544 IEEE TRANSACTIONS ON ELECTROMAGNETIC...

544 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 46, NO. 4, NOVEMBER 2004

Construction of Broadband Passive MacromodelsFrom Frequency Data for Simulation of Distributed

Interconnect NetworksSung-Hwan Min, Student Member, IEEE, and Madhavan Swaminathan, Senior Member, IEEE

Abstract—This paper discusses a method for the constructionof multiport broadband passive macromodels using frequencydata obtained from an electromagnetic simulation or measure-ments. This data could represent the frequency response of adistributed interconnect system. The macromodels are generatedusing rational functions by solving an eigenvalue problem. Fornumerical computation, the macromodels are represented as asummation of rational functions consisting of low-pass, band-pass,high-pass, and all-pass filters. The stability and passivity of themacromodels are enforced through constraints on the poles andresidues of rational functions. To enable the construction of broad-band macromodels, methods based on band division, selector,subband reordering, subband dilation, and pole replacement havebeen used. Two test cases that describe the performance of theproposed algorithm, and three test cases that are representative ofdistributed systems have been analyzed to verify the efficiency ofthe method.

Index Terms—Broadband multiport passive macromodels, cir-cuit simulation, distributed interconnect networks, frequency de-pendent data, macromodeling, passivity, rational functions.

I. INTRODUCTION

FOR SYSTEMS operating at high frequencies, the dis-tributed behavior of interconnects becomes very important.This parasitic behavior of the structure can be extracted usingan electromagnetic tool or from measurements. The electro-magnetic tool discretizes Maxwell’s equations and computesthe frequency response of the structure. Similarly, a vector net-work analyzer can be used to measure the frequency responseof the structure. In either case, the response of the device isavailable as frequency dependent data that represent scattering,admittance, or impedance parameters of the structure. Thisinformation can be represented as a black box, which capturesthe behavior of the device at the input–output ports. This blackbox representation is called a macromodel. This paper focuseson the construction of broadband passive macromodels usingrational functions that can be embedded into a circuit simulatorsuch as simulation program with integrated circuit emphasis(SPICE). The macromodels can be combined with a larger

Manuscript received December 23, 2002; revised August 12, 2003. Thiswork was supported in part by the Defense Advanced Research Projects Agency(DARPA) as part of the NEOCAD program under Contract S-101329-B and inpart by Government Prime under Contract N66001-01-C-8042.

The authors are with the Department of Electrical and Computer Engineering,Packaging Research Center, Georgia Institute of Technology, Atlanta, GA30332-0250 USA (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TEMC.2004.837683

circuit by synthesizing SPICE net lists for both time-domainand frequency-domain simulation.

For a linear time-invariant passive circuit, the macromodelsconstructed using rational functions have to satisfy the stabilityand passivity conditions [1]. The stability condition requires thatthe poles of the macromodel lie on the left half of the s plane.The passivity condition represents the inability of a passive cir-cuit to generate energy. If the passivity condition is not satisfiedat all frequency points, a stable macromodel combined with astable circuit can generate an unstable time-domain response[1], [3], [4]. Hence, a passive circuit can behave as an ampli-fier or an oscillator during time-domain simulation. Though thefrequency response of a linear circuit may be band limited, thepassivity condition needs to be enforced over infinite frequency.This is because the macromodels satisfying passivity at discretefrequency samples may violate passivity at other frequenciesboth within and outside the frequency bands of interest. Thisis especially true for distributed networks containing multipleresonances in the frequency response.

In [2], a method has been discussed for developing stablemacromodels based on the orthogonal Chebyshev polynomialexpansion for improving the accuracy of the macromodels,but this method is not guaranteed to be passive. In [3], aneigenvalue approach has been discussed for enforcing pas-sivity of the circuit by compensating the poles and residues ofnonpassive macromodels using linearization and constrainedminimization through quadratic programming. In [6], a methodhas been discussed that computes the frequency response of themacromodels, searches the frequency band of violation, andcompensates nonpassive macromodels by inserting additionalrational functions. Both methods in [3] and [6] use discrete,band-limited frequency samples for enforcing passivity. Hence,the generated macromodels can still violate passivity overcontinuous frequency and outside the band-limited frequencyresponse. This can create problems since excitation of unstablepoles outside the frequency band of interest can result in a non-passive transient response. In this paper, simplified formulaefor enforcing passivity of multiport circuits have been analyti-cally derived from the pole-residue form of rational functionsconsisting of low-pass, band-pass, high-pass, and all-passfilters using the maximum modulus theorem [14]. Based on thepassivity formulae, this paper directly compensates the polesand residues of nonpassive macromodels. Since the compensa-tion mechanism is frequency independent, the passivity of themacromodels is satisfied over infinite frequency.

0018-9375/04$20.00 © 2004 IEEE

MIN AND SWAMINATHAN: CONSTRUCTION OF BROADBAND PASSIVE MACROMODELS FROM FREQUENCY DATA 545

Over the last decade, various macromodeling methodshave been developed for capturing the distributed behavior ofmulti-GHz networks by using the Pade approximation [4]–[6],least squares approximation [7]–[9], orthogonal polynomials[2], [10], and vector fitting [11]. However, since the methodsin [2]–[11] use a single matrix to approximate the frequencyresponse of distributed networks having many resonances, theycould have an ill-conditioned matrix problem and often requirelarge computational memory and long computational time.Therefore, the macromodels developed in [2]–[11] were limitedto low-order systems, and hence, may not be relevant for realisticdistributed networks that often contain hundreds of poles. In thispaper, new methods have been proposed for the construction ofbroadband passive macromodels for multiport distributed net-works using the property that a passive system can be representedas a summation of passive subsystems. These methods are: 1)the band division method, which divides the computationaldomain into smaller subcomputational domains or subbands;2) the selector method, which selects the filters; 3) the subbandreordering method, which determines the construction sequenceof the subbands; 4) the subband dilation method, which dilatesthe bandwidth of each subband; and 5) the pole replacementmethod, which increases the accuracy of macromodels over thesubbands and the entire frequency band. Since submacromodelsapproximate the frequency response of low-order systems, andthe band division method alleviates the ill-conditioned matrixproblem. Using the remaining methods, broadband macro-models are constructed by combining macromodels from eachsubband and updating the results on a global scale.

To enable circuit simulation, broadband passive macromodelshave been synthesized into SPICE net lists using resistors, in-ductors, capacitors, and controlled sources [12]. These methodshave been realized in the Broadband Efficient MacromodelingProgram (BEMP) developed at Georgia Institute of Technology,Atlanta, GA, which uses C/C++ and is executable on a Win-dows operating system. BEMP uses admittance parameters togenerate the macromodels and SPICE net lists.

The methods discussed in this paper have been applied to twotest cases, namely: 1) the frequency response derived from aknown transfer function and 2) frequency response of a one-porttransmission line. These test cases provide information on theapplication of the methods described in this paper. In addition,three test cases that are representative of distributed networkshave been analyzed, namely: 3) four-port lossy transmissionline; 4) four-port transmission line data from a vector networkanalyzer measurement; and 5) four-port power plane pair withdecoupling capacitors.

This paper is organized as follows. Section II discussesthe pole-residue form of rational functions representing alinear time-invariant passive network. Section III discussesanalytical formulae of multiport passive circuits based on thepole-residue form of rational functions for enforcing passivity.In Section IV, a method for the construction of macromodelsusing least squares approximation is discussed. Section Vpresents band division, subband reordering, subband dila-tion, and pole replacement methods for the construction ofbroadband macromodels containing a large number of poles.Section VI presents the synthesis of macromodels followed by

five examples for demonstrating the accuracy and validity ofthe proposed methods in Section VII.

II. RATIONAL FUNCTIONS

Using the rational function, the frequency response ofany linear time-invariant passive network can be represented as

(1)

where , is the angular frequency in radians per second,and and are unknown real numerator and denominatorcoefficients, respectively. In (1), can be scattering, admit-tance, or impedance parameters generated from an electromag-netic simulation or measurements. The goal of solving (1) is torepresent the frequency response using a rational functionby computing the orders of the numerator and denominator,and , respectively, and the real vectors and consistingof the numerator and denominator coefficients, respectively

Using filter theory, in (1) can also be represented as asummation of filters in the form

(2)

where , , , and are low- pass,band-pass, high-pass, and all-pass filters, respectively. The fil-ters shown in (2) represent a complete set for representing anytransfer function. The rational function in (1) can be rewrittenin a pole-residue form as

(3)

where and are the poles and residues of the rational func-tion, respectively, and is the order of the rational function.Combining (2) and (3) results in

(4)

where , , , , , , , , , and are real andthe poles are simple or nonoverlapping poles. By combining thecomplex-conjugate poles, (4) can be rewritten as

(5)


where the superscripts , , and are the numberof low-pass, band-pass, and high-pass filters, respectively. In(2), the low-pass filter consists of a real pole withresidue ; the band-pass filter consists of complex-conjugate poles with residues ; the high-passfilter consists of a real pole with residue ; and theall-pass filter consists of residues and .

III. STABILITY AND PASSIVITY

Macromodels constructed using rational functions need tosatisfy the stability and passivity conditions for a linear time-in-variant passive system. The stability condition requires that fora stable system, the output response be bounded for a boundedinput excitation [15]. Hence, the rational function representinga stable system has to satisfy the following stability constraints:1) the poles lie on the left half of the s plane, implying that

, , and in the pole-residue formof the rational function; 2) the rational function does not con-tain multiple poles along the imaginary axis of the s plane; and3) the difference between the numerator and denominator or-ders of the rational function does not exceed unity, implyingthat . Among the above constraints, the secondand third constraints are enforced by construction while the firstconstraint is satisfied using a search procedure in this paper.

The passivity condition requires that a passive circuit does notcreate energy. Since nonpassive macromodels combined with astable circuit can generate an unstable time-domain response,this condition becomes important when the macromodels needto be combined with a larger circuit for time-domain simulation.Unlike the stability condition, it is more difficult to satisfy thepassivity condition during the construction of the macromodels[1], [3], [6], [13]. The passivity conditions for a multiport net-work are twofold, namely, 1)

for all , where is the complex-conjugate operator and2) is a positive real matrix, i.e., the product

, for all with and any arbitrary vector .When the rational function matrix is used, these con-ditions are translated into the following passivity constraints:1) does not contain poles on the right half of the s-plane;2) does not have multiple poles on the imaginary axis ofthe s plane; 3) the coefficients of are all real; and 4) thereal part of must be positive semidefinite for all frequen-cies, implying that the eigenvalues of are positiveor zero for all frequencies. The last constraint is derived fromthe maximum modulus theorem [14]. Among the above con-straints, the first and second constraints are included as part ofthe stability condition and the third constraint is satisfied by con-struction while the fourth constraint is enforced analytically inthis paper.

Based on the maximum modulus theorem [14], the passivitycondition for a one-port network can be written as

(6)

It is important to note that in (6) (and not )simplifies the derivation of the analytical formulae for satisfying

passivity of the macromodels. The basic idea behind the con-struction of passive macromodels for a passive system is thatthe summation of passive subnetworks is passive [7]. The ra-tional function in (5) can be regarded as a summation ofpassive subnetworks consisting of complex-conjugate poles andreal poles with corresponding residues, , and . If every subnet-work in (5) satisfies the passivity condition, the rational function

satisfies the passivity condition as well.Substituting into (5), the rational function

can be separated into the real and imaginary parts as

(7)

By regrouping terms, the real and imaginaryparts of the rational function are shown in (8) and

(9), respectively

(8)

(9)

The passivity of each subnetwork in (5) can be satisfied usingthe following formulas [7]:

(10)

In (10), the analytically derived passivity formulae can be ob-tained from (8) by satisfying (6). Equation (5) can be gener-alized for a distributed multiport network containing commonpoles as

(11)


where the residue matrices , , , , , are amatrix for a P-port network. In (11), is a rational

function matrix, which has to be positive semi-definite at allfrequencies. Using the property of positive semi-definiteness,the passivity formulae in (10) for a multiport network can berewritten as

(12)

The following properties of multiport passivity formulae in (12)are apparent during the construction of the passive macromodel.These properties have been used for the construction of broad-band passive macromodels in later sections.

1) The multiport passivity formulae only depend on the polesand residue matrices, which are independent of frequency.Hence, the passivity of the circuit is satisfied over infinitefrequency.

2) The multiport passivity formulae are only enforced oneach subnetwork of , and there is no relationshipfor passivity between subnetworks except that they con-tribute to the overall response of the macromodel. Thismakes the method simple to use.

3) For compensating negative eigenvalues in (12), there aretwo free matrix variables and related to two freevariables of complex-conjugate poles , a freematrix variable related to a real pole , a free ma-trix variable related to a real pole , and a free ma-trix variable . These can be suitably changed.

4) There are no constraints enforced on the residue matrix.

The nonpassive macromodel that does not satisfy the pas-sivity condition needs to be appropriately modified so that themacromodel becomes passive. For enforcing passivity of themacromodel in this paper, fixed common poles and symmetricresidue matrices are assumed during compensation. If negativeeigenvalues are obtained in the residue matrices , , in(12), negative eigenvalues are set equal to zero or changed toa small positive value and then a new residue matrix is recon-structed. If the passivity formulae for complex-conjugate poleswith two residue matrix variables and are violated, thenegative eigenvalues of are set equal to zero or changed to asmall positive value and then a new residue matrix is recon-structed. Based on the compensated matrix , the residue ma-trix is iteratively found for satisfying the passivity formulaein (12). A small positive value is used to ensure that the macro-model does not violate the passivity condition even though thismay cause small numerical errors in the solution.

Though the summation of passive subnetworks results in apassive network, the enforcement of passivity on subnetworksrepresents a sufficient but not necessary condition. Since thesummation of nonpassive subnetworks may result in a passivenetwork, a concern is that the passivity constraints discussedin this paper may not apply for all frequency data. For a net-work containing only complex-conjugate poles, the passivity

constraints enforced on subnetworks can be easily satisfied dueto the form of (5). The problem arises when the network con-tains real poles. As an example, consider the transfer function

(13)

which is stable and passive but contains a nonpassive subnet-work. Using (5), the above transfer function can be rewritten inthe form

(14)

where the subnetworks are passive and the resulting transferfunction is the same as (13). The passivity constraints can nowbe enforced on each subnetwork. In certain rare cases, when thefrequency response cannot be represented as a summation ofpassive subnetworks as in (5), enforcement of (12) may lead tosome loss of accuracy. In addition, it is important to note that fora network containing overlapping real poles, constraints (12) ap-plied to subnetworks are not always valid.

IV. LEAST SQUARES APPROXIMATION

As mentioned earlier, (1) needs to be solved to compute theorders, and , and the real coefficient vectors, and .Equation (1) can be rewritten in the form

(15)

For a given from either measured or simulated data, whichrepresents the frequency response of a one-port network, (15)can be written as a matrix equation in the form

(16)

where the matrix is given by

......

The vectors and in (16) are real coefficient vectors of thenumerator and denominator, respectively. After premultiplying(16) with the transpose of , (16) becomes

(17)

which can be written as an eigenvalue equation [8], [13]

(18)


Fig. 1. Minimum eigenvalue versus the order NS = DS � 1.

where is the minimum eigenvalue of the matrix ,where is the transpose operator. From (18), the computationof the real coefficient vector requires the estimation of theinteger orders, and . The orders in (1) can be estimatedusing the minimum eigenvalue tracking method discussed in [8]and shown in Fig. 1. In Fig. 1, the minimum eigenvalue hasbeen plotted as a function of the orders, and . Basedon (18), a nontrivial solution exists when , as shownin Fig. 1. Hence, has been used as a parameter for trackingthe optimum solution in this paper. In Fig. 1, the solution arearepresents the region with a valid solution, which correspondsto .

From Fig. 1, for the specific example, the orders,and , result in the eigenvector corresponding tothe minimum eigenvalue. The eigenvector contains the co-efficient vectors and of the rational function. The stabilitycan now be enforced on the denominator coefficient vector ofthe eigenvector in (18) by applying the constraints ,

, and . This has been done in this paper by usingthe root finding method [16] to compute the poles and discardingthe unstable poles. For computing the residues corresponding tothe stable poles, (19) is solved using the eigenvalue method dis-cussed earlier. The matrix structure in (19) depends on the ap-plication and can be arbitrary [7]. For a multiport network con-taining common poles, the residues for each port are constructedindependently by solving (19). Finally, the passivity formulae in(12) are used to enforce passivity of the rational function

...

...

...

...

...

...

...

...

(19)

where

V. BROADBAND MACROMODELS

Realistic distributed networks such as interconnects oper-ating over a broad frequency range often contain a large numberof poles and the amplitude variation of the frequency responsecould be large. This can create numerical problems in (18) and(19) since the matrix can become an ill-conditionedmatrix [1], [2]. This is apparent in (1), where the power se-ries expansion can have a large dynamic range. For instance,if the frequency response ranging from dc to 2 GHz needs tobe approximated using 20 complex-conjugate poles and tworeal poles, the dynamic range of elements in the matrixis from 1 to , which causes the matrix tobecome ill conditioned. This problem can be improved usingfrequency scaling and the computed poles and residues can bereconstructed using the scaling factor . Equation (15) can bewritten in the form

(20)

However, it has been shown that the scaling factor in (20) forapproximating the frequency response does not result in signifi-cant improvement in the approximation beyond 20–30 poles, asillustrated in [2]. Hence, the author in [2] has used the Cheby-shev polynomial expansion to approximate the frequency re-sponse using the orthogonal property of the Chebyshev poly-nomials. However, it is important to note that the power seriesexpansion with frequency scaling is comparable to the Cheby-shev polynomial expansion for low-order systems, as discussed


in [10]. In addition, a major problem with the Chebyshev poly-nomial expansion is that it needs to be finally converted into thepole-residue representation for implementation in SPICE [2],[10]. It is important to note that the size of the matrixhas not been changed and the ill-conditioned matrix problemstill exists when the frequency response having a large numberof poles is approximated. This paper proposes a method to re-duce the size of the matrix and , which alleviates theill-conditioned matrix problem.

As shown in (11), the rational function matrix is rep-resented in the pole-residue form as a summation of subnet-works. Both the stability and passivity conditions in each sub-network are satisfied using stability constraints and multiportpassivity formulae described earlier. Since stability constraintsand passivity formulae are only enforced on each subnetwork,there is no relationship for the stability and passivity condi-tions between subnetworks except that they contribute to theoverall response of the macromodels. This enables the entirefrequency response to be divided into subfrequency bands (orsubbands).

The basic idea for the construction of broadband macro-models is that if complex-conjugate poles and real poles canbe extracted from a localized region of the frequency response,then the original frequency response can be divided into sub-bands, as shown in Fig. 2(a) and (b). In Fig. 2(a) and (b), polesfrom the localized region within a subband can be extracted.The process of dividing the entire frequency band of interestinto smaller subbands is called a band division in this paper.The subbands are either of uniform or nonuniform width andcontain a set of sampled frequency data, as shown in the figure.The criterion for choosing the width of each subband dependson the nature of the frequency response, but it is desirable todivide the entire frequency band into subbands where reso-nant peaks exist. In areas where no resonant peaks exist, thesubbands are overlapped to minimize the number of subbandsrequired. The orders and for each subband can beestimated using the minimum eigenvalue tracking method [8]or the number of resonant peaks.

Using the band division method, the macromodels for eachsubband can be extracted in parallel. However, macromodelsfrom each subband can interact with each other since the extrap-olation of each macromodel outside the subband of interest canproduce a nonzero frequency response. This interaction can re-sult in an erroneous frequency response when the macromodelsare combined. During the least squares approximation processdiscussed earlier, poles located outside the subband can be fre-quently found since there are no constraints that limit the posi-tion of the poles. After removing the unstable poles, stable poleslocated outside the subband have to be handled since they am-plify the interactions between subbands. Although these polesincrease the accuracy in the subband of interest, they usuallyreduce the accuracy in adjacent frequency bands. Therefore, se-lectors are used to restrict the poles within the frequency band ofinterest during the generation of macromodels in each subband.This ensures smooth extrapolation and the absence of resonantpeaks in adjacent bands. In this paper, four selectors have beenused, namely, low-pass, band-pass, high-pass and band-rejectselectors.

Fig. 2. (a) and (b) band division. (c) Subband reordering. (d) Subband dilation.

To compensate for the inaccuracy in the subbands after theremoval of poles outside the subbands, three methods, namely,subband reordering, subband dilation, and pole replacementmethods have been used in this paper.

In the subband reordering method, the macromodels foreach subband are constructed after subtracting the frequencyresponse of the previous macromodels from the original fre-quency response. As discussed earlier, selectors are applied tohandle the poles located inside or outside the frequency band.At every stage of this process, the subbands are reordered basedon the magnitude of the frequency response in decreasing order.The macromodel for the subband with the largest magnitudeis always constructed first, prior to the other subbands. Thisis shown in Fig. 2(c), where subbands 2 and 3 in Fig. 2(a)are interchanged during the macromodel construction process.Subband reordering reduces numerical errors caused by domi-nant poles in an adjacent subband.

In subband dilation method, the subbands are dilated to pro-vide local correction in the region between adjacent subbands.This method results in an overlap between subbands, as shownin Fig. 2(d). In the figure, subband “ ” is dilated to overlap sub-bands “ ” and “ .” The amount of overlap is determinedby the position of poles in each subband. If the poles are locatedat the boundary between bands, then these poles have the max-imum effect on both frequency bands. As the poles are locatedfarther away from the boundary, the effect of these poles on ad-jacent bands is minimized. Using this criterion, the subbandsare suitably dilated such that the poles at the boundary lie in theoverlap region. The macromodels of three subbands are then it-eratively corrected by monitoring the error in three subbandsusing the pole replacement method discussed later.

Though subband reordering and subband dilation methodsminimize the interaction between subbands, correction mayonce again be necessary since these methods may sometimesmiss the poles at the boundary between subbands or generatespurious poles within the subband. This paper uses a polereplacement method to improve the accuracy of the constructedmacromodels by discarding spurious poles and extractingaccurate poles, as shown in Fig. 3. The pole replacement


Fig. 3. Pole replacement.

method is applied after the macromodels for each subband areconstructed. The details of the pole replacement method aredescribed below.

1) The process begins with the comparison between the fre-quency response from the macromodels and theoriginal frequency response .

2) Based on the error criterion, the maximum differencevalue and location between and

are calculated and stored.3) The algorithm searches for a set of poles around the loca-

tion and determines the width of the subband to berecalculated.

4) The RMS error between and inthe subband is calculated and stored. The frequency re-sponse associated with the poles in the subband is thensubtracted from in order to recalculate the fre-quency response of the subband. Then, the difference be-tween and is calculated and stored in

.5) Using , new poles and residues are recalculated in

the subband.6) The frequency response using new poles and residues

is computed and the RMS error betweenand in the subband is again calculated

and stored.7) If , then old poles and residues are re-

placed with new poles and residues. If ,then old poles and residues are retained.

8) Steps 1–7 are repeated until the error is minimized.The location and size of each subband is recalculated itera-

tively in the pole replacement method. The error criterion deter-mines the maximum error value , the location , andthe width of the subband being recalculated. In this paper, theRMS error has been used as the error criterion and the

subband width around the location is determined based onthe position of poles being replaced. Assuming the number ofpoles being replaced is , the algorithm searches forpoles around the location . The poles are then re-ordered in increasing frequency such that pole correspondsto the lowest frequency and pole corresponds to the highestfrequency. The left and right boundaries of the subband are thendetermined as the midfrequency points between poles ,and , , respectively. If the width and position of thesubband remains the same as before, then the subband is suitablydilated to minimize error. If the location is not changedafter dilating the subband, then the location is stored andignored in subsequent iterations.

With the use of band division, subband reordering and sub-band dilation methods initially, the required number of itera-tions for the pole replacement method locally is minimum. Eventhough the pole replacement method was originally intended forincreasing the accuracy of poles and residues of passive macro-models, it can be used to compensate for negative eigenvaluesby inserting additional poles and residues, as discussed earlier.After the poles and residues in the entire frequency band or sub-bands are calculated, the residue matrices and can be cal-culated.

It is important to note that broadband macromodels can beconstructed using band division, subband reordering, subbanddilation, and pole replacement methods along with frequencyscaling, without having an ill-conditioned matrix problem. Inaddition, since the number of required poles is reduced and theorders and become small within a subband, the size ofthe matrix becomes small and the required computa-tional memory and CPU time can be reduced.

The methods discussed in this paper have been incorporatedinto the Broadband Efficient Macromodeling Program (BEMP)developed at Georgia Tech. The algorithm for BEMP is shown


Fig. 4. Flow chart of BEMP.

Fig. 5. Illustration of the band division and subband reordering methods.

in Fig. 4. The program was developed using C++ language andis executable on a Windows operating system. The applicationof the discussed methods is graphically illustrated in Fig. 5. Theexact poles of the network are shown at the top of the figure.It has been assumed that all the poles are complex-conjugatepoles. In Fig. 5, the good poles are the exact poles of the net-work. All poles except the good poles are regarded as spurious

poles. Using the band division method, the entire frequency re-sponse has been divided into seven nonuniform subbands suchthat each subband has between one to four resonant peaks. Thebands are numbered from 1 to 7 horizontally, and named A–Gvertically. This has been done intentionally to differentiate banddivision from subband reordering. The steps illustrated belowassume that after the extraction of the poles from each sub-band, the corresponding residues are extracted. The frequencyresponse of the subband macromodel is then subtracted fromthe overall response prior to the macromodel construction forthe next subband. The various steps are described below.

Step 1) During the construction of submacromodels fromsubband A, three good poles and a spurious polelocated outside subband A are extracted using theeigenvalue method discussed earlier. After applyingthe low-pass selector with bandwidth equal to sub-band 1, three good poles are extracted.

Step 2) After subtracting the frequency response of theabove macromodel from the overall response, aspurious pole located outside subband B and twogood poles are extracted from subband B. Afterapplying the band-pass selector on subband B, thespurious pole is removed and two good poles areretrieved. Note that there is an exact pole near theboundary between subbands B and C, which is notincluded.

Step 3) During the calculation in subband C, a spuriouspole located outside subband C and two good polesare extracted. After applying the band-pass selector,two good poles are retained. Up to this point, sevengood poles have been extracted and an exact polenear the boundary of subbands B and C has beenmissed.

Step 4) The next computational domain moves to subbandD corresponding to subband 5 instead of subband4 (subband reordering). This is because the magni-tude of the frequency response is larger in subband5 than in subband 4. Submacromodels constructedfrom subband D result in two spurious poles andtwo good poles. After applying the band-pass se-lector to remove poles located outside subband D, aspurious pole and two good poles are extracted.

Step 5) From subband E corresponding to subband 4, a spu-rious pole and two good poles are extracted. Afterapplying the band-pass selector, a spurious pole andtwo good poles are retained.

Step 6) From subband F corresponding to subband 6, twospurious poles located outside subband F and twogood poles are extracted. After applying the band-pass selector, two good poles are retained.

Step 7) From subband G, a spurious pole and a good poleare extracted after applying the high-pass selector.After collecting submacromodels from each sub-band, 14 good poles, and 3 spurious poles arefound using the band division and subband re-ordering methods. The pole replacement method isnow applied to correct the good poles and eliminatethe spurious poles.


Fig. 6. Electrical network configurations for macromodels using admittanceparameters.

Fig. 7. Values of elements (electrical representations of macromodels).

VI. NETWORK SYNTHESIS

Using the pole-residue representation of the rational functionrepresenting the admittance parameters, electrical networksconsisting of resistors, inductors, capacitors, and controlledsources can be constructed [12]. In (5) and (11), RLC networkscan be used to represent complex-conjugate poles and residuesof the band-pass filter, RL networks can be used to representreal poles and residues of the low-pass filter, and RC networkscan be used to represent real poles and residues of the high-passfilter. A resistor and a capacitor can be used to represent theresidues of the all pass filter. The electrical network configu-rations are shown in Fig. 6 and the values of the componentsare shown in Fig. 7. It is important to note that the frequencyscaling in (20) and the local ground for SPICE subcircuits havebeen used in the circuit implementation.

VII. RESULTS

To demonstrate the validity of the methods discussed in thispaper, results from five test cases have been presented.

TABLE IWITHOUT USING SELECTORS

A. Frequency Response From a Known Transfer Function

To demonstrate the application of the band division andsubband reordering methods using selectors, the frequencyresponse derived from a known transfer function has been used.The transfer function has a real pole of with residue 2 andfour complex-conjugate poles of andwith corresponding residues and . The transferfunction was used to extract the frequency response from0.001 rad/s to 10 rad/s with 10 000 frequency samples. Thefrequency response was divided into three irregularly spacedsubbands, – rad/s, – rad/s, and

– rad/s. The criterion for choosing the subbands wasbased on given frequency data whereby at least one resonantpeak existed in the subbands. Using (18), the first and secondsubbands resulted in the orders of 6 and of 7 and thethird subband resulted in the orders of 10 and of 11.

Initially, no selectors were used to remove spurious poles. Thepoles and residues of the constructed macromodels are shown inTable I. The frequency responses of the macromodels are shownin Fig. 8. As is evident in Fig. 8, the macromodels constructedfrom three subbands result in an erroneous frequency responsebecause of the interactions between subbands.

Next, selectors were applied to each subband. From the firstsubband, a real pole and two complex-conjugate poles were ex-tracted. Of these poles, only the real pole was located withinsubband 1. Using a low-pass selector with bandwidth equal tosubband 1, only one real pole was retained. The frequency re-sponse of the macromodel generated from subband 1 is shown inFig. 9(a). After subtracting this macromodel from the originalfrequency response, the poles were extracted from subband 2.Four complex-conjugate poles were extracted. Two complex-conjugate poles were located within subband 2. Using a band-pass selector with bandwidth equal to subband 2, two complex-conjugate poles were retained. The frequency responses fromthe macromodels generated from subbands 1 and 2 are shownin Fig. 9(b). After subtracting these macromodels from the orig-inal frequency response, two complex-conjugate poles were ex-tracted from subband 3. Using a band-pass selector with band-width equal to subband 3, two complex-conjugate poles wereretained.


Fig. 8. The magnitude comparison between original data and the response of macromodels without using selectors. [Solid: original. Dash: Macromodels. Dot:deviation.] (a) The first subband. (b) The first and second subbands. (c) The entire frequency band.

Fig. 9. The magnitude comparison between original data and the response of macromodels with selectors. [Solid: original. Dash: macromodels. Dot: deviation.](a) First subband. (b) First and second subbands. (c) Entire frequency band.

TABLE IIUSING SELECTORS (BEFORE)

The poles and residues before and after the construction ofsubmacromodels are shown in Tables II and III, respectively.In Table II, the poles (second complex-conjugate pole pair) ex-tracted from subband 1 inaccurately capture the poles of sub-band 2 and the poles extracted from subband 2 accurately cap-ture the poles of subband 3, resulting in an erroneous frequencyresponse without the use of selectors. The comparison betweenoriginal frequency data and the response of broadband macro-models constructed from subbands 1, 2, and 3 using selectors isshown in Fig. 9(c) indicating the accuracy of the macromodels.

TABLE IIIUSING SELECTORS (AFTER)

It is interesting to note that for this example, the macromodelof subband 3 is required even if frequency data only up to sub-band 2 are available. This is possible through the use of a high-pass selector in subband 2.

B. One-Port Lossy Transmission Line

To demonstrate the performance of the pole replacementmethod, one-port admittance parameter for a lossy transmissionline up to 2.5 GHz was considered. The frequency responsewas equally divided into ten subbands having a bandwidth of


Fig. 10. Passive macromodels without using the pole replacement method(NS = 4 andDS = 5 in each subband, number of iterations of the local polereplacement method = 0, number of iterations of the global pole replacementmethod = 0, number of complex-conjugate poles = 30, and number of realpoles = 1).

Fig. 11. Passive macromodels using the pole replacement method (NS = 4andDS = 5 in each subband, number of iterations of the local pole replacementmethod = 50, number of iterations of the global pole replacement method =100, number of complex-conjugate poles = 66, and number of real poles = 4).

250 MHz and was approximated using the estimated orderof 4 within each subband. Fig. 10 shows the comparison

between original data and the frequency response of the macro-models. It is important to note that the number of originalresonance peaks is roughly 32 and the estimated number ofcomplex-conjugate poles is 64. However, the number of com-plex-conjugate poles and real poles extracted was 30 and 1,respectively, because the lower estimated order of 4 wasused in each subband. It is obvious that there are many missingpoles and spurious poles based on the comparison in Fig. 10.After using the pole replacement method in each subband (thenumber of iteration ) and over the entire computationaldomain (the number of iteration ), the missing poles wereextracted and the spurious poles were discarded. The number ofcomplex-conjugate poles and real poles extracted was 66 and 4,respectively. Fig. 11 shows the comparison between originalfrequency data and the response of the macromodels after usingthe pole replacement method.

Fig. 12. Lossy coupled transmission line.

C. Four-Port Lossy Coupled Transmission Line

The third test case is a four-port lossy coupled transmissionline. Fig. 12 shows the transmission line with its self andcoupling parameters, which was modeled using the W-elementtransmission lines available in HSPICE, which is a commer-cially available circuit simulator [17]. The frequency response(admittance parameters) for a four-port coupled transmissionline with length of 0.43 cm, which has uniformly distributedfrequency samples from 0.1 to 1 MHz (5 samples) and from10 MHz to 10 GHz (4000 samples), was generated usingHSPICE. Using the band-division method, the entire frequencydomain was divided into 80 uniform subbands without over-lapping subbands. Each subband had 50 frequency samples.The first subband was again divided into two smaller subbandshaving 10 and 40 samples, respectively. The number of localand global iterations for the pole replacement method was 3 and50, respectively. Using the estimated orders of 4 andof 5 within each subband, the number of complex-conjugatepoles and real poles extracted was 240 and 3, respectively. Thecomparison between original data and the frequency responseof constructed macromodels for Y14 admittance parameterover a bandwidth of 10 GHz is shown in Figs. 13 and 14.

To demonstrate the passivity of the macromodels in the fre-quency-domain, four eigenvalues are shown in Fig. 15, whichare all greater than zero.

To show the difference between stable and passive macro-models, nonpassive macromodels of the transmission line inFig. 12 were also constructed without enforcing the passivityformulae. These macromodels were synthesized into an equiv-alent circuit, and simulated in HSPICE. A trapezoidal currentsource with 0.1-ns rising and 0.2-ns falling times was excitedat port 1. Resistors with a 30- value were terminated at port 2and 4. A 30- resistor and a transmission line having a 50characteristic impedance and 1.2-ns delay were terminated atport 3 in series. The macromodels violated the passivity condi-tion resulting in a diverging result in the time-domain simula-tion, as shown in Fig. 16.

Using the constructed passive macromodels, the time-domainwaveforms of the original network and macromodels have beencompared up to 200 ns in Fig. 17, which shows a good agree-ment with HSPICE. The authors believe that the slight disagree-


Fig. 13. Comparison of real Y14 (admittance).

Fig. 14. Comparison of imaginary Y14 (admittance).

Fig. 15. Comparison of four eigenvalues versus frequency between originaldata and the frequency response calculated from the macromodel.

ment is because of the manner in which the transmission lineequations having an infinite number of resonances are modeledand solved using the lumped elements in the circuit simulator.

Fig. 16. Time-domain simulation of nonpassive macromodels.

Fig. 17. Time-domain comparison between a lossy coupled transmission lineand passive macromodels.

D. Four-Port Transmission Line From Measurements

The uniformly distributed 799 frequency samples from50 MHz to 20 GHz were measured using a vector networkanalyzer. The entire frequency band was divided into 16 sub-bands having a bandwidth of 1.25 GHz. The macromodels wereconstructed from the lower subband using the orders of 8and of 9 in all subbands. The number of local and globaliterations for the pole replacement method was 2 and 200,respectively. The number of complex-conjugate poles and realpoles extracted were 224 and 1, respectively. Figs. 18 and 19show the comparison between original frequency data and thefrequency response of constructed macromodels for real (Y11)and imaginary (Y12) admittance parameters, respectively. Themacromodels were connected to a 400 mV source having 50-psrising/falling times and a 625-ps period with a 50- terminationat port 1. Resistors with a 50- value were terminated at ports2, 3, and 4, as shown in Fig. 20. The time-domain waveformsof port 1 and 2 are shown up to 30 ns, demonstrating passivity.


Fig. 18. Comparison of real Y11 (admittance).

Fig. 19. Comparison of imaginary Y12 (admittance).

Fig. 20. (a) Black box representation of the transmission line. (b) Time-domain waveforms.

Fig. 21. (a) Power plane pair with decoupling capacitors. (b) Circuit simulated.

E. Four-Port Power Plane Pair With Decoupling Capacitors

A 4 in 6 in power plane pair with a dielectric thickness of 62mil is shown in Fig. 21. Six decoupling capacitors representedas RLC circuit elements, where and are the series equiv-alent resistance and inductance of the capacitor, respectively,were connected to the power plane, as shown in Fig. 21. A totalof 1000 uniformly spaced samples were generated from 1 MHzto 4 GHz using the cavity resonator method described in [18].Although the frequency response of the power plane pair withdecoupling capacitors is highly irregular, the entire frequencydomain for admittance parameters was divided into 20 uniformsubbands having 50 samples per subband without overlappingsubbands. The number of local and global iterations for the polereplacement method was 1 and 200, respectively. Using the or-ders of 4 and of 5 within each subband, the numberof complex-conjugate poles and real poles extracted was 78and 1, respectively. The comparison between original data andthe frequency response of the constructed macromodels for ad-mittance parameters over a bandwidth of 4 GHz is shown inFigs. 22 and 23. The macromodels of the power plane with de-coupling capacitors were simulated to get the voltage fluctuationon the power plane using the configuration shown in Fig. 21. Themacromodels were connected to a 2.5 V dc source (port 1) andthe node (port 4) of a linear driver with 0.7-ns rising and0.3-ns falling times. A transmission line having a 50- charac-


Fig. 22. Comparison of the real part of admittance parameters for a powerplane with decoupling capacitors.

Fig. 23. Comparison of the imaginary part of admittance parameters for apower plane with decoupling capacitors.

Fig. 24. Time-domain simulation of a power plane pair with decouplingcapacitors.

teristic impedance and 1.2-ns delay and a 50- resistor wereconnected in series between the output of the driver and thecommon ground of the power plane. The time-domain voltagefluctuations at port 2 and 4 are shown in Fig. 24, which demon-strates that the macromodel is stable and passive.

VIII. CONCLUSION

An efficient method for constructing broadband passivemacromodels for multiport circuits has been presented usingband division, subband reordering, subband dilation, and polereplacement methods. The major advantage of this method isthat it can be applied to distributed interconnect networks thatoften require many poles for approximation. The stability andpassivity of broadband macromodels were satisfied throughstability constraints and multiport passivity formulae. Theseconstraints were enforced on the poles and residues of rationalfunctions. Broadband macromodels were constructed using theband division method, which divides the entire computationaldomain into subcomputational domains, the selector method,which selects the filters, the subband reordering method, whichprovides a construction sequence for reducing the interac-tions between subbands, the subband dilation method, whichprovides local correction in the boundary region betweensubbands, and the pole replacement method, which increasesthe accuracy of macromodels. Numerically, these methodsalleviated the ill-conditioned matrix problem of least squaresapproximation. In addition, these methods reduced the memoryrequirement and computational time. The efficiency of themethod was demonstrated through several test cases represen-tative of distributed interconnect networks.

ACKNOWLEDGMENT

The authors would like to thank Dr. W. Beyene of Rambusfor supplying the measured S-parameters of interconnects,Dr. S. Chun of IBM for providing frequency data of powerplanes, and Dr. S. Dalmia of Georgia Institute of Technologyfor many useful discussions during the course of this work.

REFERENCES

[1] R. Achar and M. S. Nakhla, “Simulation of high-speed interconnects,”Proc. IEEE, vol. 89, May 2001.

[2] W. T. Beyene, “Improving time-domain measurements with a networkanalyzer using a robust rational interpolation technique,” IEEE Trans.Microwave Theory Tech., vol. 49, pp. 500–508, Mar. 2001.

[3] B. Gustavsen and A. Semlyen, “Enforcing passivity for admittance ma-trices approximated by rational functions,” IEEE Trans. Power Syst., vol.16, pp. 97–104, Feb. 2001.

[4] L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation fortiming analysis,” IEEE Trans. Computer-Aided Design, vol. 9, pp.352–366, Apr. 1990.

[5] E. Chiprout and M. S. Nakhla, “Analysis of interconnect networks usingcomplex frequency hopping (CFH),” IEEE Trans. Computer-Aided De-sign, vol. 14, Feb. 1995.

[6] R. Achar, P. K. Gunupudi, M. S. Nakhla, and E. Chiprout, “Passive inter-connect reduction algorithm for distributed/measured networks,” IEEETrans. Circuits Syst. II, vol. 47, pp. 287–301, Apr. 2000.

[7] S. H. Min and M. Swaminathan, “Efficient construction of two-port pas-sive macromodels for resonant networks,” presented at the IEEE 10thTopical Meeting Electrical Performance of Electronic Packaging, Oct.2001.

[8] K. L. Choi and M. Swaminathan, “Development of model libraries ofembedded passives using network synthesis,” IEEE Trans. Circuits Syst.II, vol. 47, pp. 249–260, Apr. 2000.

[9] W. T. Beyene and J. E. Schutt-Aine, “Efficient transient simulation ofhigh-speed interconnects characterized by sampled data,” IEEE Trans.Comp., Packag, Manufact. Technol. A, vol. 21, Feb. 1998.

[10] S. Chakravorty, S. H. Min, and M. Swaminathan, “Comparison betweenChebyshev and power series expansion for interpolating data,” in IEEE10th Topical Meeting Electrical Performance Electronic Packaging,Oct. 2001.


[11] B. Gustavsen and A. Semlyen, “Rational approximation of frequencydomain responses by vector fitting,” IEEE Trans. Power Delivery, vol.14, pp. 1052–1061, July 1999.

[12] M. J. Choi and A. C. Cangellaris, “A quasi three-dimensional distributedelectromagnetic model for complex power distribution networks,” inProc. IEEE 51th ECTC, May 2001, pp. 1111–1116.

[13] N. Na, “Modeling of simulation of planes in electronic packaging,”Ph.D. dissertation, Georgia Inst. Technol., Atlanta, GA, 2001.

[14] S. Kami, Network Theory: Analysis and Synthesis. Boston, MA: Allynand Bacon, 1966, pp. 96–98.

[15] F. F. Kuo, Network Analysis and Synthesis. New York: Wiley, 1962.[16] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Nu-

merical Recipes in C: The Art of Scientific Computing, 2nd ed. Oxford,U. K.: Cambridge Univ. Press, 1992.

[17] Star-HSPICE Manual, Avant! Corporation, Fremont, CA, Dec. 1999.[18] S. Chun, “Methodologies of modeling in simultaneous switching noise

in multi-layered packages and boards,” Ph.D. dissertation, Georgia Inst.Technol., Atlanta, 2002.

Sung-Hwan Min (S’01) received the B.S. degreein electronic engineering from Hanyang University,Ansan, Korea, in 1996 and the M.S. degree inelectrical and computer engineering from GeorgiaInstitute of Technology, Atlanta, in 2001, where heis currently pursuing the Ph.D. degree in electricaland computer engineering.

Currently, he is a Graduate Research Assistant atthe Packaging Research Center, Georgia Institute ofTechnology. His research interests are in electromag-netic design, modeling, and analysis of high-speed

mixed signal systems and packages using the macromodeling technique.

Madhavan Swaminathan (SM’99) received theM.S.E.E. and Ph.D. degrees in electrical engineeringfrom Syracuse University, Syracuse, NY, in 1989and 1991, respectively.

He is currently a Professor in the School ofElectrical and Computer Engineering and a ResearchDirector at the Packaging Research Center, GeorgiaInstitute of Technology, Atlanta. Before joiningGeorgia Institute of Technology, he was with theAdvanced Technology Division of the PackagingLaboratory at IBM, E. Fishkill, NY, where he was

involved with the design, analysis, measurement, and characterization forhigh-performance computer systems. While at IBM, he also reached the secondinvention plateau. He has published more than 150 publications in refereedjournals and conferences, has coauthored three book chapters, has eight issuedpatents, and seven patents pending. His collaborations with industry, govern-ment labs, and universities include IBM, Sun, Rambus, Hughes Research Labs,AMD, Intel, Sarnoff Corporation, Rochester Institute of Technology, DARPA,3M, National Semiconductor, Semiconductor Research Corporation, Cadence,and Ansoft Corporation. His research interests are in design, electromagneticmodeling, circuit modeling, characterization and testing of high-requencydigital, and mixed signal ICs and packages.

Dr. Swaminathan served as the cochair for the 1998 and 1999 IEEE TopicalMeeting on Electrical Performance of Electronic Packaging (EPEP); as theTechnical and General Chair of the IMAPS Next Generation IC and PackageDesign Workshop, and was the cochair for 2001 IEEE Future Directions in ICand Package Design Workshop. He currently serves as the chair of T-12, theTechnical Committee of Electrical Design, Modeling, and Simulation withinthe IEEE Components, Packaging, and Manufacturing Technology (CPMT)Society, and also serves on the technical program committees of EPEP, SignalPropagation on Interconnects Workshop, Solid State Devices and MaterialsConference, and Interpack. He was the recipient of the 2002 OutstandingGraduate Research Advisor Award from the School of Electrical and ComputerEngineering, and the 2003 Outstanding Faculty Leadership Award for theDevelopment of Graduate Research Assistants, from Georgia Institute ofTechnology. He was the coauthor of the Best Student Award at EPEP’00 andEPEP’02, Outstanding Student Paper Award at ECTC’96, and the IMAPSEducation Award in 1998. Dr. Swaminathan has been a Guest Editor for theIEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURINGTECHNOLOGY—ADVANCED PACKAGING, and IEEE TRANSACTIONS ONMICROWAVE THEORY AND TECHNIQUES. He is currently an Associate Editor ofthe IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGY.

tocConstruction of Broadband Passive Macromodels From Frequency DatSung-Hwan Min, Student Member, IEEE, and Madhavan Swaminathan, SI. I NTRODUCTIONII. R ATIONAL F UNCTIONSIII. S TABILITY AND P ASSIVITYIV. L EAST S QUARES A PPROXIMATION

Fig.€1. Minimum eigenvalue versus the order $NS=DS-1$ .V. B ROADbAND M ACROMODELS

Fig.€2. (a) and (b) band division. (c) Subband reordering. (d) SFig.€3. Pole replacement.Fig.€4. Flow chart of BEMP.Fig.€5. Illustration of the band division and subband reorderingFig.€6. Electrical network configurations for macromodels using Fig.€7. Values of elements (electrical representations of macromVI. N ETWORK S YNTHESISVII. R ESULTS

TABLE I W ITHOUT U SING S ELECTORSA. Frequency Response From a Known Transfer Function

Fig.€8. The magnitude comparison between original data and the rFig.€9. The magnitude comparison between original data and the rTABLE II U SING S ELECTORS (B EFORE )TABLE III U SING S ELECTORS (A FTER )B. One-Port Lossy Transmission Line

Fig.€10. Passive macromodels without using the pole replacement Fig.€11. Passive macromodels using the pole replacement method (Fig.€12. Lossy coupled transmission line.C. Four-Port Lossy Coupled Transmission Line

Fig.€13. Comparison of real Y14 (admittance).Fig.€14. Comparison of imaginary Y14 (admittance).Fig.€15. Comparison of four eigenvalues versus frequency betweenFig.€16. Time-domain simulation of nonpassive macromodels.Fig.€17. Time-domain comparison between a lossy coupled transmisD. Four-Port Transmission Line From MeasurementsFig.€18. Comparison of real Y11 (admittance).Fig.€19. Comparison of imaginary Y12 (admittance).Fig.€20. (a) Black box representation of the transmission line. Fig.€21. (a) Power plane pair with decoupling capacitors. (b) Ci

E. Four-Port Power Plane Pair With Decoupling Capacitors

Fig.€22. Comparison of the real part of admittance parameters foFig.€23. Comparison of the imaginary part of admittance parameteFig.€24. Time-domain simulation of a power plane pair with decouVIII. C ONCLUSIONR. Achar and M. S. Nakhla, Simulation of high-speed interconnectW. T. Beyene, Improving time-domain measurements with a network B. Gustavsen and A. Semlyen, Enforcing passivity for admittance L. T. Pillage and R. A. Rohrer, Asymptotic waveform evaluation fE. Chiprout and M. S. Nakhla, Analysis of interconnect networks R. Achar, P. K. Gunupudi, M. S. Nakhla, and E. Chiprout, PassiveS. H. Min and M. Swaminathan, Efficient construction of two-portK. L. Choi and M. Swaminathan, Development of model libraries ofW. T. Beyene and J. E. Schutt-Aine, Efficient transient simulatiS. Chakravorty, S. H. Min, and M. Swaminathan, Comparison betweeB. Gustavsen and A. Semlyen, Rational approximation of frequencyM. J. Choi and A. C. Cangellaris, A quasi three-dimensional distN. Na, Modeling of simulation of planes in electronic packaging,S. Kami, Network Theory: Analysis and Synthesis . Boston, MA: AlF. F. Kuo, Network Analysis and Synthesis . New York: Wiley, 196W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanne

Star-HSPICE Manual, Avant! Corporation, Fremont, CA, Dec. 1999.S. Chun, Methodologies of modeling in simultaneous switching noi

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