Incorporation of conductor losses into the stripline simulator UA-FWLIS by using a surface impedance...

Incorporation of Conductor Losses into theStripline Simulator UA-FWLIS by Usinga Surface Impedance Formulation

Yi Cao,1 Xing Wang,2 Zhaohui Zhu,3 Steven L. Dvorak,1 John L. Prince1

1 Department of Electrical and Computer Engineering, The University of Arizona, Tucson, AZ 857212 NVIDIA Corp, 2701 San Tomas Expressway, MS: 12, Santa Clara, CA 950503 Intel Corp., 5000 W Chandler Blvd., MS: CH5-157, Chandler, AZ 85226

Received 2 November 2006; accepted 4 May 2007

ABSTRACT: We use a surface impedance formulation to enable the MoM-based full-wave

layered interconnect simulator, UA-FWLIS, to handle conductor losses for stripline inter-

connects. Because these approaches are fully compatible with the previously developed ana-

lytical calculations for the reaction matrix elements, the computational efficiency of UA-

FWLIS is not affected by including conductor losses. VVC 2008 Wiley Periodicals, Inc. Int J RF and

Microwave CAE 18: 187–194, 2008.

Keywords: method of moments; full-wave; surface impedance; conductor losses

I. INTRODUCTION

With increasing clock rates in high-performance

VLSI systems, it is becoming more and more impor-

tant to accurately model the performance of intercon-

nects, especially at high frequencies. The MoM-

based, full-wave layered interconnect simulator (UA-

FWLIS) is an efficient interconnect simulation tool

that has been developed for the purpose of modeling

stripline interconnects [1–6]. Because the closely

spaced ground planes cut off the propagation of all

the higher-order modes in the parallel-plate wave-

guide in stripline packaging structures, the reaction

matrix in UA-FWLIS is very sparse. Thus, by using

sparse matrix solution techniques and other accelerat-

ing techniques, UA-FWLIS can achieve much higher

computational efficiency when compared with Agi-

lent Momentum [7].

In the early versions of UA-FWLIS, all conduc-

tors were assumed to be perfect electric conductors

(PEC). However, conductor losses have to be

accounted for in high-frequency interconnect simu-

lations. Surface impedance formulations are the

simplest way to handle conductor losses in inter-

connect and packaging structures, and are widely

used [8–12]. In this article, we incorporate a sur-

face impedance formulation into UA-FWLIS. In

addition to reducing the number of unknowns, we

also show that the surface impedance formulation

avoids the branch-cut singularities that appear

when the ground planes are modeled as finite con-

ductivity regions. The absence of these branch-cut

singularities allows us to directly apply our previ-

ously developed techniques for obtaining closed-

form representations for the reaction elements. It

also allows us to apply the sparse matrix techni-

ques that are developed in [7].

In the following sections, we first validate the use

of surface impedance boundary conditions in lossy

stripline structures. We show that for practical inter-

connect applications, the electric field expressions

Correspondence to: S. L. Dvorak; e-mail: [email protected]

DOI 10.1002/mmce.20277Published online 19 February 2008 in Wiley InterScience (www.

interscience.wiley.com).

VVC 2008 Wiley Periodicals, Inc.

187

that result from the surface impedance approach

accurately approximate the exact solution for the

ideal case where an analytical solution can be easily

obtained. We also discuss the branch-cut singularities

for these two cases. The details of the surface imped-

ance approach for modeling the losses on the traces

and ground planes are also given.

II. SURFACE IMPEDANCEFORMULATION FOR THE TRACES

Because UA-FWLIS employs a surface current for-

mulation, an effective surface current is defined first.

This allows the effects of conductor losses to be eas-

ily included into UA-FWLIS by adding an extra term

to every reaction element. According to Ampere’s

law, the magnetic field and the total current through

the cross section of a long conductor line satisfy the

following equation

ZC

H � dl ¼ZC

n̂ 3 Htan � ðn̂ 3 dlÞ

¼ZC

Jeffs � ðn̂ 3 dlÞ ¼ I; ð1Þ

where C is the perimeter of the cross section and the

unit vector n̂ is chosen normal to the perimeter. Here,

we can see that n̂ 3 Htan can be considered as an

effective surface current density that plays the same

role as the surface current on a PEC. We then use

this effective surface current as the unknown in the

problem and apply the general boundary condition

Zeffs n̂ 3 Htan ¼ Etan ¼ Zeff

s Jeffs ð2Þ

on the nonideal conductor surface, where Zeffs is a

properly defined surface impedance.

On the surfaces of the lossy traces, the integral

equation that is to be solved by the MoM becomes

Zeffs Jeff

s � Eitan ¼ Es

tan

��surface of non-PEC

¼ � �L � Jeffs :

ð3Þ

Therefore, the conductor losses can be included in

the reaction elements by writing

Zmn ¼ Tm; �L � Jeffns

� �þ Tm; Zeff

s Jeffns

� �: ð4Þ

Note that the reaction elements for the lossy traces

consist of a PEC term plus an additional term that

models the effects of the conductor losses on the

traces. Because the additional term is very easy to

compute when the surface impedance is a constant

along the surface of every cell, the reaction elements

can still be calculated efficiently by the previously

developed residue series representation [1].

In general, the surface impedance Zeffs varies

with the location on the perimeter of the cross sec-

tion and for different frequencies, thus, making it

difficult to find an analytical expression for Zeffs .

However, we found that for frequencies higher

than 1 GHz, the wave impedance of a conductive

half space, Z0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffijxl=r

p, is an appropriate approx-

imation for the surface impedance in most off-chip

cases. In most PCB and packaging structures, the

thickness of a typical wire is on the order of

20 lm, and the width is much larger than the thick-

ness. For copper conductors, the skin depth at

1 GHz is 2.1 lm, which is a small fraction of the

thickness of an off-chip line. Therefore, the dimen-

sions of the cross section for off-chip interconnects

are large enough to accurately apply Z0 as the sur-

face impedance at frequencies higher than 1 GHz.

If simulations at lower frequencies are needed,

where the current is distributed relatively uniformly

over the cross section, then a location-dependent

surface impedance [12], can be used. When Z0 is

applied as the surface impedance, we can see that

for a fixed frequency, Zeffs is a constant and the

term including the losses in expression (4) is just

the inner product of the expansion and testing func-

tions. For the case of rooftop expansion and testing

functions, this lossy term has a nonzero value only

for the cases of self-reactions and the reactions

between overlapping cells.

III. SURFACE IMPEDANCEFORMULATION FOR THE POWER/GROUND PLANES IN STRIPLINESTRUCTURES

In this section, we validate the use of a surface im-

pedance formulation for stripline configurations with

lossy ground planes. Here, we first study a simple

homogeneously filled stripline. Such structures can

be modeled as a three-layer dielectric structure as

shown in Figure 1. In this model, the lossy upper and

lower ground planes are considered as semiinfinite

regions with high conductivity, i.e., good conductors.

After investigating this general model we found that

the surface impedance approach provides a good ap-

proximate solution for the electric fields as compared

with the three-layer approach when the inverse Fou-

rier transform variable |kq| is not too large and r is

188 Cao et al.

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

large enough. This condition is always met for the

packaging problems where UA-FWLIS is applicable.

After defining the following variables

k1 ¼ k2 ¼ k ¼ xffiffiffiffiffile

p; k0 ¼ k3 ¼ x

ffiffiffiffiffiffile0

p;

kz1 ¼ kz2 ¼ kz; kz0 ¼ kz3 ¼ k0z; e0 � �j

rx; ð5Þ

kq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � k2

z

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik02 � k02z

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2

y

q;

we can obtain expressions for the tangential spectral

domain electric fields in regions 1 and 2 (for the pur-

pose of saving space, only the TE mode results are

given)

Figure 1. Three-layer dielectric media model for lossy

ground planes.

TE~EðiÞx ¼ �

~Jsk2k2

y ½ejkzznið1 þ k0z=kzÞ2 þ e�jkzznið1 � k02z =k

2z Þ�

2xek2qkz½ejkzdð1 þ k0z=kzÞ

2 � e�jkzdð1 � k0z=kzÞ2�ejkzzi

�~Jsk

2k2y ½e�jkzznið1 � k0z=kzÞ

2 þ ejkzznið1 � k02z =k2z Þ�


2 � e�jkzdð1 � k0z=kzÞ2�e�jkzzi

TE~EðiÞy ¼

~Jsk2kxky½ejkzznið1 þ k0z=kzÞ

2 þ e�jkzznið1 � k02z =k2z Þ�


2 � e�jkzdð1 � k0z=kzÞ2�

ejkzzi

þ~Jsk

2kxky½e�jkzznið1 � k0z=kzÞ2 þ ejkzznið1 � k02z =k

2z Þ�


2 � e�jkzdð1 � k0z=kzÞ2�

e�jkzzi

ð6Þ

where k02z 5 2jxlr 2 k2q and the variables zi and zni

are defined as

i ¼ 1 : z > zn; z1 ¼ d � z; zn1 ¼ zn

i ¼ 2 : z � zn; z2 ¼ z; zn2 ¼ d � znð7Þ

Next, we employ surface impedance boundary

conditions to model the lossy ground planes at z 5 0

and z 5 d.

The expressions for the electric field components

can now be written as follows:

TE~EðiÞx ¼ �

~Jsk2k2

y ½ejkzzniðZeffs þ xl=kzÞ2 þ e�jkzzniðZeff2

s � x2l2=k2z Þ�

2xek2qkz½ejkzdðZeff

s þ xl=kzÞ2 � e�jkzdðZeffs � xl=kzÞ2�

ejkzzi

�~Jsk

2k2y ½e�jkzzniðZeff

s � xl=kzÞ2 þ ejkzzniðZeff2s � x2l2=k2

z Þ�2xek2

qkz½ejkzdðZeffs þ xl=kzÞ2 � e�jkzdðZeff

s � xl=kzÞ2�e�jkzzi

TE~EðiÞy ¼

~Jsk2kxky½ejkzzniðZeff

s þ xl=kzÞ2 þ e�jkzzniðZeff2s � x2l2=k2

z Þ�2xek2

qkz½ejkzdðZeffs þ xl=kzÞ2 � e�jkzdðZeff

s � xl=kzÞ2�ejkzzi

þ~Jsk

2kxky½e�jkzzniðZeffs þ xl=kzÞ2 þ ejkzzniðZeff2

s � x2l2=k2z Þ�

2xek2qkz½ejkzdðZeff

s þ xl=kzÞ2 � e�jkzdðZeffs � xl=kzÞ2�

e�jkzzi

ð8Þ

where zi and zni are defined in eq. (7).

If we compare the general expressions in eq. (6)

with those in eqs. (8), we find that when kq 5 0 and

k02z 5 2jxlr, the expressions for the electric field for

the three-layer and the surface impedance models are

exactly the same provided we let Zeffs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffijxl=r

p.

This is expected, since the special case of kq 5 0 is

associated with the normal incidence plane wave case

and Z0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffijxl=r

pis the wave impedance in the con-

ductive regions. We can also show that eqs. (8) are

Incorporation of Losses into UA-FWLIS 189


good approximations to eq. (6) when |kq / k| is not too

large and r � xe0, i.e., it is a good conductor. In

Figure 2, we plot the relative error for the x-compo-

nent of the TE spectral domain electric field TE~Ex for

the surface impedance formulation, as compared with

the three-layer model formulation, as a function of

kq/k and r/xe, where we assume that the results for

the three-layer model are the exact results. From

Figure 2, we can see that for a good conductor (r �xe0) and for the range of kq that UA-FWLIS encoun-

ters, the use of the surface impedance as a lossy

boundary condition provides a very accurate model

for the lossy ground planes even when we chose the

simplest form for Zeffs . Note that the case |kq| � k is

associated with highly attenuated modes, which do

not contribute appreciably to the reaction elements.

This conclusion is also supported by the study on the

time-domain surface impedance of a homogeneous

lossy half-space [13], which indicated that when the

conductivity of the half-space s is large, the effect of

the different incident angles only appears at very

early-time of the reflected field. For the case of highly

conductive lossy ground planes, this means that only

frequencies that are well beyond hundreds of GHz will

see such effects and subsequently the surface imped-

ance approach would cause errors. In UA-FWLIS, the

value of kq depends on the pole locations and |kq| will

not be too large if the residue series for calculating of

the reaction elements converge quickly. This has been

found to be true for all the inspected cases and the

computational efficiency of UA-FWLIS also largely

depends on this important feature.

Another advantage of the expressions in eqs. (8)

when compared with eq. (6) involves the potential

branch-cut singularities. The branch-cut singularities

are associated with the multivalued square root func-

tions. Because it can be shown that eqs. (8) are even

functions of the variable kz, there are no branch-cut

singularities in these expressions. Therefore, we can

directly apply the techniques that were developed in

[1–6] to obtain closed-form expressions for the reac-

tion elements when the surface impedance formula-

tion is employed. In a similar manner, it can also be

shown that eq. (6) are even functions of kz, so they do

not contain branch-cut singularities associated with

this square root. However, eq. (6) do contain branch-

cut singularities associated with the square root

k0z ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik02 � k2

q

q. When performing a contour integra-

tion analysis of the integrals, one would have to

deform the contour around this branch cut. Fortu-

nately, since k0 would have a large negative imagi-

nary part, this branch-cut contribution would be small

and could be ignored.

Recall that the expression for the reaction element

between two horizontal cells can be written as [14]

Zmn ¼j

4p2xe

Z 1

0

Z p

�p

~Jnsxe��ze¼zen

~Tmsxtð�kq cos bmn;

� kq sin bmnÞ��zt¼ztm

�GTM

��ze¼ztm

cos h cos bmn

þ GTE��ze¼ztm

sin h sinbmn�kqdhdkq; ð9Þ

for the case of PEC ground/power planes. In the case

of lossy ground/power planes, only the terms GTE and

GTM will be affected by the use of the surface imped-

ance boundary conditions. However, since the Gfunctions are not functions of q (the angular integra-

tion variable), the surface impedance formulation

will not have an effect on the analytical calculation

of the angular integral, and the efficient ILHI based

algorithm can still be used. To apply residue theory

to compute the semiinfinite integral over kq, we need

to find all the poles for the function in the reaction

integrand. Unfortunately, unlike the PEC case, the

TE and TM modes have different poles when there

are lossy conductor planes. In addition, these poles

are now the roots of transcendental equations. In con-

trast, the poles in the PEC ground plane case are ana-

lytically known, i.e., kz 5 np/d, n 5 1,2, . . . . We

employ a Newton iteration method for complex func-

tions [15] to find the poles for the lossy plane case.

The roots for the PEC case can be used as the initial

values for the Newton’s iteration method. In all the

cases we investigated, convergence results in three or

four iterations, since we are assuming good conduc-

tors. In fact, since Z0 is a relatively small value in the

case of good conductors, the roots kz, especially those

associated with the TM mode, are almost identical to

Figure 2. Relative error for the electric field from the

three-layer model and the surface impedance approach.

190 Cao et al.


np/d. Therefore, if the ground planes have high con-

ductivity, then the effects of lossy ground planes

should be very small.

IV. SIMULATION RESULTS

To validate UA-FWLIS with the conductor loss

enhancement, we investigate several examples and

compare the S-parameters from UA-FWLIS with

those obtained by the commercial MoM-based simu-

lation tool, Agilent Momentum. Note that the S-pa-

rameters in this section are all calculated using the

characteristic impedance of the transmission line as

the reference impedance at each port.

A. Example 1—Multi-Line Filter

The first example we tested is the multiline, stripline

filter, which is shown in Figure 3.

The substrate is taken to be FR4, which has a rela-

tive permittivity of 4.4 and a loss tangent of 0.0001.

The spacing between the two ground planes is 600

lm, and the conductor traces are centered in the mid-

dle of the substrate. All traces are 30 mm long and

400 lm wide, and they are separated by 30 lm from

each other. The ground planes and traces are taken to

be copper, which has a conductivity of 5.8 3

107 S/m. In Figure 4, the S-parameters are plotted

over a frequency range from 5 to 12 GHz. For com-

parison purposes, the results for PEC lines, and lossy

lines with lossless ground planes, are also shown in

these figures. It can be seen that the results from both

simulators agree very well when losses are included.

Slight increases in the insertion loss and return loss

can be observed when the lossy results are compared

with the PEC line results. It should also be noticed

that the effect associated with lossy ground planes is

quite small compared with the effects caused by the

losses on the interconnect lines, as previously antici-

pated.

B. Example 2—Crossover Lines

The next example that we tested is a stripline cross-

over structure, which is shown in Figure 5.

Figure 4. S-parameter results for the filter. (a) S11 magnitude. (b) S21 magnitude.

Figure 3. Multiline filter structure.

Figure 5. Stripline crossover structure.



The substrate is the same as in Example 1. The

spacing between the two ground planes is 400 lm,

and each line is 22.5 mm long and 120 lm wide. The

ends of the lines without ports were left as open cir-

cuits. Layers 1 and 3 are both 160 lm thick, and the

thickness of the middle layer is 80 lm. The material

for ground planes and the traces is copper with a con-

ductivity of 5.8 3 107 S/m. In Figure 6, we plot S11

and S21 obtained by both agilent momentum and UA-

FWLIS over a frequency range from 4 to 20 GHz.

Once again. we see that the results from both simula-

tors agree very well, and the effect of finite-conduc-

tivity planes is small compared with the effects of the

lossy interconnect lines.

V. CONCLUSIONS

In this article, the capability of handling conductor

losses, from both traces and ground planes, was built

into the full-wave MoM-based stripline simulator,

UA-FWLIS. By introducing an effective surface cur-

rent and using surface impedance boundary condi-

tions, the losses on the conductor traces and ground

planes are easily incorporated into UA-FWLIS by

adding an extra term to the reaction elements and

modifying the electric fields, respectively. These

approaches did not affect the angular integral in the

expression of reaction elements, so the analytical cal-

culation of the reaction matrix elements, which is

based on the use of ILHI and residue theory, was

maintained after including the conductor losses, thus,

also maintaining the computational efficiency of UA-

FWLIS. A comparison between the S-parameter

results from UA-FWLIS with those from Agilent

Momentum validated the accuracy of the enhanced

version of UA-FWLIS. We also observed that the

effects of including finite-conductivity power/ground

planes is relatively small compared with those of the

lossy interconnect lines.

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BIOGRAPHIES

Yi Cao received the B.S.E.E., M.S., and

Ph.D. degree from Shanghai Jiao Tong Uni-

versity, Shanghai, China, in 1994, 1996,

and 1999. From 2004, he is a postdoctoral

researcher at the University of Arizona. His

main research area is the modeling for high-

speed interconnects and packaging, includ-

ing the numerical techniques for solving

field problems and circuit simulation.

Xing Wang received his B.S. degree in

1999 in the Dept. of Electronics in

Tsinghua University, Beijing, China. He

received both M.S. and Ph.D. degrees in

the Dept. of Electrical and Computer

Engineering at The University of Ari-

zona, USA, in 2002 and 2006, respec-

tively. His major is in the interconnect

packaging simulation/design area, espe-

cially in the techniques of full-wave simulation methods. He is

now working in the signal integrity research lab at nVIDIA

Corp., Santa Clara, USA. His interest of research is in packaging

design and simulation, high speed/frequency IC interconnect mod-

eling, simulation, and methodologies.

Zhaohui Zhu (M’06) received the B.S.

degree in electrical engineering from the

University of Science and Technology of

China in 1991, the M.S. degree in signal

processing from the Institute of Electron-

ics, Academia Sinica in 1994, and Ph.D

degree in the Department of Electrical

and Computer Engineering at the Univer-

sity of Arizona, Tucson in 2005. From

1994 to 2000, she was with Telecommunication Department of

China National Clearing Center, where she worked as a system

engineer. She is currently working as a packaging engineer for

Design Process Development in ATD, Intel Corporation. Her

research interests include signal integrity, electromagnetic model-

ing of high-speed circuits, electromagnetic transients, wave propa-

gation, and theoretical and computational electromagnetics.

Steven L. Dvorak received his B.S.

(1984) and Ph.D. (1989) degrees in Elec-

trical Engineering from the University of

Colorado, Boulder. Dr. Dvorak is cur-

rently a Professor in the Department of

Electrical and Computer Engineering at

the University of Arizona, Tucson, Ari-

zona. He served as an Assistant Professor

in this department from 1989 to 1996 and

an Associate Professor from 1996 to 2004. Dvorak previously

held a position with TRW Space and Technology Group from

1984 to 1989. His principal interests include electromagnetic mod-

eling of high-speed interconnects, electromagnetic transients, wave

propagation, theoretical and computational electromagnetics, optics,

geophysical applications of electromagnetics, applied mathematics,

and microwave measurements. Dr. Dvorak is an elected member of

the International Union of Radio Science Commissions B and F,

and a member of the IEEE and Tau Beta Pi. Dr. Dvorak received

the Antennas and Propagation Society S. A. Schelkunoff Prize Pa-

per Award in 1997 and the URSI Young Scientist Award in 1996.

He was also awarded the Department of Electrical and Computer

Engineering IEEE and HKN Outstanding Teaching Award and the

Andersen Consulting Teaching Award in 1994.

John L. Prince (S’65-M’68-SM’78-

F’90) received the BSEE degree from

Southern Methodist University, and as

an NSF Graduate Fellow received the

MSEE and Ph.D. degrees in electrical

engineering from North Carolina State

University. He was a Professor of Elec-

trical and Computer Engineering and

Director of the Center for Electronic



Packaging Research at the University of Arizona. He came to

the University of Arizona in 1983. He was the Principal Investi-

gator of the Semiconductor Research Corporation (SRC) Pro-

gram in VLSI Packaging and Interconnection Research at the

university from 1984 until his death in December of 2005. In

1991–1992, he was Acting Director, Packaging Sciences at

SRC. He had extensive industrial experience. He was active in

consulting work in both the reliability and packaging areas. He

was co-author on two books in the field of electronic packag-

ing, Simultaneous Switching Noise of CMOS Devices and Sys-

tems, by Senthinathan and Prince, and Electronic Packaging:

Design, Materials, Processing and Reliability, by Lau, Wong,

Prince and Nakayama. He taught courses in electronic packag-

ing and the University of Arizona. His research interests cen-

tered on developing modeling and simulation techniques for

switching noise in packages and MCMs, on modeling and simu-

lation techniques for mixed-signal system packaging, and on

the development of high frequency measurements on packaging

structures. He authored or co-authored over 210 papers in the

field of electronic packaging and 35 papers in the fields of

semiconductor device physics, process development, and reli-

ability.

194 Cao et al.


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