A Multigrid-Algorithm Solves the Laplace-Equation in 2D-Electrostatic Structures for the Microwave Circuit Design R. Kulke, I. Wolff IMST Institute of Mobile and Satellite Communication Techniques Carl-Friedrich-Gauss-Str. 2 D-47475 Kamp-Lintfort, Germany Tel. #49-2842/981 214 Fax #49-2842/981 299 e-mail:
[email protected]
Abstract A multigrid solver has been developed to determine the static potential distribution of 2dimensional microwave quasi TEM1-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. The aim for the new solver is to achieve more flexibility in finding and defining novel structures for high frequency circuits. The paper introduces 2 new procedures for the grid refinement. This results in homogenous grids with nearly equidistant gridlines. An example of a 9-conductor waveguide will be given. The iteration results show an improved convergence behavior in comparison with the conventional bisection method. An equivalent circuit will be calculated from the potential distribution. This makes the solver applicable in time- or frequency-domain circuit design tools. Motivation The IMST supports the high frequency design of analog circuits with various simulation tools. One of these activities is a library with models for more than 14 coplanar elements [1]. The characteristic of coplanar circuits is, that all metallization for the conductors are on the top of a carrier substrate. The advantage of this technique is mainly a cost reduction for the fabrication (e.g. no vias through the substrate, no backside processing, etc.). The electrical advantage lies in poor dispersion behavior, because the electromagnetical field is concentrated in the small slots among the signal and the ground lines. Though, an accurate modelling technique has been developed by Naghed [2], which determines the static potential distribution among the conductor lines. Many applications have shown, that these models are valid up to 70 GHz or even more. The potential distribution is described by the Laplace equation, that’s why this problem is solved with the Gauss-Seidel method including the option of sucessive over-relaxation. Nevertheless, two drawbacks led to the decision to develop a new solver with multigrid techniques. On the one hand the user has no flexibility to define new structures, and on the other hand the simulation time increases from a moment to a number of minutes, if 3d and complex structures will be calculated. At the time being, the multigrid solver for arbitrary 2d rectangular TEM wave-guides is available. Some basic ideas have been taken from Schuster [3]. 1
TEM: Transversal-Electro-Magnetical wave-guide: the field components of the electromagnetical field in the direction of the wave propagation can be neglected
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2D Multigrid Algorithm The heart of the present simulator combines the Gauß-Seidel method including under- and over-relaxation (GS/SOR) as well as the multigrid (MG) and full-multigrid (FMG) techniques. The basic routines have been derived from Hackbusch [4] and modified to fit to this potential problem. Table 1 gives an overview of the available simulation parameters. The user can choose one of the methods: GS/SOR, MG or FMG.
LEVELS ORDERING
CYCLING N_PRE N_POST PROLONG RESTRICT
N_CYCLES F_SOR Table 1:
Value 0 ... 10 Red-Black Lexico. Rot-Lexico. Zebra-Line V, W, ... 1 ... 3 (or more) 0 ... 2 (or more) 9-Point Injection, 5-Point, 9 Point 1 ... 0 < F_SOR < 2
GS / SOR X
MG X
FMG X
X
X
X
X X X X
X X X X
X
X
X X
X X
X X
Available methods and parameters for the solver
The X’s indicate, which parameter influences the iteration method. It is planed, that some of the parameters will no longer be fixed to one value, but will be a function of the level (e.g. F_SOR(level), N_PRE(level) etc.). The other interface to the simulator is the 2d gridgenerator. The geometry of the structure is described by a simple language, where the box around the domain and the rectangular objects with material parameters are defined. In the coarsest discretization level the grid lines lie on the edges of each object. Special refinement techniques for the higher levels will be discussed in the following chapter. The output of the solver is the potential distribution Φ, which is used to determine the equivalent circuit parameters (C’ = capacitance’s and L’ = inductance’s). These elements or the S-, Y- or Z-matrix (S = scattering, Y = admittance and Z = impedance) can be the interface to any time- or frequency-domain circuit simulator. Figure 1 shows the organization of this project. The tasks in light gray colored boxes have already be implemented in C and C++, while the dark boxes show the simulation parts for 3d-structures as well as the improved tools for the input and output interface to the simulator. This will be the future activities.
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extern
available
future
Circuit Simulator
Element Paramters Simulator Paramters
e.g. Y-Matrix
MG-Control File
Comfortable Language for Parametric Elements
*.2d Simple Language for Geometry
*.3d
2D-Grid Generator
2DSimulator
Equivalent Circuit
3D-Grid Generator
3DSimulator
Equivalent Circuit
Input Figure 1:
MG-Work Bench
Interface to Timeor FrequencyDomain Simulator
Output
Simulator
Project organization
The Difference Equation Figure 2 illustrates the general configuration around one grid crossing point with the potential Φ. The neighbored points have the potentials ΦX, ΦX’, ΦY and ΦY’, while the materials (dielectric layers or blocks) are defined by the relative permittivities εXY, εXY’, εX’Y’ and εX’Y. The distances to the neighbors are x, x’, y and y’. This makes clear, that non-equidistant grids are allowed and typical for our applications. The derivation of the difference equation (1) is made in [3]. It is important to point out, that the coefficients CX, CX’, CY and CY’ depend only on the distances to the neighbors and the bordered materials. Therefore, these parameters can be determined from the grid generator and can be stored to each grid point. Φ = Cx Φ x + Cx' Φ x' + C y Φ y + C y' Φ y + C f f =
(
)
1 B Φ + Bx' Φ x' + By Φ y + B y' Φ y + f Bs x x (1)
with Bx = (ε xy y + ε xy ' y ') x
(2a)
Bx ' = ( ε x ' y y + ε x ' y ' y ') x '
(2b)
By = ( ε xy x + ε x ' y ' x ' ) y
(2c)
By ' = ( ε xy ' x + ε x ' y ' x ') y '
(2d)
Bs = Bx + Bx ' + By + By '
(2e)
εx'y
Φy y
x'
Φx'
εx'y'
εxy
Φ y' Φy'
x
Φx εxy'
Figure 2: Configuration around a grid crossing
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The boundaries of the domain can be electrical walls, which are equivalent to Dirichlet boundaries, or magnetical walls = Neumann boundaries. The conditions & & for the tangential and normal components of the electrical ( E ) and the magnetical field ( H ) are illustrated in figure 3 and 4.
Etan = 0 Enorm
Etan Enorm = 0
Htan Hnorm = 0
Htan = 0 Hnorm
Figure 3: Field components close to an electrical wall
Figure 4: Field components close to a magnetical wall
Multi-Conductor Waveguide The configuration in figure 5 shows the crossview of a 9-conductor waveguide surrounded by electrical walsl. The substrate consists of 4 dielectric layers, which are the carriers of the conductors. Additionally, the grid lines of the coarsest discretization level (L0) are drawn into the structure. As mentioned, the x- and y-gridlines of L0 cross the edges of each block, layer or conductor. Thus, the grid exactly projects the geometry of the structure in each level. To determine the capacitor among two lines or one line and ground, this lines has to be set to 1V, while all other conductors are set to 0V. The resulting potential distribution inside the dielectric areas, which is described by the Laplace equation, is solved with the current multigrid algorithm.
Φ=1V
Φ=0V
Φ=0V
Φ=0V
Φ=0V
Φ=0V
Φ=0V
Φ=0V
Φ=0V
∆Φ = 0 Level 0
Ground: Φ = 0V Figure 5:
9-conductor waveguide in an electrical box
The difficulty now is to find grids for the finer discretization levels. A common method, which has been implemented into the software, is the bisection of two neighbored gridlines per level. Usually, the procedure starts in level 0 with a strong non-equidistant grid. This will R. Kulke, I. Wolff
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lead to a concentration of gridlines in the higher levels and to an increase of unknowns for the solver. Typically, the small dimensions come from the metallization thickness or bridging structures. From the physical point of view a high resolution of grid points is needed around the edges of the conductors, since this are the critical areas for the electro-magentical field. The bisection method cannot reflect these properties. Moreover, tests with simple PDE’s in a homogenous domain, where the analytical solution is known, show that a strong nonequidistant grid may lead to poor convergence or even divergence. To overcome this problem, two automatic grid techniques have been implemented. The aim was to achieve homogenous grids in each level. The first variant is called AUTO1. In this procedure the smallest distance of two gridlines is determined from the maximal distance of two gridlines in level 0 (DXL0 and DYL0) and the maximal number of grid refinements (Lmax): DX L =
DX L0 2
Lmax
,
DYL =
DYL0 2 Lmax
15 x 15
DX=14.25, DY=11.88, 10 x 9
29 x 23
DX=7.13, DY=5.94, 19 x 14
57 x 39
DX=3.56, DY=2.97, 37 x 24
73 x 47
DX=1.78, DY=1.49, 73 x 47
(3a,b)
Figure 6: Grid refinements from AUTO1 (DX = 1.87, DY = 1.49) and AUOT2
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With this method, a lot of gridlines of the bisection are left out. Nevertheless, the drawback of inhomogenous grids remain for lower levels. That’s why the procedure AUTO2 has been introduced. Here, the smallest distance of two gridlines is calculated for each level by: DX L =
DX L0 2L
,
DYL =
DYL0
with L = Level = 1,2,3 Lmax
2L
(4a,b)
where L is the current level. In L0 and Lmax AUTO1 and AUT2 generate the same grid, but in AUTO2 the number of unknowns is lower for 0 < levels < Lmax. Figure 6 compares the grids from AUTO1 and AUTO2 for the 9-conductor waveguide application. Table 2 gives an overview of the number of gridpoints per level, the simulation time (on a Sun Sparc 20 with 50 MHz) as well as the capacitance C’11 and the Euclidean norm of the residuum. Method BISECTION AUTO1 AUTO2
Number of Grid-Pkts. in Level Time C’11 |Residuum| L0 L1 L2 L3 L4 [sec] [pF/m] E2-Norm 64 225 841 3249 12769 3.3 108.9 8.29 E-3 64 225 667 2223 3431 1.7 86.241 5.86 E-9 64 90 266 888 3431 1.1 86.244 3.31 E-6
Table 2: Comparison of the grids and results from BISECTION, AUTO1 and AUTO2 The gain of the new grid generators is obvious. A better result than C’11 = 86.241 pF/m could not be achieved with 5 levels. The slope in figure 7 illustrates the convergence rates as a function of the time. The graphs prove the considerations, which have been made. 1.0E-01 1.0E-02 1.0E-03 1.0E-04 1.0E-05 1.0E-06 1.0E-07 1.0E-08 1.0E-09
|Residuum| after 10 W-Cycles
0
0.5
1
Bisection
Auto1
Auto2
1.5
2
2.5/ sec Time
3
3.5
Figure 7: Convergence of the 3 iteration procedures The capacitances of the equivalent circuit where calculated from the electrical potential distribution among one line and the other conductors. If the structure consists of n lines, the potential has to be determined n-times. Only the conductor under consideration is set to 1V, while all other are fixed to 0V. Finally, this results in a capacitance matrix C’ with nxn coefficients. The C’ii are the capacitors from conductor i to ground, while C’ij is the capacitor from line i to j. C’ij= C’ji is a statement for the quality of the numerical result. Figure 8 shows the upper left corner of the 9-conductor waveguide and the positions of the capacitors, which are given by the equation (5). Q’ is the total charge of the conductor, while U is the difference between &1V and 0V. The charge can be calculated by the integration of the electrical flux density D around the metallization (6). R. Kulke, I. Wolff
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C′ =
Q' F U m
(5)
&& ∆Φ i ∆si with Q' = ∫ Dn d s = ∫ εE norm d s = ∑ ε C
C
i
∆n i
(6)
This proves, that the potential distribution Φ(x,y) is needed to determine the equivalent capacitances of an n-conductor waveguide. Φ(x,y) as a result from the MG-algorithm is shown in figure 9.
∆si Enorm
∆ni
C'11
#1 = 1V
C'12
#2 = 0V
C'14 C'15 #4 = 0V
#5 = 0V
Figure 8: Positions of the equivalent capacitors for conductor 1
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1
1
0.8
0.8
0.6
0.6
0.4 0.4 0.2 0.2
0
0 Legende 80 1+ 0.8 bis 1 0.6 bis 0.8 0.4 bis 0.6 0.2 bis 0.4 0 bis 0.2
140 120 100 60
y
80 60
40
x
40 20
20 0
0
Figure 10:Distribution of the static electrical potential in volts on the 9-conductor waveguide calculated by the MG-solver with the paramters: 4 Levels (AUTO2=73x47 pkts); Lexicogr. Ord.; W-Cycles; N_PRE=2; N_POST=1; 9-Point Prol.; Injection; 50 Cycles; F_SOR=1.0
References [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] M. Naghed: „Analyse koplanarer Mikrowellenstrukturen mit der Methode der quasistatischen Finiten-Differenzen“, Dissertation im Fachbereich Elektrotechnik der Universität-Gesamthochschule-Duisburg, Duisburg 1992 [3] M. Schuster: "Lösung dreidimensionaler elektrostatischer Randwertprobleme mit dem Multi-Grid-Verfahren", Diplomarbeit an der Uni-Duisburg, Fachgebiet Allgemeine und Theoretische Elektrotechnik, März 1993 [4] W. Hackbusch: "Multi-Grid Methods and Applications", Springer Verlag New York Tokyo, 1985
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