Proximity Tmplats for Modling of Skin and Proximity Effcts on Packags and High Frquncy Intrconnct Luca Danil Massachustts Instit. of Tch. dluca@mit.du Albrto Sangiovanni-incntlli Univ. of California, Brkly albrto@cs.brkly.du Jacob Whit Massachustts Instit. of Tch. whit@mit.du ABSTRACT Modling th xponntially varying currnt distributions in conductor intriors associatd with high frquncy intrconnct bhavior causs a rapid incras in th computation tim and mmory rquird vn by rcntly dvlopd fast lctromagntic analysis programs. In this papr w dscrib a procdur to gnrat numrically a st of basis functions which fficintly rprsnt conductor currnt variation, and thus improving solvr fficincy. Th mthod is basd on solving a squnc of tmplat problms, and is asily gnralizd to arbitrary conductor cross-sctions. Rsults ar prsntd to dmonstrat that th numrically computd basis functions ar svn to twnty tims mor fficint than th commonly usd pic-wis constant basis functions. Catgoris & Subjct Dscriptors: B.7. Simulation, B.8. Prformanc Analysis & Dsign Aids, I.6 Simulation & Modling. Gnral Trms: Algorithms, Prformanc. Kywords: Skin Effct, Proximity Effct, Parasitic xtraction, Intrconnct analysis.. INTRODUCTION Th nw gnration of fast lctromagntic analysis programs, basd on acclratd intgral quation mthods, has rducd from days to minuts th tim rquird to analyz thousands of simultanously intracting conductors [,, 3,, 5]. As good as ths fast solvrs ar, thy ar ithr inappropriat for, or ar vry infficint at, analyzing intrconnct xhibiting high frquncy ffcts. With procssor clock spds now xcding two gigahrtz and harmonics xcding twnty gigahrtz, it is no longr possibl to ignor ths high frquncy ffcts. Th high frquncy ffcts that ar most troublsom for fast solvrs ar skin and proximity ffcts. Nvrthlss such phnomna can significantly affct intrconnct prformanc and should not b nglctd, in particular whn ithr wir width or thicknss ar qual to, or largr than two skin-dpths. As clock frquncis ris, skin ffcts hav bcom mor significant in printd circuit This work was supportd by th MARCO Intrconnct Focus Cntr and th Smiconductor Rsarch Corporation. Prmission to mak digital or hard copis of all or part of this work for prsonal or classroom us is grantd without f providd that copis ar not mad or distributd for profit or commrcial advantag and that copis bar this notic and th full citation on th first pag. To copy othrwis, to rpublish, to post on srvrs or to rdistribut to lists, rquirs prior spcific prmission and/or a f. ICCAD Novmbr -,, San Jos, California, USA. Copyright ACM...$5.. boards (PCB) and IC packags, and such ffcts ar vn bcoming important for on-chip intrconnct. Whn on considrs that th skin dpth in aluminum drops blow a micron at approximatly tn gigahrtz, it is not surprising that skin ffcts ar sn in IC s with multi-gigahrtz clocks. Skin and proximity ffcts ar troublsom for prsnt fast solvr bcaus thy gnrat an xponntially varying currnt distribution insid ach conductor. Trying to rprsnt that currnt variation using th pic-wis constant [6, 7] or pic-wis linar basis functions commonly availabl in fast solvrs [] rquirs a larg numbr of unknowns. Sinc th computation tim for fast solvrs is supposd to incras only linarly with th total numbr of basis functions usd in th problm, it may sm that th incras in unknowns to rprsnt currnt variation is not that problmatic. Howvr, whn many basis functions ar usd to rprsnt th currnt variation in a cross-sction of a conductor, thos basis functions dnsly intract in a way that can not b rducd by th algorithms usd in most fast solvrs. For this rason, th computation tim for modling high frquncy ffcts incrass with th squar of th numbr of unknowns rquird to modl th currnt variation within conductors vn for fast solvrs. Th rapid incras in fast solvr computation tim associatd with modling high frquncy ffcts has focusd rsarch fforts on finding mthods to ithr avoid rprsnting currnts in conductor intriors [8, 9,,,, 3], or to gnrat spcializd basis functions which mor asily captur th xponntial variation of th conductor currnt [, 5]. In this papr w dmonstrat that it is possibl to gnrat numrically a st of basis functions which fficintly rprsnt conductor currnt variation. Our mthod is basd on solving a squnc of simpl tmplat problms for th typical gomtris associatd with a givn intrconnct tchnology. Ths tmplat problm solutions ar thn usd as rplacmnt for th pic-wis constant basis functions in an intgral quation mthod basd on th Galrkin discrtization [6] of th Mixd-Potntial Intgral Equation (MPIE) [6, 7, ]. As our rsults will dmonstrat, th numrically computd basis functions rquir 7 to tims fwr unknowns than pic-wis constant basis functions. It should b notd that similar prformanc was achivd by gnrating basis functions using -D conduction mods [, 5], but unlik th conduction mod approach, th tmplat approach is asily xtndd to gnral shap cross-sctions (.g. trapzoidal). In th nxt sction, w summariz th intgral quation mthod basd on th Galrkin discrtization of th Mixd-Potntial Intgral Equation (MPIE). In Sction 3 w dscrib th stps for th prcomputation of our tmplat basis functions. In Sction w show how to us th tmplats in th Galrkin intgral quation mthod undrlining som numrical implmntation issus. Finally in Sction 5 w prsnt svral xampl rsults on typical IC, packag
' '... and PCB simpl intrconnct structurs.. BACKGROUND. Th Mixd Potntial Intgral Equation Formulation (MPIE) As in [6, 7, ] th following st of intgral quations can b usd for th solution of th conductor currnt distribution, J, and of th conductor surfac charg, ρ, J r σ jω µ π πε S J r j ω c r r r r dr φ () ρ r j ω c r r r r dr φ r () J r (3) ˆn J r jωρ r () whr and S ar th union of th conductor volums and surfacs, φ is th scalar potntial on th conductor surfacs, µ is th magntic prmability, ε is th dilctric constant in fr spac, σ is th conductors conductivity, ω is th angular frquncy of th conductor xcitation and c is th spd of light.. Discrtization and Galrkin procdur In ordr to solv ()-() numrically, it is common to introduc an approximat rprsntation of th volum currnts and surfac chargs as a wightd sum of a finit st of basis functions w j C 3 and v m C as in J r w j r I j (5) j ρ r m v m r q m (6) whr I j and q m ar th basis function wights. According to th Galrkin procdur [6], on can substitut th rprsntations (5) and (6) into () and (), and thn rquir that th quation rsiduals ar orthogonal to th basis functions. In this way on gnrats a systm of quations for th wights, j w j r I j σ jωµ π w j j r j ω c r r I j r! r dr φ " w i#%$ v m r j ω c r r q m r! r dr! φ r&" v l# $ " πε S m whr th innr products ar dfind as f r ( w i r *) g r, v l r *) w i r + f r dr (7) v l r g r dr- (8) S As in [7], a matrix quation is obtaind of th form R jωl P / I q / φ / (9) whr I and q ar vctors of currnt and charg basis function wights, rspctivly, and φ and ar th vctors gnratd by innr products of th surfac potntial or th volum potntial gradint with th basis functions. From th Galrkin condition, using th sam notation as for instanc in [] th matrics R, L and P ar R i j σ L i j P lm µ π πε w i r + w j r dr () r + w j r j w c r r dr dr () S w i v l S r r r v m r j w c r r r r dr dr- () Equation (9) can b combind with th rmaining quations (3) and () in svral ways [6, 7, 8], to gnrat a dns systm of quations for th basis function wights. Itrativ mthods combind with fast matrix-vctor product tchniqus ar thn commonly usd to solv such systm [,, 3,, 5]..3 Th classical pic-wis constant basis functions approach A classical choic for th basis functions is to us pic-wis constant functions. For instanc small panls with a uniform charg distribution can b usd for th surfac charg []. Short thin filamnts with a uniform currnt distribution can b usd for th volum currnts [6, 7, ]. Th currnt basis functions ar gnratd by first subdividing long wirs into a larg numbr of sctions that ar short compard to th wavlngth of th highst frquncy of intrst. Th intriors of ach conductor sction is thn dividd into a bundl of paralll thin filamnts. Most fast mthods for solving intgral quations ar basd on approximating distant intractions. Such mthods can not b usd to rduc th cost of computing th intraction btwn th thin filamnts in ach conductor sction, as ths filamnts ar not distant from ach othr. Th intractions btwn filamnts in a sction must b rsolvd dirctly, and it thn follows that th cost of using many filamnts for ach sction grows as th squar of th numbr of filamnts pr sction. Thrfor, finding a diffrnt basis which uss fwr functions to rprsnt currnt distribution in ach conductor sction is an important fficincy considration, vn whn using fast solvr. 3. PRE-COMPUTATION OF THE PROXIM- ITY TEMPLATE BASIS FUNCTIONS In this Sction, w dscrib our procdur to construct a st of tmplat basis functions for th discrtization of th conductor volums within th contxt of an intgral quation lctromagntic fild solvr. As in th classical pic-wis constant approach [6, 7, ] dscribd in Sction.3, or as in th conduction mods approach [, 5], w assum that th currnt flows only along th lngth of th conductors, and that long conductors ar subdividd into sctions short compard to th smallst wavlngth of intrst. W thn catgoriz and labl ach conductor sction according to its cross-sction typ. Each typ is uniquly idntifid by its cross-sction dimnsions and shap. For instanc, for wirs with a trapzoidal cross-sction: width, thicknss and tching slop could b usd as idntifying paramtrs. Oftn whn prforming an lctromagntic analysis, on is intrstd in th currnt (or filds) distribution at a particular xcitation frquncy, or in th impdanc at som trminals for svral xcitation frquncis. For ach frquncy of intrst and for ach wir cross-sction typ, w pr-comput off-lin a st of proximity tmplat basis functions. Each basis function is constructd by solving a small simulation xprimnt:
x 5 x 5 x 5 x 6.5.5 5 5 x 5 Figur : Exampl of th first proximity tmplat basis function. On th lft, w show a cross-sction of th simulation xprimnt stup usd to pr-comput th basis function. Th basis function is dfind as th currnt dnsity (shown on th right) rsulting on th cross-sction of th wir.. Givn a cross-sction typ, for th construction of th first tmplat basis function w considr on wir not intracting with any othr wir, and xcitd with a unity currnt sourc at th frquncy of intrst. For th solution of this simpl problm w us a vry fin pic-wis constant thin filamnt discrtization mthod [6, 7, ]. W thn choos as basis function th currnt dnsity profil drivd on th ntir cross-sction by this analysis. W show on th lft of Fig. th thin filamnt discrtization of th wir cross-sction and to its right th rsulting cross-sction currnt dnsity that w us as first basis function. In Fig. th wir cross-sction is rctangular, but in gnral cross-sction shaps can b handld in th sam way by our procdur. From an intuitiv point of viw, th tmplat basis function dscribd hr is usd to captur skin ffct phnomna.. Othr basis functions ar thn constructd to captur proximity ffct phnomna. In ordr to captur proximity ffct phnomna du to wirs on th sid, w construct a scond tmplat basis function by solving a scond simpl xprimnt. In this scond xprimnt w considr two wirs not intracting with any othr wir. Th cross-sction of th main wir is chosn qual to th cross-sction shap and dimnsions for th typ undr considration. A scond auxiliary wir is locatd on on of th two sids of th main wir, as clos to th main wir as th tchnology fabrication procss would allow. Th auxiliary wir is chosn with th minimum width and thicknss allowd by th tchnology fabrication procss. Fig. shows on th lft th cross-sction configuration of th two wirs. For th analysis of this problm w us a classical and vry fin pic-wis constant discrtization for both such wirs. W short thm togthr on on sid, apply a unity currnt sourc at th rmaining two trminals, and solv for th currnt dnsity within th conductors. W finally dfin as scond proximity tmplat basis function th currnt dnsity profil obsrvd on th main wir. On th right in Fig. w show th cross-sctional currnt dnsity of th scond basis function. 3. W procd constructing additional proximity tmplat basis functions using th procdur dscribd in point abov, but vry tim moving th auxiliary wir in a diffrnt location around th main wir, always as clos to th main wir as th tchnology fabrication procss allows. Fig. 3 shows othr two xampls of tmplat basis functions with thir corrsponding xprimnt stups for th sam cross-sction as in Fig. and. 3. Choosing th numbr of tmplat basis functions pr wir cross-sction Mor spcifically, th total numbr of tmplat basic functions prcomputd for ach cross-sction typ can b dcidd according to th following considrations. In som cass, on only nds to us a total of thr proximity tmplats for ach cross-sction typ: a skin ffct tmplat constructd as in Fig., and two sid proximity tmplats, on for th right sid as in Fig. and on for th lft sid (typically symmtric to th on in Fig. ). This choic is typically appropriat for wirs on most Printd Circuit Board (PCB) applications, whr th sparation btwn diffrnt layrs is particularly larg, and proximity ffcts ar only obsrvd in corrspondnc of sid by sid wirs, and not in corrspondnc of wirs on diffrnt layrs. Whn sparation btwn mtalization layrs is small as in packags and in intgratd circuits, on nds to b abl to account for proximity ffcts du not only to wirs sid by sid but also du to wirs on uppr and lowr layrs. In this cas, for thin wirs w us a total of nin tmplats: a skin ffct tmplat, four proximity tmplats constructd using an auxiliary wir movd to ach of th four sids of th main wir, and four proximity tmplats with th auxiliary wir movd to ach of th cornrs around th main wir. Fig.,, and 3, show four out of nin of such tmplats. If appropriat, symmtry of th cross-sction can b xploitd to infr th othr fiv tmplat rsults. Finally, in th cas of considrably wid wirs, in addition to th nin tmplats prviously dscribd, on nds to us a fw mor proximity tmplats to captur appropriatly proximity ffcts du to thin wirs in any location abov or blow such wid wir. In our implmntation w prcomput tmplats using an auxiliary wir that for ach tmplat is movd in diffrnt locations around th main wir. W rmind th radr that th auxiliary wir width an thicknss ar chosn as th minimum allowd by th tchnology procss, and that th auxiliary wir is locatd at th minimum distanc from th main wir allowd by th tchnology procss. For ach tmplat w thn mov th auxiliary wir to locations ach sparatd by tims th auxiliary wir width. 3. Accuracy and basis function rachnss
x 5 x 5 x 5 x 6.5.5 5 5 x 5 Figur : Exampl of th scond proximity tmplat basis function. On th lft, a cross-sction of th simulation xprimnt stup usd to pr-comput th basis function. On th right th basis function itslf: i.. th currnt dnsity rsulting on th main wir cross-sction. Th accuracy of th final solution is rlatd to th ability of th chosn basis functions to xplain most of th cross-sctional currnt dnsity capturing currnt crowing in diffrnt parts of ach cross-sction du to th spcific locations of narby wirs. Mor prcisly, in linar algbra trms: whn considring th cross-sctional currnt dnsity as a vctor, th accuracy of th final solution is rlatd to th ability of th chosn basis functions to span most of th subspac gnratd by all practical currnt dnsity vctors. In gnral th accuracy of th final solution can b arbitrarily improvd if th st of all basis functions that on can choos from is sufficintly rach to span th ntir subspac of all practical solutions. In our cas, in thory th basis function st is quit rach sinc on could incras th accuracy of th final solution by simply adding mor and mor basis functions, on for ach possibl practical location of narby wirs. Howvr w hav obsrvd xprimntally (s Exampls 5. and Exampl 5.) that th procdur in 3 and 3. constructs a much smallr st of basis functions at th sam tim still allowing of a good final solution accuracy. 3.3 Advantags and disadvantags of th proximity tmplat basis functions From th construction procdur dscribd abov in 3 and 3. on can notic svral advantags and disadvantags of our tmplat basis functions in particular whn compard to othr highr ordr basis function choics such as th conduction mods dscribd in [, 5]. Among th advantags it can b noticd that: our tmplat basis functions can handl any wir cross-sction shap, i.. th common trapzoidal cross-sctions du to chmical tching slops. In th conduction mods basis function approach [, 5], instad, only cross-sction shaps for which analytic solutions of th diffusion quation ar availabl can b handld, i.. mainly rctangular and cylindrical cross-sctions. our tmplat basis functions can captur proximity ffcts du to thin wirs abov vry wid wirs (as shown latr in Exampl 5. and Fig. 5) that ar not capturd by th conduction mod approach. Among th disadvantags of our tmplat basis functions, w rmind th radr that: a complt st of tmplat basis functions nd to b prcomputd for ach wir cross-sction typ (i.. shap and dimnsions). Fortunatly, on can furthr obsrv that oftn th actual numbr of wir cross-sction typs on a typical PCB, packag or IC is quit limitd. For instanc th tching slop can b assumd constant for all cross-sctions for a givn procss. Th variability of th wir thicknss is limitd to th numbr of mtalization layrs in th dsign. Also th variability of th wir width paramtr in practical dsigns is oftn limitd to a finit and small st of admissibl valus by dsign ruls or CAD tools. It is also worth noticing that onc th tmplat basis functions ar computd thy can b stord in a fil and r-usd for subsqunt dsigns basd on th sam procss tchnology. Anothr disadvantag of our proximity tmplats compard to th conduction mods is that a complt st of tmplat basis functions nd to b pr-computd for ach frquncy of intrst. Typically on is not intrstd in a larg numbr of frquncis. For instanc in digital intrconnct on is typically only intrstd in th clock frquncy and its first to 5 harmonics. Onc again, on can furthr notic that onc th tmplat functions ar calculatd for a particular frquncy, thy can b stord and r-usd in subsqunt dsigns for analysis at that sam frquncy. Howvr admittdly a significant advantag of th conduction mod basis functions ovr our proximity tmplats is th availability of th conduction mods in analytical form which can b xploitd whn prforming modl ordr rduction. 3. Rprsntation of basis functions As just obsrvd in th prvious Sction, w rprsnt our basis functions with a picwis constant valus of th currnt dnsity on ach small cross-sctional filamnt. In this way for ach basis function w only nd to stor som information on th discrtization schm from which on can asily driv filamnt gomtris (.g. width of cornr filamnt and incrmntal ratio btwn narby filamnts), and a vctor with th valus of currnt dnsity on ach filamnt. On could think of using mor fficint rprsntations in trms of som intrpolation functions in ordr to sav som storag mmory and som computation tim in th Galrkin intgral computations. W xpct howvr to obtain th most advantag by fitting our tmplats basis functions to intrpolation function not only to rprsnt thir shap along th wir cross-sction but also and abov all to captur thir dpndncy from frquncy. In fact, this
x 5 x 5 x 5 x 6.5.5 5 5 x 5 x 5 x 5 x 5 x 6.5.5 5 5 x 5 Figur 3: Exampl of othr two proximity tmplat basis function. On th lft, a cross-sction of th simulation xprimnt stups usd to pr-comput th basis functions. On th right th basis functions thmslvs: i.. th currnt dnsitis rsulting on th main wir cross-sctions. could allow us to prform modl ordr rduction with our tmplat basis functions as fficintly as with th conduction mods. Howvr, w hav not vrifid yt th practical fasibility of such procdur.. PARASITIC EXTRACTION FOR A LARGE COLLECTION OF INTERCONNECT Givn a larg collction of wirs, for a givn frquncy of intrst, ach wir is associatd with th st of pr-computd proximity tmplat basis functions corrsponding to its cross-sction typ. Th basis functions chosn in this way, togthr with a standard Galrkin procdur [6], ar usd to discrtiz th Mixd Potntial Intgral Equation (MPIE) and calculat th ovrall rsistanc R and th partial inductanc L matrics in q. () and () as shown in Sction.. Accumulation of charg on th surfacs of th conductors can still b handld for xampl using th classical picwis constant discrtization of such surfac into small panls as dscribd in Sction.3. A msh analysis tchniqu [7] is thn finally usd to st up a linar systm of quations that can b solvd to find th wights w j and v m associatd with ach singl basis function. From a numrical implmntation prospctiv on can obsrv that th proximity tmplat basis functions as constructd in Sction 3 ar not orthogonal. Th rsistanc matrix for instanc is block diagonal. In gnral, whn th basis functions ar almost linarly dpndnt, thir associatd cofficints rprsnting th final solution may rsult vry larg, similar in magnitud, and possibly of opposit phass partially cancling ach othrs, which may produc rrors whn using a finit prcision rprsntation. On can avoid this problm and achiv bttr numrical stability by ortho-normalizing th basis functions bfor using thm with for instanc a Modifid Gramm-Schmidt procdur [9]. Anothr advantag of orthonormalizing th basis functions is that a compltly diagonal rsistanc matrix is producd, which is convnint for instanc whn prforming a subsqunt modl ordr rduction stp that may rquir an invrsion of such matrix. Th orthogonalization procdur is quit fast and most importantly it is part of th prcomputation phas, hnc it dos not affct th spd and mmory prformanc during th analysis of a vry larg collction of intrconnct. As a final rmark, it can b noticd that our proximity tmplats basis functions can b usd in combination with fast matrix solvrs [,, 3,, 5]. 5. EXAMPLES 5. Capturing proximity ffct btwn two wirs at arbitrary distanc In this Sction, w intnd to show with an xampl that although our proximity tmplats ar constructd using an auxiliary wir vry clos to th main wir, such tmplat basis functions can succssfully captur proximity ffcts du to wirs at any arbitrary distanc. Lt us considr for instanc a typical PCB wir 5µm wid, and 35µm thick. In this xampl w usd a st of thr tmplat basis functions pr cross-sction. On tmplat was constructd using on wir alon with a 5µm x 35µm cross-
x 5 x 3 3 5 3 5 6 7 x.3 3 proximity tmplats pr cross sction 3 thin filamnts pr cross sction 5 x 9 3 proximity tmplats pr cross sction 3 thin filamnts pr cross sction.5.5 R = R(Z) Rsistanc [Ohms]..5. L = Im(Z) / jw Inductanc [H] 3.5.5 3 6 7 8 9 frquncy [Hz].5 6 7 8 9 frquncy [Hz] Figur : Frquncy rspons of two shortd PCB wirs: ral part of th impdanc on th lft, and imaginary part dividd by ω on th right. Th diffrnt curvs rprsnt diffrnt distancs btwn th two wirs: µm, 9µm, 35µm, 8µm, and 69µm (from top to bottom on th lft and from bottom to top on th right). Th continuous lins ar th rsults obtaind using th classical thin filamnt mthod. Th small crosss ar th rsults obtaind using thr tmplat basis functions pr-computd for a minimum sparation distanc µm. sction. A scond tmplat was constructd using on main wir (cross-sction: 5µm x 35µm) in th cntr and on auxiliary wir (5µm x 35µm) on on sid at a sparation distanc of µm. A third tmplat was constructd by moving th auxiliary wir to th othr sid at th sam sparation distanc. For th construction of th basis functions, w discrtizd ach wir into x=3 thin filamnts. Aftr th thr tmplat basis functions hav bn constructd, w usd thm in th intgral quation Galrkin procdur dscribd in Sction. to calculat th frquncy rspons of two wirs with th sam cross-sction at diffrnt sparation distancs: µm, 9µm, 35µm, 8µm, and 69µm. W compar in Fig. th rsult obtaind using our thr proximity tmplat basis functions pr cross-sction with th rsult obtaind using 3 thin filamnts basis functions pr cross sction. Of cours on can xpct a ngligibl rror whn th wirs sparation is xactly qual to th sparation usd for th construction of th basis functions (µm). Howvr, w also obsrvd an qually vry small rror (worst cas.7% rror for th ral part of th impdanc, and a.% rror for th imaginary part dividd by ω) for th cas in which th sparation btwn th two wirs incrasd to an arbitrary distanc and did not coincid anymor with th sparation usd during th construction of th basis functions. 5. Capturing proximity ffct btwn a thin wir in an arbitrary location abov a wid wir From th prvious xampl w hav sn that th proximity tmplats is an approach at last as fficint as th conduction mods approach [] in trms of usd numbr of unknowns. In addition, w show in this xampl that th proximity tmplats can succssfully captur on particular cas not capturd by th conduction mods approach: proximity ffcts btwn a thin wir abov and clos to a vry wid wir. Lt us considr for instanc a packag wir µm wid and µm thick. Lt us pr-comput a st of nin proximity ffcts basis functions for this wir. Fig.,, and 3 show four of such nin basis functions for th cross-sction typ dscribd in this xampl. Th auxiliary wir is µm wid and is movd into svral locations all around th main wir all at a distanc of µm. Aftr th computation of th basis functions, w hav stup th xprimnt on top of Fig. 5. Th small wir is µm x µm, th widr wir right blow it is µm x µm at a µm sparation. W can also notic that th small wir is off cntr by µm so that its location dos not coincid with on of th locations usd for th basis function construction (compar th cross-sction in Fig. 3 with th cross-sction of th gomtry in Fig. 5). Th two rmaining picturs in Fig. 5 compar th cross sctional currnt dnsity rsulting from using our st of nin pr-computd proximity tmplat basis functions (lft), with th rsult (on th right) obtaind using a st of 6x9 = thin filamnts basis functions. W conclud that th proximity tmplats provid accurat rsults not only for wirs at an arbitrary distanc as shown in Exampl 5., but also for wirs locatd in btwn th original locations usd for th basis functions construction. 5.3 A packag powr and ground distribution xampl Finally w show hr a packag powr and ground distribution grid Exampl (Fig. 6). Wirs ar µm wid and 5µm thick. rtical sparation btwn layrs is 5µm. Sid sparation btwn Gnd and dd lins is mm. Th total packag siz is mm x mm. W assumd bond wirs connctions shorting Gnd and dd wirs to an undrnath PCB on all cornrs of th packag. W can no-
x 5...3 x 3..5.6.7.8.9 x 5 x 6 x 6 x 6 8 6 x 5 x 6 8 6 x 5 Figur 5: Packag trac cross sctional currnt dnsity rconstructd from a st of nin pr-computd proximity tmplat basis functions (pictur on th lft), compard to th currnt dnsity (pictur on th right) from using a st of 6x9 = thin filamnts basis functions. A largr currnt distribution can b noticd on on dg and on on cornr of th cross-sction du to th proximity of th small wir. Not that th location of th small wir is off cntr, hnc it dos not coincid with any of th locations usd to pr-comput th nin tmplat basis functions. On can notic that th currnt dnsity is still capturd accuratly. tic that only on typ of cross-sction is prsnt in this dsign. For that cross-sction typ, w hav pr-computd a st of thr basis functions as in Exampl 5.. W hav thn usd our thr basis functions pr sgmnt to discrtiz th ntir gomtry and find th frquncy rspons at on particular nod of th grid: th nod at x = mm and y = mm (s Fig. 6). In our simulations w hav also includd th ffcts of charg accumulation on th surfac of th conductors using a pic-wis constant discrtization into small panls. In Fig. 6 w compar th frquncy rspons of th grid at th nod indicatd abov according to our thr proximity tmplats pr wir vrsus th frquncy rspons obtaind using a thin filamnt discrtization with 5x = thin filamnts pr ach wir sgmnt of th grid. Our approach rquird a total of 8x3= unknowns for th conductor currnts, whil to gt a similar accuracy with th thin filamnt approach w had to us a total of 8x=96 unknowns. In particular, for our proximity tmplats approach w obsrvd from th admittanc phas vs. frquncy curv in Fig. 6 a worst cas rror of.5% in th position of th rsonancs. W obsrvd from th admittanc amplitud vs. frquncy curv a worst cas 7% rror in amplitud at th rsonancs, whr th impdanc is mainly dtrmind by skin ffcts and proximity ffcts. 6. CONCLUSIONS In this papr w hav dscribd a procdur to construct a st of tmplat basis functions for th discrtization of conductor volums in an intgral quation mthod. Th tmplat basis functions ar pr-computd off-lin using small simulation xprimnts. Th tmplats can captur succssfully both skin ffct and proximity ffcts. Our xampls show that compard to th thin filamnt mthods thy provid th sam 7 to improvmnt factors in trms of numbr of unknowns rportd by th conduction mods approach []. In addition th proximity tmplats can b mployd in applications with wir cross-sctions of arbitrary shap, and with proximity ffcts on wid wirs du to abov and clos thin wirs. 7. ACKNOWLEDGMENTS Th authors would lik to thank Dr. Jol Phillips (with Cadnc Brkly Labs) for usful fdback and discussion. 8. REFERENCES [] K. Nabors and J. Whit. Fastcap: a multipol acclratd 3-d capacitanc xtraction program. IEEE Trans. on Computr-Aidd Dsign of Intgratd Circuits and Systms, ():7 59, Novmbr 99. [] M. Kamon, M. J. Tsuk, and J. K. Whit. Fasthnry: A multipol-acclratd 3-d inductanc xtraction program. IEEE Trans. on Microwav Thory and Tchniqus, (9):75 8, Sptmbr 99. [3] J. R. Phillips and J. Whit. A prcorrctd-fft mthod for lctrostatic analysis of complicatd 3-d structurs. IEEE Trans. on Computr-Aidd Dsign of Intgratd Circuits and Systms, 6():59 7, Octobr 997. [] J. Tausch and J. Whit. A multiscal mthod for fast capacitanc xtraction. In Dsign Automation Confrnc, pags 537, Jun 999. [5] S. Kapur and D. Long. Larg scal capacitanc calculations. In Dsign Automation Confrnc, pags 7 9, Jun. [6] A. E. Ruhli. Equivalnt circuit modls for thr dimnsional
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