Characterization of Normal Propagation Delay for Delay Degradation Model (DDM)

Cha ac e iza ion o No mal P opaga ion Delay o Delay Deg ada ion Model (DDM)⋆ Alejand o Mill´an, Jo ge Juan, Manuel J. Bellido, Paulino Ruiz-de-Cla ijo, and Da id Gue e o Ins i u o de Mic oelec onica de Se illa - Cen o Nacional de Mic oelec onica A . Reina Me cedes, s/n (Edi icio CICA) - 41012 Se illa (Spain) Tel.: +34 955056666 - Fax: +34 955056686 h p://www.imse.cnm.es Depa amen o de Tecnologia Elec onica - Uni e sidad de Se illa A . Reina Me cedes, s/n (E. T. S. Ingenie ia In o ma ica) - 41012 Se illa (Spain) Tel.: +34 954550974 - Fax: +34 954552764 h p://www.d e.us.es {amillan, jjchico, bellido, paulino, gue e}@imse.cnm.es Abs ac . In p e ious pape s we ha e p esen ed a e y accu a e model ha handles he gene a ion and p opaga ion o gli ches, which makes an impo an headway in logic iming simula ion. This model is called Delay Deg ada ion Model (DDM). Cha ac e izing DDM comple ely also implies he cha ac e iza ion o he no mal p opaga ion delay. In his pa- pe , we p opose a simple heu is ic model ha includes i s dependence on he ou pu load and he inpu ansi ion ime. We ha e es ed his model and ound a mean de ia ion lowe han 4%. Also, we p esen a cha ac e - iza ion p ocess o his model ha is ully in eg a ed in o AUTODDM wi hou a ec ing he o al simula ion ime needed o cha ac e ize a s an- da d cell. 1 In oduc ion In he ield o logic simula ion o digi al CMOS ci cui s, delay models exis ha ake in o accoun mos issues a ec ing accu acy [1–4]: low ol age, submic on and deep submic on de ices, ansi ion wa e o m, e c. The e a e also dynamic e ec s, he mos impo an being he so-called inpu collisions [5], which hap- pens when wo o mo e inpu signals change almos simul aneously. The ype o inpu collision ha mo e no ably a ec s he beha iou o digi al ci cui s a e he gli ch collisions, o hose ha may cause na ow pulses o gli ches. In p e- ious pape s [6–8] we ha e p esen ed a e y accu a e model ha handles he gene a ion and p opaga ion o gli ches, which makes an impo an headway in logic- iming simula ion. This model is called Delay Deg ada ion Model (DDM). One impo an poin in any delay model (including he DDM) is he de ini ion o he model pa ame e s and he se up o a use ul cha ac e iza ion p ocess ⋆This wo k has been pa ially suppo ed by he MCYT MODEL p ojec TIC 2000- 1350 and MCYT VERDI p ojec TIC 2002-2283 o he Spanish Go e nmen . The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48 ha desc ibes how he model pa ame e alues a e ob ained. This in o ma ion is necessa y o be able o ep oduce simula ion esul s by o he s and also o check he iabili y o he app oach: a model ha is e y ha d o expensi e o cha ac e ize may be useless. In p e ious pape s [8, 9] we ha e desc ibed he cha ac e iza ion p ocess o he deg ada ion pa ame e s o DDM and we ha e p esen ed a ool ha au oma es he p ocess, called AUTODDM. The men ioned DDM is compa ible wi h any model o he no mal p opa- ga ion delay, whe e “no mal” means he con en ional delay conside ed by mos logic- iming simula o s when deg ada ion e ec is no aken in o accoun . In he specialized li e a u e he e a e di e en pape s [1, 2] whe e au ho s p esen accu- a e no mal delay models and i would be possible o selec one o hese models o p o ide a no mal p opaga ion delay model o he DDM, hough, since hey a e models ocused on he geome ic le el, hey a e no sui ed o ou aims. A his ime, he DDM ocuses on ci cui s desc ibed a he ga e-le el, and is being implemen ed in a logic- iming simula o based on s anda d cells, called HALOTIS [10]. F om his pe spec i e, an app op ia e no mal p opaga ion delay model ha complemen s he DDM should be desc ibed a he same le el. I should be simple enough o be as and easy o implemen wi hou signi ican loss o accu acy, and mus be also easy o cha ac e ize, possibly using he same da a ex ac ed om he DDM cha ac e iza ion. In his wo k, we ha e ob ained such a model o he no mal p opaga ion delay, sui ed o he DDM, ha includes i s dependence on he ou pu load and he inpu ansi ion ime a he ga e-le el. We ha e also de eloped a cha ac e iza ion p ocess and included i in he p e iously de eloped ool AUTODDM [9]. The analysis is ca ied ou in a 0.35 µm CMOS echnology using he s anda d cell lib a y p o ided by The Found y. The o ganiza ion o he pape is as ollows: in Sec . 2 he cha ac e iza ion p ocess o he deg ada ion pa ame e s is p esen ed; in Sec . 3 we p esen he esul s o he no mal p opaga ion delay e alua ion and we p opose a simple model ha i s he eal beha iou e y well; Sec . 4 p esen s he cha ac e iza ion p ocess o he p oposed model; inally we will inish wi h he main conclusions o his wo k. 2 Cha ac e iza ion P ocess o he Deg ada ion Pa ame e s The equa ion o e alua e he p opaga ion delay acco ding o he DDM is: p= p01−exp −T−T0 τ (1) whe e Tis he ime elapsed since he las ou pu ansi ion, p0is he no mal p opaga ion delay and T0and τa e he deg ada ion pa ame e s. Fo each ga e, τand Todepend on he ou pu load (CL), he supply ol age (VDD), he inpu ansi ion ime (τin) and he posi ion o he inpu ha is The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48 changing s a e (i). I has been ob ained [8] ha his dependence can be exp essed as: τxVDD =Axi +BxiCL(2) T0x=1 2−Cxi VDD τin (3) whe e xs ands o o depending on he sense o he ou pu ansi ion ( ise o all espec i ely). A CMOS ga e is ully cha ac e ized wi h espec o he deg ada ion e ec when he se {Axi, Bxi, Cxi}is ob ained o each ga e inpu . So, he objec i e o he cha ac e iza ion p ocess is o ob ain he alues o he se o deg ada ion pa ame e s o (2) and (3) o a pa icula ga e, i.e.: {Axi, Bxi, Cxi}x= , i = 1...n (4) The cha ac e iza ion p ocess is composed o h ee main asks [9]: (a) ob ain p s. Tcu es co esponding o (1); (b) ob ain τ s. CLcu es co esponding o (2); and (c) ob ain T0 s. τin cu es co esponding o (3). The main idea in his p ocess is o es ablish he adequa e a ia ion anges o CLand τin in o de o ob ain accu a e alues o A,B, and C. Wi h espec o he a ia ion o CL, he ange depends on he ga e’s inpu capaci ance (Cin) a ying be ween 2Cin and 10Cin, while he ange o τin is calcula ed as a unc ion o he no mal p opaga ion delay when he inpu ansi ion ime is ze o ( his pa ame e is called ps). So, an adequa e ange o τin a ies be ween 0.1 ps (co espond- ing o su icien ly as ansi ions) and 10 ps (co esponding o su icien ly slow ansi ions). 3 No mal P opaga ion Delay Analysis and Modeling o DDM Ac ually, cha ac e izing he DDM comple ely also implies he cha ac e iza ion o he no mal p opaga ion delay ( p0), and he alue o p0depends on bo h CL and τin [1, 4]. Ou main objec i e is o analyse he beha iou o p0in o de o implemen i , as pa o DDM, in a logic iming simula o (HALOTIS) ocused on he simula ion o ci cui s based on s anda d cell lib a ies. The model o p0should be simple and as in e ms o compu a ion ime, hough i mus be accu a e enough inside he CLand τin a ia ion anges exposed in he p e ious sec ion. This model should also be de eloped a he same le el han he DDM (a he ga e-le el) p o iding a se o cha ac e is ic ga e pa ame e s. We ha e s udied he alue o p0wi h espec o CLand τin o h ee di e en ga es: an in e e (INV), a wo-inpu s NAND ga e (NAND2), and a wo-inpu s NOR ga e (NOR2). Fo hese ga es we ha e measu ed he delay om each inpu o he ou pu o he ga e o bo h alling and aising ou pu ansi ions. We will no e each case as GATE i-R/F, whe e GATE is INV (in e e ), NAND2 ( wo-inpu s NAND ga e), o NOR2 ( wo-inpu s NOR ga e); iis he numbe o The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48 he inpu changing; R means a ising ou pu ansi ion; and F means a alling one. Figu e 1.a p esen s he h ee-dimensional ep esen a ion o p0wi h espec o CLand τin ob ained by elec ic simula ion wi h HSPICE [11] (in subsequen pa ag aphs, we will e e o hese da a as a HSPICE-g id) o he case o INV 1-R (in e e , inpu 1 changing wi h aising ou pu ). As we can obse e, he g id su ace con o ms p ac ically o a plane. Figu es 1.b and 1.c show he co e- sponding g id ob ained o he cases o NAND2 2-R and NOR2 1-F espec i ely. In hese las wo igu es i can be seen he same beha iou as in he i s one. Due o hese esul s, we p opose he nex simple heu is ic model in o de o i he no mal p opaga ion delay: p0=DxiCL+Exiτin +Fxi (5) whe e Dxi,Exi, and Fxi a e he model pa ame e s. An indi idual pa ame e alue is ob ained o each ype o ou pu ansi ion ( o , no ed by x) and each inpu o he ga e (no ed by i). This model elies on he men ioned se o pa ame e s {Dxi, Exi, Fxi}which ha e o be cha ac e ized o each ga e and ansi ion ype. In o de o e i y ha his simple model co ec ly adjus s he ga es beha iou , we ha e i ed hese pa ame e s using mul iple linea eg ession o e he HSPICE-g id. Figu e 2 shows he same ep esen a ions o Fig. 1’s bu , in his case, p0is calcula ed applying (5). I is clea ha he beha iou o his simple model co ec ly adjus s he HSPICE-g id. The men ioned esul has been ob ained o he whole se o s udied cases. In able 1 we can see, o each case: he alue o he pa ame e s (D,E, and F), he mean absolu e e o (e ) in ps, and he mean de ia ion (de ) exp essed in o pe cen ages. This e o measu es a e calcula ed con as ing he alue o p0in he HSPICE-g id wi h he alue ob ained om he p oposed model (5). I shows clea ly ha he app oxima ion is adequa e, since he mean de ia ion is always lowe han 4%. 4 Cha ac e iza ion P ocess o No mal P opaga ion Delay Once we ha e es ablished a linea model o he alue o p0, we ha e o de elop a cha ac e iza ion p ocess o be included in AUTODDM. Ou in en ion is also o educe he impac on he o al cha ac e iza ion ime as much as possible. Ac ually, i is possible o pe o m an adequa e cha ac e iza ion o he D, E, and Fpa ame e s using he same da a epo ed by AUTODDM. This ool pe o ms wo g oups o simula ions: one o a se o CL alues and a ixed ypical τin and he o he o a se o τin alues and a ixed ypical CL. So, da a epo ed by AUTODDM p o ide wo lines in he HSPICE-g id (Fig. 3). Figu e 4 shows he app oxima ion ob ained s a ing om AUTODDM da a. As we can see, hese alues a e p ac ically he ones ob ained o Fig. 2. In able 2 we p esen he cha ac e iza ion da a ob ained om AUTODDM esul s o he The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48 10 15 20 25 30 35 40 45 0 500 1000 1500 2000 2500 0 100 200 300 400 500 600 τin (ps) CL ( F) p0 (ps) (a) 510 15 20 25 30 35 0 500 1000 1500 2000 100 200 300 400 500 600 700 τin (ps) CL ( F) p0 (ps) (b) 510 15 20 25 30 35 40 45 0 500 1000 1500 2000 2500 100 200 300 400 500 600 τin (ps) CL ( F) p0 (ps) (c) Fig. 1. HSPICE-g ids o : (a) INV 1-R, (b) NAND2 2-R, and (c) NOR2 1-F The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48 10 15 20 25 30 35 40 45 0 500 1000 1500 2000 2500 0 100 200 300 400 500 600 τin (ps) CL ( F) p0 (ps) D= 4.27 E= 0.180 F= 30.0 (a) 510 15 20 25 30 35 0 500 1000 1500 2000 100 200 300 400 500 600 700 τin (ps) CL ( F) p0 (ps) D= 4.16 E= 0.213 F= 93.1 (b) 510 15 20 25 30 35 40 45 0 500 1000 1500 2000 2500 100 200 300 400 500 600 τin (ps) CL ( F) p0 (ps) D= 3.61 E= 0.135 F= 114.9 (c) Fig. 2. G ids ob ained wi h (5) applying mul iple linea eg ession o HSPICE da a o : (a) INV 1-R, (b) NAND2 2-R, and (c) NOR2 1-F The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48 10 15 20 25 30 35 40 45 0 100 200 300 400 500 100 150 200 250 300 τin (ps) CL ( F) p0 (ps) (a) 510 15 20 25 30 35 0 100 200 300 400 500 100 150 200 250 300 350 τin (ps) CL ( F) p0 (ps) (b) 510 15 20 25 30 35 40 45 0 200 400 600 800 1000 150 200 250 300 350 τin (ps) CL ( F) p0 (ps) (c) Fig. 3. Poin s ob ained wi h AUTODDM o : (a) INV 1-R, (b) NAND2 2-R, and (c) NOR2 1-F The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48 10 15 20 25 30 35 40 45 0 500 1000 1500 2000 2500 0 100 200 300 400 500 600 700 τin (ps) CL ( F) p0 (ps) D= 4.03 E= 0.200 F= 30.9 (a) 10 15 20 25 30 35 40 45 0 500 1000 1500 2000 2500 0 100 200 300 400 500 600 700 τin (ps) CL ( F) p0 (ps) D= 4.25 E= 0.193 F= 95.0 (b) 10 15 20 25 30 35 40 45 0 500 1000 1500 2000 2500 0 100 200 300 400 500 600 700 τin (ps) CL ( F) p0 (ps) D= 3.51 E= 0.144 F= 113.6 (c) Fig. 4. G ids ob ained wi h (5) applying mul iple linea eg ession o AUTODDM da a o : (a) INV 1-R, (b) NAND2 2-R, and (c) NOR2 1-F The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48 Table 1. Cha ac e iza ion o no mal p opaga ion delay pa ame e s using HSPICE da a Ga e Case D (ps/ F) E F (ps) e (ps) de (%) INV 1-R 4.27 0.180 30.0 4.6 1.74 INV 1-F 3.57 0.100 31.5 6.8 3.75 NAND2 1-R 4.21 0.202 55.4 2.2 1.03 NAND2 2-R 4.16 0.213 93.1 3.0 1.20 NAND2 1-F 2.77 0.083 42.4 3.9 3.27 NAND2 2-F 2.75 0.019 61.1 3.8 3.34 NOR2 1-R 4.07 0.087 99.7 3.8 1.63 NOR2 2-R 3.95 0.160 43.9 2.6 1.24 NOR2 1-F 3.61 0.135 114.9 4.8 1.58 NOR2 2-F 3.47 0.125 59.7 4.6 2.29 same cases con empla ed in able 1. The e o alues shown in his second able ha e been calcula ed in e e ence o he HSPICE-g id. So, on he one hand, he use o his simple heu is ic model allows us o include he whole calculus in o AUTODDM wi hou a ec ing he o al simula ion ime needed o cha ac e ize a s anda d cell. On he o he hand, hese da a allow us o ob ain p ac ically he same alues o he pa ame e s D,E, and F, and o keep he mean de ia ion unde 4%. Table 2. Cha ac e iza ion o no mal p opaga ion pa ame e s using AUTODDM da a Ga e Case D (ps/ F) E F (ps) e (ps) de (%) INV 1-R 4.03 0.200 30.9 8.0 2.47 INV 1-F 3.53 0.109 31.3 7.4 3.89 NAND2 1-R 4.22 0.188 57.0 5.2 1.90 NAND2 2-R 4.25 0.193 95.0 9.8 2.72 NAND2 1-F 2.88 0.078 42.7 4.1 3.57 NAND2 2-F 2.86 0.014 62.1 3.9 3.47 NOR2 1-R 3.97 0.103 99.1 7.8 2.87 NOR2 2-R 3.89 0.167 43.6 2.7 1.15 NOR2 1-F 3.51 0.144 113.6 5.2 1.60 NOR2 2-F 3.53 0.120 61.1 4.9 2.59 5 Conclusions Cha ac e izing he DDM comple ely also implies he cha ac e iza ion o he no mal p opaga ion delay ( p0), and he alue o p0depends on bo h CLand τin. This pape p esen s he analysis we ha e ca ied ou abou he alue o p0in a 0.35 µm CMOS echnology. In his way, we ha e p oposed a simple heu is ic The inal publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/3-540-45716-X 48