Enhanced
Numerical
Integration
Technique
Markus Biicker
INCASES Engineering
GmbH
Paderborn, Germany
ABSTRACT - An enhanced
integration
technique
is
presented to compute the elements of the system
matrix of the EFIE-MOM
using triangular
surface
patches.
It is shown, that spending
a lot more
efforts in the accuracy of the single matrix element,
the needed number of unknowns
can be reduced
significantly
without
a loss of precision.
Therefore,
the total computation
time, consisting
mainly of
matrix filling time and solving time, can also be
reduced. This technique
is helpful when e.g. power
supply structures
of modern Printed Circuit Boards
shall be examined efficiently.
Introduction
In computational
electromagnetics
the Method of Moments
(MOM) is in widespread use in order to solve the Electric
Field Integral Equation (EFIE) [5]. This method has been
successfully applied in antenna design and in scattering problems where the dimensions of the structure are at least in the
same order of magnitude as the wavelength of the electromagnetic fields. These structures can be represented very
well by triangular surface patches, for which Rao has introduced an integration technique for the calculation of the coupling impedances between all triangular patches [8].
The application of the EFIE-MOM on Printed Circuit Boards
(PCBs) causes additional
numerical efforts because of the
electrical small size of the structures, like the small distances
between power planes on different PCB layers [7]. Applying
Rao’s formulation
on these kind of problems, a very dense
mesh of patches is necessary to obtain a sufficient accuracy.
This leads to a large number of unknowns within the system
of linear equations, and accordingly to a huge computation
time.
This paper describes an enhanced integration
technique,
which generates a very stable algorithm requiring much less
patches, and therefore less computational
efforts. In order
to show the differences between the Rao formulation
and
the enhanced formulation
here, the derivation of the integrals which have to be computed is performed first in the
following section.
Mathematical
The surface current distribution
Formulation
& on a perfectly
conductive
0-7803-5057-X/99/$10.00 © 1999 IEEE
328
body excited by an incident electric field gi must be adjusted
in a way, that the tangential
electric field strength on the
surface S is vanishing:
i3x(13-~)~s=o
,
(1)
where 5 is the scattered field of the conductive body,+which
can be derived from the surface current distribution
JS and
from the surface charge u:
P(&,
u) = -jwiq&)
- Vaqu)
)
(2)
with the vector potential Land the scalar potential a. These
potentials are functions of the current distribution or the surface charge, respectively, multiplied with the Green’s function
of the free space G:
ii(F)
=
-& /
a(3
=
-&
=
,-j&i
-
G(r’, ?‘)
S
J
R
f&r”)
u(F)
. G(F, F’) dS’
- G(r’,?‘)
with
and
dS’
(3)
(4)
R= ]?-?I
.
Here, k denotes the wavenumber, which is Ic = 2n/X, and X
is the wavelength.
R is distance between source and observation point. The surface charge cr is related to the surface
divergence of & by the equation of continuity
-jwu
Substituting
equations
-dx~~s=iix
= Vl, - fs
.
(5)
(2) to (5) into (1) the EFIE is derived:
(%@.G(r;?‘)dS’-
&v Js v; .& .G(r’, F’)dS’
The unknown function
tions fn(q,
originally
gulized surface:
*
)I
(6)
S
& is expanded in a set of basis funcproposed by Glisson [a] for a trian-
Each basis function
gles Tn+ and TnB:
f?Z(fl=
1
fn is defined on a pair of adjacent
&.fl$(.(r3
,
if r’is in T,+
&-P;;(F)
,
if r’is in Tny
trian-
(8)
7 otherwise
no
since the system matrix is bad conditioned, the convergence
may be bad and the direct solution can be faster.
For the further evaluation of equation (lo), we focus on the
integration
over the vector potential of the positive observation triangle A dzf sT,+i.
p’,s dS. The results can be
transfered to all other integrals appearing in equation (10)
as well. With the inserted vector potential and basisfunction
we obtain for the positive triangle observation TA:
where A,f is the area of the respective triangle, 1, is the
length of the nth inner edge and p? is a vector from the free
vertex to r’ (see figure 1).
--P 1,
,7;(q)
41r 2A,+ ssT,+ T,+
--P L
&$‘)
4n 2A,
T,+ T,
JJ
. G(r’, S’) dS’ ep’,+(?) dS
+
. G(r’, F’) dS’ . &‘#)
(12)
dS
Rao’s Approximation
Figure 1: Pair of triangles
In order to obtain a system of linear equations, a testing
procedure is applied to the EFIE (6). Defining a symmetric
product of test- and basisfunctions:
< f, s’ >dzf
sS
f.j’dS
In order to solve this integral expression conveniently,
Rao
introduced an approximation
by assuming a constant vector
potential over the whole observation triangle T,f. Therefore,
the testing procedure may be applied to the centroid (?&*)
of the respective triangle only:
,
equation (6) is tested at observation triangles T,,$ and T;.
Choosing test- and basisfunctions equally leads with the testfunction fnz to:
Herewith, the integration over the observation triangle simplifies to the constant value of the area of the respective
triangle A,$.
The vector potential at the centroid A(?;+)
for r” within
the positive source triangle Tnr can be evaluated as:
,-jkR
AZ F’) . R
(10)
Replacing the potentials A’ and @ by their definitions (3,4),
and inserting the basis function (8), this results in a quite
complex integral expression, consisting of double integrals
over each combination
of positive and negative source- and
observation triangles. This resulting equation can be written
in matrix form as:
Z*J=V
R
with
Hence, equation
A
M
329
I+-+:++
.
(12) simplifies
--P LA,+
+c+
4n 2A,$ pm J
with R
=
I?‘-?;+[
to:
,-jkR
T,+ p’,+(P) +R
J
--P lnA,+ +c+
4n 2A, pm
(11)
with the symmetric and dense system matrix Z, the unknown
current vector J and the excitation vector V. This matrix
equation can be solved by applying standard solvers for linear systems of equations, like Gaussian elemination, or LUdecomposition.
Iterative solvers may be applyied as well, but
=
dS’ (14)
T,-
dS +
,-jkR
p’,-(P)
.R
dS’
(15)
.
The evaluation of the remaining integrals is described by
Rao [8], where two cases are distinguished:
i) source and
observation triangle are identical (m = n), and ii) they are
-~
different (n # m). In the first case, the occuring singularity
at R + 0 is treated by splitting the integrand into a nonsingular and a singular part. The integral over the singular
part is evaluated analytically
and has a nonsingular
value.
The remaining second integral over the nonsingular integrand
is computed quite accurate using a Gaussian quadrature formula [l]. This pure numerical method is also proposed for all
integrals occuring in the second case, where R > 0, Vm, n.
This technique (called here: “conventiot~al
Rao”) can be
used with little errors whenever only one surface (e.g. a
single plate or a sphere) is considered.
In cases of parallel
planes, where the vertical separation may become very small
compared to the wavelength,
R will be very small although
n # m! In order to improve the numerical stability, the analytical solution of the singular part should be extended to
all cases whenever the distance R is smaller than a given E.
This method is refered here as “~Rao”.
To demonstrate the numerical problems of the conventional
Rao method, a configuration
shown in Figure 2 was used
in order to examine the integral values. The observation
point is moved at vertical distances d above the triangle
starting at (z = 0,~ = 0) along the x-axis.
The source
points are located according to a 7-point Gaussian quadrature formula [l]. All values are plotted normalized to the
wavelength X. The normalized edge length of the triangle is
set to + = 2 . 10m4.
Y
lE+O 1
0
I
I
0.2
I
1
0.4
I
I
0.6
,
I
I
I
*lE-3
1.0
x/lambda ->
Figure 3: Norm. Integral value; conv. Rao (---); E-Rao (-)
(cl): 6 = lpm/X
(v): 6 =lOOpm/A
(*): S = 500bm//\
algorithm very instable and sensitive to the exact location
of the integration
points, and therefore the results of the
conventional Rao method are quite mesh-sensitive.
Increasing the vertical distance, the differences are vanishing
at around 6 = 500,um/X.
This indicates the lower limit of
the distance between two planes, where the conventional Rao
method can be applied.
When the horizontal distance becomes large enough, both
approaches give the same integral value. As a conclusion of
this examination,
a distance-depending
rule for the decision
between the analytical and the numerical solution can be
found. The use of the analytical solution for the self coupling
integrals only (m = n), as proposed by Rao is not always
sufficient.
0: Sourcepoints
Observationpoint
0:
I
Although the numerical accuracy can be improved using the
c-Rao method, the approximation
in (13) still leads to instable results in configurations
with two or more surfaces
in small distances, because the distance R = IF' - ?&+I in
(14) depends strongly on the relative position of the triangles and therefore on the meshing of the surfaces. To clear-
X
Figure 2: Source points within the integration domain,
observation point is moved alon the x-axis
In Figure 3 the integral values are plotted vs. horizontal
x-coordinate,
for 3 different, normalized vertical distances
S = d/)r (6 = lpm/X,
lOO,um/X, and 500,um/X).
The
dashed lines are indicating the results of the conventional
Rao method and the solid lines the values gained with the
c-Rao method.
Between the (6 = l,um/X)- curves, the deviation becomes extremely large whenever the observation point is exacty above
of one of the source points. This behaviour makes the whole
330
(a)
(W
Figure 4: Vertical cross-section through parallel triangles,
(a): not shifted;
(b): shifted
ify this, two identical triangles with egde length iA =lOmm
above each other with a vertical separation of d = 1OOpm
are examined as shown in figure 4(a).
Note, that the
schematic plot is not scaled. The minimal distance between
the source points (e) and the centroid of the observation
triangle (0) is Rmin = d = 100pm.
This value has a
significant effect on the integral value, due to the Green’s
function.
A slight horizontal shift in the relative position
(e.g. s = lmm), figure 4(b), will increase the minimal distance Rmin = &q7
= 1.005 mm, which is quite different from the “real” vertical distance of the triangles. Therefore, the integration
will give a complete different value, although the geometry has changed only slightly.
Enhanced
Approach
To improve the stability, the approximation
in equation
is replaced by a Gaussian quadrature formula:
(13)
0
0.2
0.4
Figure 5: Norm. Integral
value; Galerkin
(Cl): 6 = lpm/X
(*):
This evaluation
of the integral is very accurate (depending on the order N), since the integrand A(?) . J&(c)
is
a non-singular
function over the integration
domainl.
The
possible singularities
are handled analytically,
as described
above.
Now both, the basis and the testing function are
evaluated by the same numerical integration scheme, therefore this method is refered here as “Galerkin’s Method”.
For
equation (15) this results in
A
M
s T,+
+
p L
--cui
47~f&i-
R
N
Numerical
dS’
with R
=
p-iq
pfa(F’)
.R
dS
c-Rao (---)
not as
as the
result
shifted
sensitive to the relative
E-Rao method is. The
in quite similar integral
case.
Results
capacitor
c.f ormula
,-jkR
s T,
Plate
.;;(q
i=l
(-);
S = lOOpm/X
Subject of the examination is a square shaped planar capacitor of edge length a and distance d. it is fed by a voltage
source located on a connecting via between the planes, which
is placed at the center of the planes. The (quasi-stationary)
capacitance can be calculated by an analytical formula,
,-jkR
$2(F”).
(v):
1.0
-s-
6 = 500pm/X
effect, the Galerkin method is
position between two triangles
example shown in figure 4 will
values for the original and the
Example:
*lE-3
x/lambda
0.6
(17)
.
For each part of the sum, the remaining integral is either
evaluated analytically or computed numerically, according to
the c-Rao method depending on the distance R.
Figure 5 shows a comparison of the computed integral values d/A, obtained by Rao’s approximation
and by Galerkin’s
method. In the Galerkin case, the single observation point is
replaced by a triangle of same size and orientation, which is
moved across the source triangle at distance d.
Galerkin’s method results in slightly smaller values, because
of the averaging effect of the sum. Due to this averaging
lAn order of N=7 was chosen to obtain the results throughout
this contribution.
331
=
~OE,(U + d)’
d
’
(18)
which can cover edge effects approximatively
only, by adding
the vertical distance d to the edge length a [6]. Using the
EFIE-MOM-solver
COMORAN
[2], the capacitance is computed applying both techniques, c-Rao and Galerkin. The
vertical distance was chosen large enough allowing the ERao method to gain accurate results as well. The results
are summarized in table 1, where the matrix filling time -&fill
and the memory requirements for matrix storage are listed
as well.
While increasing the edge length of the triangles (la), the
discrepancy between the analytical value and the results of
the c-Rao method are increasing as well, until the numerical results can only be refered as “noise”.
On the other
hand, using Galerkin’s method, the results are quite stable.
The deviation from the analytical value is mainly due to the
bad approximation
of the edge effect in equation (18). To
a =50mm, d = lmm, E,.=4.0 =k Cfo,.mu~o= 92pF
Memory
= [A
2mm 1 94.lpF
3mm
93.9pF
5mm
959pF
1Omm
1506pF
15mm 1 noise
.
985s 1 94.4nF
312s
94.4pF
23s 94.4pF
1.5s 94.2pF
0.6s 1 94.lpF
Table 1: Numerical
accuracy
39431s 1 464 MB
97 MB
12331s
948s
10 MB
60s I 682 kB
13s 1 185 kB
vs. edge length
achieve a required accuracy, Galerkin’s Method needs much
less memory and computation
time, in spite of the larger effort in computing a single matrix element.
In the next step, the accuracy versus the vertical distance
is examined. The square plate capacitor of 50mm x 50mm
is meshed with an edge length of lA = 5mm. Again, the
computed capacitance of both methods is compared to the
analytical value gained with equation (18). Note, that this
formula becomes inaccurate at large distances d. In table 2
the results are listed.
I
a = 5omm, &r = 4.0 , lA = 5mm
1 brn 1 283.50’fF
Table 2: Numerical
(
88.55 nF I
accuracy
For verification purposes, they examined a simple power bus
structure and performed measurements on it [3] 2. The configuration is shown in Figure 6.
vs. vertical
c
TI .024mm
65mm
Figure 6: Power bus configuration
Two square, parallel power planes are connected by a shorting
pin. The dielectric carrier is described by E,. = 4.7 (FR-4).
A wide band signal is fed to the structure through a coaxcable attached to a SMA-connector
at port 1. Using a second SMA-connector
(port 2), the transmitted
signal can be
measured. Within COMORAN these vertical discontinuities
are considered using a “thin wire model”.
Figure 7 shows
the transmittance
5’21 between the two ports.
88.55 nF /
distance
At large distances, both numerical methods have similar results. Decreasing the vertical distance between the planes,
the e-Rao method results deviate from the analytical solution. Below d = 1 mm, these results are not usable anymore,
while the Galerkin results are still very accurate, even in the
dense cases, like d = 1 pm.
0
400
800
1200
1600
2000
2800
*1E+6
f [Hz] -a
Figure 7: Comparison between 5’21 measured (---) and
computed using COMORAN
(-)
These examples proove the accuracy and numerical stability
of the enhanced method and show the potential of reducing
computation
time, despite the increased integration effort.
The dashed line indicates
the measured results, while
the solid line represents the COMORAN
results (Galerkin
method). In order to show the stabilty and accuracy of the
enhanced method, the mesh for the numerical computations
Example:
‘The ]Szl] measurements were conducted by the Electromagnetic Compatibility Laboratory, University of Missouri-Rolla for
DC power bus modeling development. An HP8753D network analyzer was used and the reference planes were at the 3.5 mm test
cable connectors. A simple 12-term error correction model using
an open, short, and load was used in the calibration. Port extension was used to move the measurement planes to the coaxial
cable feed terminals.
Power
bus
The following example shows the application of the enhanced
algorithm in the area of power bus analysis. The design of
complex power plane systems and the placement of decoupling capacitors can be tackled with the described approach.
A circuit extraction approach is currently under development
at the EMC-Lab. of the University of Missouri-Rolla
(UMR).
332
was chosen
quite
rough,
so that
there
are not
more
than
at
least 10 segments
The
per wavelength.
of the numerical
computation
accuracy
measurement
results
is quite
good.
compared
An improved
the
agreement
Using
the
enhanced
can be achieved
by spending
more triangles
per wavelength.
An explanation
of the deviations
between numerical
and mea-
terns, including
efficiently
from
sued
range.
results
connectors
can be found
in the simple
and the attached
of the proposed
8 the dominant
is shown
mittance
at a frequency
off
5’21 shows a significant
and the shorting
be seen,
the feeding
port
is quite
real part
with
=
almost
1.3 GHz,
minimum.
to the second
distribution
port
J>
[l]
dots.
It can
is transmitted
or the shorting
M. Abramovitz
matical
where the transThe connectors
by the white
no current
References
sufficient.
of current
pin are indicated
that
applications
DeCaps.
can be performed
by COMORAN
quasi-statical
frequencies
up to the gigahertz
the influences
of different
DeCap-placements,
the ac-
algorithm
In figure
clearly
for the SMA-
method,
coax-cable.
Nevertheless,
in order to study
power bus systems and different
curacy
model
integration
two or more very dense neighboured
conductive
planes can
be computed
with good accuracy
and within reasonable
comput&ion
time. The analysis of e.g. complex
power bus sys-
and
I.A.
Functions.
Handbook
Stegun.
Dover
Publications,
of Mathe-
Inc.. New York,
1970.
[2]
from
Bicker,
Markus
and
Oing,
Stefan.
Simulator
Coupling
‘IEEE 1998 International
Symposium
on Electromagnetic
Compatibility: Denuer, Colorado, (ISA, pages 656-661, August
Technique
pin.
for Complex
PCB Structures.
24-28 1998.
[3]
Fan,
J.;
Shi,
Modeling
continuities
H.;
Orlandi,
A.;
and
Drewniak,
Power
Bus Structures
with
Mixed
Potential
Integral
J.’ L.
Vertical
Equation
DisFor-
mulation
with
Circuit
Extraction.
internal
EMI
modeling progress summary,
Electromagnetic
Comof Missouri-Rolla,
patibility
Laboratory,
University
http://www.emclab.umr.edu,
:, January, 1999.
[4]
[5]
Glisson,
Allen
Wilburn
merical
Techniques
faces.
1978.
The
Harrington.
Univiversity
Roger
ment Methods.
reprinted
[6]
Meinke;
Figure
8: Current
distribution
on the upper
(real
part)
at f = 1.3 GHr
[7]
Oing.
Gundlach.
Stefan;
Radiation
plane
An enhanced
triangular
derived.
integration
technique
for the
EFIE-MOM
with
patches
based upon Rae’s formulation
has been
The accuracy
of the new method
has been prooved
on a simple
example,
is available.
computing
In spite
each single
for
which
also
an analytical
solution
of the increased
numerical
effort
in
matrix element,
the total computaion
time can be reduced significantly,
because much
are necessary
to achieve the required
accuracy.
less elements
333
[8]
Rae. Sad&a
by Mo-
1968;
FI.. 1982.
Line
New
York,
Methods
for
Structures
on
Ziirich
Symposium
on ElecZiirich, Switzerland..
sup-
249-255,
1997.
Electromagnetic
Radiation
of Arbitrarily-Shaped
Patch Modeling.
The liniversity
1980.
June
York,
M&bar.
Simulation
of Transmission
Madiraju.
New
Heidelberg,
Holger.
Sur-
Ph.D.,
der Hochfrequenrtech-
Berlin.
12th Internattonal
tromagnetic
Compatibility,
plement(l8W2):pp
Co.,
Taschenbuch
PCB.
Conclusions
of Nu-
Computation
Company,
Publishing
Eckardt.
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Field
F.
V&g.
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On the
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by Krieger
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Springer
Tokio..
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JR.
for Treating
Surfaces
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and
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Ph.D.,