IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 12, DECEMBER 2004 3357
Evaluation of Green’s Function Integrals in
Conducting Media
Swagato Chakraborty, Student Member, IEEE, and Vikram Jandhyala, Senior Member, IEEE
Abstract—This paper presents an accurate integration method
for computing Green’s function operators related to lossy con-
ducting media. The presented approach is ultrawideband, 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 toward permitting exact ultrawide-band fre-
quency domain surface-only-based integral-equation simulation
of arbitrarily-shaped three-dimensional conductors, and toward
obviating the need for volume-based explicit frequency-depen-
dent 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.
Index Terms—Boundary element methods, conducting bodies,
electromagnetic (EM) scattering, integral equations, skin effect.
I. INTRODUCTION
SURFACE and volumetric integral equation techniquesare powerful paradigms for modeling electromagnetic
(EM) interactions in integrated circuit (IC) and packaging
problems. While coupled electromagnetic and circuit analyzes
have been successfully realized through the popular volumetric
partial element equivalent circuit (PEEC) 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]–[12] it has been shown that surface integral
equations and method of moments (MoM) formulations can
be interpreted and applied as generalizations of volumetric
EFIE—based PEEC. At high frequencies, surface impedance
approximations are sufficiently accurate to model losses and
inductive behavior caused by skin effects. However, at lower
frequencies, standard surface impedance approximations are
invalid. Therefore, for broadband simulation as necessitated in
digital or ultrawide-band systems, a volumetric formulation is
typically required at low frequencies. However in a volumetric
formulation, the skin effect needs to be modeled explicitly
through a volume meshing. It is noted that some recent efforts
Manuscript received October 28, 2002; revised October 29, 2003. This work
was supported in part by the Defense Advanced Research Projects Agency-Mi-
crosystems Technology Office (DARPA-MTO) NeoCAD grant N66001-01-1-
8920, the National Science Foundation (NSF) CAREER grant ECS-0093102,
NSF-SRC Mixed-Signal Initiative grant CCR-0120371, and in part by a grant
from Ansoft Corporation.
The authors are with the Department of Electrical Engineering, University of
Washington, Seattle, WA 98195 USA (e-mail: jandhyala@ee.washington.edu;
swagato@u.washington.edu).
Digital Object Identifier 10.1109/TAP.2004.836430
have been aimed at obtaining new surface impedance approx-
imations [8].
Handling a mix of full-wave and skin-like effects with a sur-
face-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 microelec-
tronic structures where geometry detail and not wavelength is
the guiding factor in mesh discretization. To accomplish a sur-
face-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
[10], and such a formulation requires explicit computation of
the Green’s function integrals in the interior lossy medium, in
contrary to the volumetric formulation, where the Green’s func-
tion integrals are always computed in the background medium.
This paper presents an exact formulation and accurate numer-
ical quadrature scheme to efficiently compute highly damped
Green’s functions in lossy conductors. The presented method
is general in terms of geometries, frequencies, material param-
eters, and relative separation and orientation of source and ob-
server regions, and potentially forms an important step toward
the realization of a surface-only ultrawide-band integral equa-
tion formulation.
It should be noted that the low frequency-dependence and
modeling issue being addressed here is distinct from the clas-
sical low frequency ill-conditioning of an EFIE formulation. 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 com-
plementary to advances in improving EFIE conditioning [9] at
low frequencies.
The presented quadrature scheme, discussing computation
of the relevant Green’s function integrals in lossy media using
RWG functions in a PMCHW formulation, 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, the remaining one-dimensional
(1-D) integral is computed as a summation of several super-
posed integrals over different bands in the integration coordi-
nate.
Section II of this paper presents the two-region formulation
that utilizes the integrals that are the subject of this paper. Ex-
isting quadrature schemes are discussed in Section III. The spe-
cific frequency dependence of the integrals under study is out-
lined in Section IV. Section V presents the polar-coordinate-
based integration schemes. Numerical results, self-consistency
0018-926X/04$20.00 © 2004 IEEE