Driver Modeling and Alignment for Worst-Case Delay Noise

Transcription

1 Drier Modeling and Alignmen for Wors-Case Noise Supamas Sirichoiyakul, Daid Blaauw, Chanhee Oh, Rafi Ley*, Vladimir Zoloo, Jingyan Zuo Moorola Inc. Ausin, TX, *Moorola Semiconducor Israel Ld. Tel Ai, Israel Absrac In his paper, we presen a new approach o model he impac of cross-coupling noise on inerconnec delay. We inroduce a new linear drier model ha accuraely models he noise pulse induced on a swiching signal ne due o cross coupling capaciance. The proposed model effeciely capures he non-linear behaior of he icim drier gae during he ransiion and has an aerage error below 8% whereas he radiional approach using a Theenin model incurs an aerage error of 48%. We also discuss he wors case alignmen of he aggressor ne ransiions wih respec o he icim ne ransiion, emphasizing he need o maximize no merely he delay of he inerconnec alone bu he combined delay of he inerconnec and receier gae. We show ha he wors case alignmen of an aggressor ne ransiion is a funcion of he receier gae oupu loading, icim ransiion edge rae, and he noise pulse widh and heigh and hence propose a pre-characerizaion approach o efficienly predic he wors-case alignmen. The proposed mehods were implemened in an indusrial noise analysis ool called ClariNe. Resuls on indusrial designs are presened o demonsrae he effecieness of our approach. 1 Inroducion and Preious Work Due o process scaling, cross-coupling capaciance has become a dominan porion of he oal parasiic inerconnec capaciance. As preiously obsered [1], he inerconnec delay of such nes are srongly dependen on wheher heir neighboring nes are simulaneously swiching or no. The ne under consideraion is referred o as he icim ne, and he neighboring nes ha are capaciiely coupled o i are referred o as aggressor nes. When he aggressor nes swich, hey induce a noise pulse on he icim ne. If he icim ne is sable when he aggressors swich, he resuling noise pulse can cause a funcional failure. This siuaion is referred o as funcional noise. If he icim ne iself is also swiching when he aggressors swich, is delay can eiher increase or decrease depending on he aggressor and icim swiching direcions. This is referred o as delay noise and is he focus of his paper. daid_blaauw@ .mo.com Since he inerconnec is conneced o non-linear drier and receier gaes, he coupled delay analysis is an inherenly non-linear problem. Howeer, non-linear simulaion is no pracical due o he large size of he inerconnec model which can consis of many housands of elemens for a single icim ne. To efficienly address he analysis of large designs, linear models of he drier and receier gaes are ypically consruced o allow he use of efficien linear simulaion and superposiion. The drier gae is radiionally modeled wih a Theenin model consising of a Theenin resisance and a olage source wih a ramp ransiion. The receier gae loading is modeled wih a grounded capacior. Figure 1(a) shows he linear model for a icim ne wih wo aggressor nes. Using superposiion, each of he drier gaes is simulaed in urn, while oher Theenin olage sources are shored. Figure 1(b) shows he simulaion model when aggressor drier A is simulaed, i.e. swiching. A similar model is used o simulae aggressor drier B. Figure 1(c) shows he simulaion model used o simulae he icim drier. The olage waeforms obsered a he receier inpu from all simulaions are added ogeher using superposiion o obain he noisy waeform as shown in Figure 1(d). Typically, he amoun of delay noise has been measured as he difference of he 5% Vdd crossing imes of his noisy waeform and he noiseless waeform obained from he simulaion in Figure 1(c). Alernaiely, he receier gae can be simulaed wih he noisy waeform as shown in Figure 1(d) and he delay noise is hen measured a he receier oupu insead. Using linear drier and receier models has he adanage ha a reduced-order model of he nework needs o be creaed only once wih mehods such as PRIMA [2], and is hen reused in all differen drier simulaions., d, d, d, d RAh RVh RBh A V B RAh A RVh V RBh B 65 (a) Linear Model 65 (c) Vicim Swiching receier, d icim RAh A RVh V 65 RBh B aggressor A aggressor B This use of he Theenin model is based on he sandard approach in he analysis for non-coupled inerconnecs. Theenin gae model parameers (, d, and R h ) are a funcion of he effecie load ha he drier gae sees. I reflecs he fac ha he drier is acually a non-linear deice, which is approximaed wih differen Theenin models a differen effecie gae loading alues. The effecie icim (b) Aggressor A Swiching receier (d) Receier Simulaion Figure 1. Inerconnec analysis using linear simulaion and superposiion a aggressor A

2 loading of a gae is calculaed using so-called C-effecie ieraions [3][4] and capures he resisie shielding of he inerconnec. For a paricular effecie load, he Theenin model parameers are opimized o obain a good correspondence wih he non-linear drier simulaion a he 1%, 5%, and 9% ransiion imes. The described approach for a coupled inerconnec nework simulaes he icim drier and each aggressor drier in urn. When an aggressor drier is simulaed (Figure 1(b)), he icim and oher aggressor driers are modeled wih heir Theenin olage source grounded, i.e. heir Theenin resisances are conneced o ground. These grounded resisances, or holding resisances, represen he abiliy of he hese driers o hold heir signal lines seady while he simulaed aggressor gae injecs noise hrough he coupling capaciances. Howeer, he Theenin resisance has been calculaed o model he aggregae resisance of he drier oer an enire ransiion of a gae whereas he noise from he simulaed aggressor is injeced for only a shor period of ime while he icim is swiching. Since he small signal conducance of he drier gae aries dramaically during he ransiion, an accurae holding resisance is a funcion of he duraion of he injeced noise and is alignmen relaie o he gae ransiion. I is hus clear ha he sandard Theenin resisance is no a good approximaion o model he grounded driers in he superposiion flow for coupled inerconnecs. In Figure 2, we show an example of a coupled icim and aggressor nework whose noise pulse compued using he Theenin resisance for icim drier significanly underesimaes he acual noise injeced on he icim ne. V dd /2 Using Theenin drier and non-linear drier Using Theenin drier Using non-linear drier Using Theenin drier Using non-linear drier Figure 2. Simulaion resuls using Theenin model To address his issue, a modified C-effecie calculaion approach has been proposed in [5], [6]. I accouns for he addiional charge ha a swiching drier gae sees when oher gae driers are swiching simulaneously. In his approach, he Theenin model parameers are updaed using a modified effecie loading capaciance ha accouns for he charge injeced due o he swiching aggressor nes. Since his charge is aeraged oer he enire ransiion, he exac small signal conducance of he non-linear icim drier gae during he shor period ha he aggressor swiches canno be accouned for. We propose a new approach which models he icim drier gae wih a differen resisance when is olage source is shored in he superposiion flow (Figure 1(b)). This new resisance model is referred o as he ransien holding resisance -R r. The ransien holding resisance is a funcion of he noise widh, heigh, and alignmen relaie o he icim ransiion. I is compued wih a single non-linear simulaion of he drier gae using a single effecie oupu load. For efficiency, i can be precharacerized and sored in a able similar o ha for he Theenin model. Noe ha a similar problem occurs when he icim drier is swiching and he aggressor driers are shored in he superposiion flow (Figure 1(c)). In his case, he noise pulse injeced on he aggressor ne by he icim will be underesimaed due o he Theenin resisance used for he aggressor drier. Howeer, he olage on he aggressor ne is no of direc ineres o our analysis and has only an indirec effec on he icim ne. This explains why he noiseless icim ransiion using a sandard Theenin model shown in Figure 2 is quie accurae. Howeer, he proposed approach can also be exended o he shored aggressor drier models o calculae heir ransien holding resisances if needed. Afer all linear models are calculaed for he drier gaes, he remaining issue is how o align he ransiion of he aggressor nes relaie o he ransiion of he icim ne. The aggressor nes mus be aligned wihin he consrains of he swiching iming windows ha are calculaed during iming analysis [1]. One difficuly is ha he iming windows are a funcion of he added delay due o cross coupling noise, and his added delay is in urn a funcion of he aggressor iming windows. In [8][9], i was shown ha ieraiely calculaing he iming windows and he added noise delay will conerge on he correc soluion. In pracice, ery few ieraions are needed for conergence. The ask ha we examine in his paper is o deermine he swiching ime wihin he consrains of he iming windows ha produces he wors case icim delay in each ieraion. We approach his problem in wo seps: Firs, we deermine he wors alignmen of he aggressors relaie o each oher. This will produce a composie noise pulse which is he superposiion of all aggressor induced noise pulses. Second, we deermine he wors case alignmen of he composie noise pulse wih respec o he icim ransiion ime. Typically, he objecie has been o maximize inerconnec delay, which is measured from 5% crossing ime of he icim drier oupu o he 5% crossing ime of he icim receier gae inpu. In [6] i was shown ha under reasonable assumpions, his delay is maximized by aligning all aggressor noise pulses such ha heir peaks occur a he same ime. The peak of his composie noise pulse is hen aligned a he poin where he noiseless icim ransiion reaches Vdd/2 + V n for a rising ransiion, where V n is he heigh of he composie noise pulse, as shown in Figure 3. In iming analysis, howeer, he rue objecie is no o maximize only he inerconnec delay, bu he combined delay of he inerconnec and he receier gae, measured from 5% crossing ime of he icim drier oupu o he 5% crossing ime of he icim receier gae oupu. Figure 3 also shows ha aligning he composie noise pulse for he wors inerconnec delay may resul in a combined inerconnec and receier delay no being increased a V dd /2+V n V dd /2 V n V n receier inpu receier oupu noise waeform Figure 3. Wors case alignmen a receier inpu

3 all. This occurs when he alignmen for maximizing he inerconnec delay places he aggressor ransiion oo lae and he receier gae has already compleed is ransiion. In his siuaion he noise pulse a he receier inpu is quie large and a correc alignmen of he aggressor would hae significanly increased he delay a he receier oupu. Noe ha in his case, due o he effecie filering of he receier gae, he noise pulse a he receier oupu is less han 1mV and does no consiue a funcional noise failure. I is herefore clear ha aligning he aggressor ransiion based solely on maximizing he inerconnec delay is no alid and ha he effec of he alignmen on he receier oupu ransiion mus be considered. When he receier delay is included in he aggressor alignmen objecie, he wors case aggressor alignmen becomes a funcion of he receier gae ype, size, P/N raio, and oupu load. Furhermore, he receier gae is highly non-linear, making efficien closed form soluions difficul. In his paper, we firs examine he wors case alignmen of he aggressor ransiions wih respec o each oher. We show ha he wors case alignmen does no always occur when all aggressor noise pulses hae coinciden peaks. In hese cases, howeer, he receier delay is relaiely insensiie o he exac alignmen of he aggressor peaks, and we show ha using aligned noise peaks inroduces only a ery small amoun of error. Second, we propose an effecie pre-characerizaion approach o calculae he alignmen of he composie noise pulse relaie o he icim ransiion ime. Alhough he wors case alignmen is a funcion of he noiseless ransiion ime, he noise pulse heigh and widh, and he receier gae loading, we can predic he wors case alignmen of any possible condiion using he precalculaed alignmen of a small se of condiions. This pre-characerizaion approach has he adanage ha for a paricular ype of receier gae, we precalculae he wors case alignmen for a small se of condiions, afer which he alignmen for any insaniaion of he gae is obained easily hrough able lookup and inerpolaion. In a coupled inerconnec nework, he linear drier models are a funcion of aggressor alignmen and, conersely, he alignmen is a funcion of he noise widh and heigh, which are in urn a funcion of he linear drier models. Hence, i is impossible o deermine one wihou firs deermining he oher. In he oerall approach, we ierae beween he linear model calculaion and he alignmen calculaion o reach conergence. The oerhead in each ieraion is relaiely small because he linear model calculaion inoles only one non-linear simulaion of he icim drier circui and he alignmen calculaion inoles only able lookup and inerpolaion operaions. In pracice we find ha only one or wo ieraions are needed. The remaining of his paper is organized as follows. Secion 2 presens he mehod for calculaing he ransien holding resisance needed o model he grounded icim drier model in he superposiion flow. Secion 3 presens he mehods for calculaing aggressor alignmen. Secion 4 presens he resuls of he proposed approach, and Secion 5 presens our conclusions. 2 Vicim Drier Model In he proposed superposiion flow, he olage source of he icim drier model is shored when simulaing he noise injeced by an aggressor drier as shown in Figure 1(b). The icim drier is hen represened only by he Theenin resisance R h. This model inroduces a significan error as i does no represen he gae conducance during he ime of he noise injecion, as illusraed earlier in Figure 2. We propose a more accurae model by replacing he sandard Theenin resisance R h wih a ransien holding resisance R r. We deermine his ransien holding resisance such ha i produces a maching noise waeform wih noise injeced on he nonlinear icim drier. Our approach is oulined as follows. Firs, we obain he aggressor noise on he icim ne by performing a linear simulaion using he sandard Theenin resisance for he shored icim drier as in he original approach shown in Figure 1(b). From he simulaion resul, we calculae he associaed noise curren ha is injeced ino he icim drier oupu. Then, we simulae he non-linear icim drier swiching wih an effecie load, boh wih and wihou his injeced noise curren. Since he icim drier is swiching when he aggressor drier injecs noise on he icim, we canno direcly obsere he noise injeced on he icim line. We can only consruc i from he difference of he drier responses wih and wihou injeced noise. Thus, we subrac he wo drier oupu waeforms o obain he noise waeform a he icim drier oupu. We hen calculae a ransien holding resisance ha yields a maching noise pulse. Each of hese seps is explained in more deail below: 1. Using Theenin models for he icim and aggressor driers, we simulae one aggressor drier a a ime while grounding he icim and all oher aggressor models. In each simulaion we record he olage waeform a he icim drier oupu and hen calculae he oal noise olage V n as he sum of all olage waeforms. 2. Using he simplified model shown in Figure 4(a), we calculae he curren waeform I n injeced ino he drier gae as follows: I n = V n R h + C load ( V n ) ( ), where R h is he icim drier Theenin resisance, and C load is he effecie load capaciance as calculaed wih C-effecie ieraions. R h (R r ) I n V n (V n ) C load (a) icim drier gae V 1 (V 2 ) C Load Figure 4. Transien holding resisance (R r ) calculaion 3. We perform a non-linear simulaion of he icim drier gae wih C load a he oupu o obain a noiseless ransiion V 1. We repea his simulaion wih he added curren source I n obained from Sep 2 conneced a he gae oupu, and measure he olage waeform V 2 a he gae oupu, as shown in Figure 4(b). 4. We calculae he noise olage response of he non-linear model, V n, by subracing he wo non-linear simulaion resuls: V n = V 1 - V Finally, we consruc he equialen linear model wih he ransien holding resisance R r by replacing R h in Figure 4(a) wih R r. We deermine he alue of R r such ha he area under he resuling noise olage waeform V n maches he area under V n. The alue of R r is calculaed as follows: (I n ) (b) 1 2 ' n

4 I n = V n R r + C load ( V n ) ( ) In d 1 = end = R r V n d + C load V n = Since V n is a noise waeform which will reurn o is original alue a end, V n ( ) = V n ( end ), i.e. V n =. = = end Also, o mach he area of V n and V n, we replace V n d wih V n d. Thus: R V n d r = In d where In d and V n d are obained from Sep 2 and Sep 4, respeciely. 6. We calculae he noise waeform by performing a linear simulaion using R r in place of he icim drier Theenin resisance R h in he circui shown in Figure 1(b). Then we coninue wih he radiional linear simulaion and superposiion approach described earlier. Noe ha afer Sep 6, he noise curren I n has changed requiring a recalculaion of R r. Howeer, in pracice a single or a mos wo ieraions are necessary. Also, when he alignmen of he aggressor ransiion changes wih respec o he icim ransiion, he non-linear noise waeform will be affeced, and R r mus be recalculaed. In Figure 5, we show he simulaion resuls when he proposed approach is applied on he circui producing he waeforms shown in Figure 2. The resul shows ha he olage waeforms mach closely wih he full non-linear simulaion resuls. In his case, he calculaed ransien holding resisance, R r is 1463 Ohms, whereas he original heenin resisance was 123 Ohms. V dd /2 Using Theenin drier and non-linear drier Using Theenin drier Using R r Using non-linear drier Using Theenin drier Using R r Using non-linear drier Figure 5. Linear noise simulaion using R r 3 Aggressor Alignmen for Wors-Case The inerconnec and receier delay are srongly dependen on how he noise waeforms are aligned wih respec o he icim ransiion. We approach his problem in wo seps. Firs we deermine he alignmen of he aggressor wih respec o each oher, forming a composie noise waeform. Then, we align his composie noise waeform wih respec o he icim ransiion. We discuss each of hese wo issues in more deail below. 3.1 Alignmen Among Aggressors Tradiionally, he noise waeforms induced on he icim ne are aligned such ha heir peaks coincide. Such an alignmen will produce a composie noise pulse wih a maximum pulse heigh and minimum noise pulse widh. Conersely, shifing he alignmen of he indiidual noise peaks will resul in a wider and less high composie noise pulse. When considering only he inerconnec delay, a composie noise waeform wih aligned aggressor noise pulses will ypically resul in he maximum delay. Howeer, as we hae discussed earlier ha considering only he inerconnec delay is no meaningful, and he receier delay mus be included. Since he receier gae acs as a low pass filer, he maximum heigh composie noise pulse may no always resul in he maximum response a he receier oupu. Especially when he receier gae has a large capaciie load, a composie noise pulse wih a lower peak olage and wider widh can resul in a sronger response a he oupu. Figure 6 shows he combined inerconnec and receier delay of a circui wih wo aggressor nes similar o Figure 1(a) under arying alignmens. In one case, he receier gae has a small oupu load, allowing i o pass a high frequency noise pulse relaiely well. In his case, he wors aggressor alignmen occurs when he noise peaks of boh aggressor nes coincide. Howeer, when he same receier gae has a large oupu load, i acs more srongly as a low pass filer, and he wors aggressor alignmen occurs when heir peaks are no aligned and a wider and less high composie noise pulse is presened o he receier gae (ns) Load = 15ff Load = 5ff Aggressor Alignmen (ns) Aggressor aggressors aligned wors case delay Haing o consider non-aligned aggressor peaks grealy expands he search space for he wors case aggressor alignmen and makes he problem more complicaed. Howeer, in he cases where he wors case delay occurs wih non-aligned aggressor noise peaks, he error inroduced by using aligned noise peaks is ery small. In Figure 6, for example, he delay difference a he receier oupu is only 2.7 ps. In general, he cases where he wors case delay is produced by non-aligned aggressor noise peaks are when he icim ransiion is relaiely fas, he aggressor ransiion is relaiely slow, or he receier oupu load is large. In all of hese cases, he exra delay is relaiely small and insensiie o he alignmen. Therefore, we can align all aggressor peaks ogeher wihou incurring a large error in he delay calculaion. In all our simulaions, he error inroduced by his approximaion is less han 5 percen. 3.2 Alignmen wih Respec o he Vicim Transiion Afer he composie noise pulse has been consruced, is alignmen relaie o he icim ransiion mus be deermined. The wors case alignmen is complicaed by he non-linear behaior of 1.5 Alignmen (ns) Figure 6. s. alignmen for 2 aggressors

5 he receier gae and by he fac ha i is a funcion of he receier gae ype, gae size, oupu load, and he composie noise waeform and noiseless ransiion edge rae a he receier inpu. Finding he acual wors case alignmen inoles performing an expensie search using a large number of non-linear simulaions. This is clearly oo expensie o perform during iming analysis. We herefore propose a new pre-characerizaion mehod where he wors case noise alignmen is sored in a lookup able. Since he number of parameers ha influence he wors case alignmen is ery high, he number of daa poins needed o build a simple linearly inerpolaed lookup able would be unaccepable. For insance, if for a paricular gae he four dimensions (oupu load, inpu noise pulse widh / heigh, and inpu edge rae) were precharacerized a 1 poins each, a oal of 1, pre-characerizaion poins would be required, which is unaccepable. In our approach, howeer, we were able o simplify he lookup able, such ha only 8 pre-characerizaion poins are required, while mainaining an accuracy wihin 1%. The dependence of he wors case alignmen on all four parameers is discussed in more deail below. Receier oupu load capaciance. To undersand he behaior of he receier gae wih respec o is oupu load, Figure 7(a) shows he oal delay (he combined inerconnec and receier delay) as a funcion of he composie noise pulse alignmen for differen receier oupu load capaciance alues. The simulaion shows ha for small receier loads, he alignmen is ery sensiie and een a small shif in alignmen can produce a dramaic change in he delay. Howeer, for large oupu loads, he delay is relaiely insensiie o he alignmen and a deiaion in he alignmen resuls in only a small error in he added delay. In our approach we herefore use he wors case alignmen a minimum receier oupu load for all loading condiions for he receier gae. From Figure 7(a), i is clear ha his will inroduce only a small error for he case where he receier gae has a large capaciie load. Vicim Edge Rae. The wors-case alignmen exhibis a non-linear relaionship as a funcion of he edge rae, if he alignmen is measured from he sar of he noiseless icim ransiion. Howeer, when we measure he alignmen wih respec o he 5% crossing ime of he icim ransiion, he relaionship closely approximaes a linear funcion. To illusrae his, Figure 7(b) shows he oal delay as a funcion of he composie noise pulse alignmen for differen icim ransiion imes wih he alignmen measured relaie o he 5% Vdd crossing ime of he icim ransiion. Since he wors case alignmen is nearly linear wih respec o he icim ransiion ime, we need o precharacerize a gae for only minimum and maximum icim ransiion ime and linearly inerpolae for poins in beween. To deermine he wors case alignmen for differen icim slopes and receier oupu loads, we herefore need only wo pre-characerizaion poins, one a maximum icim slope and one a minimum icim slope. Boh pre-characerizaions are performed wih minimum receier oupu load. Figure 9(a) shows he accuracy of his alignmen predicion approach for all possible icim slopes and receier loads for a ypical gae. In all cases he error is less han 7%. Noise heigh and widh. The wors case noise alignmen ime is no a linear funcion of he noise pulse heigh and widh, complicaing he generaion of an efficien able. Insead, we use he receier inpu olage a he ime poin when he noise pulse reaches is peak, which we refer o as he alignmen olage (olage V a in Figure 1(d)). When considering he noise alignmen in erms of is alignmen olage, he noise pulse widh and heigh is linearly dependen on he alignmen. Figure 8(a) and Figure 8(b) show he oal delay as a funcion of he alignmen olage for arying noise pulse widhs and heighs, respeciely. We precharacerize he alignmen olage a he four condiions corresponding o he minimum and maximum pulse widh and pulse heigh. Noe ha we can always calculae he alignmen ime from he alignmen olage and he icim ransiion ime. Figure 9(b) shows he error in he calculaed delay using his alignmen predicion approach for all possible noise pulse widhs and heighs. In all cases he error is less han 8% increasing noise widh Volage a Noise Peak (a) For arious pulse widhs (b) For arious pulse heighs Figure 8. as a funcion of alignmen olage The oerall pre-characerizaion approach uses 8 receier gae condiions - 2 poins in each of he pulse widh, pulse heigh, and icim slope dimensions, all wih he minimum receier oupu load. For each case, a wors case alignmen olage is recorded in a able. During he acual noise analysis run, he acual alignmen is calculaed by performing linear inerpolaions of he alignmen olages in noise widh and heigh dimensions, mapping he resuling alignmen olages o alignmen imes, and hen linearly inerpolaing he alignmen ime in he receier inpu dimension increasing noise heigh Volage a Noise Peak.45.4 increasing load % Error % Error Alignmen (a) For differen receier loads (b) For differen icim slopes Figure 7. as a funcion of noise alignmen increasing slope Alignmen Vicim Slope (ns) Receier Load (ff) Noise Heigh.4 (V) Figure 9. Error plo for prediced alignmen wih (a) arying receier load and icim slope (b) arying noise heigh and widh Noise Widh (ns) 2

6 4 Resuls The proposed algorihms were implemened in an indusrial noise analysis ool called ClariNe, which has been used on a number of chip designs [7]. The resuls for 3 nes from a high performance microprocessor block are shown in Figure 13 and Figure 14. Figure 13 shows he accuracy obained wih he proposed ransien holding resisance calculaion. The calculaed delay using linear simulaion wih eiher he original Theenin resisance or our proposed ransien hold resisance are ploed on he Y-axis, and are compared wih he delay obained using Spice simulaion of he full non-linear circui, ploed on he X-axis. The resuls show ha he ransien holding resisance has a significanly higher accuracy, wih an aerage error of 7.41% compared o he Theenin resisance, wih a aerage error of 48.63%. Moreoer, he Theenin resisance incurs a higher error for nes wih a larger delay and in all cases underesimaes he delay, which is undesirable for noise analysis. Linear Model Exra Resuls (ps) * Theenin R resuls + Transien holding R resuls - Perfec mach Non-linear model exra delay resuls (ps) Figure13. Linear model resuls s. non-linear simulaion In Figure 14, he delay using prediced alignmen (ploed on he Y-axis) is compared wih he delay using an exhausie search of he wors case alignmen (ploed on he X-axis). The prediced alignmen ha maximizes he delay a he receier inpu using he mehod presened in [5] is compared wih our mehod which aims o maximize he delay a he receier gae oupu. Our proposed mehod has a significanly higher accuracy, wih a wors case error of 15ps, compared o he approach in [5] which incurs a wors case error of 31ps. 5 Conclusions In his paper, we hae presened a new approach o accuraely calculae he exra delay due o cross coupled noise injecion. We hae proposed a new linear model ha accuraely capures he nonlinear behaior of he icim drier gae when noise is injeced from aggressor nes. Resuls show ha his model significanly reduces he error in he calculaed noise. The model is obained hrough a simple simulaion of he drier gae and can be precharacerized for gaes prior o noise analysis. For deermining he alignmen of he aggressor noise pulses relaie o he icim ransiion, we hae demonsraed he need o include he icim receier gae delay in he alignmen objecie funcion. We hae shown ha while in some cases non-aligned aggressor noise peaks will resul in he wors case delay noise, aligned aggressor noise peaks can be used wih a small error. To deermine he alignmen of he composie noise pulse relaie o he icim ransiion, we proposed an effecie pre-characerizaion mehod ha requires only 8 pre-characerizaion poins and allows for accurae calculaion of he exra delay. Finally, resuls were shown on indusrial circuis demonsraing ha he proposed mehods significanly increase he accuracy of he analysis. 6 References [1] K. L. Shepard, V. Narayanan, P. C. Elemendorf and G. Zheng, Global Harmony: Coupled noise analysis for full-chip RC inerconnec neworks, Proc. Inl. Conf. Compuer-Aided Design, pp , [2] A. Odabasioglu, M. Celik and L. T. Pileggi, PRIMA: Passie reduced-order inerconnec macromodeling algorihm, Proc. Inl. Conf. Compuer-Aided Design, pp , [3] F. Daru, N. Menezes, and L. T. Pileggi, Performance Compuaion for Precharacerized CMOS Gaes wih RC Loads, IEEE Transacion on Compuer-Aided Design of Inegraed Circuis and Sysems, Vol.15, No. 5, pp , May 1996 [4] J. Qian, S. Pullela and L. T. Pillage, Modeling he effecie capaciance for he RC inerconnec of CMOS gaes, IEEE Trans. Compuer-Aided Design, pp , December [5] F. Daru, L. T. Pileggi, Calculaing Wors-Case Gae s Due o Dominan Capaciance Coupling, Proc. DAC, pp , June [6] P.D. Gross, R. Arunachalam, K. Rajagopal, L.T. Pileggi, Deerminaion of wors-case aggressor alignmen for delay calculaion, Proc. ICCAD, pp , Noember [7] R. Ley, D. Blaauw, G. Braca, A. Dasgupa, A. Grinshpon, C. Oh, B. Orsha, S. Sirichoiyakul, V. Zoloo, Clarine: A noise analysis ool for deep submicron design, Proc. DAC, pp , June 2. [8] S. Sapanekar, Capuring he Effec of Crossalk on, Proc. VLSI Design 2, pp , January 2. [9] R. Arunachalam, K. Rajagopal, L. T. Pileggi, TACO: iming analysis wih coupling Proc. Design Auomaion Conference, pp , June 2. Exra From Prediced Alignmen (ps) * Align a.5 Vdd resuls + Our alignmen predicion resuls - Perfec mach Exra From Wors Case Alignmen (ps) Figure 14. Alignmen predicion resuls