electromagnetic analysis for inhomogeneous interconnect and packaging structures based on...

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IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY 1

Electromagnetic Analysis for InhomogeneousInterconnect and Packaging Structures Based on

Volume-Surface Integral EquationsKuo Yang, Wei Tian Sheng, Zhen Ying Zhu, and Mei Song Tong

Abstract— Electromagnetic analysis for interconnect and pack-aging structures usually relies on the solutions of surface integralequations (SIEs) in integral equation solvers. Though the SIEs arenecessary for the conductors in the structures, one has to assumea homogeneity of material for each layer of a substrate if SIEs areused for the substrate. When the inhomogeneity of materials inthe substrate has to be taken into account, then volume integralequations (VIEs) are indispensable. In this paper, we consider theinhomogeneous materials of substrate and replace the SIEs withthe VIEs to form volume-surface integral equations (VSIEs) forthe entire structures. Also, the use of VIEs could alleviate low-frequency effects, remove the need of selecting a basis function formagnetic current, and facilitate geometric discretization in somescenarios. The VSIEs are solved with the method of momentsby using the Rao–Wilton–Glisson basis function to represent thesurface current on the conductors and Schaubert–Wilton–Glissonbasis function to expand the volume current inside the substrate.To avoid the inconvenience of charge density in the traditionalimplementation of the VIEs, we suggest that the dyadic Green’sfunction be kept in its original form without moving the gradientoperator onto the basis and testing functions. Numerical examplesare presented to demonstrate the effectiveness of the approach.

Index Terms— Electromagnetic (EM) analysis, inhomogeneoussubstrate, interconnect and packaging structure, volume-surfaceintegral equation(VSIEs).

I. INTRODUCTION

EFFICIENT and accurate electromagnetic (EM) analysisplays a central role for understanding the electrical per-

formance of interconnect and packaging structures in micro-electronic or nanoelectronic devices [1]. However, performingEM analysis for such structures is a nontrivial task due to theinherent structural characteristics. The structures are usuallyvery small compared with the wavelength within a certainfrequency range, so an extra care is required for numericalaccuracy and double-precision variables should be used inprogramming. Also, the structures include multiscale features,namely, some parts are much smaller than others in dimen-sions [2], [3]. Moreover, the EM analysis usually requiresto cover a wide range of frequency with significant low-frequency components and it will suffer from low-frequency

Manuscript received August 12, 2012; revised November 30, 2012; acceptedJanuary 12, 2013. Recommended for publication by Associate Editor E.-P. Liupon evaluation of reviewers’ comments.

The authors are with the Department of Electronic Science and Tech-nology, Tongji University, Shanghai 200092, China (e-mail: [email protected]; [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TCPMT.2013.2241436

effects [4]. These factors remarkably distinguish the packagingproblems from other EM problems and seriously deterioratethe conditioning of system matrix in the numerical procedure.Consequently, it is essential to choose appropriate governingequations, robust numerical method, and wise implementationscheme so that the best numerical solutions can be achieved.In practice, there were two choices for describing the involvedEM nature of the structures, namely, the differential equationapproaches and integral equation approaches. Although theformer have been widely applied to the EM modeling andsimulation of interconnect and packaging structures [5]–[11],the latter are preferred in many scenarios due to their distinc-tive merits and have also been extensively employed in theEM analysis of those structures [2]–[4], [12]. The integralequation approaches include the surface integral equations(SIEs) for both impenetrable and penetrable materials andthe volume integral equations (VIEs) for penetrable materialsas governing equations, but the SIEs are usually preferredwhenever available because they require a less number ofunknowns in the geometric discretization [13].

However, the SIEs for penetrable materials require ahomogeneity of materials and their scope of applicationscould be confined. The SIEs include the electric fieldintegral equation (EFIE), magnetic field integral equa-tion (MFIE), combined field integral equation (CFIE),Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) for-mulation [14], and Müller formulation [15]. All these equa-tions include both electric current and magnetic current asunknown functions on material interfaces or boundaries andeach has to represent both currents with appropriate basis func-tions in the method of moments (MoM) [16] solution. Thoughthe well-known Rao–Wilton–Glisson (RWG) basis function[17] is a natural choice to represent the electric current, howone should represent the magnetic current is less obvious. Onecould employ the RWG again or n̂ × RWG basis function,where n̂ is a unit normal vector on material interfaces, torepresent the magnetic current, but both choices present someproblems [18]. One could also use the dual basis functionproposed by Chen and Wilton in 1990 to remedy the problem[19], but its implementation is quite sophisticated. The goodrepresentation of magnetic current is very important because itis related to the conditioning of resultant system matrix, whichseriously affects the iterative-method-based fast algorithms,such as multilevel fast multipole algorithm (MLFMA) [20].In the contrast, the VIEs are more generalized and allow

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2 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY

the involvement of inhomogeneous materials. Moreover, theyare more insensitive for low-frequency effects because theybelong to the second kind of integral equations whose sys-tem matrix is diagonally dominant and well-conditioned ingeneral. Also, only one unknown function, i.e., the electricflux density, needs to be represented in general and thewell-known Schaubert–Wilton–Glisson (SWG) basis function[21] is enough to expand it. In addition, using the VIEs mightfacilitate geometric discretization for the structures with manymaterial interfaces because the SIEs need to individually dis-cretize two sides with conforming meshes for each interface.

Owing to the above reason, we use the VIEs to replace theSIEs for the substrate and they are coupled with the SIEs forthe conductors to form the volume-surface integral equations(VSIEs) for describing the EM features of interconnect andpackaging structures in this paper. The VSIEs are usuallysolved with the MoM in which the RWG basis function isused to expand the surface current on the conductors with atriangular tessellation while the SWG basis function is appliedto represent the flux density in the substrate with a tetrahedraldiscretization. The VSIEs have been extensively studied andapplied to solve EM problems [22]–[31], but they have neverbeen used to analyze interconnect and packaging problems,which include multiscale features and low-frequency effects.Since the traditional MoM implementation of the VIEs has toassume a homogeneous material in each tetrahedral elementand also requires to take care of the surface charge densityon the common faces of paired tetrahedrons, we suggestthat the dyadic Green’s function be kept in its original formwithout moving the gradient operator onto the basis and testingfunctions. We can avoid the inconvenience caused by thecharge density in this way though requiring a good treatmentfor the hypersingularity in the dyadic Green’s function. Withour technique of singularity treatment [32], the suggestedscheme is feasible and can show certain merits of implementa-tion. Numerical examples for inhomogeneous interconnect andpackaging structures are presented to illustrate the approachand good results are observed.

II. VOLUME-SURFACE INTEGRAL EQUATIONS (VSIES)

The interconnect and packaging structures include bothconducting signal lines (transmission lines) and ground anddielectric substrates as sketched in Fig. 1. We assume thatthe signal lines and ground are perfectly electric conductors(PECs) and the involved EM features can be described by thefollowing EFIE:

−n̂ ×Eex (r) = n̂ × iωμ0

∫S

G(r, r′) ·JS(r′) d S′, r ∈ S (1)

where JS(r′) is the electric current induced on the conductorsurface S whose unit normal vector is n̂ and Eex (r) representsa delta-gap excitation at an appropriate position on the con-ductors. Also, G(r, r′) is the dyadic Green’s function definedby

G(r, r′) =(

I + ∇∇k2

0

)g(r, r′) (2)

Fig. 1. Typical interconnect and packaging structure. The signal lines(transmission lines) and ground are conductors while the substrates aredielectric.

where I is the identity dyad, g(r, r′) = eik0 R/(4π R)is the (3-D) scalar Green’s function in which R = |r − r′|is the distance between an observation point r and a sourcepoint r′, and k0 is the wavenumber of the free space with apermittivity ε0 and a permeability μ0. For the substrate whichcould consist of inhomogeneous materials with a permittivityε(r′) and a permeability μ(r′), we can use the VIEs to catchup its EM characteristics, i.e.,

E(r) = Eex (r) + iωμ0

∫V

G(r, r′) · JV (r′)dr′

−∇ ×∫

VG(r, r′) · MV (r′)dr′, r ∈ V (3)

H(r) = Hex (r) + iωε0

∫V

G(r, r′) · MV (r′)dr′

+∇ ×∫

VG(r, r′) · JV (r′)dr′, r ∈ V (4)

where Eex (r) = Hex (r) = 0 in the substrate in general, and

JV (r′) = iω[ε0 − ε(r′)]E(r′)MV (r′) = iω[μ0 − μ(r′)]H(r′) (5)

are the induced volumetric electric and magnetic currentsinside the substrate, respectively. If the substrate is nonmag-netic or has the same permeability as the background, whichis usually true, then MV (r′) = 0 and only (3) is needed, whichcan be reduced to

E(r) = iωμ0

∫V

G(r, r′) · JV (r′)dr′, r ∈ V . (6)

When considering the coupling of fields produced by thesurface current on the conductors and the volume currentinside the substrate, we can form the following VSIEs:

0 = n̂ ×[

Eex(r) + iωμ0

∫S

G(r, r′) · JS(r′) d S′

+iωμ0

∫V

G(r, r′) · JV (r′)dr′]

, r ∈ S (7)

E(r) = iωμ0

∫S

G(r, r′) · JS(r′) d S′

+iωμ0

∫V

G(r, r′) · JV (r′)dr′, r ∈ V (8)


YANG et al.: EM ANALYSIS FOR INHOMOGENEOUS INTERCONNECT AND PACKAGING STRUCTURES 3

from which the unknown currents can be solved and equiv-alent circuit parameters of the structure can subsequently beextracted.

III. METHOD OF MOMENTS SOLUTION

The above VSIEs can be solved by the traditional MoM inwhich the surface current on the conductors is expanded bythe RWG basis function while the electric flux density insidethe substrate is represented with the SWG basis function, i.e.,

JS(r′) =Nc∑

n=1

Jnen(r′) (9)

D(r′) =Nd∑

n=1

Dnfn(r′) (10)

where en(r′) is the RWG basis function and Nc is the numberof RWG triangle pairs while fn(r′) is the SWG basis functionand Nd is the number of SWG tetrahedron pairs. Also, Jn andDn are the expansion coefficients for the two basis functions,respectively. The flux density instead of current density ischosen as an unknown function in the substrate because itis normally continuous across material interfaces. The currentdensity is related to the flux density through

J(r′) = −iωκ(r′)D(r′) (11)

whereκ(r′) = [ε(r′) − ε0]/ε(r′) (12)

is the contrast ratio of permittivity. Using the RWG andSWG basis function as a testing function to test (7) and (8),respectively, we can obtain the following matrix equation:

−⟨em(r),

Eex (r)iωμ0

⟩=

Nc∑n=1

Jn〈em(r), G(r, r′), en(r′)〉

+Nd∑

n=1

Dn〈em(r), G0(r, r′), fn(r′)〉, m = 1, 2, . . . , Nc (13)

Dm

⟨fm(r),

fm(r)iωμ0ε(r)

⟩=

Nc∑n=1

Jn〈fm(r), G(r, r′), en(r′)〉

+Nd∑

n=1

Dn〈fm(r), G0(r, r′), fn(r′)〉, m = 1, 2, . . . , Nd (14)

where G0(r, r′) = −iωκ(r′)G(r, r′). The above matrix equa-tion can be solved with any matrix solver in principle but theimplementation of system matrix could be quite different withdifferent strategies.

IV. INCONVENIENCE OF TRADITIONAL IMPLEMENTATION

SCHEME

To generate the matrix elements of system matrix in theabove, it is conventional to move the gradient operator ofdyadic Green’s function to the basis function and testingfunction with a divergence operation so that the degree ofsingularity in the kernel can be reduced to a 1/R weak level,yielding much ease of treatment. However, this practice will

Fig. 2. SWG basis function is defined over a pair of tetrahedrons T ±n with

a common face an . There are surface charges at the common face when thepaired tetrahedrons are dissimilar in material.

produce a surface charge density on the common faces ofpaired tetrahedrons in the VIEs as shown in Fig. 2 and itis inconvenient to take care of it. The integral kernel, namely,the dyadic Green’s function includes two terms correspondingto the vector potential and scalar potential, respectively, andthe VIE can be equivalently written as

E(r) = Eex (r) + iωμ0

∫V

g(r, r′)JV (r′)dr′

− 1

ε0∇

∫V

ρ(r′)g(r, r′)dr′, r ∈ V (15)

where ρ(r′) is the charge density, which is related to thecurrent density by

ρ(r′) = 1

iω∇′ · JV (r′). (16)

If the flux density is represented with the SWG basis functionas in (10), then the charge density can be found by [21]

ρ(r′) = −Nd∑

n=1

Dnκ(r′)∇′·fn(r′)−Nd∑

n=1

Dnfn(r′)·∇′κ(r′). (17)

In the above, we can see that the charge density includes twoterms and the first term represents the induced volume chargedensity inside tetrahedrons while the second term denotesthe induced surface charge density on the common facesseparating dissimilar media. The volume charge density canbe viewed as a different unknown function from the currentdensity and it is equivalently represented with a lower orderbasis function, i.e., the divergence of the SWG basis function.From the definition of the SWG basis function, we know that itis actually a first-order polynomial and its divergence is just azeroth-order polynomial (constant), which may not be accurateenough to represent an unknown function. Also, the surfacecharge density is generated by the gradient of κ(r′), which isa delta-like generalized function and is approximated with thedifference of κ(r′) in the paired tetrahedrons when their mate-rials are dissimilar. If the material of object is continuouslyinhomogeneous, the scheme has to assume a homogeneous



Fig. 3. Evaluation of hypersingular integrals over a tetrahedral elementin the MoM implementation of VIEs. The observation point is inside thetetrahedron and a local coordinate system (u, v, w) is established over oneof the tetrahedral faces by choosing the projection of the observation pointon the face as the origin and the normal vector of the face pointing to theobservation point as the w axis.

material in each tetrahedron which is inaccurate and willproduce a fictitious surface charge density at the common facesof paired tetrahedrons. Overall, the charge density is worserepresented with the basis function compared with the fluxdensity. Moreover, when the SWG basis function is chosenas a testing function, the similar problem exists because wechange the gradient operation on the scalar potential �(r) intoa divergence operation on the testing function to lower thedegree of singularity in the kernel, i.e.,

〈∇�(r), fm(r)〉 =∫

�Sm

�(r)fm(r) · n̂m(r)dr

−∫

�Vm

�(r)∇ · fm(r)dr (18)

where �Sm is the boundary of �Vm in which fm(r) is definedand n̂m(r) is its unit normal vector. Since the scalar potentialis much more important than the vector potential at lowfrequencies, the testing scheme could cause more numericalerrors at low frequencies.

V. ALTERNATIVE IMPLEMENTATION SCHEME

To avoid the problems in the traditional implementationscheme, we suggest that the original form of the dyadicGreen’s function be kept in the VIEs, and the SWG basis func-tion only represents the flux density without being involvedin the charge density. In this way, we can get rid of theinconvenience resulting from the charge density and allowa continuous inhomogeneity of material in each tetrahedralelement. This alternative scheme of implementation may notbe feasible when one cannot handle the hypersingularity inthe dyadic Green’s function. However, the development ofthe robust technique of treating the hypersingularity in recentyears has made it feasible. Although the original form andthe divergence form of the VIEs are strictly equivalent intheory and the transformation between the two forms shouldnot introduce extra errors in formulations, the original formis more friendly in numerical implementation now when the

hypersingular dyadic Green’s function can be accurately eval-uated. In fact, we have transformed the inconvenience of thecharge density in the traditional scheme into the complexityof evaluating the dyadic Green’s function in the alternativescheme. Though the treatment of hypersingularity for thedyadic Green’s function is complicated, the use of its finalformulations is simple, so we can obtain a merit from such atransformation. Note that we can account for the continuousinhomogeneity of material within each tetrahedron in the inte-gration without the need of taking an excessive approximation,i.e., constant permittivity, as done in the traditional scheme.For the evaluation of hypersingular dyadic Green’s function,we have built a special subroutine for it in our code library andusing it has become easy now. The involved related techniquecan be briefly described as follows by referring to Fig. 3.

The dyadic Green’s function has nine components but onlysix are independent due to its symmetry and the SWG basisfunction can be treated as a polynomial function in terms ofits definition. The hypersingularity appears in the kernel thatmultiplies the dyadic Green’s function by the constant termof the polynomial. The combination of the dyadic Green’sfunction with other higher order terms of the polynomial onlyleads to a weaker singularity and many mature techniquescan handle it [32]. We first perform a singularity extractionprocedure by subtracting and adding back the hypersingularterms in the kernel so as to regularize the original kernel andallow the use of numerical quadrature rules to evaluate itsintegral. We then individually handle the added-back termsfor their integrals by deriving the analytical formulationsunder the Cauchy-principal-value (CPV) sense. After applyingthis procedure, we can obtain the following six added-backhypersingular integrals:

I1 =∫

�V

(3u2

R5− 1

R3

)dV ; I2 =

∫�V

uv

R5dV ;

I3 =∫

�V

(3v2

R5 − 1

R3

)dV ; I4 =

∫�V

w0(−u)

R5 dV ;

I5 =∫

�V

w0(−v)

R5dV ; I6 =

∫�V

3w20

R5dV . (19)

The above volume integrals can be decomposed into a lineintegral along the w axis and a surface integral over thetriangle �S as shown in Fig. 3. As an example, we considerthe first integral in the above, i.e.,

I1 =∫ h

0dw

∫�S

(3u2

R5− 1

R3

)d S. (20)

The outer line integral with respect to w is regular whenthe inner surface integral has a closed-form formulation andcan be evaluated by a numerical quadrature rule (Gaussianquadrature rule). The inner surface integral could be regular,near-singular, or singular, depending on the spatial relationbetween the observation point P0 and the contributing triangle�S = �P1 P2 P3, which is dependent on w. However, nomatter how the observation point P0 is close to the contributingtriangle �S, the inner surface integral is always integrableunder the CPV sense and can be derived in a closed-formformulation as shown in [32]. Thus, the solutions of above



(a) (b)

(c)

Fig. 4. Geometries of typical interconnect and packaging structures.(a) One conducting signal line with a bridge at the top of a one-layer dielectricsubstrate. (b) One cross-shaped conducting signal line at the top of a one-layer dielectric substrate. (c) Two straight conducting signal lines at the topof a two-layer dielectric substrate.

0 0.2 0.4 0.6 0.8 1−45

−40

−35

−30

−25

−20

−15

Frequency (GHz)

Magnitude of S Parameters (dB)

S11, SIEs

S11, VSIEs

0 0.2 0.4 0.6 0.8 1−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Frequency (GHz)


S12, SIEs

S12, VSIEs

(a) (b)

Fig. 5. S parameters for the interconnect and packaging structure, includingone signal line with a bridge and a one-layer dielectric substrate. The geometryis defined with l = 10.0, w = 5.0, d = 0.1, s = 0.2, t = 0.05, a = 1.0,b = 0.2, and h = 0.3, all in millimeters, and the substrate has a relativepermittivity εr = 3.0. (a) S11. (b) S12.

hypersingular integrals over a tetrahedron can be obtainedconveniently. Although the alternative scheme requires theabove efficient treatment for the hypersingularity, we canstill benefit from such a change because using the resultantformulations is simple.

VI. NUMERICAL EXAMPLES

To illustrate the proposed approach, we present severalnumerical examples for the EM analysis of typical intercon-nect and packaging structures, which might include inhomo-geneous materials. The geometries of structures are shown inFig. 4 and we consider three cases, i.e., (a) one conductingsignal line with a bridge at the top of a one-layer dielectricsubstrate, (b) one cross-shaped conducting signal line at thetop of a one-layer dielectric substrate, and (c) two straightconducting signal lines at the top of a two-layer dielectricsubstrate. It is assumed that the signal lines and ground aremade up of PECs and the dielectric substrates are also losslessbut could be inhomogeneous. The structures are strictly in 3Dand both the interconnects and ground have a nonnegligiblethickness.

0 0.2 0.4 0.6 0.8 1−55

−50

−45

−40

−35

−30

−25

−20

−15

Frequency (GHz)


S11, FEM

S11, VSIEs

0 0.2 0.4 0.6 0.8 1−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Frequency (GHz)


S12, FEM

S12, VSIEs

(a) (b)

Fig. 6. S parameters for the interconnect and packaging structure, includingone cross-shaped signal line and a one-layer dielectric substrate. The geometryis defined with l = 10.0, w = 5.0, d = 0.1, s = 0.2, t = 0.05, a = 1.0,b = 0.2, and h = 0.3, all in millimeters, and the substrate has a relativepermittivity εr = 3.0. (a) S11. (b) S12.

Fig. 7. S parameters for the interconnect and packaging structure, includingtwo straight signal lines and a two-layer dielectric substrate. The geometry isdefined with l = 10.0, w = 5.0, d = 0.1, s = s1 = s2 = 0.2, t1 = t2 = 0.05,and h1 = h2 = 0.3, all in millimeters. The upper and lower layers of thesubstrate have a relative permittivity εr1 = 3.0 and εr2 = 5.0, respectively.

TABLE I

COMPARISON OF COMPUTATIONAL COSTS BETWEEN VSIE APPROACH

AND FEM FOR THREE INTERCONNECT STRUCTURES

Costs Structure (a) Structure (b) Structure (c)

VSIECPU time (S) 187 196 267

Memory (MB) 552 587 791

FEMCPU time (S) 225 238 372

Memory (MB) 435 468 612

The dimensions of the geometries are characterized asfollows (all in millimeters). In the first case (a), the length,width, and height of the substrate are l = 10.0, w = 5.0,and h = 0.3, respectively, and the thickness of the ground isd = 0.1. The signal line has a width s = 0.2, a thicknesst = 0.05, and a bridge with a length a = 1.0 and a heightb = 0.2. In the second case (b), the profile of the substrateand ground as well as the width and thickness of the signalline are the same as in the first case (a). The extension ofthe signal line on the two sides in the y direction has alength a = 1.0 and a width b = 0.2. In the third case (c),the width and thickness of the two straight signal lines ares1 = s2 = 0.2 and t1 = t2 = 0.05, respectively, and the



Fig. 8. S parameters at low frequencies for the interconnect and packagingstructure, including two straight signal lines and a two-layer dielectricsubstrate. The geometry is defined with l = 10.0, w = 5.0, d = 0.1,s = s1 = s2 = 0.2, t1 = t2 = 0.05, and h1 = h2 = 0.3, all in millimeters. Theupper and lower layers of the substrate have a relative permittivity εr1 = 3.0and εr2 = 5.0, respectively.

0 0.2 0.4 0.6 0.8 1−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)


S11, FEM

S11, VSIEs

0 0.2 0.4 0.6 0.8 1−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Frequency (GHz)


S12, FEM

S12, VSIEs

(a) (b)

Fig. 9. S parameters for the interconnect and packaging structure,including one signal line with a bridge and a one-layer dielectric sub-strate. The geometry is defined with l = 10.0, w = 5.0, d = 0.1,s = 0.2, t = 0.05, a = 1.0, b = 0.2, and h = 0.3, all inmillimeters, and the substrate has an inhomogeneous relative permittivityεr (y) = cos(2.0π |y|/w) + 3.0 in the y direction. (a) S11. (b) S12.

spacing between the two lines is s = 0.2. The profile ofthe substrate and ground is the same as that in the previouscases, but there are two dielectric layers with an equal heighth1 = h2 = 0.3 in the substrate now. The relative permittivityof the substrate is εr = 3.0 in the first and second case whileit is εr1 = 3.0 for the upper layer and εr2 = 5.0 for the lowerlayer in the third case (the relative permeability μr = 1.0is assumed in all cases). All the signal lines are locatedsymmetrically at the top of the substrate in the y directionand the bridge or extension of the signal line is centeredat the signal line in the x direction. The signal propagatingin the lines is excited with a delta-gap source located at thecenter of the side wall of the substrate (in the y-z plane). Wediscretize the surfaces of conductors into triangular patchesand volumes of substrates into tetrahedral elements for thethree structures, resulting in 5278, 5642, and 7165 unknowns,respectively. Figs. [5]–[7] plot the solutions of S parametersfor those structures based on the proposed approach andthey are in good agreement with the corresponding solutionsobtained from the traditional MoM of SIEs. Note that theproposed approach can still give rise to good solutions even

0 0.2 0.4 0.6 0.8 1−55

−50

−45

−40

−35

−30

−25

−20

Frequency (GHz)


S11, FEM

S11, VSIEs

0 0.2 0.4 0.6 0.8 1−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Frequency (GHz)


S12, FEM

S12, VSIEs

(a) (b)

Fig. 10. S parameters for the interconnect and packaging structure with onecross-shaped signal line and a one-layer dielectric substrate. The geometry isdefined with l = 10.0, w = 5.0, d = 0.1, s = 0.2, t = 0.05, a = 1.0, b = 0.2,and h = 0.3, all in millimeters, and the substrate has an inhomogeneousrelative permittivity εr (y) = cos(2.0π |y|/w)+3.0 in the y direction. (a) S11.(b) S12.

Fig. 11. S parameters for the interconnect and packaging structure withtwo straight signal lines and a two-layer dielectric substrate. The geometry isdefined with l = 10.0, w = 5.0, d = 0.1, s = s1 = s2 = 0.2, t1 = t2 = 0.05,and h1 = h2 = 0.3, all in millimeters. The upper and lower layers of thesubstrate have an inhomogeneous relative permittivity defined by εr1(y) =cos(2.0π |y|/w) + 3.0 and εr2(y) = cos(2.0π |y|/w) + 5.0, respectively, inthe y direction.

at the relatively low frequencies between f = 0.01 GHz andf = 0.1 GHz while the traditional MoM of SIEs may notproduce correct solutions as shown in Fig. 8 if the dualbasis is not used [18]. Since the low-frequency problem isapproximately scale-invariant, obtaining the accurate solutionat the frequency roughly f = 0.1 GHz is a key [4].

The SIEs are invalid if the substrate includes inhomoge-neous materials and the VSIEs are indispensable. We con-sider this case to demonstrate the robustness of the proposedapproach now. We assume that the relative permittivity of thedielectric substrate obeys a basin function defined by

εr (y) = cos(2.0π |y|/w) + 3.0 (21)

in the y direction for the first and second structures. In the thirdstructure, the upper layer has the same relative permittivity asthe above, namely, εr1(y) = cos(2.0π |y|/w) + 3.0, but thelower layer possesses a relative permittivity defined by

εr2(y) = cos(2.0π |y|/w) + 5.0. (22)



Figs. [9]–[11] displays the solutions of the correspondingS parameters for those structures with the inhomogeneousmaterials in the substrate and these results are verified witha finite element method (FEM) solver. Usually, it is notnecessary to compare the integral equation solver with thedifferential equation solver like FEM because their advantagesand disadvantages have already been commonly recognized[12]. Nevertheless, we provide a comparison between theseapproaches for their computational costs, which are shown inTable I and it can be seen that the proposed approach requiresless CPU time but more memory usage.

VII. CONCLUSION

EM modeling and simulation are essential for analyzing anddesigning interconnect and packaging structures. The SIEs ofintegral equation approaches were widely employed in formu-lating the solvers due to their less number of unknowns in thediscretization of formulations. Nevertheless, the SIEs requirea homogeneity of materials in the substrate and may notbe convenient for discretizing structures with many materialinterfaces. Furthermore, the SIEs in the substrate require twobasis functions to individually represent the unknown electriccurrent and magnetic current on the material interfaces andhow to represent the magnetic current is less obvious. TheSIEs are also sensitive to the low-frequency effects and theregular remedy strategy by using loop-tree and loop-star basisfunctions may not be suitable here due to the complexity of theinterconnect and packaging structures. In this paper, we devel-oped a VSIE approach by replacing the SIEs with the VIEsin the substrate for the EM analysis of interconnect and pack-aging structures. This replacement allows the inhomogeneityof materials in the substrate and removes the drawbacks ofthe SIEs. The resultant VSIEs were solved with the MoM inwhich the surface current on the conductors was expandedwith the RWG basis function while the flux density insidethe substrate was represented by the SWG basis function. Toavoid the inconvenience caused by the charge density in theVIEs, we proposed an alternative implementation scheme, i.e.,keeping the dyadic Green’s function for the substrate in itsoriginal form without moving the gradient operator onto thebasis function and testing function. Such a change also allowsthe inhomogeneity of materials in each tetrahedral elementin addition to the convenience of implementation thoughaccurate evaluation of hypersingular integrals was required.Numerical examples for the EM analysis of homogeneousor inhomogeneous interconnect and packaging structures havedemonstrated the effectiveness of the proposed approach.

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Kuo Yang received the B.S. degree in electricalengineering from the Binhai College, Nankai Uni-versity, Tianjin, China, in 2011. She is currentlypursuing the M.S. degree with the Department ofElectronic Science and Technology, Tongji Univer-sity, Shanghai, China.

Her current research interests include antennas andpropagation, electromagnetic analysis of intercon-nect structures, and computational electromagnetics.

Wei Tian Sheng is currently pursuing the degreewith the Department of Electronic Science and Tech-nology, Tongji University, Shanghai, China.

His current research interests include modelingand simulation for electromagnetic problems, elec-tromagnetic analysis of interconnect and packagingstructures, and design of microwave devices.

Mr. Sheng was a recipient of the Meritorious Win-ner Award at the Mathematical Contest in Modelingfor U.S. Undergraduate Students in 2012.

Zhen Ying Zhu is currently pursuing the degreewith the Department of Electronic Science and Tech-nology, Tongji University, Shanghai, China.

His current research interests include modelingand simulation for electromagnetic problems, elec-tromagnetic analysis of interconnect and packagingstructures, and antenna designs.

Mr. Zhu was a recipient of the Special-ClassAward at the 7th National Information TechnologyApplication Contest hosted by the Ministry of Edu-cation of China in 2012.

Mei Song Tong received the Ph.D. degree in elec-trical engineering from Arizona State University,Tempe, in 2004.

He is currently a Distinguished Professor and theChair of the Department of Electronic Science andTechnology, School of Electronics and InformationEngineering, Tongji University, Shanghai, China. Heis currently a Visiting Professor with the Universityof Illinois at Urbana-Champaign and an HonoraryProfessor with the University of Hong Kong, HongKong. He was a Research Scientist with the Cen-

ter for Computational Electromagnetics and Electromagnetics Laboratory,Department of Electrical and Computer Engineering, University of Illinois atUrbana-Champaign, Urbana. He has authored or co-authored more than 100papers in refereed journals and conference proceedings, and has co-authoreda book. His current research interests include electromagnetic field theory,antenna theory and design, simulation and design of radio frequency andmicrowave circuits and devices, interconnect and packaging analysis, inverseelectromagnetic scattering for imaging, and computational electromagnetics.

Prof. Tong was a recipient of the Visiting Professor Award from KyotoUniversity, Japan, in 2012. He is a fellow of the Electromagnetics Academy,a Full Member (Commission B) of the U.S. National Committee for the Inter-national Union of Radio Science, and a member of the Applied ComputationalElectromagnetics Society and the Sigma Xi Honor Society. He is an AssociateEditor or a Guest Editor of several well-known international journals, includ-ing the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEETRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING

TECHNOLOGY, the International Journal of Numerical Modeling: ElectronicNetworks, Devices and Fields, Progress in Electromagnetics Research, andthe Journal of Electromagnetic Waves and Applications.

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