Explct Formulas and Effcent Algorthm for Moment Computaton of Coupled RC Trees wth Lumped and Dstrbuted Elements Qngan Yu and Ernest S.Kuh Electroncs Research Lab. Unv. of Calforna at Berkeley Berkeley CA947 USA Emal: qyukuh @eecs.berkeley.edu Abstract In today s deep submcron technology the couplng capactances among ndvdual on-chp RC trees have essental effect on the sgnal delay and crosstalk and the nterconnects should be modeled as coupled RC trees. We provde smple explct formulas for the Elmore delay and hgher order voltage moments and a lnear order recursve algorthm for the voltage moment computaton for lumped and dstrbuted coupled RC trees. By usng the formulas and algorthms the moment matchng method can be effcently mplemented to deal wth delay and crosstalk estmaton model order reducton and optmal desgn of nterconnects. Introducton The RC tree s a typcal model for on-chp nterconnects. The Elmore delay and the moment matchng method have been successfully used for the delay estmaton model order reducton and nterconnect optmzaton []. These tasks can be very effcently mplemented as there are explct formulas and a lnear order algorthm for the moment computaton of such RC trees []. In today s deep submcron technology due to the dense placement of the nterconnect wres and the large aspect rato of wre heght over wre wdth the couplng capactance between two wres may be even larger than the ground capactance of each wre and can never be neglected. In ths case the nterconnects should be modeled by several RC trees wth floatng capactors connected among the tree nodes. We call such knd of nterconnect model coupled RC trees. It s well known that the capactve couplng has sgnfcant effects on the sgnal delay and crosstalks. Unfortunately no smple formulas and lnear order algorthms have been publshed for the computaton of the Elmore delay and hgher order voltage moments of such coupled RC trees so that these effects cannot be as effcently dealt wth as n the sngle RC tree case. In the lterature three ways are often used. The frst one relys on the decouplng technque whch s typcally restrcted to some very smple cases [6 7]. The second one relys on the general methods for RC nterconnect networks [8 9] where the advantage of the tree structure s not taken so s not effcent enough. The thrd one uses very rough model e.g. usng the length of overlap between two wres as the metrc of the crosstalk[] whch s far from accurate. In ths paper we provde explct formulas and a lnear order recursve algorthm for the computaton of the Elmore delay and hgher order voltage moments of coupled RC trees consstng of lumped andor dstrbuted elements. They are based on the moment model of a capactor whch s a current source wth a known value. We use the technque of source splttng to splt each floatng current source nto two ground sources so that the coupled RC trees are decoupled durng moment computaton and the algorthm for a sngle RC tree can be appled. For dstrbuted RC lnes by usng the same model and technque for dstrbuted capactances we show that the moments of voltage and capactve current along a lne are polynomals of the coordnate of the lne. We derve recursve formulas for the coeffcent of these polynomals and set up an exact moment model for RC lnes. Wth these models we extend the lnear moment computaton algorthm for a sngle RC tree wth lumped elements to coupled RC trees wth both lumped and dstrbuted elements wthout the need of dscretzaton of the dstrbuted lnes. Ths algorthm s exact and very effcent. It s useful n the delay and crosstalk estmaton model order reducton and optmal desgn of such type of nterconnects. Coupled RC trees wth lumped elements. Defntons A set of coupled RC trees s an RC network whch conssts of a number of RC trees wth floatng capactors connected among the non-ground nodes of ndvdual trees. Let be the node voltage at the -th node of the -th tree
* and ts Laplace transform. Let then s called the k-th order moment of moments of currents are smlarly defned.. The Elmore delay.. Formula. The It s well known that when a unt mpulse voltage s appled to the root of the -th tree and all other roots are grounded the negatve of the frst order moment s called the Elmore delay of node voltage. From the -v characterstc of a capactance : and let and be the k-th order moment of and respectvely we have and.e. the capactor behaves as an open crcut when the -th order moment s concerned and a current source when the k-th ( ) order moment s concerned. Therefore when we are concerned about the Elmore delay of all the -th order voltage moments on the trees are equal to ther root voltages. Let be the tree node set of the -th tree and be the p-th node n then and! #"%$ '&. For the frst order moment each capactor connected to ether grounded or floatng behaves as a current source valued. Let ( be the total capactance connected to the node and ) be the total capactve current flowng out of then ts frst order moment ) +* (. Let be the path from the root of the -th tree to node.- and the total resstance on the path. Then the current ) * flows along path and causes a component ( for. Therefore we have 4. Also note that ( () Denote the Elmore delay of as then ( () It can be seen that the couplng capactances n the actve tree have the same contrbutons to the Elmore delay as the ground capactances do and the formulas of the Elmore delay are the same for both the sngle tree and coupled RC trees... Effect of non-zero ntal states on sgnal delay It s well known that the sgnal delay n a net s affected by the sgnals at ts neghbour[] especally when the two nets are suppled by complement dgtal sgnals at the same tme.e. one net s source voltage goes up from to and another one s from to zero. In the extreme case that the source voltage of the second net s a step down functon t s equvalent to the case that the voltage supply for the second net s zero but ts ntal states are. Now we gve a formula to show the effect of the non-zero ntal states on the Elmore delay of the sgnal propagaton n the frst net. Suppose that the -th tree s n the zero ntal state and suppled by a unt mpulse voltage and let 7 be ts capactvely coupled trees wth unt valued ntal states and zero valued voltage supples. Recall that from the -v characterstc of a capactor C: &8 when 49#: =>? 9< => % where a current source valued and n the opposte drecton of the capactor voltage represents the effect of the ntal state. Let @A be the total couplng capactance connected among and the nodes n the trees n 7.e. @A BDCFE # # s the couplng capactance connected between and then there are total capactve current @A gong along the path from ts source to node where nodes whose contrbuton to the zero-th order moment of @A and the component of the zero-th order moment of the coupled neghbourng trees s s caused by the effect of the nonzero ntal states of H @A () and the component of caused by the source s * I ( KMLNLOL (4) where the superscrpts "nt" and "s" refer to the effect of the ntal states and the source voltage respectvely. Now suppose that a unt step voltage s appled to the -th tree then by superposton * * P ( %LOLNL @A MLNLOL ( @A QMLNLOL (5) Now the negatve coeffcent of n the parentheses ( s an "equvalent Elmore delay" of @A # @A t s n the set of (. Note that for each too and the effect of the couplng capactance to the Elmore delay due to the nonzero ntal states n the worst case s equvalent to doublng the capactance value. Example. For the coupled trees show n Fg. tree s suppled by and tree s suppled by. When tree s n
zero ntal states the Elmore delays of and are as follows: 7 8 9 7 Q 8 9 7 8 9 When the ntal states of tree are the "equvalent Elmore delays" of and are as follows: 7 8 9 7 8 9 7 8 9 Compared wth the frst case t can be seen that the couplng capactances 7 8 and 9 are doubled n the second case.. Hgher order moments Consder the k-th order voltage moments wth %. As mentoned n Sec.. each ground capactance connected to node s equvalent to a current source * and each couplng capactance connected between nodes and s equvalent to a current source * * wth drecton from node to node. Usng the source splttng technque n crcut theory ths floatng current source can be splt nto two ground sources wth one from node to ground valued * * and the other from node to ground valued * * as shown n Fg.. It s obvous that the two sources contrbute to the k-th order voltage moments n the & -th and & -th tree respectvely n the way smlar to that the ground sources * and * do. The splttng of the floatng current source nto two ground sources s equvalent to decouplng the coupled RC trees durng moment computaton and for each decoupled RC tree the moment formulas and moment computaton algorthm for a sngle RC tree can be appled. Based on ths reasonng we have the formulas for the moments of a set of coupled trees as follows where s the set of couplng capactances connected to node : & +* ) E E B ) ) E # (6) +* * ( +* ) E E ) ) # * B E (7) In the last expresson of the above equaton the frst term n the summaton ( * s the same as n the sngle RC tree case and n addton each couplng capactance has a contrbuton # * to the moment * whch s partcular for the coupled RC trees. Example. For the coupled RC trees shown n Fg. for we have * 7 * 7 4* 8 * 8 5* 9 * I 9 6* * 7 * 7 4* 8 * 8 5* 9 * 9 6* * 7 * 7 4* 8 * 8 5* 9 * 9 6* Coupled RC trees wth dstrbuted lnes. Moment model of dstrbuted RC lnes An RC lne located n Tree & and connected between nodes and ts father node s denoted by & For smplcty we consder two coupled lnes & and & and the result s easly extended to multple coupled lnes. Assumng that the length of the two lnes s normalzed to and and correspond to the near and far end of each lne. For & let and be ts total ground capactance and resstance respectvely the -th order voltage moment at coordnate ) * the total k-th order moment of capactve current at coordnate z wth an nfntesmal nterval and the k-th order current enterng and leavng the lne. Let be the total couplng capactance between &. and & We frst show by nducton that and. ) * are polynomals of and gve recursve formulas to compute the coeffcents of the polynomals. Startng from order t s known that ) * whch s denoted by *. Smlarly s a constant and s denoted by *. For the " order moment wth for & we have ) * * * * (8)
where when & %& and versa and & %& and ) * ) * and vce (9) It can be seen from the formulas that when s a polynomal of of order ) * * s a polynomal of z of the same order and * s of order. It can be derved that. Let ) * * and * *. From Eq(8) we have * and from Eq(9) we have we have and * * * * * () () * * () () Eqs()-() form a set of recursve formulas to compute the parameters s and s. When the k-th order s s are known where (4) ) * * (5) s the total k-th order capactve current moment contrbuted by & and where * (6) ) * * (7) s the total k-th order moment of the voltage drop on the lne contrbuted by ts k-th order capactve current moment. From the above two equatons the moment model of an RC lne can be presented by Fg... Equaton of voltage moments The voltage moment s contrbuted by both lumped and dstrbuted elements. The component correspondng to the contrbuton of lumped elements s gven by Eq(7). From Fg. t can be seen that the current source s connected to node so that the contrbuton of the current source s * and for each & on the path the voltage source contrbutes an amount of. We ntroduce a functon & & such that & A& ff the branch connected between and s a lne and & A& otherwse. We defne a functon " & such that " A& f and " A& otherwse. Then we have the formula for the voltage moments wth order as follows where s the lumped ground capactance connected to. * & A& +* ) E E * B ) ) E # +* * " A& (8) 4 Algorthm for moment computaton From the prncples descrbed n the prevous sectons we provde an effcent recursve algorthm for the computaton of moments n coupled RC trees. The computaton s done from order up to some specfed maxmum order wth the use of KCL and KVL of the moment model of the crcut. Wth order the computaton s carred out by computng current moments from the leaves of each tree upstream to ts root then computng the voltage moments from the root of each tree downstream to ts leaves. The algorthm s descrbed as follows. & "!! "! # $! f(rc lnes exst) for each tree 5 do!%! A& A& & (' A& & O "!! "! for each tree 5 wth source voltage do for each node n 5 do for each & do Compute * for each & do LNLOL
Compte * LOLOL Compute and!%! (! &!%! = f( $ A& )!%!! "! ) * for each couplng capactance! * * for each of the son node of!"!!%! & %!%! f( & & )!"! return(!%! ) (' (! & & ) f( else * f( & & ) for each of the son node of & ' A& % do! "! ) do do When applyng the algorthm to lumped RC trees for each order each grounded capactor s vsted once and each floatng capactor vssted twce n the call of functon!%! and each floatng node s vsted once n the call of functon ('. Therefore the computaton cost s proportonal to the total number of capactors tmes the maxmum order of nterest. In practcal cases the number of capactors connected to each node s lmted by a constant and the computaton complexty of the algorthm can be expressed as where s the number of the nodes n the network and s the maxmum order requred. Ths s a lnear order algorthm and s very effcent. When applyng the algorthm to dstrbuted RC trees suppose that there s an RC lne connected between each floatng node and ts father node then the computaton complexty s as the number of s and s grows lnearly wth the order. However f dscrete model s used to represent each RC lne as used n RICE [] f for each lne there are sectons n the model then the computaton cost wll be. It has been shown [] that n order to get exact moment matchng by a nonunform dscrete model and t s often seen n the lterature e.g. n [] that a large number of unform RC sectons are used to model a lne and the number of sectons s proportonal to the length of the lne but n our algorthm the computaton cost s ndependent of the lne length. Therefore ths algorthm runs both more accurately and much faster than the usng of dscrete model for RC lnes. error A SP NEW max(%) 5.9 8.4 ( ) 67. 65. 5. average(%) 7.5 8.9 ( ).4 7.7 5.6 large tems 9 (5) 9 7 Table. Test data for coupled RC lnes error A SP NEW max(%) 66.4 4.7 ( ) 9. 54. 9.9 average(%)..57 ( ) 5. 7. 7.4 large tems 45 () 4 56 4 Table. Test data for coupled RC trees 5 Examples Based on the effcent algorthm for moment computaton we developed a new crosstalk model for coupled RC trees. The model s based on the moment matchng model wth slght modfcatons so that the model s more accurate and always stable. Because of the lmtaton of the paper there s no space for ts detaled descrpton whch can be found n [5]. We have tested 4 examples and the results are summerzed n Table and for coupled RC lnes and trees respectvely. In the frst lne of each table "" and "" refer to the one pole and two pole model generated by approxmaton wth moment matchng up to the order of and. " " refers to the model by [] whch s an approxmated second order model. "7 " refers to the model of [] and " " refers to our new model. The tem "large tems" refers to the number of tests that the absolute error exceedng %. It can be seen that our new model works better than the models marked "" "A" and "SP". Compared wth the model marked "" (the rd order model) note that there are 5 cases n the RC lne tests and 8 cases n the RC tree tests that the -pole model s unstable the data lsted outsde the parentheses only refer to the stable tests and those nsde the parentheses refer to all the tests. It can be seen that the new model performs better than exstng models thus far. 6 Conclusons We have provded smple explct formulas for Elmore delay and hgher order voltage moments and a lnear order recursve algorthm for the moment computaton for coupled RC trees wth lumped andor dstrbuted elements. As there s no dscretzaton for dstrbuted EC lnes t s exact and
V s R R R C C C V F()k I k E k R () I () + - k V k V s C C C 7 8 9 R 4 4 R 5 5 R 6 6 k C C C 4 Fgure. Moment Model of RC Lne n n I Fgure. Coupled RC Trees C n n n (a) k n (b) Ik -I (C) Fgure. Moment Model of Floatng Capactor very effcent. Wth the explct formulas the formulas for the senstvty of moments w.r.t. the crcut parameters can be easly derved whch wll be very useful when nterconnect desgn and optmzaton s concerned. These formulas and algorthms provde an effcent way to deal wth the delay and crosstalk estmaton and model order reducton and wll be benefcal for the nterconnect desgn optmzaton and smulaton. These formulas and algorthms can also be easly extended to RLC trees wth both capactve and nductve couplng. References [].Cong et al "Interconnect desgn for deep submcron IC s" Proc. ICCAD 97 Nov. 997 pp.478-485. [] Q.Yu & E.S.Kuh "Exact moment matchng model of transmsson lnes and applcaton to nterconnect delay estmaton" IEEE Trans. on VLSI Sys. vol. pp.- une 995. [].S.Ym & C.M.Kyung "Reducng cross-couplng among nterconnect wres n deep-submcron datapath desgn" Proc. of 6-th DAC une 999 pp.485-49. k [4] K..Kerns et al "Stable and effcent reducton of substrate model networks usng congruence transforms" Proc. ICCAD 95 Nov. 995 pp.7-4. [5] Q.Yu and E.S.Kuh "New effcent and accurate moment matchng based model for crosstalk estmaton n coupled RC trees" Proc. ISQED March. [6] T.Sakura"Closed form expressons for nterconnecton delay couplng and crosstalk n VLSIs" IEEE Trans. on Electron Devces vol.4 No. pp.8-4 99. [7] H.Kawaguch and T.Sakura "Delay and nose formulas for capactvely coupled dstrbuted RC lnes" Proc. ASP-DAC 98 pp.5-4 Feb. 998. [8] P.Feldmann & R.W.Fruend "Crcut nose evaluaton by approxmaton based model order-reducton technques" Proc. ICCAD 97 pp.-8 Nov. 997. [9] K.L.Shepard et al "lobal Harmony: Coupled nose analyss for full-chp RC nterconnect networks" Proc. ICCAD 97 pp.9-46 Nov.997. [] H.Zhou and D.F.Wang"An optmal algorthm for rver routng wth crosstalk constrants" Proc. ICCAD 96 pp.-5 Nov. 996. [] C.L.Ratzlaff and L.T.Pllage "RICE: rapd nterconnect evaluaton usng AWE" IEEE TCAD vol. pp.76-76 une 994. [] E.Acar et al "SP: a stable -pole RC delay and couplng nose metrc" Proc. 9-th reat Lakes Symp. on VLSI pp.6- March 999. [] A.Vttal et al "Crosstalk n VLSI nterconnects" IEEE Trans. on CAD vol.8 No. pp.87-4 Dec.999.