahren", Diplomarbeit an der Uni-Duisburg, Fachgebiet Allgemeine und Theoretische Elektrotechnik, Mrz 1993