Tehniques for Inluding Dieletris when Extrating Passive Low-Order Models of High Speed Interonnet Lua Daniel University of California, Berkeley dlua@ees.berkeley.edu Alberto Sangiovanni-Vinentelli Univ. of California, Berkeley alberto@ees.berkeley.edu Jaob White Massahusetts Institute of Tehnology white@mit.edu ABSTRACT Interonnet strutures inluding dieletris an be modeled by an integral equation method using volume urrents and surfae harges for the ondutors, and volume polarization urrents and surfae harges for the dieletris. In this paper we desribe a mesh analysis approah for omputing the disretized urrents in both the ondutors and the dieletris. We then show that this fully meshbased formulation an be ast into a form using provably positive semidefinite matries, making for easy appliation of Krylovsubspae based model-redution shemes to generate aurate guaranteed passive redued-order models. Several printed iruit board examples are given to demonstrate the effetiveness of the strategy. 1. INTRODUCTION Dieletri materials are present in almost all modern eletroni iruits: from Printed Ciruit Boards PCBs), to pakages, Multi- Chips Modules MCMs), and Integrated Ciruits. Dieletris an signifiantly affet both the performane and the funtionality of eletroni iruits. For instane, they an hange interonnet delays, as well as the positions of frequeny response resonanes. Ignoring dieletris an therefore potentially lead to very inaurate results both in timing analysis tools and in signal integrity tools. Integral equation methods have proved to be very effetive tools for analyzing on-hip and off-hip interonnet strutures, and there are several approahes for inluding dieletri interfaes in integral formulations. For problems whih an be viewed as flat interfaes of infinite extent, suh as multilayer printed iruit boards, the dieletri interfae onditions an be satisfied by an appropriate hoie of Green s funtion [1, 2, 3, 4, 5]. For general shape or finite-size dieletri bodies, it is possible to replae the dieletris with equivalent fititious eletri and/or magneti surfae urrents [6, 7]. General dieletri shapes an also be handled by a volume integral equation VIE) approah, in whih ase the polarization urrents are introdued in the volume of the dieletris, and This work was supported by the MARCO Interonnet Fous Center, and by the Semiondutor Researh Corporation. The authors would also like to thank Matt Kamon for useful feedbak and disussion. Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies bear this notie and the full itation on the first page. To opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speifi permission and/or a fee. ICCAD 2001, November 4-8, 2001, San Jose, California, USA. Copyright 2001 ACM...$5.00. harges are introdued on their surfaes [8, 9]. As the last deade has made lear, detailed eletromagneti analysis is a vastly more effetive tool if it an be used to automatially generate aurate iruit-level models of the interonnet. In this paper we show that areful disretization of the VIE method leads to, at least in the low frequeny regime, a linear dynamial system with positive semi-definite matries. This positive-definite result is important beause it makes possible the straight-forward appliation of the Krylov-subspae based guaranteed passive model-order redution MOR) tehniques [10, 11, 12, 13, 14, 15, 16, 17]. The paper is organized as follows: in Setion 2 we summarize Maxwell equations in VIE form. In Setion 3, we desribe the mesh analysis formulation for dieletris. In Setion 4 we show that the matries resulting from the VIE full mesh analysis approah an be ast into a positive semidefinite form, hene allowing guaranteed passive low order models extration. Finally, in Setions 6 and 7 we show the results of our implementation on simple examples of interonnet strutures inluding dieletris. 2. BACKGROUND Maxwell equations in Mixed Potential Integral Equation MPIE) form are as shown in 1)-4), where V and are the union of the ondutor and dieletri volumes respetively, r and r d are vetors indiating points in V and respetively. µ is the magneti permeability, ε 0 is the permittivity, ε r is the dieletri relative permittivity, σ is the ondutivity of the metal, and ω is the angular frequeny of the ondutor exitation. J is the urrent density in the ondutors. J d jω ε ε 0 E is the polarization urrent density in the interior of the dieletris, and E is the eletri field. The kernel K for a full-wave formulation is a frequeny dependent funtion K r 1 r 2 e jω ε 0µ r 1 r 2 5) r 1 r 2 When the relevant length sales are muh smaller than a wavelength, the above kernel an be approximated with the frequeny independent funtion K r 1 r 2 1 r 1 r 2 6) The salar potentials φ and φ d, an be related to the surfae harge ρ and ρ d, on both the ondutor and dieletri surfaes as shown in 3)-4), where S is the union of the ondutor surfaes, S d is the union of the dieletri surfaes, r s is a vetor indiating a point in S, and r s is a vetor indiating a point in S d. Within eah ondutor, and within eah homogeneous blok of dieletri, J r 0 7) 0 8) J d r d
J r jω µ K r r σ 4π V J r dr J d r d jω jω ε µ K r d r ε 0 4π V J r dr 1 K r s 4πε 0 S rs ρ rs drs 1 K r ds S rs ρ rs drs 4πε 0 K r r d J d r d dr d φ 1) K r d r d J d r d dr d φ d 2) K r s S d rds ρ d rds drds φ r s 3) K r ds S d rds ρ d rds drds φ d r ds 4) for all points r and r d in the interior of V and respetively. In addition, the urrent normal to the ondutor and dieletri surfaes is responsible for the aumulation of surfae harge, ˆn J ˆn J d r s r ds jωρ jωρ d r s 9) r ds 10) where ˆn is the unit normal to S and S d at the points r s and r s respetively. The main unknowns, J, J d, ρ, and ρ d an be approximated by a weighted sum of a finite set of basis funtions. One lassial hoie for the basis funtions is to over the surfae of eah ondutor and of eah dieletri with panels, eah of whih hold a onstant harge density. To model urrent flow, the interiors of all ondutors and dieletris are divided into a 3-D grid of filaments. Fig. 3 shows an example of the 3D volume disretization of a dieletri parallelepiped. Eah filament arries a onstant urrent. Other basis funtions hoies are possible for the interior of the ondutors [18]. A Galerkin method [19] an be used to transform the Mixed Potentials Integral Equations 1)-4) into an algebrai form R sl 1 s 0 Pol P I I d q q d V φ φ d 11) where I, I d, q and q d are vetors of basis funtion weights for the ondutor urrents, dieletri polarization urrents, ondutor harges and dieletri harges respetively. V,, φ C and φ d are the vetors generated by inner produts of the basis funtions with the potential gradient and with the potential itself. The resistane matrix R, the indutane matrix L and the oeffiients of potential matrix P are all derived diretly from the Galerkin ondition [19], R L P R 0 L L d L d L dd P P d 12) 13) P d P dd 14) L and P are frequeny dependent when using a full-wave kernel as in 5), and frequeny independent when using a quasi-stati kernel as in 6). Matrix Pol in 11) is a diagonal matrix arrying the polarization oeffiients Pol i i A i l i ε ε 0 15) where l i and A i are the length and the ross-setional area of dieletri filament i respetively. 3. THE MESH FORMULATION Imposing harge onservation 9)-10) and urrent onservation 7)-8) on surfae and on interior of both ondutors and dieletris makes it possible to use a mesh analysis approah. As a summary, a omplete mesh formulation for strutures inluding both ondutors and dieletris, after the Galerkin transformation an be written simply as: M Z d M T I m V ms 16) where I m are the unknown mesh urrents, V ms is the vetor of known mesh voltage soures, non zero only on the rows assoiated with the external iruit terminals. Z d is the Galerkin impedane matrix Z d R sl 1 s M is a very sparse mesh analysis matrix, 0 Pol 0 0 1 s P M M f M d M p M pd! 17) 18) where submatries M f and M p are the KVL s mesh matries for the ondutors filaments and panels as desribed in [20]. In a very similar way to [20], we an onstrut also M f d and M pd, the KVL s mesh matries for the dieletri filaments and panels. In fat, as for the ondutors, dieletri panel harges an be treated as displaement urrents flowing on iruit branhes to the node at infinity. A set of independent meshes for the three dimensional disretization of the blok of dieletri an be found using a minimum spanning tree. 4. PASSIVE MODEL ORDER REDUCTION In this Setion, we will limit ourself to the usage of the quasistati kernel in 6) whih produes frequeny independent L and P matries in 13) and 14). The tehnique to handle dieletris in [21] uses a similar quasi-stati assumption, and seems more advantageous requiring fewer unknowns. However, not only magneti oupling between ondutive and polarization urrents are negleted by that formulation, but also the matries used in that formulation are not in the form required for Krylov-subspae based passive model-redution shemes [10]. In this Setion, we show instead an easy way to ast our mesh analysis approah into a form suitable for passive redued order modeling. Let us hoose as state vetor for a linear system representation: x #" I m Q s Q ds Q dv $ T 20) In view of this hoie, we an rewrite 16) as shown in 19) on next page, where Q s Q ds Q dv! T M p M pd M f d! T I m s 21)
- M f M f d! R sl! M f M f d! T I m M p M pd! P Q s Q ds! T M f d Pol! Q dv V ms 19) Or finally in linear system terms: ˆL dx dt y t where matries ˆL and ˆR are defined as ˆRx t Bu t 22) Cx t 23) M f M f d* L M f M f d* T ˆL % &') 0 P T 0 Pol* T +, M f M f d* R M f M f d* T M p M pd* P ˆR % &' - P T M p M pd* T Pol* T M T f d +, M f d Pol* 24) 25) V ms. Vetor u t ontains the exitation voltage soures, Bu t Vetor y t ontains the observed output urrents, derived through matrix C from the mesh urrents I m in the state vetor x t. THEOREM 1. Matries ˆL and ˆR in 24) and 25) are positive semidefinite. PROOF. The polarization matrix Pol! is diagonal with positive oeffiients, hene it is positive semidefinite. When using a Galerkin tehnique [19], the oeffiient of potential matrix P in 14) and the indutane matrix L in 13), are both positive semidefinite. The matrix M f M f d! L M f M f d! T is then also positive semidefinite. Sine all the three bloks of the blok-diagonal matrix ˆL in 24) are positive semidefinite, ˆL is positive semidefinite. This onludes the first part of the proof. To prove that ˆR in 25) is positive semidefinite let us alulate: 2 M f M f d! R M f M T f d! ˆR Rˆ T 0 0 26) The resistane matrix R is positive semidefinite, hene the submatrix 2 M f M f d! R M f M f d! T is positive semidefinite and so is ˆR Rˆ T. Sine x T ˆR Rˆ T x 2x T ˆRx, we an then finally onlude that ˆR is positive semidefinite. OBSERVATION 1. When modeling the input impedanes and the transfer funtions of a 3D struture, we apply input voltages at some ports, and we measure the resulting urrents on the same set of ports, hene we are hoosing C B T in eq 22) and 23). OBSERVATION 2. From Theorem 1 and from Observation 1, one an onlude that the formulation in 22)-25) satisfies to the onditions for guaranteed passive Krylov subspae based model redution in [10]. 5. SUMMARY OF OUR PROCEDURE We summarize here briefly for the onveniene of the reader our entire proposed proedure in its final form: 1. First, we disretize both the volumes and the surfaes of the ondutors and dieletris. An example is shown in Fig. 3. 2. We use a standard Galerkin tehnique [19] to onstrut matries R L P Pol in eq.12) to 15). 3. A mesh analysis approah is used to onstrut the sparse KVL s matries M f M f d M p and M pd in 18). More details on how to handle ondutors are in [20]. For the dieletris, we use a minimum spanning tree to find a set of independent meshes. 4. A Krylov subspae based model redution algorithm suh as [10] is then used to produe redued order linear system models. At eah step of the algorithm the quantity ˆR s 0 ˆL! 1 ˆLv, ould be omputed using fast matrix vetor produts and Krylov subspae iterative methods. 5. The redued order model is then used to obtain a plot of the frequeny response as in Fig. 1, and Fig. 4 or to produe an equivalent SPICE iruit for a time domain simulation inluding the non-linear iruitry. The overall omplexity of this proedure is O N m Nlog N, where N m is the total number of moments mathed by the model redution algorithm at all frequeny expansion points. N is the size of the original full linear system model in 22)-23), or about the number of basis funtions used in the volume and surfae disretization. 6. A TRANSMISSION LINE EXAMPLE Two PCB traes are onsidered in this example in a transmission line onfiguration. Traes are loated on opposite sides of a dieletri substrate, and shorted at one end. Fig 1 shows the frequeny response of suh transmission line struture. In Fig 1 we also show the response of the alulated redued order model. Admittane [1/Ohm] 1 10 1 10 2 10 3 Shorted T Line with dieletri in between. epsr=4. lenght=30m. mesh analysis, redued order model mesh analysis, full model 10 4 0 1 2 3 frequeny [Hz] 4 5 6 x 10 8 Figure 1: Redue order modeling of a shorted PCB transmission line. Traes dimensions are 250µm x 35µm x 30m. A 100µm thik dieletri layer ε r. 4) is present between the two traes. The ontinuous line is the admittane vs. frequeny of the alulated redued model. The irles are the response of the original system. At the frequenies where the frequeny independent kernel in 6) yields aurate results, it may also be reasonable to neglet magneti oupling between ondutors and dieletri polarization urrents. However there are ases where even with a non-fullwave
kernel one might observe some effets of the magneti oupling between / dieletri polarization urrents and ondutors. One of suh ases is illustrated in Fig. 2. A via is loated in proximity of the shorted PCB transmission line. The line is then exited at a frequeny lose to the first quarter-wavelength resonane. In this situation most of the urrent loses its path through the dieletri layer in the form of polarization urrents. If a nearby via orresponds to a quiet vitim line, some oupling an be observed between the vertial polarization urrents and the via. Figure 3: Two traes part of an MCM interonnet system figure above). A dieletri layer εr 4 is present underneath the traes and the hips. The figure below shows the volume polarization urrents inside the dieletri layer at the 3 GHz resonane. For visualization purposes, the axes in this piture are not to-sale. Traes are 2m long, 4mm far apart, 250µm wide and 40µm thik. Figure 2: Via loated near a PCB transmission line. In this piture we do not show the dieletri layer whih is loated between the two dark PCB transmission line traes. Shadings orrespond to urrent density amplitudes. On top we show the urrent densities orresponding to the ase where magneti oupling between polarization urrents and ondutors is aounted for. On the bottom we show the same example but setting Ld 0, Ld 0 and Ldd 0 in 13) whih orresponds to negleting magneti oupling between polarization urrents and ondutors. 7. MCM INTERCONNECT EXAMPLE In a seond example, we have applied our tehnique to analyzing two wires of an interonnet bus on an Multi-Chip Module MCM), as shown in Fig. 3. A dieletri layer εr 4) is present underneath the traes and the hips. In Fig. 4 we show the frequeny response of the two interonnets when shorted on one side and driven on the other. We show the frequeny response with and without the dieletri substrate. A signifiant differene in the resonane position an be observed. Fig. 3 shows the polarization volume urrents at the first resonane f 3 GHz. In Fig. 4 we ompare the redued order model to the full model for the ase when the dieletri substrate is present. The redued order model has been built mathing four moments around eah of the following expansion points: s1 j2π 100 KHz, s2 j2π 3 GHz, and s3 j2π 6 GHz. In order to inlude also the point s0 0 among the other expansion points, a non-singular R in 25) would be neessary. It an been shown [22] that a matrix of the form suh as in 25) is non-singular under the ondition that there are no ut-sets of only apaitors. Unfortunately, eah node in our dieletri disretization is suh a ut-set when dieletri losses are negligible. Therefore, for lossless dieletris the point s0 0 annot be inluded in the multipoint expansion algorithm, and a non-zero low frequeny expansion point is used instead. From our experiments, we have observed that this expansion point restrition is not interfering with auray. For instane in this partiular example, the zero frequeny behavior of the struture has been aurately aptured as shown in Fig. 5, whih is a magnified view of the low frequeny part of the plot in Fig. 4, 8. CONCLUSIONS In this paper we desribed applying the mesh analysis approah to solving for the disretized urrents and harges in a VIE formulation. We showed that the approah leads to a system with provably positive semidefinite matries, making for easy appliation of Krylov-subspae based model-redution shemes to generate aurate guaranteed passive redued-order models. Several printed iruit board examples demonstrated the effetiveness of the strategy. Arguably, it is tempting to assume that the VIE approah is a step bakward, as it involves disretizing volumes instead of surfaes. However, volume integral equation methods are used for magneti analysis of ondutor problems, beause ondutors oupy a vanishingly small region of the problem domain. The same vanishingly small oupany argument an be made for dieletris as well. In addition, sine polarization urrents are not outputs, it
Admittane [1/Ohm] 10 1 1 10 1 10 2 10 3 10 4 10 5 MCM interonnet with dieletri, redued order model with dieletri, full model without dieletri 0 1 2 3 4 5 6 7 frequeny [Hz] x 10 9 Figure 4: Admittane vs. frequeny for the two traes in Fig. 3. Admittane [1/Ohm] 12.749 12.748 12.747 12.746 12.745 12.744 12.743 MCM interonnets. Zoom at DC. mesh analysis, redued order model mesh analysis, full model 12.742 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 frequeny [Hz] Figure 5: Magnified view of the low frequeny part of the plot in Fig. 4, to verify that the redued model ontinuous line) aptures orretly the DC behavior of the original system irles). might be possible to align them with a regular grid. Suh an alignment might improve the performane of fast solvers, suh as the Conjugate Gradient FFT CGFFT) [23] or Preorreted-FFT [24] methods. This is an important onsideration sine suh solvers are required when using any integral formulation on models with ompliated geometries. 9. 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