4302 IEEE TRANSACTIONS ON MICROWAVE THEORY AND …djiao/publications/SaadImp.pdf · is both a...

4302 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 12, DECEMBER 2013

A New Volume Integral Formulation for Broadband3-D Circuit Extraction in Inhomogeneous MaterialsWith and Without External Electromagnetic Fields

Saad Omar, Student Member, IEEE, and Dan Jiao, Senior Member, IEEE

Abstract—A new first-principles-based volume integral equa-tion (VIE) formulation is developed for the broadband full-waveextraction of general 3-D circuits, containing arbitrarily shapedlossy conductors with inhomogeneous dielectrics. The proposedformulation accentuates all the advantages of the VIE formulationtraditionally developed for solving wave-related problems, whileallowing for the extraction of multiport circuit parameters suchas impedance -, admittance -, and scattering -parameters atports located anywhere in the physical structure of a circuit. Itsfirst-principles-based formulation without circuit-based simpli-fications and approximations can also be utilized to analyze theperformance of a circuit in adverse ambient conditions, such asthe exposure to strong external electromagnetic fields. In addi-tion, the magneto-quasi-static and electro-magneto-quasi-staticcounterparts of the proposed full-wave formulation are also givenfor low-frequency applications. Numerical experiments havevalidated the accuracy and capability of the proposed new VIEformulation.

Index Terms—Broadband analysis, circuit modeling, externalelectromagnetic fields, full-wave analysis, impedance extraction,-parameter extraction, 3-D structures, volume integral equations(VIEs).

I. INTRODUCTION

E LECTRONIC circuits operating in a densely integratedenvironment, military integrated circuits in the battlefield,

microwave, and RF circuits in communication satellites can beexposed to strong external fields. Circuit modeling techniquesignoring such severe, yet practical ambient conditions can po-tentially lead toward catastrophic design failures. The circuitsused for military applications require a high order of sensitivityand accuracy. These circuits cannot sustain those operating en-vironments that were not catered during their design process.Although there have been many developments in circuit extrac-tion techniques, little work has been reported in circuit param-eter extraction with (explicitly defined) external electromag-netic fields taken into consideration. The underlying problem

Manuscript received July 04, 2013; revised September 08, 2013; acceptedSeptember 30, 2013. Date of publication October 25, 2013; date of current ver-sion December 02, 2013. This work was supported by the National ScienceFoundation (NSF) under Grant 0747578, the Semiconductor Research Corpo-ration (SRC) under a grant (Task 1292.073), and the Office of Naval Research(ONR) under Grant N00014-10-1-0482. This paper is an expanded paper fromthe IEEEMTT-S InternationalMicrowave Symposium, Seattle,WA, USA, June2–7, 2013.The authors are with the School of Electrical and Computer Engineering,

Purdue University, West Lafayette, IN 47907 USA.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMTT.2013.2285355

is both a circuit and a scattering problem, thus challenging tosolve. The physical layout of the problem under analysis can beviewed as a “circuit” only to the internal circuit sources. How-ever, the same physical layout behaves as a “scatterer” to ex-ternal electromagnetic fields.The problem of a circuit exposed to external electromagnetic

fields is an open-region problem, which is different from tradi-tional problems considered in circuit design. To model open-re-gion problems, integral-equation (IE)-based solvers are moreefficient than partial-differential-equation-based solvers sincethey avoid the use of any artificial absorbing boundary condi-tion and they handle the open-region radiation condition ana-lytically. There are two major classes of IE-based solvers: sur-face IE-based solvers and volume integral equation (VIE)-basedones. Compared to surface IE-based solvers [1]–[5] for broad-band circuit modeling, VIE-based solvers have a greater capa-bility in handling inhomogeneous dielectric and conductive ma-terials, both of which are common in integrated circuits.The VIE has been used to derive an equivalent circuit model

from physical geometry in the partial-element equivalent-cir-cuit (PEEC) method [6]–[9]. It has also been employed to per-form a quasi-static analysis to extract frequency-dependent re-sistances and inductances of 3-D lossy conductor networks [10],[11]. The PEEC-based VIE formulation has also been combinedwith the equivalent charge method for impedance extraction inmultiple dielectrics [12]. These circuit-orientated VIE methodsfor circuit modeling involve certain treatments and simplifica-tions that are not used in a VIE-based solver developed foranalyzing wave-related problems such as propagation, radia-tion, and scattering problems [13]. In addition, they have nottaken external electromagnetic fields into consideration in cir-cuit model generation.The VIE solver for wave problems [13]–[18] entails no

theoretical approximations or simplifications. It is theoreticallyvalid in a full electromagnetic spectrum. It formulates a VIE forboth fields inside nonperfect conductors and fields outside con-ductors in the inhomogeneous materials that are different fromthe background material. With a tetrahedron-element-baseddiscretization, it offers great flexibility in modeling arbitrarilyshaped conductors and dielectrics. With vector basis functions,it is capable of capturing both conduction and displacementcurrents flowing along an arbitrary direction inside conductornetworks and dielectric materials. Despite the aforementionedadvantages, the VIE solvers developed for wave problems havebeen found not amenable for solving circuit problems. This isbecause for solving wave-related problems, an incident fieldor a delta-gap voltage source [21] is generally used in the VIE

0018-9480 © 2013 IEEE

OMAR AND JIAO: NEW VOLUME INTEGRAL FORMULATION FOR BROADBAND 3-D CIRCUIT EXTRACTION IN INHOMOGENEOUS MATERIALS 4303

solver as the excitation. Although the formulation is convenientfor scattering and radiation analysis, it is difficult to be adoptedfor circuit parameter extraction at ports that can be locatedanywhere in the physical layout of a circuit. For port-parameterextraction, the delta-gap source model is generally used. Thismodel requires two ports to be very close to each other, andalso it introduces a piece of conductor to fill the gap betweenthe two ports. Although there have been many enhancementsto the delta-gap source model such as [22], [23], this sourcemodel cannot be adopted for circuit parameter extraction atports located anywhere in the physical layout of a circuit.The contribution of this work is the development of a

new first-principles-based full-wave VIE formulation for theextraction of general 3-D circuits having arbitrarily shapedlossy conductors and inhomogeneous dielectrics exposed toexternal electromagnetic fields or without external fields. Thisformulation retains the first-principles-based accuracy and allthe inherent flexibility of the VIE formulation developed forsolving wave problems, while allowing for the extraction ofmultiport circuit parameters, such as impedance, admittance,and scattering parameters, at any port of interest, and from lowto any electrodynamic frequency. Moreover, the first-principlesaccuracy of the proposed new formulation permits the analysisof circuit performance in a severe ambient environment, suchas strong external fields, which has rarely been studied before.This formulation can also be straightforwardly modified to anelectro-magneto-quasi-static (EMQS) or magneto-quasi-static(MQS) formulation for circuit parameter extraction at lowfrequencies. Numerical experiments have demonstrated itsaccuracy and added capability. We have presented the basicidea of this work in [19]. In this paper, we complete it fromboth theoretical and numerical perspectives.The remainder of this paper is organized as follows. In

Section II, we formulate the proposed new VIE that is appli-cable to circuit, scattering, and simultaneous circuit-scatteringanalysis. In Section III, the discretization of the proposed VIEformulation is given and the construction of the system matrixis detailed. In Section IV, we demonstrate the accuracy andcapability of the proposed new VIE formulation through a suiteof on-chip and package examples without external electro-magnetic fields or exposed to external fields. The concludingremarks are included in Section V.

II. PROPOSED NEW VIE FORMULATION FOR BROADBANDCIRCUIT MODELING IN INHOMOGENEOUS MATERIALS

WITH AND WITHOUT EXTERNAL FIELDS

Consider a general 3-D circuit with a union of arbitrarilyshaped conductors of finite conductivity , and hence, acomplex permittivity , and permeability .These conductors are embedded in inhomogeneous dielectricsthat can be lossless, lossy, and dispersive. Both conductorsand dielectrics are characterized by space-dependent complexpermittivity with being thedielectric constant. The background material is assumed tobe free space having permittivity . It can also be anothermaterial. The circuit can be simultaneously exposed to internalcircuit sources and external electromagnetic fields, such as anincident plane wave in a broad band of frequencies.

A. Full-Wave Formulation

When a circuit is exposed to an incident field , based onthe volume equivalence principle, the equivalent volume current

radiating in the background material producesthe scattered field. Thus, the total field at any point is equalto the sum of the incident field and the scattered field

(1)

which is expressed in the following form of the VIE [13]:

(2)

where Green’s function ,being the angular frequency, is the free-space wavenumber,is the contrast ratio defined as

(3)

and is

(4)

which is related to the equivalent volume current by. From the third term in (2), it can be seen that the

scattering potential , at any point , is given by

(5)

which can further be written as

(6)

where is the density of equivalent volume charges (volumepolarization charges), and is the density of the equivalentsurface charges (surface polarization charges) at the materialdiscontinuity where is discontinuous. and have arelationship with , which can be derived from (5) as follows:

(7)

in which superscripts and are the indices of the two ma-terials at a material discontinuity, and denotes a unit vectornormal to the material interface and pointing from materialto material . It is worth mentioning that although the diver-gence of is zero in a source-free region, this condition is notexplicitly satisfied in the VIE formulation since this conditionhas been satisfied implicitly by the Ampere’s law used to buildthe VIE formulation. The vector basis functions employed toexpand such as the Schaubert–Wilton–Glisson (SWG) basesalso have a nonzero divergence.In analyzing wave-related problems, the incident field

is chosen either as an incident plane wave or a delta-gap voltagesource. Neither of them is suitable for multiport circuit param-eter extraction where ports can be physically located anywhere


in the layout of an integrated circuit. In contrast, an electric po-tential -based excitation facilitates the extraction of multiportcircuit parameters. We can attach each port in turn to an elec-tric potential, while grounding all the other ports, to obtain thecurrent at each port, from which the circuit parameters such asadmittance ( ), scattering ( ), and impedance ( ) parameterscan be obtained. Surface IE-based methods in [1] and [2] arecapable of accommodating a -based excitation by introducingpotential and charge as additional unknowns. However, as canbe seen from (2), the only unknowns in the traditional VIE for-mulation for solving Maxwell’s equations is , and the poten-tial and charge unknowns are not necessary. In order to incor-porate the electric-potential-based excitation while preservingthe first-principles-based nature of the original wave-based VIEformulation shown in (2), we propose a new VIE formulationas follows. The essential idea of the proposed formulation canalso be applied to the surface IEs to incorporate an electric po-tential-based excitation.When the circuit is attached to a potential source, the total

potential is known at the contact surface where the source isattached. Instead of solving (1), we solve

(8)

subject to

(9)

where denotes a point on the contact surface, where potentialis applied. Equation (8) can further be written as

(10)

where the potential is generated by equivalent volumecharges and equivalent surface charges at the material dis-continuity, which is the same as that shown in (6). However,one has to realize one key difference in order to develop acorrect first-principles-based VIE formulation for modelinga potential-based excitation. To explain, in the case of theincident field -based excitation, the relationship betweencharge densities ( and ) and is known as shown in (7).Therefore, we only need to solve for to obtain the circuitresponse to the incident field. In other words, the only unknowncontained in (1), and hence (2), is . In contrast, in the caseof the potential-based excitation, since the contact surface isnow attached to potential , the equivalent surface chargedensity on the contact surface, denoted by , is unknown,whereas the relationship between and other charge densities,i.e., and not belonging to the contact surfaces, remainsintact as that shown in (7). Therefore, for circuits attached toa potential-based source, we solve for . To be morespecific, the in (10) can be written as

(11)

where denotes the union of all the interfaces at the materialdiscontinuity, while represents the contact surface. As a re-sult, the unknowns contained in (8), and hence (10), are bothand . Although are additional unknowns, in(9) provides the same number of additional equations to com-plete the numerical system. To summarize, when a circuit is at-tached to a potential source, we solve the following two equa-tions simultaneously:

(12)

(13)

As can be seen from the above derivation, in the proposedVIE formulation, we do not introduce potential , volumecharge density , noncontact surface charge density as addi-tional unknowns for circuit parameter extraction. Instead, weonly introduce as an extra unknown, which is necessaryto satisfy the known potential condition at the contact surface.As a result, all the first-principles features of the originalwave-based VIE are preserved for the following reasons.Firstly, the formulation is truly full wave, which is valid ina full electromagnetic spectrum from low to electrodynamicfrequencies. Secondly, the formulation is equally applicable toproblems with a uniform material and problems with compli-cated inhomogeneous materials without any need for change.Thirdly, not only the current inside the nonperfect conductors ismodeled, but also the displacement current outside conductorsis captured in the inhomogeneous materials different from thebackground material. Lastly, the formulation can be used tocapture currents flowing along an arbitrary direction both insideconductor networks and outside conductors in the inhomoge-neous materials. Moreover, the proposed new VIE formulationfacilitates the circuit network parameter extraction at portslocated anywhere in the physical layout of the circuit. This isbecause the potential source can be attached to any physicalpoint without any difficulty in numerical simulation. Owingto its first-principles-based accuracy without circuit-basedsimplifications and approximations, in conjunction with (2),the proposed formulation also permits the analysis of a circuitexposed to both internal circuit sources and external electro-magnetic fields, the problem that is both a scattering and acircuit problem.


B. MQS and EMQS Formulation

The full-wave VIE formulation developed in Section II-A canalso be modified to perform analyses in MQS and EMQS set-tings for applications with relatively small electric sizes. Whenthe entire structure being simulated is electrically small, thestatic and/or quasi-static physics can be utilized to simplify theanalysis and generate a better-conditioned numerical system. Inan EMQS analysis, we reduce the Green’s function in the pro-posed full-wave formulation to its static form. In a MQS anal-ysis, since the displacement current is ignored and , allthe charges become zero. As a result, in (6) is zero and thereis no relationship shown in (11) anymore. In this case,for the incident-field-based excitation, we set to be zero in(1), and hence (2), to perform a MQS analysis. For the poten-tial-based excitation, we solve the following equations:

(14)

(15)

(16)

(17)

where denotes the static counterpart of the Green’s func-tion , denotes a point on the noncontact surface, the thirdequation ensures the volume polarization charge density tobe zero, while the fourth equation sets the surface polarizationcharges on the noncontact surfaces to be zero. Since nowbecomes just conduction current as the displacement currentis ignored, the computational domain that includes both dielec-tric and conductor regions is also reduced to conductor regionsonly in a MQS analysis. Notice that in the above equation,and are solved instead of and solved in the full-wavecase. This is because the charges are zero in the MQS analysis,the relationship between and charges does not exist. Hence,in the VIE, also becomes unknowns to be solved, except forthe on the contact surfaces (points), but the condition of zerovolume and surface charges yields the same number of equa-tions as the number of unknowns, thus completing the nu-merical system. In addition to EMQS and MQS analyses, theproposed full-wave formulations can also be straightforwardlyreduced to static formulations.

C. Circuit Parameter Extraction

For multiport circuit parameter extraction where ports can belocated anywhere in a circuit, we attach each port in turn to thegiven potential by setting at the port, whileattaching the rest of the ports to . Owing to theflexibility of the proposed formulation, can be set at alocation as small as a point. Therefore, one can either letbe satisfied at one point , a few points, or the whole area ofthe port depending on the port area that is actually contacted bythe potential source. With the potential-based excitation, issolved from the proposed VIE formulations. The current at anyport of the circuit can then be computed as follows:

(18)

where represents the cross-sectional area of the conductorat the port, and is a unit vector normal to . can beas small as a single triangular patch. From the known potentialand the current computed at each port, we can obtain any kind ofnetwork parameters such as admittance ( )-, impedance ( )-,and scattering ( )-parameters.

III. SYSTEM MATRIX CONSTRUCTION

We discretize the computational domain into tetrahedralelements. In each tetrahedral element, the unknown electricdisplacement is expanded into SWG basis functions

[13], the coefficient of which is denoted by . Each ofthe SWG basis function is defined for a face of a tetrahedron.The basis function associated with the th face common totetrahedra and is defined as follows:

otherwise

A. Full-Wave Formulation

1) External Field Excitation: For the external-field-basedexcitation, we solve (2). By expanding the unknown in eachelement by the SWG basis, and also testing (2) using the samebasis, we obtain

(19)

Rewriting the above in a matrix equation format, we obtain

(20)

where the th entry of vector is

(21)

and the -matrix element at the th row and the th column,, is the term shown in the square bracket in (19).


It is worth noting that the parenthesized third term in ,which itself has four components, denotes the potential ob-served at either an outermost boundary surface triangular patch(obtained by integrating over ) or an internal patch (obtainedby integrating over ) due to contributions from both volumepolarization charges and surface polarization charges .Another important fact is that the material discontinuity alsooccurs at the outermost boundary of the structure.2) Potential-Based Excitation: When the circuit is attached

to an electric potential, the new unknown is expanded by thepulse basis functions, i.e., being a constant on each triangularpatch. The Galerkin method is applied to test (12), while thecentroid collocation method is applied to test (13). The resultingsystem of linear equations can be written as

(22)

where matrices and can be viewed as the segregated ver-sions of in (20). The contributions from the charges on thecontact surface, , appear as entries of , while all remainingcontributions are carried by . Mathematically,

(23)

and

(24)

The contribution from all contact-surface patches is contained inthe right-hand-side vector since the potential on the contactsurface is known according to (9). Thus, we have

(25)

The second row of (22) is nothing but the discretized equiv-alent of (9) with computed from (11). Its right-hand side is

simply the potential on the contact-surface patches. Thus, ’sentries are

(26)

The potential generated by the charges on the contact surface isshown by the third term in (11), from which in (22) can beobtained as

(27)

The potential contributed by all other charges is given by thefirst and second term in (11), from which we obtain

(28)

The numerical system (22) can be used for full-wave circuitparameter extraction with a circuit source only, while the inclu-sion of (20) permits the full-wave analysis of a circuit exposedto external fields, interferences, radiations, etc. It is worth men-tioning that for the latter problem, we cannot simply set the

in the right-hand side of (22) to be the incident field forentries not belonging to the contact surface, while lettingbe (25) for contact-surface entries. This is because for the inci-dent-field-based excitation, the surface charge density at anymaterial discontinuity is related to , as shown in (7). However,the same relationship does not hold true for the potential-basedexcitation at the contact surface. Therefore, instead of (6), (11)is used in (10) for solving the VIE for the potential-based excita-tion. In other words, the VIE equations to be solved are differentfor the two different excitations. Hence, we need to superposethe solutions obtained from (20) and (22) to analyze a circuit ex-posed to both internal circuit sources and external electromag-netic fields instead of keeping the left-hand-side matrix of (22)the same while using both excitations in the right-hand side.

B. Quasi-Static Analysis

For the VIE (14)–(17) derived for an MQS analysis, we solvefor . Since the on the contact surface is known, we onlyneed to solve for , where and are the po-tentials on the centroid of each tetrahedral element, and at thecenter of each noncontact surface patch, respectively. The nu-merical system for (14)–(17) is again obtained by expandingin each tetrahedral element by using SWG vector bases and

then using Galerkin testing. and are expanded by pulsebasis functions, and the centroid collocation method is used totest (16), whereas (17) is applied at the center of each noncon-tact surface patch. Therefore, we obtain the following numericalsystem:

(29)


TABLE IEXTRACTED IMPEDANCE PARAMETERS AT FULL-WAVE FREQUENCY POINTS

where

(30)

(31)

and is the same as that shown in (25). The matrix is asparse matrix of dimension , where and are thenumber of tetrahedrons, and the total number of SWG bases, re-spectively. is nonzero only for those four bases that con-stitute the th tetrahedron and given by

(32)

The matrix is also a sparse matrix. Its dimension is ,where is the number of noncontact surface patches. The

is 1 only when the patch corresponding to the th basisis a noncontact surface patch, i.e.,

(33)

in which denotes a noncontact surface.

IV. NUMERICAL RESULTS

In order to validate the proposed VIE formulations, we ex-tract the impedance -parameters of a straight conductor wireand those of a 1 5 on-chip bus structure. We also extract the-parameters of a package interconnect provided by a semicon-ductor company with and without inhomogeneous materials in abroad band of frequencies. In addition, we simulate circuits ex-posed to both external fields and internal circuit sources in orderto demonstrate the capability of the proposed VIE solver in an-alyzing such problems. An irregularly shaped spiral inductor isalso simulated to show the flexibility of the proposed VIE for-mulation in geometrical modeling.

A. Impedance Extraction of a Straight Conductor Wire

A conductor wire of dimension 1 mm 1mm 4mm is sim-ulated in free space from 1 Hz to 1 GHz. The conductivity ofthe wire is 5.8 10 S m. In order to capture the skin effect,the structure is discretized into 1200 tetrahedrons. We attach thenear end of the wire to , and the far end to . In other

Fig. 1. Comparison of the impedance parameter of a straight wire conductorextracted from the proposed VIE solver and that from FastHenry. (a) Resistance( ). (b) Inductance (nH).

Fig. 2. Geometrical details of a 1 5 bus structure.

Fig. 3. Cross-sectional view of a 3-D package interconnect.


Fig. 4. -parameters of a 3-D package interconnect with a uniform material simulated by the proposed full-wave VIE solver in comparison with reference data.(a) (dB). (b) phase (degrees). (c) (dB). (d) phase (degrees).

words, we let (9), and hence (13), be satisfied at the points lo-cated at the near end of the wire with , and let (13)satisfied at the far-end points with . After the current issolved from the proposed VIE formulation, the input impedanceof the wire can be obtained by dividing the potential differencebetween the two wire ends by the current computed at the nearend. The resistance and inductance can then be readily identi-fied from the input impedance. The proposed VIE formulationin its MQS form is employed to carry out the simulation. Theresults are compared with the reference data generated by theMQS-based FastHenry [10] in Fig. 1 from 1 Hz to 1 GHz. Ex-cellent agreement in impedance parameter can be observed, val-idating the proposed MQS-based VIE formulation.The proposed full-wave VIE formulation is also used to sim-

ulate this example. It shows that at the last two frequency points(0.1 and 1 GHz), full-wave effects are actually pronounced,which cannot be captured by an MQS analysis. Hence, we com-pare the results generated from the proposed full-wave formu-lation with those obtained from a full-wave solver in [4]. Asshown in Table I, the inductance value is the same, while thereis a small difference in resistance, which can be attributed to

Fig. 5. Cross-sectional view of a multiple-dielectric package interconnect.

the volume-based discretization used in the proposed solver incontrast to the discretization used in the surface IE solver [4]. Acomparison between the full-wave results shown in Table I andthe MQS results depicted in Fig. 1 also reveals the inaccuracyof the MQS approximations at 0.1 and 1 GHz.

B. Impedance Extraction of a 1 5 On-Chip Bus

The second example is a 1 5 bus structure having typicalon-chip dimensions in a single material (air). As illustrated inFig. 2, each bus has a dimension of 2 m 2 m 20 m, while


Fig. 6. -parameters of a 3-D package interconnect with nonuniform materials simulated by the proposed full-wave VIE solver in comparison with referencedata. (a) . (b) . (c) . (d) .

the spacing between two adjacent buses is 5 m. The metal con-ductivity is again 5.8 10 S m, and the frequency for extrac-tion is 10 GHz. The discretization results in 6000 tetrahedrons.The impedance parameter matrix extracted from the proposedfull-wave VIE formulation is given in the matrix shown at thebottom of this page. All impedances are in 10 . The errorof the above impedance matrix is shown to be 0.29%, computedfrom ,where is obtained fromFas-tHenry [10] with 12 000 filaments, is from the proposed full-wave VIE solver, and Frobenius norm is used. Since the full-wave effect is not dominant in this on-chip example due to itssmall electric size even at 10 GHz, we found that both MQS andfull-wave solvers can produce accurate results for this example.

C. -Parameter Extraction of a Package Interconnect

A 3-D package interconnect provided by IBM, the cross sec-tion of which is shown in Fig. 3, is extracted in a broad band of

frequencies from 1 to 30 GHz. The interconnect has one metalplane on the top and one at the bottom, each of which is of width0.88 mm. A center strip is of width 0.025 mm and its distanceto the left boundary is 0.33 mm. The length of the structure is1 cm. The background material is air. The conductivity of themetal is 5.8 10 S m.In Fig. 4, we plot the -parameters extracted from the pro-

posed full-wave VIE solver in comparison with the referencedata provided by IBM in the entire simulated frequency band.Excellent agreement is observed in both magnitude and phasefor all -parameters.

D. -Parameter Extraction of a Package Interconnect WithMultiple Dielectrics

A 3-D package interconnect with multiple dielectrics, whosedetailed geometrical and material data are shown in Fig. 5, isextracted in a broad band of frequencies from 1 to 30 GHz. The


Fig. 7. Currents of a 3-D package interconnect in nonuniform materials with and without external electromagnetic fields. (a) (A). (b) Phase ( ) (degrees).(c) (A). (d) Phase ( ) (degrees).

length of the structure is 1 cm. The conductivity of the metal is5.8 10 S m.In Fig. 6, we plot the -parameters extracted from the pro-

posed VIE solver in comparison with the reference data pro-vided by the Intel Corporation. Once again, excellent agreementis observed in the entire frequency band for all -parameters.

E. Simulation of Multiple-Dielectric Package InterconnectExposed to External Fields

Next, we study the circuit performance in the presence of anexternal electromagnetic field. The package interconnect simu-lated in Section IV-D is illuminated by an incident plane wave

, as shown in Fig. 5, and meanwhile the cir-cuit is attached to circuit sources. In Fig. 7, we plot the currentsat each port obtained without an incident field, with an incidentfield, and with a strong incident field, when port 1 is attachedto 1 V, while port 2 is grounded. Port 1 is located at the nearend of the conductor in the middle layer, while port 2 is at thefar end. It is evident that when a circuit is exposed to a strongexternal field, its performance can be significantly altered andthe circuit can even fail. The currents obtained at each port withone port attached to 1 V and other ports grounded are definedas the -parameters of a circuit. When there are external elec-tromagnetic fields, from Fig. 7, it can be seen that the circuit-parameters can be significantly changed not only in values,

but also in frequency dependence. is chosen to be compa-

Fig. 8. 3-D view of an irregularly shaped inductor.

rable to the electric field induced by the circuit source, whichis 1 V m for the “with an incident field” case, and 10 V mfor the “with a strong incident field” case, respectively. Since,in this example, the polarization direction of the aligns wellwith the -field generated by the circuit source, there is a sig-nificant impact of the external field on the circuit response whenthe external field is strong. The current received at port 2 due tothe circuit excitation at port 1 can become almost zero at rela-tively high frequencies since the current induced by the externalelectromagnetic fields counteracts that generated by the circuitsource.

F. Simulation of an Irregularly Shaped Spiral Inductor

A two-layer irregularly shaped spiral inductor in free space,as illustrated in Fig. 8, is simulated to demonstrate the flexi-


Fig. 9. -parameters of an irregularly shaped inductor. (a) and magnitude. (b) and phase.

bility of the proposed VIE formulation in geometrical modeling.The outer radius of the inductor is 100 m. The thickness of themetallic wire, the via height , and the port length are all2 m. The conductivity of the metal is 5.8 10 S m. Port 1is located from to at the lower layer, wheredenotes the aziumthal angle. Port 2 is located fromto in the upper layer. The structure is discretized into348 tetrahedrons. The -parameters of the inductor from 1 to10 GHz are simulated by the proposed full-wave VIE formula-tion. The results are shown in Fig. 9.

V. CONCLUSIONS

A new first-principles-based VIE formulation has been de-veloped for the broadband full-wave extraction of general 3-Dcircuits, containing arbitrarily shaped lossy conductors with in-homogeneous dielectrics. In addition to performing broadbandcircuit parameter extraction, the proposed VIE formulation alsopermits the analysis of circuits in the presence of both internalcircuit sources and external electromagnetic fields. For such a si-multaneous scattering-circuit analysis, as analyzed in this work,the equations that the equivalent currents and charges have tosatisfy when the circuit is attached to a circuit source are dif-ferent from those equations they have to satisfy when the cir-cuit is exposed to an external electromagnetic field. Therefore,one cannot keep the system matrix the same while using bothexcitations (circuit sources and external electromagnetic fields)in the right-hand side.The proposed formulation accentuates all the inherent advan-

tages of the VIE formulation traditionally developed for solvingwave-related problems, while facilitating circuit parameter ex-traction such as impedance and scattering parameters at portslocated anywhere in the physical structure of a circuit. Besidesthe VIEs, the method proposed in this paper for incorporatingan electric potential-based excitation can also be applied to thesurface IEs. In addition to the general setting where ports canbe arbitrarily located and be far from each other, the proposedpotential-based source model is equally applicable to the settingwhere the delta-gap source model is valid for use. In the lattercase, the proposed potential-based source model helps removethe inaccuracy of the delta-gap source model due to the finitegap width and the artificially introduced conductor for filling

the gap. Last, but not least, because of its conformity with thewave-based VIE formulation, the proposed formulation also fa-cilitates the acceleration of its direct solution based on [20] forlarge-scale computation, which will be explored in the future.

ACKNOWLEDGMENT

The authors would like to thank Dr. J. Morsey, IBM, andDr. S. Chakravarty, Intel Corporation, for providing intercon-nect structures and reference data for validation.

REFERENCES[1] Z. Zhu, B. Song, and J. K. White, “Algorithms in FastImp: A fast and

wideband impedance extraction program for complicated 3-D geome-tries,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol.24, no. 7, pp. 981–998, Jul. 2005.

[2] B. Song, Z. Zhu, J. Rockway, and J.White, “A new surface integral for-mulation for wideband impedance extraction of 3-D structures,” in Int.Comput.-Aided Design Conf., San Jose, CA, USA, 2003, pp. 843–847.

[3] Y. Wang, D. Gope, V. Jandhyala, and C. J. R. Shi, “Generalized Kirch-hoff’s current and voltage law formulation for coupled circuit-electro-magnetic simulation with surface integral equations,” IEEE Trans. Mi-crow. Theory Techn., vol. 52, no. 7, pp. 1673–1682, Jul. 2004.

[4] W. Chai and D. Jiao, “Direct matrix solution of linear complexityfor surface integral-equation based impedance extraction of highbandwidth interconnects,” in 48th ACM/EDAC/IEEE Design Automat.Conf., San Diego, CA, USA, 2011, pp. 206–211.

[5] W. Chai and D. Jiao, “Direct matrix solution of linear complexity forsurface integral-equation based impedance extraction of complicated3-D structures,” Proc. IEEE (Special Issue), vol. 101, no. 2, pp.372–388, Feb. 2013.

[6] A. E. Ruehli, “Equivalent circuit models for three-dimensionalmulticonductor systems,” IEEE Trans. Microw. Theory Techn., vol.MTT-22, no. 3, pp. 216–221, Mar. 1974.

[7] A. Rong and A. C. Cangellaris, “Generalized PEEC models for threedimensional interconnect structures and integrated passives of arbitraryshapes,” inProc. Elect. Perform. Electron. Packag. Conf., Boston,MA,USA, 2001, pp. 225–228.

[8] A. E. Ruehli, G. Antonini, J. Esch, J. Ekman, A. Mayo, and A. Or-landi, “Nonorthogonal PEEC formulation for time-and frequency-do-main EM and circuit modeling,” IEEE Trans. Electromagn. Compat.,vol. 45, no. 5, pp. 167–176, May 2003.

[9] P. J. Restle, A. E. Ruehli, S. G. Walker, and G. Papadopoulos, “Full-wave PEEC time-domain method for the modeling of on-chip intercon-nects,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol.20, no. 7, pp. 877–887, Jul. 2001.

[10] M. Kamon, M. J. Tsuk, and J. K. White, “FastHenry,” MIT, Cam-bridge, MA, USA. [Online]. Available: http://www.rle.mit.edu/cpg/re-search_codes.htm

[11] M. Kamon, M. J. Tsuk, and J. K. White, “FASTHENRY: A multipole-accelerated 3-D inductance extraction program,” IEEE Trans. Microw.Theory Techn., vol. 42, no. 9, pp. 1750–1758, Sep. 1994.


[12] Y. Yi, P. Li, V. Sarin, and W. Shi, “A preconditioned hierarchical al-gorithm for impedance extraction of three-dimensional structures withmultiple dielectrics,” IEEE Trans. Comput.-Aided Design Integr. Cir-cuits Syst., vol. 27, no. 11, pp. 1918–1927, Nov. 2008.

[13] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral mod-eling method for electromagnetic scattering by arbitrarily shaped inho-mogeneous dielectric bodies,” IEEE Trans. Antennas Propag., vol. 32,no. 1, pp. 77–85, Jan. 1984.

[14] M. M. Botha, “Solving the volume integral equations of electromag-netic scattering,” J. Comput. Phys., vol. 218, no. 1, pp. 141–158, 2006.

[15] L. E. Sun and W. C. Chew, “A novel formulation of the volume inte-gral equation for general large electromagnetic scattering problems,”in IEEE Int. Antennas Propag. Symp., San Diego, CA, USA, 2008, 4pp.

[16] C.-C. Lu, P. Yla-Oijala, M. Taskinen, and J. Sarvas, “Comparison oftwo volume integral equation formulations for solving electromagneticscattering by inhomogeneous dielectric objects,” in IEEE Int. AntennasPropag. Symp., Charleston, SC, USA, 2009, 4 pp.

[17] C. Pelletti, G. Bianconi, R. Mittra, and A. Monorchio, “Volume inte-gral equation analysis of thin dielectric sheet using sinusoidal macro-basis functions,” IEEE Antennas Wireless Propag. Lett., vol. 12, pp.441–444, Jul. 2009.

[18] M. I. Sancer, K. Sertel, J. L. Volakis, and P. Van Alstine, “On volumeintegral equations,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp.1488–1495, May 2006.

[19] S. Omar and D. Jiao, “A new volume integral equation formulationfor analyzing 3-D circuits in inhomogeneous dielectrics exposed to ex-ternal fields,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA,USA, 2013, 3 pp.

[20] S. Omar and D. Jiao, “An -matrix based fast direct volume integralequation solver for electrodynamic analysis,” in Int. Annu. Rev. Progr.Appl. Comput. Electromagn., Columbus, OH, USA, 2012, 6 pp.

[21] R. F. Harrington, Field Computation by Moment Methods. Piscat-away, NJ, USA: IEEE Press, 1993.

[22] G. P. Junker, A. A. Kishk, and A. W. Glisson, “A novel delta gapsource model for center fed cylindrical dipoles,” IEEE Trans. AntennasPropag., vol. 43, no. 5, pp. 537–540, May 1995.

[23] Y. H. Lo, S. He, L. Jiang, and W. C. Chew, “Finite-width gap exci-tation and impedance models,” in IEEE Int. Antennas Propag. Symp.,Spokane, WA, USA, 2011, pp. 1297–1230.

Saad Omar (S’13) received the B.S.E.E degree(with highest distinction) from the University ofEngineering and Technology, Lahore, Pakistan, in2009, the Masters degree in electrical and computerengineering from Purdue University, West Lafayette,IN, USA, in 2011, and is currently working towardthe Ph.D. degree in electrical and computer engi-neering at Purdue University.Since 2009, he has been a Research Assistant with

the On-Chip Electromagnetics Laboratory, PurdueUniversity, West Lafayette, IN, USA. His current

research interests include computational and applied electromagnetics, directintegral equation solvers, inverse scattering problems, fast and high-capacity

numerical methods, high-performance very large scale integration (VLSI)computer-aided design (CAD) tools, high-frequency VLSI circuit design andanalysis, microwave and millimeter-wave circuits, and bio-electromagnetics.Dr. Omar is an active member of the IEEE Microwave Theory and Tech-

niques Society (IEEEMTT-S), IEEEAntennas and Propagation Society (AP-S),and Golden Key International Honour Society. He was the recipient of Pak-istan’s most prestigious Presidential Award, 15 Gold Medals, and the NationalTalent Scholarship for his record-breaking academic performances both in pre-engineering and engineering schools.

Dan Jiao (S’00–M’02–SM’06) received the Ph.D.degree in electrical engineering from the Universityof Illinois at Urbana-Champaign, Urbana, IL, USA,in 2001.From 2001 to 2005, she was with the Technology

Computer-Aided Design (CAD) Division, Intel Cor-poration, as a Senior CAD Engineer, Staff Engineer,and Senior Staff Engineer. In September 2005, shejoined Purdue University, West Lafayette, IN, as anAssistant Professor with the School of Electrical andComputer Engineering, where she is currently a Pro-

fessor. She has authored two book chapters and over 180 papers in refereedjournals and international conferences. Her current research interests includecomputational electromagnetics, high-frequency digital, analog, mixed-signal,and RF integrated circuit (IC) design and analysis, high-performance very large-scale integration (VLSI) computer-aided design (CAD), modeling of microscaleand nanoscale circuits, applied electromagnetics, fast and high-capacity numer-ical methods, fast time-domain analysis, scattering and antenna analysis, RF,mi-crowave, and millimeter-wave circuits, wireless communication, and bio-elec-tromagnetics.Dr. Jiao has been a reviewer for many IEEE journals and conferences.

She is an associate editor for the IEEE TRANSATCTIONS ON COMPONENTS,PACKAGING, AND MANUFACTURING TECHNOLOGY. She was the recipient ofthe 2013 S. A. Schelkunoff Prize Paper Award of the IEEE Antennas andPropagation Society, which recognizes the Best Paper published in the IEEETRANSACTIONS ON ANTENNAS AND PROPAGATION from the previous year.Since 2013, she has been a University Faculty Scholar of Purdue University.She was among the 85 engineers selected throughout the nation for the NationalAcademy of Engineering’s 2011 U.S. Frontiers of Engineering Symposium.She was the recipient of the 2010 Ruth and Joel Spira Outstanding TeachingAward, the 2008 National Science Foundation (NSF) CAREER Award, the2006 Jack and Cathie Kozik Faculty Start Up Award (which recognizes anoutstanding new faculty member of the School of Electrical and ComputerEngineering, Purdue University), a 2006 Office of Naval Research (ONR)Award under the Young Investigator Program, the 2004 Best Paper Awardpresented at Intel Corporation’s annual corporate-wide technology conference(Design and Test Technology Conference) for her work on a generic broadbandmodel of high-speed circuits, the 2003 Intel Corporation Logic TechnologyDevelopment (LTD) Divisional Achievement Award, the Intel CorporationTechnology CAD Divisional Achievement Award, the 2002 Intel CorporationComponents Research the Intel Hero Award (Intel-wide she was the tenthrecipient), the Intel Corporation LTD Team Quality Award, and the 2000 RajMittra Outstanding Research Award presented by the University of Illinois atUrbana-Champaign.

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