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 .
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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
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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
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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
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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
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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
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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
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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
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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