Electric Field Integral Equation Combined with CylindricalConduction Mode Basis Functions for Electrical Modeling

of Three-dimensional Interconnects

Ki Jin Han, Madhavan Swaminathan, and Ege EnginSchool of Electrical and Computer Engineering, Georgia Institute of Technology

266 Ferst Drive, Atlanta, Georgia 30332{kjhan, madhavan, engin}@ece.gatech.edu

ABSTRACTFor the modeling of interconnect in three-dimensional pack-agings, this paper proposes a method based on the electricfield integral equation (EFIE) with cylindrical conductionmode basis functions (CMBF). The bases are defined todescribe arbitrary skin and proximity effects, and partialimpedances are obtained from the formulation of the EFIEwith CMBFs. Examples of several 3-D interconnects verifythat the proposed method is efficient in speed and memory.

Categories and Subject DescriptorsI.6 [Computing Methodologies]: Simulation and Model-ing

General TermsAlgorithms, Design

Keywords3-D integration, conduction mode basis function, electricfield integral equation, proximity effect, skin effect, system-in-package

1. INTRODUCTIONA major trend of current electronic packaging is integrat-

ing the multifunction in a three-dimensional (3-D) structuresuch as system-in-package (SiP). As the density of the 3-Dintegrations is higher, the electrical modeling of intercon-nects becomes more difficult due to the increasing numberof interconnects. In fact, the number of bond wires in typ-ical 3-D integrations is over one thousand, which also ac-companies stronger electrical coupling between neighboringinterconnects. With the need for a wide-band model to re-alize various functions in SiP, constructing the full couplingmodel of a large number of interconnects for a wide band isa main design requirement for advanced packaging designs.

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.DAC 2008, June 813, 2008, Anaheim, California, USA.Copyright 2008 ACM ACM 978-1-60558-115-6/08/0006 ...$5.00.

For the purpose of finding a large coupling model in 3-D integrations, modeling methods should manage the issueof accuracy, speed, and memory more deliberately. One ef-ficient modeling method proposed by Daniel et al. [1] isto use global basis functions on a conductor cross section tosolve the electric field integral equation (EFIE). A few globalconduction mode basis functions (CMBF) capture skin andproximity effects, so the CMBF-based method reduces thesize of the partial impedance matrix considerably comparedto the classical partial element equivalent circuit (PEEC)method [7]. However, the original CMBF-based method,which was for the modeling of micro-strip type intercon-nects on a printed circuit board (PCB), is not geometricallysuitable for the modeling of wire bonds or other 3-D struc-tures with circular cross sections. A more important issue inusing CMBFs on Cartesian coordinates is the required addi-tional procedure of constructing basis functions to describearbitrary current crowding.

For interconnect modeling in various 3-D integrations, weintroduced a new type of CMBFs constructed on cylindricalcoordinates [3]. A main advantage of using the cylindricalCMBFs is its fitness to interconnects with the circular crosssection. Building upon the previous research, this paperproposes using a set of two orthogonal higher-order cylin-drical CNBFs to describe arbitrary current crowding. Thispaper also discusses the EFIE formulation with cylindricalCMBFs, the calculation of a partial impedance matrix, andthe construction of an equivalent circuit, which were lackingin the previous paper.

2. CYLINDRICAL CMBFSA basic idea of the CMBF-based method is to skip the pro-

cedure of generating conductor filaments and to reduce thesize of the partial impedance matrix by using a small num-ber of global basis functions that describe the current densitydistribution on a conductor cross section. The global basesare constructed from the diffusion equation of the currentdensity, as in the case of CMBFs in Cartesian coordinates[1]. With assumptions that the media is a good conductorand that axial current density is dominant, we can obtainthe following equation in cylindrical coordinates.

1

[Jz

] +1

22Jz2

+ 2Jz = 0, (1)

where 2 = j = (

1+j

)2, = 2f is angular fre-

quency (rad/sec), = 4 107 is the permeability, is

421

24.4

the conductivity, and = 1/f is the skin depth. From

the solutions of the above diffusion equation, the followingform of the basis functions is obtained [3].

cos(n( 0))Jn() n = 0, 1, 2, . . . , (2)where Jn() is the n

th order Bessel function, asymptoticbehavior of which is exponential function of . For the useof (2) as basis functions, the order n and the orientation 0are defined as follows.One way of classifying cylindrical CMBFs is according

to their orders. If n is zero, the behavior of the cylindricalCMBF is the same as that of the current distribution due tothe skin effect, which is exponential in the radial variationand uniform in the angular variation. If n is higher thanzero, the current distribution has a sinusoidal variation inthe angular variation that can be used for the description ofcurrent crowding due to the proximity effect. Therefore, wecan define the 0th and the higher (n > 0) order bases as theskin-effect mode and the proximity-effect mode, respectively.The other way of classification is necessary for describ-

ing the orientation of current crowding. The main conceptcomes from the fact that the current density distribution isalways a periodic function of , which can be expressed withfollowing Fourier series expansion.

Jz(, ) = A0J0(k)+n=1

(An cos(n)+Bn sin(n))Jn(k).

(3)For the nth(n > 0) order, we can define two parts relatedto cos(n) and sin(n) as d-(direct) and q-(quadratic)modes, respectively. Since these orthogonal bases can cap-ture current crowding in any orientation, any procedure todetermine proximity bases, depending on the geometric con-figuration of conductors, is not necessary.In summary, the following set of bases can sufficiently de-

scribe the entire current density distribution over the crosssection of the ith conductor in the global coordinates.Skin-effect mode (n = 0):

wi0 =

{ziAi0

J0((r ri) i) r Vi0 elsewhere

(4)

Proximity-effect, direct mode (n > 0):

wind =

{zi

AinJn((r ri) i) cos(ni) r Vi

0 elsewhere(5)

Proximity-effect, quadratic mode (n > 0):

winq =

{zi

AinJn((r ri) i) sin(ni) r Vi

0 elsewhere, (6)

where r = xx+ yy+ zz is a point in the ith conductor, ri =xix+yimy+zimz is the center point of the i

th conductor, andAin is the effective area. Ain is a constant that normalizesthe basis function so that the integration of the function overthe cross section equals one. Figure 1 shows normalized plotsof the skin-effect and proximity-effect basis functions on thecross section diameter of 30 m at 1 GHz.

3. FORMULATION OF EFIE WITH CYLIN-DRICAL CMBFS

With the defined cylindrical CMBFs, this section outlinesthe basic procedure of formulating a voltage equation from

-1-0.5

00.5

1

-1

-0.5

0

0.5

1-1

0

1

2

3

-1-0.5

00.5

1

-1

-0.5

0

0.5

1-1

0

1

2

3

-1-0.5

00.5

1

-1

-0.5

0

0.5

1-1

0

1

2

3

Figure 1: Skin-effect (left) , the first- (middle) andthe second- (right) order proximity-effect CMBFs.Real parts (upper) are biased for separation withimaginary parts (lower).

EFIE and discusses the calculation of the partial impedancesand the construction of the equivalent network.

Both the classical PEEC method [7] and the CMBF-basedmethod [1] are based on the following EFIE.

J(r, )

+ j

V

G(r, r) J(r, )dV = (r, ), (7)

where G(r, r) = 1/|r rj |. In this paper, the retardationterm is neglected as in the (Lp, R) PEEC method. In gen-eral, the global basis functions are defined over the entire do-main, but CMBFs are defined over a conductor cross sectionas shown in (4), (5) and (6). That is, CMBFs are localizedon a conductor cross section. Thus, we can define the fol-lowing approximate current density of a conductor segmentj.

Jj(r, ) =n,q

Ijnq wjnq(r, ). (8)

By inserting (8) into (7) and applying inner product basedon Galerkins method we obtain the voltage equation con-taining the following partial resistance, partial inductance,and modal voltage difference.

n,q

IjnqRp,imd,jnq + jn,q

IjnqLp,imd,jnq = Vimd. (9)

Combining similar equations, interactions between all modesin two conductors i and j are expressed by the followingsubmatrix equation

(Rij + jLij)Ij = Vi. (10)

Finally, the submatrix equations between two conductor seg-ments are combined with the global matrix equation thatmodels the coupling in the entire conductor system. Sincetwo or three higher-order modes are found to be sufficientto describe the accurate current density distribution in prac-tice, the required memory of the proposed method is muchless than that of the classical PEEC method.

Since the calculation of partial resistances and inductancesin (9) are involved with multiple integrals of integrands thatare not simple low-order functions, required numerical in-tegrals can be a computational burden. Reduction of thecalculation time is achieved by finding analytic expressionsand using several techniques. Firstly, integrals related tothe partial resistances have their indefinite integrals. In ad-dition, the global matrix of partial resistances is diagonalbecause of the local and orthogonal property of the cylindri-cal CMBFs. Partial self inductances are obtained from the

422

Conductor #1 Conductor #2

imqjnpqnj

jnqjnqimpimpL ILjV,,

,,Loops from proximity-effect modes

10,LVdLV 11,dLV 12, dLV 22,dLV 21,20,LV qLV 22,qLV 21,qLV 11,qLV 12,

10,10R

10,10L

ddR 11,11

ddL 11,11

ddR 12,12

ddL 12,12

ddR 22,22

ddL 22,22

ddR 21,21

ddL 21,21

20,20R

20,20L

qqR 22,22

qqL 22,22

qqR 21,21

qqL 21,21

qqR 11,11

qqL 11,11

qqR 12,12

qqL 12,12

Figure 2: Equivalent circuit example of two conduc-tor segments. Two proximity-effect basis functionsare included for each conductor.

numerical integration of the following formula.

Lp,imd,inq =

8

,

20

Jm(

)Aim

Jn()

AinIIzdd

d,

(11)where I() and Iz(D) are indefinite integrals over an-gular and axial variables, respectively. For the integrationover angular variables, a coordinate transform is useful forreducing computational cost and avoiding singularity duringnumerical integrations.In calculation of partial mutual inductances, the orien-

tation between two conductor segments should be defined.With the defined orientation, we can obtain indefinite inte-grals for axial variables. The analytical expression is simi-lar to that of the two thin conductor filaments [6], but therevised expression is more general since it contains local co-ordinate variables that express inside of the two conductorsegments. Since the analytic integral and other harmonicfunctions are independent of the frequency, pre-computationof these parts prior to a frequency sweep simulation can re-duce the total sweep time. For integrals over the other fourvariables, several quadrature techniques can accelerate thecomputation.With the calculation of partial resistances and inductances,

the modal voltage difference in (9) should be considered togenerate the global impedance matrix equation. In case thebasis function is in the skin effect mode, the modal poten-tial difference is the actual voltage difference between thetwo nodes. On the other hand, if the bases are in the prox-imity effect modes, the modal potential difference becomeszero since the higher-order bases contain harmonic functions.Therefore, the global impedance matrix equation can be ex-pressed as follows.(

Zss ZspZps Zpp

) (IsIp

)=

(Vi0

), (12)

where Zss, Zsp, Zps, and Zpp are partial impedances groupedby skin- and proximity-effect modes, Is and Ip are skin- andproximity-effect currents, and Vi is the voltage difference ofa conductor segment. From the viewpoint of circuit topol-ogy, the equivalent circuit generated from the proximity ef-fect basis function forms a closed loop that is inductivelycoupled with the other circuits. An equivalent circuit exam-ple of the two conductor segments is shown in Figure 2.

4. SIMULATION EXAMPLESThis section validates the established modeling procedure

in the previous sections by describing a simple interconnect

Table 1: Comparison of two simulation methodswhen the rotation angle is zero degrees.

FASTHENRY CMBF method# bases/cross section 152 5CPU time 42.25 hrs 20.67 mins

example (three parallel cylinders) and two close-to-real ex-amples (pin array and wire bonds).

Figure 3 shows the configuration of three parallel cylin-ders, where conductor # 1 is rotating with respect to con-ductor # 2. Both ends of conductor # 3 is grounded, andother two conductors are excited differentially. By testingseveral rotation angles, we can validate how two orthogonalhigher-order bases capture current crowding regardless oftheir absolute orientations. In the plots of loop resistancesand inductances for the rotation angles of 0 and 120 degrees(Figure 3), inductances in low frequencies are identical tothe loop inductances calculated with the analytic method[2]. On the other hand, inductances in high frequencies arelower than those in the analytic AC values, and the valuewith the unbalanced ground (0 degrees) is higher than thatwith the balanced ground conductor (120 degrees). Dueto the higher proximity of the ground conductor, high fre-quency resistances with the balanced ground conductor arehigher than those with the unbalanced ground. Lower leftpart in Figure 3 shows current density distributions on con-ductor cross sections at 100 MHz for the rotation angles of 0,60, and 120 degrees. Although the absolute orientations ofdirect and quadratic modes are fixed for every case, currentcrowding due to the proximity effect is accurately captured.Table 1 compares the proposed method with FASTHENRY[4]. For computing impedances of three conductors over 31frequency points on the environment of Intel Pentium 4 CPU

m40D

.deg1200

m30d

# 2

# 1# 3

mm5L

104

106

108

10101.5

2

2.5

loop

indu

ctan

ce (

nH)

frequency (Hz)10

410

610

810

100

1

2

3

4

5

loop

res

ista

nce

(ohm

)

frequency (Hz)

0 deg.120 deg.

-1 0 1 2 3 4 5 6 7 8 9

x 10-5

-2

-1

0

1

2

3

4

5

6x 10

-5

# 3(grounded)

# 2

120 deg. 60 deg.

0 deg.

Figure 3: The example of the three parallel conduc-tor segments. Upper: Configuration. Lower left:Loop resistances and inductances for different ro-tation angles (0 and 120 deg.). Lower right: TheCurrent density distribution for different rotationangles (0, 60, and 120 deg.).

423

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

104

106

108

1010

2

4

6

8

10

12

14

16x 10

-3

mut

ual l

oop

indu

ctan

ce (

nH)

frequency (Hz)

Group 1

Group 2

Group 3

Group 4

104

106

108

1010

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

self

loop

indu

ctan

ce (

nH)

frequency (Hz)

L1,1L2,2L5,5L6,6L7,7L10,10L11,11L12,12

Group 1 Group 2 Group 3 Group 4

X

Y

10 11

5 67

1 2

12

Ground

1 Testing ports

m100

m30

m60

Figure 4: Pin array example. Upper left: Configu-ration. Upper right: Inductances of pin conductors.Lower: Groups according to difference spacings.

2.60 GHz and 1.00 GB RAM, the proposed method requiresmuch less time than the FASTHENRY.Array of straight cylindrical conductor segments is struc-

ture applicable to various interconnects such as connectorpins, through-hole vias (THV), and solder bumps. An ex-ample structure in Figure 4 contains identical 5 5 con-ductors with the same patch of 60 m. Two conductorsare grounded, and eight ports indicated in Figure 4 are ob-served. Upper left part of Figure 4 shows self and mutualinductances of pin conductors. Self inductances are all iden-tical in low frequencies, but the values in high frequenciesdepend on the location of surrounding conductors. As theproximity effect is more significant, the self inductances be-comes lower. Mutual inductances are grouped into four casesaccording to their physical spacing, which is the only factordetermining mutual inductances at low frequencies. At highfrequencies, the proximity effect has the influence on lower-ing mutual inductances.One of the most popular interconnect structures in 3-D

integration is wire bonds. Since wire bonds have rather com-plicated geometry, the proper modeling of the 3-D geometryis important. In this subsection, we used an approach thatdivides a wire bond into a few straight line segments, asin [5], and then applied the CMBF-based method for everyconductor segment. Figure 5 shows an example structure ofthree wire bonds. Impedances in Figure 5 are obtained bygrounding the third conductor. Because of the asymmetryin grounding, the self inductances of the two conductors aredifferent in high frequencies. As in the previous examples,a conductor suffering the more significant proximity effect(conductor #2 in Figure 5) has higher resistance and lowerinductance.

5. CONCLUSIONSFor the modeling of the interconnect in 3-D integrations,

104

106

108

1010

0

1

2

3

4

5

6

indu

ctan

ce (

nH)

frequency (Hz)

Three bond wires (with wire #3 grounded)

L11

, R11

L22

, R22

L21

, R21

104

106

108

10100

0.5

1

1.5

2

2.5

3

resi

stan

ce (

ohm

)

frequency (Hz)

-1

-0.5

0

0.5

1

x 10-3

-10

12

34

5

x 10-3

0

0.5

1

x 10-3

~ 4 mm

200 m

300

m

wire: diameter 25 m, Au

#1

#3

#2

Figure 5: Configuration of three wire bonds and re-sistances and inductances of wires.

this paper proposed a method that solves EFIE with cylin-drical CMBFs. In defining basis functions, two orthogonalbases for proximity-effect modes were found to describe ar-bitrary current crowding. EFIE formulations and the proce-dure for constructing matrix equation were discussed in de-tail. From its application to several interconnect examples,the proposed method was validated as efficient in speed andmemory. For the purpose of modeling real structures suchas a thousand wire bonds in SiP, accelerating the proposedmethod remains as future work.

6. ACKNOWLEDGMENTThis work was supported by the Mixed Signal Design

Tools Consortium (MSDT) of the Packaging Research Cen-ter, Georgia Tech, under project number 2126Q0R.

7. REFERENCES[1] L. Daniel, J. White, and A. Sangiovanni-Vincentelli.

Interconnect Electromagnetic Modeling usingConduction Modes as Global Basis Functions. In Proc.Topical Meeting Electr. Performance Electron. Packag.,pages 8489. IEEE, Oct. 2000.

[2] F. W. Grover. Inductance Calculations - WorkingFormulas and Tables. Dover Publications, Mineola,New York, 1946.

[3] K. J. Han, E. Engin, and M. Swaminathan. CylindricalConduction Mode Basis Functions for Modeling ofInductive Couplings in System-in-Package (SiP). InProc. Topical Meeting Electr. Performance Electron.Packag., pages 361 364. IEEE, Oct. 2007.

[4] M. Kamon, M. J. Tsuk, and J. K. White.FASTHENRY: A Multipole-Accelerated 3-DInductance Extraction Program. IEEE Trans. Microw.Theory Tech., 42(9):17501758, Sep. 1994.

[5] H. Patterson. Analysis of Ground Bond Wire ArraysFor RFICs. In Proc. Radio Frequency IntegratedCircuits Symp., pages 237240. IEEE, Jun. 1997.

[6] S. Ray, A. K. Gogoi, and R. K. Mallik. Closed FormExpression for Mutual Partial Inductance between TwoSkewed Linear Current Segments. In Proc. Antennasand Propagation Society Int. Symp., pages 12781281.IEEE, Jul. 1999.

[7] A. E. Ruehli. Equivalent Circuit Models forThree-Dimensional Multiconductor Systems. IEEETrans. Microw. Theory Tech., 22(3):216221, Mar.1974.

424