Explicit Delay and Power Estimation Method for CMOS Inverter Driving on-chip RLC Interconnect Load

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1 Inrnonl Journl of Elcrcl n Elcroncs Engnrng : Explc Dly n Powr Esmon Mho for MOS Invrr Drvng on-hp R Inrconnc o Susm Shoo Mhumn D n Rjb r bsrc h rssv-nucv-cpcv bhvor of long nrconncs whch r rvn by MOS gs r prsn n hs ppr. h nlyss s bs on h π-mol of R lo n s vlop for submcron vcs. ccur n nlycl xprssons for h oupu lo volg h propgon ly n h shor crcu powr sspon hv bn propos fr solvng sysm of ffrnl quons whch ccurly scrb h bhvor of h crcu. h ffc of couplng cpcnc bwn npu n oupu n h shor crcu currn on hs prformnc prmrs r lso ncorpor n h propos mol. h sm propos ly n shor crcu powr sspon r n vry goo grmn wh h SPIE smulon wh vrg rlv rror lss hn %. ywors Dly Invrr Shor rcu Powr π-mol R Inrconnc SI I. INRODUION S h mnmum fur sz for ngr crcus s scl ownwrs h rssv componn of h nrconnc lo bcom comprbl o h g oupu mpnc n sngl lump cpcor s no longr vl g lo mol. In orr o ncorpor h ncrs rol of h rssnc n o rmn h lo bhvor n consqunly h propgon ly of h rvng MOS gs ccur quvln lo mols hv o b us. mol xprsson for MOS nvrr rvng cpcv lo ws frs nrouc by Burns [] n Hnsrn n Jppson. l. xn h work o nclu npu wvform slop ffc []. Snc boh hs works r bs on h Shockly mol xnsv sus r rqur o mol h crcu bhvor n p-submcron rgon. smpl y rlsc α-powr MOSFE mol [] n n-powr MOS mol [] r nrouc whch nclu h crrr vlocy suron ffc n clos form xprssons r rv for ly shor crcu powr sspon n rnsn volg of MOS nvrr. Bu whl rvng h oupu volg xprsson for [] [] h shor-crcu currn g-rn couplng cpcnc s nglc n oupu s rv only for fs rmp npus. Susm Shoo s wh h Dprmn of Elcroncs & ommuncon Engg. Nonl Insu of chnology Durgpur Ws Bngl (ml: susms@gml.com. Mhumn D s wh h Dprmn of Elcroncs & ommuncon Engg. Nonl Insu of chnology Durgpur Ws Bngl (-ml: mhumn_@rffml.com Rjb r s wh h Dprmn of Elcroncs & ommuncon Engg. Nonl Insu of chnology Durgpur Ws Bngl (-ml: rjbkrc@gml.com h ly mol propos n [] uss α-powr MOS mol n h shor-crcu currn hs bn consr. Bu h oupu volg n currn r ssum o b pc-ws lnr whch ls o nccurcy of hs mol. n ccur n nlycl ly mol for MOS rvr rvng cpcv lo s rv n [] for ll rgons of rmp npu whch ovrcoms ll h bov mnon wknsss. Explc xprssons for h propgon ly of MOS gs rvng R lo hv bn rv n []-[]. Bu hs mols prsn sgnfcnly lrg rror u o h fc h hy r bs on h smplf ssumpons for h rnssor opron n us smpl mols for h rprsnon of h nrconnc los. mol for ly clculon of MOS nvrr rvng R π-lo hs bn prsn n []. hs mho pr-chrcrzs n quvln rssnc for ch g n formuls wo-mnsonl look up bl for h wvform of volg sourc whch rvs h rssnc. hs chnqu howvr rqurs r-chrcrzon f h oprng conons vz supply volg n mprur rng r chng. In [] n nlycl mho wh n mphss on h crcu powr sspon hs bn propos for h MOS nvrr rvng R π lo whch rqurs lss chrcrzon ffor n lss compuonl m by rplcng ch g o smpl nvrr crcu n clculs h g ly by nlycl mho. In orr o fn nlycl xprssons for h propgon ly n h oupu wvform shp n nrconnc lo my b mol n ffrn wys []. mol for h smon of propgon ly of MOS nvrr rvng lo mol by rssor n srs wh cpcor s rv n []. Howvr h rvng rnssor hs bn consr o opr lwys n lnr mo h nflunc of shor crcu currn s lso gnor n h smplf cs of sp npu hs bn xmn hus scrfcng ccurcy. In orr o rplc h R lo n ffcv cpcnc s clcul by n ron procur bs on smplf ssumpons n h shp of h oupu rspons []. h rl oupu wvform hs bn pproxm by h chrgng / schrgng of h ffcv cpcnc unl h m nsn whr h oupu volg bcoms qul o /. purng h rmnng poron of h oupu rspons s chv by smpl rssv mol. m vryng hvnn quvln mol hs bn propos n [] for h smon of h g lys. h g s rplc by n quvln crcu mol compos of

2 Inrnonl Journl of Elcrcl n Elcroncs Engnrng : lnr volg sourc n lnr rssor whr hr vlus r rmn by usng h sm mprcl fcors n hus rucng h ccurcy spclly for submcron chnologs. goo pproxmon for n R nrconnc lo s obn wh h π mol chvng n rror s low s % n ly clculons []. nlycl xprssons for h propgon ly n h oupu wvform cn b obn onc h lo s rplc by s π quvln n h corrsponng sysm quons r solv. hs mho s xn n [] whr MOS nvrr rvng R nrconnc lo s consr n propgon ly n shor crcu powr sspon s sm wh n rror of lss hn %.Wh h ncrs n sgnl frquncy n crs n h rnssor szs h mpornc of nrconncs hv bcom omnn fcor n rmnng h crcu prformnc n p submcron chnology. low frquncs h nrconnc lns cn b mol hr s lump or srbu R crcu mol. Bu n oy s p submcron chnology s h sgnl frquncy s rply ncrsng on cn gnor h ffcs of h onchp nucnc. mul-rmp mol wh gnrl R nrconncs s lo s propos n [] whch ccurly prc boh h % ly n h ovrll oupu wvform. In [] n ffcv cpcnc of srbu R lo for smng shor-crcu powr s prsn for ffrn rnson m of rmp npu bu gnors h ffc of fs rmp n slow rmp. In hs ppr novl clos form formul hv bn propos for h smon of h ly n h shor crcu powr sspon for MOS nvrr rvng h on-chp R nrconnc for boh fs rmp n slow rmp. II. RNSIEN RESPONSE MODEING crcu compos of n nvrr rvng n R quvln π-mol s consr whr h g-o-rn couplng cpcnc m s lso kn no ccoun (Fg.. h α-powr lw mol [] whch consrs h vlocy suron ffc of shor chnnl vcs s us for h rnssor currn rprsnon: GS cuoff rgon / I D kl ( GS DS DS < DS lnr rgon ( k s ( GS DS DS suron rgon Whr D-S s h rn suron volg [] k l k s r h rns-conucnc prmrs α s h vlocy suron nx n s h zro bs hrshol volg. rsng rmp npu wh rnson m τ s ppl o h rnssor gs. h cs for fllng rmp s symmrcl. h ffrnl quons h scrb h opron of h crcu s shown n Fgur n h loop of h π sub-crcu r obn by rchoff s volg lw. R ( rchoff s currn lw n h rnssors rn no cn b xprss s n m p n n p So Whr m R ( m ( m Fg. MOS Invrr Drvng Equvln π-mol of R Inrconnc In orr o solv h ffrnl quon (Eq. nlyclly h prsc currn hrough h PMOS rnssor s nlly consr o b nglgbl. h ffc of h prsc currn componn hough PMOS rnssor ono h oupu rspons wll b rmn owrs h n of hs scon.wo ypcl css for npu rmps hv bn consr hr: ( Fs Rmp (m consn (R or /R < τ (b Slow Rmp (m consn (R or /R > τ. For fs rmp npu h NMOS rnssor s n suron whn h npu rchs o s fnl vlu. For slow rmp npu h NMOS rnssor s n lnr rgon whn h npu rchs o s fnl vlu. In orr o obn h oupu volg xprsson nlyclly four ffrn rgons of opron r consr.. Fs Rmp Inpu Rgon (<< h NMOS rnssor s n cu-off rgon bcus n < n hnc ( bcoms: ( wh h nl conons ( n. h oupu wvform xprsson cn b τ gvn s ( Whr ± ( hs xprsson scrbs h smll ovrshoo of h oupu m (

3 wvform u o h couplng cpcnc m. Gnrlly for h cs of rvng long nrconncon lns h ovrshoo vlu s lmos nglgbl snc m << n ou cn b consr o b qul o whou sgnfcn rror n hs rgon. hs rgon xns o h m nsn τ whr n. Rgon ( <<τ h NMOS vc oprs n suron n h npu sgnl s sll n suron. Equon ( bcoms: s n τ ( hs cnno b solv nlyclly. In orr o obn n nlycl xprsson for h oupu n hs rgon h currn componn s pproxm by scon orr ylor srs wh suffcn ccurcy τ/ whr n / n gvn s n h ffrnl quon cn b solv whch rsuls h followng xprsson for h oupu wvform: ( ( Whr!! n ± τ Rgon (τ<< h npu hs rch o s fnl vlu n h NMOS rnssor s sll n suron. Equon ( bcoms: s ( So h oupu wvform cn b clcul s ( ( Whr s hs rgon xns unl h m nsn whn h NMOS rnssor xs suron. h m nsn s clcul by (. R SN D ( hs cn b solv whou ny pproxmon. SN D s h rn suron volg of h NMOS vc. Rgon (> h NMOS rnssor oprs n lnr mo n hnc ( bcoms: / R l ( So h oupu rspons cn b mol s ( ( Whr n R B. Slow Rmp Inpu h oprng conons of h srucur n rgon n r h sm o h of fs npus; howvr rgon xns from m o m whr <τ. Rgon ( <<τ h NMOS rnssor oprs n lnr mo whl h npu s sll rmp on. h ffrnl quon scrbng h oupu rspons n hs rgon cn b mol s: Inrnonl Journl of Elcrcl n Elcroncs Engnrng :

4 Inrnonl Journl of Elcrcl n Elcroncs Engnrng : ( / l ( n R hs cn no b solv nlyclly. For hs rson n s ~ rplc by s vrg vlu n n. hs s vl pproxmon snc for mos of h prccl css h uron of hs rgon s vry smll n hus n ks vlus vry clos o h of vrg vlu. Hnc h soluon of ( cn b wrn s ( Whr l ~ / ( n ( ( ρ ρ ρ ( ( ( ρ ρ ρ ( ( ρ ρρ ρ ρ ( ρ ρρ ( ρ ρ ρ ρ ρ ( ( ( ( ρ ρ ρ ρ ρ ρ ρ ρ R ( ( ρ ρ ρ ρρ ρ ρ ρ n ρ ρ Rgon cn b solv n smlr mnnr s scuss for h cs of h fs npu. III. EFFE OF SHOR IRUI URREN ON PROPGION DEY In h prvous scon h currn hrough h PMOS rnssor cll shor crcu currn hs bn consr o b nglgbl. Gnrlly hs s vl ssumpon bcus h cpcv n nucv lo n long nrconncon ln s lrg nough so h h oupu volg osn chng sgnfcnly unl h m h PMOS rnssor bcoms off. hs sgnfs h h rn-o-sourc volg of h PMOS rnssor rmns smll n s currn componn lso ks smll vlus. onsqunly n orr o smplfy h mhmcl nlyss h shor crcu currn s gnor n os no hv ny sgnfcn ffc on h ccurcy of h nlyss whch hs lry bn prsn. In hs scon h ffc of h shor crcu currn on h smon of h propgon ly of gs rvng long nrconncons s scuss. h shor-crcu currn hrough h PMOS rnssor xss n h nrvl ( ov p ; whr ov s h m whr h volg ovrshoo h oupu of h nvrr css. h s u o h fc h urng h volg ovrshoo h PMOS currn s flowng owrs n hus no currn ph xss bwn n groun. m ov cn b clcul by sng h volg xprsson for h nvrr oupu n rgon qul o. s h m whn h PMOS p rnssor urns off (whn n P. h xsnc of h PMOS currn rsuls n crs of h schrgng currn n hus n ncrs n h propgon ly. I cs lk n moun of chrg Q nlly sor n h oupu no n whch hs o b rmov hrough h NMOS rnssor. onsqunly h quvln chrg cn b clcul by ngrng h currn of h PMOS vc ovr n nrvl of ov o p. onsrng h h PMOS rnssor oprs for hlf of h nrvl ( ov p n lnr mo n h h currn wvform s symmrcl roun h ml of hs nrvl [] Q cn b clcul s: Q p ov p ov p ov l τ P h PMOS rn-o-sourc volg ( rv n h prvous scon s us n hs ngrl. In hs wy h ncrs n propgon ly s foun s h m rqur o rmov h quvln chrg Q. n vrg vlu for h schrgng currn I s shoul b us. Howvr cn b pproxm by h NMOS rnssor currn m p / s I s n [ p / ] whch cn b rv from h prvous nlyss. hus h m rqur o rmov hs xr chrg whch cuss h onl propgon Q ly cn b clcul s. I I. ESIMION OF SHOR IRUI POWER DISSIPION h shor-crcu powr whch s ssp urng h oupu swchng s u o h currn s cn b foun by pplyng rchoff s currn lw h sourc of h PMOS rnssor. s p GS ( Whr GS s h currn hrough h g-o-sourc n couplng cpcnc n s gvn by GS GS. Enrgy srs o ssps m s whn s srs flowng owrs h sourc of h PMOS rnssor. m s cn b clcul by sng s usng lnr rgon xprsson for h PMOS currn. h PMOS rnssor srs s opron n lnr mo n hn nrs no h suron s

5 Inrnonl Journl of Elcrcl n Elcroncs Engnrng : ov p pproxmly s [] whr p n consqunly s rch hr mxmum vlu. ssumng h h PMOS currn n consqunly s s symmrcl roun s h ssp nrgy u o shor-crcu currn s gvn by s E ( sc s s So h shor crcu powr sspon for symmrcl rvr n for sysm clock frquncy f cn b clcul s P α ( sc fe sc Whr α s h swchng cvy of h oupu no. h logc sgs followng lrg R lo wll ssp sgnfcn moun of shor-crcu powr u o h gr wvform whch hy rcv s npu. onncng h % n % pon of h oupu wvform n ffcv rmp npu for h followng sgs s obn whch cn b us n h corrsponng formul s gvn n ( for h clculon of h shor-crcu powr sspon n hs sgs.. SIMUION RESUS ND DISUSSIONS comprson of h oupu volg rspons clcul by usng h propos mho s gvn by ( ( ( ( ( wh h SPIE rsul for boh fs rmp n slow rmp npu sgnl hs bn shown n Fgur for. μm chnology; W n nm n W p nm. h vrg rror for fs rmp s.% wh mxmum rror of.% n mnmum rror of.% n smlrly h vrg rror for slow rmp s.% wh mxmum rror of.% n mnmum rror of.% whch jusfs h ccurcy of h propos mol. Fg. omprson of Oupu Wvform bwn Smul n lcul vlus for ( Slow Rmp (τ.ns RΩ nh /.pf n (b Fs Rmp (τ.ns RΩ nh / pf. Dly lculon Snc h oupu wvform xprsson for ch of h rgons of opron s known propgon ly cn b clcul s h m from h % of h rsng /fllng npu o % of h rsng/fllng oupu wvform. Usng hs fnon h % propgon ly hs bn clcul for svrl R / vlus. h vron of lo cpcnc wh ly for ffrn vlus of R n r shown n bl for boh fs s wll s for slow rmp sgnl. h vron of lo nucnc wh propgon ly s prsn n bl II for ffrn vlus of lo rssnc n cpcnc for boh fs n slow rmps. Usng h oupu volg quons rv bov h propos ly s clcul n hs bn compr wh SPIE ly whr h vrg rlv rror s whn %. h vrons of ly wh lo cpcnc for ffrn vlus of R n for boh fs n slow rmps r grphclly rprsn n Fgur n Fgur rspcvly. For fs rmp npu h vrg rror s.% wh mxmum rror of.% n mnmum rror of.% smlrly for slow rmp h vrg rror s.% wh mxmum rror of.% n mnmum rror of.% hv bn chv whch show h h propos ly mol closly follows h SPIE ly. h vrons of ly wh lo nucnc for ffrn vlus of lo rssnc n cpcnc for fs n slow rmps r grphclly rprsn n Fgur n Fgur rspcvly. For fs rmp h vrg rror s.% wh mxmum rror of.% n mnmum rror of.% hv bn chv. Smlrly for slow rmp h vrg rror s.% wh mxmum rror of.% n mnmum rror of.% whch shows h h propos ly vrs closly wh SPIE ly. From h grphs s vn h s h m consn of R lo.. (R or /R ncrss h % ly lso ncrss. B. Shor-rcu Powr lculon Whn h npu volg wvform vs from h l sp npu n hs nonzro rs n fll m boh NMOS n PMOS rnssor wll smulnously conuc crn moun of currn urng h swchng vn snc boh h rnssors wll form conucng ph bwn o groun. hs s cll shor-crcu currn. hs shor-crcu powr sspon wll b nglgbl for hgh lo nucnc n cpcnc vlu whch s vn from bl n Fgur n Fgur. omprsons bwn h shor-crcu nrgy clcul by usng h propos mho wh SPIE for ffrn vlus of R n s shown n bl n h vrg rlv rror s.% wh mxmum rror of.% n mnmum rror of.%.

6 Inrnonl Journl of Elcrcl n Elcroncs Engnrng : Fg. o pcnc vs. Dly for Dffrn lus of o Rssnc for Fs Rmp wh nh Fg. o pcnc vs. Dly for Dffrn lus of o Rssnc for Slow Rmp wh nh (pf BE I OD PINE S PROPGION DEY FOR NH ND NH FOR FS ND SOW RMP SIGN RESPEIEY Fs Rmp Slow Rmp Propgon Dly Propgon Dly RΩ RΩ RΩ RΩ SPIE Rsul Propos Mho SPIE Rsul Propos Mho (pf SPIE Rsul Propos Mho SPIE Rsul Propos Mho (nh SPIE Rsul BE II INDUNE S PROPGION DEY FOR / D. PF ND / D PF FOR FS ND SOW RMPS RESPEIEY Fs Rmp Slow Rmp Propgon Dly Propgon Dly RΩ RΩ RΩ RΩ Propos SPIE Propos SPIE Propos SPIE Propos Mho Rsul Mho (nh Rsul Mho Rsul Mho In Fgur h vron of nrgy wh lo cpcnc for boh h propos mho n SPIE for ffrn vlus of R s shown whch shows h wh ncrs n cpcnc vlu h nrgy bcoms nglgbl. In Fgur h vron of nrgy wh lo nucnc for boh h propos mho n SPIE for ffrn vlus of R s shown whch shows h wh ncrs n nucnc vlu h nrgy bcoms nglgbl.

7 Inrnonl Journl of Elcrcl n Elcroncs Engnrng : Fg. o Inucnc vs. Dly for Dffrn lus of o Rssnc for Fs Rmp wh. pf Fg. o Inucnc vs. Dly for Dffrn lus of o Rssnc for Slow Rmp wh pf (pf SPIE Rsul BE III RIION OF SHOR-IRUI ENERGY WIH / D ND FOR DIFFEREN UES OF R Shor-rcu nrgy for nh Shor-rcu nrgy for / pf RΩ RΩ RΩ RΩ Propos Mho SPIE Rsul Propos Mho (nh SPIE Rsul Propos Mho SPIE Rsul Propos Mho Fg. pcnc vs. Enrgy for Dffrn lus of R for nh I. ONUSION h % propgon ly n shor-crcu powr sspon of MOS nvrr rvng π-mol R lo s clcul by solvng s of ffrnl quons. Fg. Inucnc vs. Enrgy for Dffrn lus of R for / pf ccur nlycl soluons of hs quons hv bn rv for ffrn rgons of opron of MOS nvrr for boh fs rmp n slow rmp npus. h % ly clcul by h propos mho hs bn compr wh h of h SPIE rsul n h vrg rror for fs rmp n slow rmp r foun o b.% n.%

8 Inrnonl Journl of Elcrcl n Elcroncs Engnrng : rspcvly. Shor crcu powr sspon clcul by h propos mho s compr wh SPIE n h vrg rror s foun o b s low s.%. hus s vn h h propos mho cn prov nw prspcv on ly n shor crcu powr sspon clculon of MOS rvr rvng R nrconnc wh hghr ccurcy. REFERENES [] J. R. Burns Swchng Rspons of omplmnry-symmry MOS rnssor logc crcus R Rv. ol. pp. - Dc.. [] N. Hsrn n. O. Jppson MOS crcu sp n buffr opmzon IEEE rns. On ompur- Dsgn ol. D- no. pp. - Mr.. []. Skur n.r. Nwon lph-powr w MOSFE Mol n s pplcons o MOS Invrr Dly n Ohr Formuls IEEE j. Sol-s crcus vol. no. pp. - prl. []. Skur. R. Nwon Dly nlyss of srs-connc MOSFE crcus IEEE J. Sol-S crcus vol. pp. - Fb.. [] S. H.. Embb R. Dmrn Dly Mols for MOS BMOS BNMOS crcus n hr pplcons for mng smulons IEEE rns. ompur- Dsgn vol. pp. - Sp.. []. Bsouns S. Nkols n O. oufopvlou nlycl rnsn Rspons n Propgon Dly Evluon of h MOS nvrr for shor-chnnl Dvcs IEEE j. Sol-s rcus vol. no. pp. - Fbrury. []. lr n E. G. Frmn Dly n Powr Exprssons for MOS Invrr rvng Rssv-pcv o Proc. of IEEE In. Symp. On rcus n sysms (ISS pp. -. [] J. Qn S. Pulll n. Pllg Molng h Effcv pcnc for h R nrconnc of MOS gs IEEE rns. ompur- Dsgn of Ingr rcus n Sysms vol. no. pp. - Dc.. [] F. Dru N. Mnzs n.. Plgg Prformnc ompuon for Pr-chrcrz MOS gs wh R los IEEE rns. ompur- Dsgn of Ingr rcus n Sysms vol. no. pp. - My. [] Flornn Dru Nol Mnzs Jssc Qn n wrnc. Pllg G-Dly mol for Hgh-Sp MOS crcus Proc. M/IEEE Dsgn uomon onfrnc pp. -. []. Hr H. Onor n. mru Esmon of Shor-crcu powr sspon for sc MOS gs rvng R π o Proc. Inrnonl Workshop Powr n mng Molng Opmzon n Smulon (PMOS pp. -. []. Skur pproxmon of Wrng Dly n MOSFE SI IEEE J. Sol-s rcus vol. S- pp. - ug. [] S. Nkols. hzgorgou n E. D. yrks-bzros Dly n Powr Esmon for MOS Invrr rvng R nrconnc os vol pp. - Jun. [] kshm. k Jn Wng nw mul-rmp rvr mol wh R Inrconnc o Proc. Inrnonl Symposum on Physcl Dsgn vol. pp. - My. [] Guoqng hn n Eby G. Frmn Effcv pcnc of R los for smng Shor-rcu Powr Proc. IEEE Inrnonl Symposum on rcus n Sysms (ISS no.