1

Evaluation of Green’s Function Integrals inConducting Media

Swagato Chakraborty and Vikram Jandhyala

{swagato,jandhyala}@ee.washington.edu

Dept of EE, University of Washington Seattle WA, 98195-2500

UWEE Technical Report Number UWEETR-2002-0016 August 2, 2002 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu

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Evaluation of Green’s Function Integrals in Conducting Media

Swagato Chakraborty and Vikram Jandhyala, Member IEEE

Department of Electrical Engineering University of Washington

Seattle WA 98195 Phone: 206-543-2186

Fax: 206-543-3842 Email: jandhyala@ee.washington.edu

Abstract

This paper presents an accurate integration method for computing Green’s function

operators related to lossy conducting media. The presented approach is ultra-wideband

i.e. the integration schemes cover the entire range of frequency behavior, from high

frequencies where skin current is prevalent to low frequencies where volume current flow

dominates. The scheme is a step towards permitting exact ultra-wideband frequency

domain surface-only-based integral-equation simulation of arbitrarily-shaped 3D

conductors, and towards obviating the need for volume-based explicit frequency-

dependent skin effect modeling. This work deals specifically with the computation of

Green’s functions and not with the unrelated but important low-frequency conditioning

issue associated with the standard electric field integral equation.

1. Introduction

Surface and volumetric integral equation techniques are powerful paradigms for

modeling electromagnetic (EM) interactions in integrated circuit (IC) and packaging

problems. While coupled electromagnetic and circuit analyses have been successfully

realized through the popular volumetric partial element equivalent circuit (PEEC)

3

approach [1,2], the search for more general approaches, especially for modeling

frequency-dependent skin effects and for arbitrarily-shaped structures, has led to circuit-

coupled surface-based electric field integral equation (EFIE) formulations [3,4]. In these

and other works [5-11] it has been shown that surface integral equations and method of

moment (MoM) formulations can be interpreted and applied as generalizations of

volumetric EFIE - based PEEC. In particular, Rao-Wilton-Glisson (RWG)-function [6]

based triangular surface tessellations permit modeling of arbitrarily-shaped structures and

arbitrarily-directed equivalent surface currents. These forms of modeling are particularly

useful for package and system-on-chip simulation and can also enable coupled circuit and

electromagnetic simulation [3].

Surface integral equation formulations are desirable for simulating packaging and

interconnect structures due to the related ease in modeling arbitrary geometries and

equivalent current flow. Also, at high frequencies, surface impedance approximations are

sufficiently accurate to model losses and inductive behavior caused by skin effects.

However, at lower frequencies, where cross sections of conductors are smaller than the

skin depth, standard surface impedance approximations are invalid. Therefore, for

broadband simulation as necessitated in digital or ultra-wideband systems, a volumetric

formulation is typically required at low frequencies. In a volumetric formulation, the skin

effect needs to be modeled explicitly. This modeling requires fine and frequency-

dependent volume meshing (Fig. 1). It is noted that some recent efforts have been aimed

at obtaining new surface impedance approximations that might be valid at low

frequencies. These are typically restricted to cases of assumed or uniform cross sections

[7], as opposed to more general 3D structures, such as packages and on-chip inductors.

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Handling a mix of full-wave and skin-like effects with a surface-only formulation

is desirable since frequency-dependent effects can be tracked without changing geometric

discretization and without taking recourse to a special volume formulation at low

frequencies. This is particularly true for small microelectronic structures where geometry

detail and not wavelength is the guiding factor in mesh discretization. To accomplish a

surface-only formulation valid for realistic conductors over a broad range of frequencies,

the interior lossy medium EM problem must be addressed and coupled to the external

medium model.

This paper presents an exact formulation and accurate numerical quadrature

scheme to efficiently compute highly damped Green’s functions in lossy conductors. The

presented method is general in terms of geometries, frequencies, material parameters, and

relative separation and orientation of source and observer regions, and potentially forms

an important step towards the realization of a surface-only ultra-wideband integral

equation formulation. The motivation behind the presented lossy medium Green’s

function quadrature is, as discussed in the previous paragraph, that a coupled integral

equation formulation, linking an exterior homogeneous medium problem to an interior

lossy medium problem is required in order to correctly predict electromagnetic behavior

of realistic conductors in specific frequency bands. As line widths of interconnects

reduce, and as progressively smaller devices and structures are integrated at the package

and chip levels, the variation in the frequency behavior becomes larger and these cannot

be handled in an ad hoc manner by mixing surface and volume formulations.

It should be noted that the low frequency-dependence and modeling issue being

addressed here is distinct from the classical low frequency ill-conditioning of an EFIE

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formulation; the problem discussed here is unrelated and not due to the relative strength

of vector and scalar potentials. In fact, depending on the conductance involved, the issue

discussed here can arise at much larger frequencies than those where the EFIE is

inherently ill-conditioned. The treatment here is complementary to advances in

improving EFIE conditioning [8] at low frequencies.

This paper presents a new integration formulation and quadrature scheme for

modeling lossy medium Green’s functions with RWG functions, including scalar and

vector (and its curl) potentials, in an accurate manner. The quadrature is initially

facilitated by transforming the Green’s function computation associated with RWG

functions into polar coordinates. Subsequently, the proper order of integration results in

one analytic integration along one coordinate. Finally, a remaining one-dimensional

integral is computed as a summation of several superposed integrals over different bands

in the integration coordinate. Each such integral is computed with an efficient quadrature

scheme described herein.

Section 2 of this paper presents the two-region formulation that utilizes the

integrals that are the subject of this paper. Existing quadrature schemes are discussed in

Section 3. The specific frequency dependence of the integrals under study is outlined in

Section 4. Section 5 presents the polar-coordinate-based integration schemes. In Section

6 the adaptive quadrature rule designed to carry out a complex one-dimensional

integration is discussed. Numerical results including behavior of the resultant one-

dimensional integrands and their sampling, the self-consistency checks and comparisons

with other techniques are detailed in Section 7, and Section 8 presents conclusions and

continuing work.

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2. Formulation and Resultant Integrals

Consider the two-region problem depicted in Fig. 2, with the two regions being a

homogeneous lossless medium, typically free space or a lossless dielectric, and the

interior of a realistic conductor. The exterior equivalent problem utilizes the

homogeneous medium Green’s function, while the lossy medium Green’s function is

required for the interior equivalent problem. For the electric field integral equation

(EFIE), scalar and vector potential integrals will be necessitated, while for the magnetic

field integral equation (MFIE), an integrand that represents the curl of the vector potential

is required. In general, for PMCHW [9] and combined field integral equation (CFIE)

formulations, all three types of integrands need to be computed.

Typically for a region characterized with arbitrary material properties, the electric

and magnetic field E and H can be represented by the equations

FAEE ×∇−∇−−= φωjinc (2.1a)

AFHH ×∇+∇−−= ψωjinc (2.1b)

where incE and incH are the incident electric and magnetic fields in the region, A and

F are the magnetic and electric vector potentials, φand ψ represent the electric and

magnetic scalar potentials, and fπω 2= where f is the frequency of operation.

The scalar and vector potentials can be written in terms of the Green’s function G and the

electric and magnetic current density, J and M as :

∫′

′′′=S

sdG )(),(4

)( rJrrrAπµ (2.2 a)

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∫′

′′′=S

sdG )(),(4

)( rMrrrFπε (2.2 b)

∫′

′′⋅∇′=S

sdGj )(),(4

)( rJrrrωεπ

φ (2.2c)

∫′

′′⋅∇′=S

sdGj )(),(4

)( rMrrrωµπ

ψ (2.2d)

where εµ , represents the material permittivity, and permeability, respectively, and

),( rr ′G for a source point r′ located in the source region S ′ , and an observation point

r is

rr

rrrr

′−=′

′−− kjeG ),( (2.3)

where k is the wave number at a frequency ω , for a material with σ , rµ , rε as the

conductivity, relative permeability and permittivity , and is given by

)(0

00 εωσεεµµω

jk rr += (2.4)

Two auxiliary potentials Π , and Γ are introduced to represent the four potentials in Eqn.

(2.2) as, ΠA µ= , ΠF ε= ; ε

φ Γ= ,µ

ψ Γ= , where

∫′

′′′=S

sdG )(),(41)( rXrrrΠπ

(2.5a)

∫′

′′⋅∇′=ΓS

sdGj )(),(4

)( rXrrrωπ

(2.5 b)

Additionally, the curl operators in Eqn. (2.1) are represented as

∫′

′′×′∇ ′−=×∇S

sdG )(),(41)( rXrrrΠπ

(2.5c)

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where Χ represents the electric or magnetic current density. The popular triangle-pair-

based Rao-Wilton-Glisson (RWG) functions [6] are used to represent )(rΧ ′ , wherein

current is modeled by edge-based piecewise linear vector functions, and the divergence

of current is represented by piecewise constant scalar functions as ±

±′=

Al

2)( ρrΧ , and

±=⋅∇Al)(rΧ [6] where ±′ρ represents the vector joining the node opposite to the edge

in question to (from) the source point r′ in the positive (negative) triangle, ±A denotes

the area of the positive(negative) triangle, and l is the length of the edge.

The generalized potential integrals Eqn. (2.5) can be written for RWG sources as

)(8

1)( scalc

vect MA

ρΜrΠ +=π

(2.6a)

scalMA

jπω4

)( =Γ r (2.6b)

)]([8

1)( scalc

vecti N

AρNRrΠ +×=×∇

π (2.6c)

where

sdR

e

T

Rjk

vect ′= ∫∫−

ρM (2.7a)

sdR

eMT

jkR

scal ′= ∫∫−

(2.7b)

sdR

jkRe

T

jkR

vect ′+= ∫∫−

3

)1(ρN (2.7c)

sdR

jkReNT

jkR

scal ′+= ∫∫−

3

)1( (2.7d)

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and iR represents the vector joining the vertex of the source triangular region T (Fig. 4)

opposite to the edge in question to the observation point, cρ is the vector from the same

vertex to the projection of the observation point onto the plane of T, and ρ is the vector

from the projection of the observation point r on the plane of T to a source point r′ on T.

rr ′−=R , is the radial distance between the source and the observation point.

3. Existing Analytical and Numerical Quadrature Schemes for Green’s Functions

In the extant literature, evaluation of the potential integral Eqn. (2.5) in free-space and

low-loss media has been done by a variety of numerical schemes. For near-field terms,

singularity extraction of the kernels in Eqn. (2.5a-b) is performed analytically [10-11] to

leave a function that can be integrated numerically with a low-order quadrature rule [12].

Recently, methods based on the Duffy transform have emerged, wherein the triangular

integration region is transformed to a rectangle with a subsequent cancellation of the

singularity. The integral in Eqn. (2.5c) has been evaluated in free space [13], and in

lossless dielectrics [9] .

When the medium is conducting, even the singularity-extracted part may exhibit a

rapid spatial decay, i.e. the extracted integral appears nearly singular when the

observation point is sufficiently close to the source triangle. Hence, standard singularity

extraction [10] fails to evaluate the integral accurately.

A suitable approach to Green’s function computation in lossy media is polar

coordinate integration, which can render the non-essential singularity cancelled through

the Jacobian of transformation. Such methods are discussed previously in [14-15] for

lossless media and in [16] for lossy media, for the restricted case of the scalar Green’s

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function in Eqn. (2.7b). However these methods are not sufficiently general for the

integrals in Eqns. (2.7a,c,d) that are related to the vector potential or its curl. Another

polar coordinate approach is proposed in [17] to evaluate the vector integral for the

specific case of self-term integration. The method is extendable to the case when the

observation point is located anywhere in the plane of the source triangle itself. This

precludes the important case of observation at a near-singular point located above or

below the source triangle, as occurs in thin conductors.

In Section 5 we propose a general method for evaluating scalar, vector, and

gradient Green’s functions in lossy media with RWG basis functions. The presented

technique works for all frequencies and for all relative positions between source triangles

and observation points. The next section discusses the frequency-dependent behavior of

the generalized potential integrals in conducting media that necessitates the specialized

quadrature presented in later sections.

4. Frequency Dependence of Green’s Functions in Conducting Media

The behavior of the Green’s functions in Eqns. (2.7) in conducting media is highly

dependent on frequency, as shown in Fig. 3. Consider a Method of Moments (MoM) [6]

matrix created for interactions between RWG functions for the interior medium

equivalent, which uses the conducting medium Green’s functions. At high frequencies,

the MoM matrix is nearly diagonal because of a very rapid exponential spatial decay of

the conducting medium Green’s function owing to the large imaginary part of the

wavenumber in Eqn. (2.4).

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At lower frequencies, the interactions between non-overlapping RWG functions

are not neglible; and the MoM matrix becomes progressively less sparse but has sections

which are numerically sparse (e.g. in double precision arithmetic) due to large

exponential decays. As the frequency is further lowered the MoM matrix is completely

full while showing a weak exponential decay with distance. Eventually, the MoM matrix

is full and the exponential decay is very weak or absent.

To summarize, at intermediate frequencies, between sharp fall-off and no fall-off

regimes, special numerical treatment is required; the integrands presented by the lossy

medium Green’s function have sharp radial decay, and non-self interactions are also

prominent. Depending on the frequency, the entire MoM matrix might be numerically

significant. Low-order Gaussian quadrature rules in [12], that are popular in RWG-based

MoM implementations will not provide accurate answers at such frequencies, owing to

rapid decays of the Green’s functions over finite distances.

5. Computation of Lossy Medium Generalized Potential Integrals

The generalized potential integrals Eqn. (2.6) for RWG sources are constituted by the

four terms in Eqn. (2.7), which can be transformed into polar coordinates as,

θρθρ

ρθρθρ

ρ ρρ

ddd

eddd

e

T

djk

T

djk

vect sinˆcosˆ22

2

22

2 2222

∫∫∫∫ ++

+=

+−+−

yxM (5.1a)

θρρ

ρ ρ

ddd

eMT

djk

scal ∫∫ +=

+−

22

22

(5.1b)

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θρθρρρ

θρθρρρ

ρ

ρ

ddd

edjk

ddd

edjk

T

djk

T

djk

vect

sin)(

)1(ˆ

cos)(

)1(ˆ

322

222

322

222

22

22

∫∫

∫∫

+

++

++

++=

+−

+−

y

xN

(5.1c)

θρρ

ρρ ρ

ddd

edjkN

T

djk

scal ∫∫+

++=

+−

322

22

)(

)1(22

(5.1d)

In the above equations, the x and y coordinates are local to the source triangle T (Fig. 4)

and define the plane in which T lies. Also, d is the perpendicular distance of the

observation point from the plane of T, and ( θρ, ) is the polar coordinate of a source point

in T, with the projection of the observation point onto the plane of T as the origin. The

scalar integrals in Eqns. (5.1b,d) and the scalar components of the vector integrals in

Eqns. (5.1a,c) can be written in a generalized form, as

∫ ∫=ρ θ

ϕχ θρθχρϕ ddI )()( (5.2)

where ϕ is one of vectM ,ϕ , scalM ,ϕ , vectN ,ϕ , scalN ,ϕ defined below as

22

2

,

22

)(d

e djk

vectM+

=+−

ρρρϕ

ρ

(5.3a)

22,

22

)(d

e djk

scalM+

=+−

ρρρϕ

ρ

(5.3b)

322

222

,)(

)1()(

22

d

edjk djk

vectN+

++=

+−

ρρρ

ρϕρ

(5.3c)

322

22

,)(

)1()(

22

d

edjk djk

scalN+

++=

+−

ρρρ

ρϕρ

(5.3d)

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Also χ is one of cχ , sχ , 0χ defined below as

θθχ cos)( =c (5.4a)

θθχ sin)( =s (5.4b)

1)(0 =θχ . (5.4c)

Owing to the simple closed form expressions for the integral of )(θχ , the integral ϕχI can

be recast as a function of ρ as

∫∫ ∫==T

dddImax

min

)()()()(ρ

ρϕχ ρρξρϕθρθχρϕ (5.5)

where minρ and maxρ are the extremal ρ for which TP ∈∋ℑ ),( θρθ , ),( θρP denotes a

point having coordinate ),( θρ (Fig. 4), and ξ is one of cξ , sξ , 0ξ with

( ) ( )

−

+−

−

=

−=

=

=

∑∑∑ ∫∑====

)()(

)(cos)(cos

)(sin)(sin

cossin

1sincos

)(

)(

)(

)()()(

minmax

minmax

minmax)(

1

)(

)(

)(

11

)(

)(0

10

max

min

max

min ρθρθρθρθ

ρθρθ

θθ

θθθ

θ

ρξρξρξ

ρξρξρξ

ρ

ρθ

ρθ

ρρ ρθ

ρθ

ρ

ii

ii

ii

K

i

K

i

K

ii

is

icK

is

c

i

i

i

i

d

(5.6)

Also, )(ρK is the number of intervals (Fig. 4) in θ , πθ 20 <≤ , for which ),( θρP lies

in T, and imaxθ and i

minθ are the limits on θ for the thi interval. The values of )(max ρθ i

and )(min ρθ i for each section are computed by obtaining the intersection of T and the

circle of radius ρ centered at the projection of the observation point onto the plane of T.

If the circle with radius ρ lies entirely in T, 1)( =ρK , πθ 21max= ,and 01

min =θ .Hence

=

πρξρξρξ

200

)()()(

0

s

c

(5.7)

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Alternatively, if for a given ρ , if the circle is completely outside T then the integral

contributions are all zero. Consequently, the constituents of the generalized potential

integrals Eqn. (5.1) can be computed using Eqn. (5.3-5.6) as

ρρξρϕρρξρϕρ

ρ

ρ

ρ

dd svectMcvectMvect )()(ˆ)()(ˆmax

min

max

min

,, ∫∫ += yxM (5.8a)

ρρξρϕρ

ρ

dM scalMscal )()(max

min

0,∫= (5.8b)

ρρξρϕρρξρϕρ

ρ

ρ

ρ

dd svectNcvectNvect )()(ˆ)()(ˆmax

min

max

min

,, ∫∫ += yxN (5.8c)

ρρξρϕρ

ρ

dN scalNscal )()(max

min

0,∫= (5.8d)

Section 6 discusses the adaptive integration rule that has been designed in order to

perform the integration in Eqns. (5.5 and 5.8) efficiently.

6. Generalized Adaptive Integration Rule for Piecewise Smooth Functions

In the one-dimensional integral Eqn. (5.5), the function )(ρϕ is smooth and continuous

over the integration interval, and the function )(ρξ is piecewise smooth and continuous.

Hence the over-all integrand is piecewise smooth and continuous. If the integrand

)()()(1 ρξρϕρ =+if , is smooth over the subintervals ),( 1+ii ρρ , where { } ,1,.....2,1,0 −∈ Li

min0 ρρ = , and maxρρ =K , the total integral is written as

ρρρ

ρϕχ dfI

L

ii

i

i

)(1

1

∑ ∫=

−

= (6.1)

In this manner the required quadrature scheme, outlined next, will not need to compute

integrals for functions that have first derivative discontinuities. The integration scheme

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works as follows. Initially a coarse estimate of estIϕχ is obtained by using a 5 point

Newton-Cotes formula based on the Bodé rule [12], over each of the individual L

segments. The rule computes an integral dxxfQb

aab )(∫= with the approximation

)}(7)4

3(32)2

(12)4

(32)(7{90

hafhafhafhafafhQab ++++++++≅ ( 6.2)

where h = b-a. Thus the total integral is initially estimated as

∑=

−=

L

i

estii

QI1

1ρρϕχ ( 6.3)

The quadrature is refined recursively by using an adaptive integration method. For

efficiency in terms of number of sample points, non-uniform sampling of the function

)(ρf is required within each subinterval ( )ρρ ,1−i . At a particular level of recursion, for a

given subinterval (a,b), an estimate of the integral abQ is obtained using Eqn. (6.2).

Subsequently, a binary split is performed on (a,b), and cbacab QQQ +=′ is obtained

where 2/)( bac += . The estimate of initIϕχ is dynamically refined as

)( abcbacinitinit QQQII −++← ϕχϕχ (6.4)

at each level of recursion. If the change in the integration result due to the binary split,

abcbac QQQ −+=∆ (6.5)

relative to the total integral initIϕχ is smaller than a pre-specified threshold tolerance tol1,

then the contribution of that split to the over-all integral is ignored. The exact stopping

criteria used for the convergence test are given by

initItol ϕχ⋅≤∆ 1 (6.6a )

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or

abQtol ⋅≤∆ 2 (6.6b )

where the criterion in Eqn. (6.6b) using the pre-specified tolerance tol2 is required in

order to expedite the convergence in the case of a nearly linear function segment.

The method described above is similar in approach to the existing popular

Matlab-based adaptive quadrature method quad[18], with some important differences. It

uses a different order rule to evaluate abQ in Eqn. (6.2). Also it uses an improved dynamic

refinement of the estimate of the total integral Eqn. (6.4), and leads to rules with smaller

number of samples for a given approximation error for the integrals in this paper. The

code/pseudocode for the rule is not included here but can be found online [19].

7. Numerical Results In this section, the proposed integration schemes are used to compute integrals for all the

cases in Eqn. (5.8). The resulting integrands and sampling quadrature points in polar

coordinates are shown here for different frequencies and accuracies. Comparisons with

two-dimensional Gaussian quadrature are also presented.

For purposes of illustration, and without loss of generality, the source triangle

sourceT for the presented results has nodes located at ( )0,1,1 − , ( )0,5.0,1 , ( )0,5.0,2− , and

the observation point obsP lies outside the plane of sourceT , at ( )1,0,0 , with all distances

measured in mm. The conductivity of the medium is that of copper 7108.5 × S/m.

The one dimensional integral in Eqn. (5.5) is evaluated using the adaptive

quadrature rule described in Section 6. Behavior of the six related integrands in Eqn.

(5.8) are shown in Figs. 5-7, for the source triangle sourceT and observation point obsP .

17

The operating frequency is 1 MHz. The adaptively sampled locations for such integrands

are also pictured. The integrands comprise piecewise smooth functions over all relevant

intervals as discussed in Section 5.

The effect on integrands of lowering frequency is depicted in Figure 8.

Specifically, the integrands )()(, ρξρφ svectM in Eqn. (5.8a) and )()( 0, ρξρφ scalM in Eqn.

(5.8b) are shown for a lower frequency 1KHz, corresponding to the higher frequency

plots in Figs. 5 and 6. It can be noted that with reduction in frequency the integrand

decays slower in distance ρ , and exhibits less oscillations, as both the imaginary and the

real parts of the wave-number k in Eqn. (2.4) becomes smaller.

A feature of the quadrature scheme outlined in this paper is the ability to vary the

tolerance and correspondingly the number of samples in order to achieve a trade-off

between efficiency and accuracy. The effect on the sample points of relaxing the

convergence criterion is shown in Fig. 9. The integrands with sample points

corresponding to higher convergence tolerances for )()(, ρξρφ svectM and )()( 0, ρξρφ scalM

are shown at 1MHz and 1 KHz respectively. The corresponding plots for lower tolerances

are presented in Figs. 5 and 8, and the difference in location and density of sample points

is evident. In general, as can be seen from the results, the scalar integrals appear to

converge to lower errors for given convergence thresholds.

A relative accuracy comparison between the proposed scheme and 7-point

Gaussian quadrature with singularity extraction is demonstrated in Figs. 10 and 11. At

low frequencies, the Green’s functions in lossy media exhibit slow decay over distance

and hence a 7-point Gaussian quadrature scheme [12] works adequately, and the relative

difference between the two methods is small. As the frequency is increased, the details

18

of the decay in the Green’s functions due to the increased imaginary part of the wave-

number (Eqn. 2.4) are not captured by the low-order Gaussian rule and the proposed

methodology of this paper is required. The fact that the discrepancy between the results

from the proposed method and from 7-point Gaussian quadrature is due to the Gaussian

quadrature becoming inaccurate is further evident from comparisons with a higher order

Gaussian quadrature rule using 25 points on a triangle. In this case the frequency at which

the 25 point quadrature breaks down increases compared to the 7 point quadrature. In

general, for any order of Gaussian quadrature, there is a frequency point beyond which

the quadrature will be inaccurate due to insufficient sampling of the details in the decay

of the Green’s function. The presented method accurately models the decay through an

analytic integration and is therefore accurate at any frequency. This is seen in both the

vector integrals (Eqns. 5.8a,c ; Fig. 10) and the scalar integrals (Eqns. 5.8 b,d; Fig. 11).

While the main aim of this work is the formulation and development of the

quadrature rules themselves, one example of the behavior of the rules when included in a

complete two-region PMCHW formulation is shown next. Figure 12 compares the

extracted resistance using a coupled circuit-EM formulation [3] and the quadrature

scheme presented in this paper, with the analytic quasi-static resistance at low frequency.

It also demonstrates the importance of radiation loss at a higher frequency, which is

modeled by a full-wave formulation. Figure 13 shows the agreement in the extracted

resistance obtained by a PMCHW formulation using the standard Gauss quadrature rule

and the proposed method at low frequency. The result matches with an impedance

boundary formulation [19] at high frequencies; the impedance boundary condition is

inaccurate at low frequencies relative to skin depth, and fails to capture the leveling off of

19

the resistance at low frequency. Conversely, the Gauss quadrature scheme becomes

inaccurate at high frequencies, which is demonstrated in Fig. 14. At such frequencies the

proposed quadrature scheme produces same result as the impedance boundary condition

formulation, while at low frequencies the two quadrature schemes produce the same

result.

8. Conclusions In this paper, a new approach to evaluate the Green’s function operators for RWG

functions in conducting media is presented. The method works for arbitrarily located

sources and observers for any frequency. This technique has been incorporated into a

broadband two-region surface formulation for accurate computation of frequency-

dependent parameters, and shows the potential to obviate the need to switch to volumetric

formulations at frequencies where volumetric current flow is dominant.

9. Acknowledgements This work was partially supported by DARPA-MTO NeoCAD grant N66001-01-1-8920,

NSF-CAREER grant ECS-0093102, NSF-SRC Mixed-Signal Initiative grant CCR-

0120371, and by a grant from Ansoft Corporation.

References

[1] H. Heeb and A.E. Ruehli, “Three-dimensional interconnect analysis using partial element equivalent circuits,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 39(11), pp. 974-982, Nov. 1992. [2] A. Rong and A.C.Cangellaris, “Generalized PEEC models for three-dimensional interconnect structures and integrated passives of arbitrary shapes,” Electrical Performance of Electronic Packaging, pp. 225 –228, Oct. 2001.

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[3] V. Jandhyala, W.Yong, D.Gope, and R. Shi, “Coupled electromagnetic-circuit simulation of arbitrarily-shaped conducting structures using triangular meshes,” Proceedings International Symposium on Quality Electronic Design, pp. 38-42, Mar. 2002. [4] S. Ponnapalli, A. Deutsch, and R. Bertin, “A package analysis tool based on a method of moments surface formulation,” IEEE Transactions on Components, Hybrids, and Manufacturing Technology, vol. 16(8), pp. 884-892, Dec. 1993. [5] J. Wang, J. Tausch, and J. White, “A wide frequency range surface integral formulation for 3-D RLC extraction,” Digest of Technical Papers International Conference on Computer-Aided Design, pp. 453-457, 1999. [6] S.M. Rao, D.R.Wilton, and A.W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Transactions on Antennas Propagation, vol. 30, pp. 409-418, 1982. [7] J.-S. Zhao and W.C. Chew, “Accurate and efficient simulation of crosstalks,” Proceedings of the Progress in Electromagnetics Research Symposium, pg. 396, Boston, July 2002. [8] S. Chen, J. S. Zhao, and W.C. Chew, “ Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Transactions on Geoscience and Remote Sensing, vol. 40(2), pp. 426-433 , Feb. 2002. [9] K.Umashankar, A.Taflove, and S.M.Rao, “Electromagnetic scattering by arbitrary shaped three dimensional homogeneous lossy dielectric objects,” IEEE Transactions on Antennas and Propagation, Vol 34(6) , pp. 758-766, June 1986. [10] R.D. Graglia, “On the numerical integration of the linear shape function times the 3-D Green’s function or its gradient on a planar triangle,” IEEE Transactions on Antennas and Propagation, vol. 41, pp. 1448-1455, 1993. [11] D.R.Wilton, S.M.Rao, A.W.Glisson, D.H.Schaubert, O.M. Al-Bundak, and C.M.Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domain,” IEEE Transactions on Antennas and Propagation, vol. AP-32, pp. 276-281, March 1984. [12] M. Abramowitz and I. Stegun, Chapter 25, Handbook of Mathematical Functions, Dover, New York, 1970. [13] R.E.Hodges and Y.Rahmat Samii, “The evaluation of MFIE integrals with the use of vector triangle basis function,” Microwave and Optical Technology Letters, vol. 14 (1), pp. 9-14, Jan.1997.

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[14] M. Gimersky, S. Amari, and J. Bornemann, “Numerical evaluation of the two-dimensional generalized exponential integral,” IEEE Transactions on Antennas and Propagation, vol. 44, pp. 1422-1425, 1996. [15] J.K.H. Gamage, “Efficient method of moments for compact large planar scatterers in homogeneous medium,” Proceedings of 11th International Conference on Antennas and Propagation, no. 480, pp. 741-744, 2001. [16] Z. Zhu, J. Huang, B. Song, and J. White, “Improving the robustness of a surface integral formulation for wideband impedance extraction of 3D structures,” Proceedings of International Conference on Computer Aided Design, pp. 592-597, 2001. [17] L. Rossi and P.J. Cullen, “On the fully numerical evaluation of the linear-shape function times the 3-D Green’s function on a planar triangle,” IEEE Transactions on Microwave Theory and Techniques, vol. 47, pp. 398-402, 1999. [18] W.Gander and W.Gautschi, “Adaptive quadrature - revisited,” Technical Report 306, Department Informatik, ETH Zürich, August 1998. [19] S. Chakraborty and V. Jandhyala, “A modified adaptive quadrature scheme,” online at http://www.ee.washington.edu/faculty/jandhyala/adaptive_quadrature, unpublished. [20] A.W.Glisson, “Electromagnetic scattering by arbitrarily shaped surfaces with impedance boundary conditions,” Radio Science, vol. 27(6) , pp. 935-943, Nov. 1992.

22

Figure 1: Volumetric gridding as a function of frequency: (top) low frequency, volumetric current flow, (middle) intermediate frequency skin effect modeling, (bottom) high frequency surface current flow. Figure 2: Classical equivalence principle modeling of a two region problem, with one region being the interior of a conductor. The subscripts denote interior and exterior parameters. Figure 3: Behavior of the MoM matrix in a lossy conducting medium. At high frequencies, surface impedance approximations are accurate and lead to a nearly diagonal matrix. As the frequency is reduced, the matrix becomes less sparse but can still have numerically zero regions where exponential decays are very large. As frequency continues to reduce, the matrix becomes dense and eventually has very little exponential decay. Note: The “structure” of the matrix above assumes that basis functions near each other are located next to each other in the matrix order. This structure is only used as a schematic guide to explain the frequency-dependent nature of the matrix for lossy media. Figure 4: Region of integration is shown for ( 321 ,, ρρρρ = ), for a triangular region T , for the projection of the observation point on the plane of triangle O . Gray sections denote intervals of θ where the source point ),( θρP lies within the triangle. Figure 5: Behavior of )()(, ρξρϕ cvectM (top) and )()(, ρξρϕ svectM (bottom) in Eqn. (5.8a) and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

1=α mm, at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in a relative integration error of 4105.1 −× . Figure 6: Behavior of )()( 0, ρξρϕ scalM (top) and )()(, ρξρϕ cvectN (bottom) in Eqns. (5.8b,c)and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

1=α mm, at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in relative integration errors of 8105.3 −× and 41028.1 −× , respectively. Figure 7: Behavior of )()(, ρξρϕ svectN (top) and )()( 0, ρξρϕ scalN (bottom) in Eqn. (5.8c,d) and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

1=α mm, at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in relative integration errors of 41028.1 −× and 13109.1 −× , respectively. Figure 8: Behavior of )()(, ρξρϕ svectM (top) and )()( 0, ρξρϕ scalM (bottom) in Eqn. (5.8a,b) and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

23

1=α mm, at a frequency of 1KHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in relative integration errors of 5105.2 −× and 121087.8 −× , respectively. Figure 9: Behavior of )()(, ρξρϕ svectM (top) and )()( 0, ρξρϕ scalM (bottom) in Eqn. (5.8a,b) and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

1=α mm, at a frequency of 1MHz and 1KHz, respectively, with 7108.5 ×=σ S/m. The stopping threshold resulted in relative integration errors of 3103.3 −× and 8106.2 −× , respectively. Figure 10: Comparison between 2D Gaussian rules with singularity extraction and proposed method for evaluation of the integral vectN and vectM in Eqns. (2.7c,2.7a) for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

1=α mm , with 7108.5 ×=σ S/m. Figure 11: Comparison between 2D Gaussian rules with singularity extraction and proposed method for evaluation of the integral scalN and scalM in Eqns. (2.7c,2.7a) for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

1=α mm , with 7108.5 ×=σ S/m. Figure 12: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm, using PMCHW formulation with the proposed quadrature scheme, for a full-wave and a quasi-static formulation and the analytic value of resistance using skin-depth. Figure 13: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm, using PMCHW formulation with the proposed quadrature scheme, Gaussian quadrature, and impedance boundary condition. Figure 14: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm using a two region PMCHW formulation with the standard Gaussian quadrature method and the method proposed in this paper, and an impedance boundary condition formulation.

24

Figure 1: Volumetric gridding as a function of frequency: (top) lowfrequency, volumetric current flow, (middle) intermediate frequency skineffect modeling, (bottom) high frequency surface current flow.

25

σint,ε int,µ int

εext,µext

Lossy Conducting Medium External Homogeneous Medium

Interior Exterior

Figure 2: Classical equivalence principle modeling of a two region problem, withone region being the interior of a conductor. The subscripts denote interior andexterior parameters.

σint,ε int,µ int

σint,ε int,µ int

εext,µext

εext,µext

26

Decreasing Frequency

Surface Impedance Nearly Diagonal

Rapid Exponential Decay Partially Sparse

Exponential Decay Dense

Slow Decay Dense

Figure 3: Behavior of the MoM matrix in a lossy conducting medium. At high frequencies, surfaceimpedance approximations are accurate and lead to a nearly diagonal matrix. As the frequency isreduced, the matrix becomes less sparse but can still have numerically zero regions whereexponential decays are very large. As frequency continues to reduce, the matrix becomes dense andeventually has very little exponential decay. Note: The “structure” of the matrix above assumes thatbasis functions near each other are located next to each other in the matrix order. This structure isonly used as a schematic guide to explain the frequency-dependent nature of the matrix for lossymedia.

27

Figure 4: Region of integration is shown for ( 321 ,, ρρρρ = ), for a triangular region T ,for the projection of the observation point on the plane of triangle O . Gray sections denote intervals of θ where the source point ),( θρP lies within the triangle.

1ρ

2ρ

3ρO

ρdT),( θρP

θ

ρ

x

y

28

Figure 5: Behavior of )()(, ρξρϕ cvectM (top) and )()(, ρξρϕ svectM (bottom) in Eqn. (5.8a) andadaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where 1=α mm,at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in a relativeintegration error of 4105.1 −× .

0.5 1 1.5 2x 10-3

-8

-6

-4

-2

0

2

4x 10-12

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

0.5 1 1.5 2x 10-3

-2

-1

0

1

2

3

4

5x 10-12

ρ(m)

Inte

gran

d X Sampled Real Part Sampled Imaginary Part

29

Figure 6: Behavior of )()( 0, ρξρϕ scalM (top) and )()(, ρξρϕ cvectN (bottom) in Eqns. (5.8b,c)andadaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where 1=α mm,at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in relativeintegration errors of 8105.3 −× and 41028.1 −× , respectively.

0 0.5 1 1.5 2 2.5x 10-3

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5 x 10-7

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

0.5 1 1.5 2x 10-3

-1

-0.5

0

0.5

1

1.5

2x 10-4

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

30

Figure 7: Behavior of )()(, ρξρϕ svectN (top) and )()( 0, ρξρϕ scalN (bottom) in Eqn. (5.8c,d)and adaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

1=α mm, at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping thresholdresulted in relative integration errors of 41028.1 −× and 13109.1 −× , respectively.

0.5 1 1.5 2x 10-3

-8

-6

-4

-2

0

2

4

6

8x 10-5

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

0 0.5 1 1.5 2 2.5x 10-3

-5

-4

-3

-2

-1

0

1

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

31

Figure 8: Behavior of )()(, ρξρϕ svectM (top) and )()( 0, ρξρϕ scalM (bottom) in Eqns. (5.8a,b) andadaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where

1=α mm, at a frequency of 1KHz, with 7108.5 ×=σ S/m. The stopping threshold resultedin relative integration errors of 5105.2 −× and 121087.8 −× , respectively.

0.5 1 1.5 2x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10-4

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

0 0.5 1 1.5 2 2.5x 10-3

-1

-0.5

0

0.5

1

1.5

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

32

Figure 9: Behavior of )()(, ρξρϕ svectM (top) and )()( 0, ρξρϕ scalM (bottom) in Eqns. (5.8a,b) andadaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where 1=α mm,at a frequency of 1MHz and 1KHz, respectively, with 7108.5 ×=σ S/m. The stopping thresholdresulted in relative integration errors of 3103.3 −× and 8106.2 −× , respectively.

0.5 1 1.5 2x 10-3

-2

-1

0

1

2

3

4

5x 10-12

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

0 0.5 1 1.5 2x 10-3

-1

-0.5

0

0.5

1

1.5

ρ(m)

Inte

gran

d

X Sampled Real Part Sampled Imaginary Part

33

Figure 10: Comparison between 2D Gaussian rules with singularity extraction and proposedmethod for evaluation of the integral vectN and vectM in Eqns. (2.7c,2.7a) for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), andobservation point located at ),0,0( α ,where 1=α mm , with 7108.5 ×=σ S/m.

101 102 103 104 105 10610-4

10-2

100

102

104

Frequency(Hz)

Rel

ativ

e D

iffer

ence

X

vectlN -25 point rule vectM -25 point rule

vectN - 7 point rule vectM - 7 point rule

34

101 102 103 104 105 10610-4

10-3

10-2

10-1

100

101

102

Frequency(Hz)

Rel

ativ

e D

iffer

ence

X

scalN - 25 point rule scalM - 25 point rule scalN - 7 point rule scalM -7 point rule

Figure 11: Comparison between 2D Gaussian rules with singularity extraction andproposed method for evaluation of the integral scalN and scalM in Eqns. (2.7c,a) for thenon-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ),and observation point located at ),0,0( α ,where 1=α mm , with 7108.5 ×=σ S/m.

35

107 108 1090

0.5

1

1.5

2

Frequency (Hz)

Res

ista

nce

(Ohm

)Full-Wave with New Quadrature Quasi-Static with New Quadrature Analytic

Figure 12: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm, usingPMCHW formulation with the proposed quadrature scheme, for a full-wave and a quasi-staticformulation and the analytic value of resistance using skin-depth.

36

Figure 13: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm,using PMCHW formulation with the proposed quadrature scheme, Gaussianquadrature, and impedance boundary condition.

107 108 1090

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Frequency (Hz)

Res

ista

nce(

Ohm

) Impedance Boundary ConditionPMCHW with the New Quadrature PMCHW with Gaussian Quadrature

37

Figure 14: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm using atwo region PMCHW formulation with the standard Gaussian quadrature method and themethod proposed in this paper, and an impedance boundary condition formulation.

108 109 1010

0

20

40

60

80

100

120

140

160

180

Frequency (Hz)

Res

ista

nce

(Ohm

)

Proposed Quadrature SchemeExisting Gauss Quadrature SchemeImpedance Boundary Condition