EE 330 Lecture 41. Digital Circuits. Propagation Delay With Multiple Levels of Logic Optimally driving large capacitive loads
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1 EE 330 Lecure Digial Circuis Propagaion Delay Wih uliple Levels of Logic Opimally driving large capaciive loads
2 Review from Las Time Propagaion Delay in uliple- Levels of Logic wih Sage Loading nalysis sraegy : Express delays in erms of hose of reference inverer Reference Inverer V DD C =C IN= COX WIN LIN I= V IN V OUT R PD V. L Tn VDD IN LIN μ C W V -V μ C W 0.8V n OX IN DD Tn n OX IN DD L n =L p =L IN ssume μ n /μ p =3 W n =W IN, W p =3W IN = HL + LH =R PDC = (RC) (RC) HL = LH In 0.5u proc =0ps, C =f,r PD =.5K
3 Propagaion Delay Through uliple Sages of Logic wih Sage Loading Review from Las Time (ll gaes sized for equal worse-case rise/fall ime as ha of reference inverer) 3k+ I NOR= 3+k I NND= G xx G x G G G 3 G n I I 3 I I (n+) G x3 G x Idenify he gae pah from o Idenify each gae I and delay: i = I (i+) Propagaion delay from o : n (i+) i= = I
4 Review from Las Time Over-drive Capabiliy V IN HL LH V OUT V IN V OUT C L C L IL Define he (symmeric) Overdrive acors of a gae o be he facors by which PU and PD resisors are scaled down relaive o hose of he reference inverer. RPD RPU R PDE = R If scaled by he same facor: PUE = HL = LH = HL R I = C = PD HL L HL = HL+ LH= I L + HL L HL R I = C = PU LH L LH L LH LH LH R = = C PD HL LH L = + LH HL = =R C PD IL IL C =C IN s may be larger or smaller han
5 Propagaion Delay Through uliple Sages of Logic wih Sage Loading Review from Las Time Each gae has symeric overdrive relaive o ha of reference inverer G x Idenify each gae s I G xx 3k i+ I NORi=i G G G 3 G n 3+k I i NNDi=i : I : I 3 3: I n: I (n+) Idenify each gae s delay G x3 G x = k I (i+) i I (i+) is he oal I a he oupu of G i Propagaion delay from o : = n (i+) i= I i
6 Propagaion Delay Through uliple Sages of Logic wih Sage Loading Review from Las Time Each gae has possibly asymmeric overdrives G x Idenify each gae s I: G xx 3ki LHi+ I HLi NORi = G G G 3 G n 3 LHi+ki I HLi NNDi = Idenify each gae s delay: HL LH I HL LH I 3 G x3 HL3 LH3 I G x HLn LHn I (n+) i = I (i+) HLi LHi I (i+) is he oal I a he oupu of G i Propagaion delay from o : n = I i (i+) HLi LHi
7 Propagaion Delay Through uliple Sages of Logic wih Sage Loading Review from Las Time inimum size sraegy: inimum sized gae has I =/. Caps sill have I=C/C 3k se i LHi + HLi I NORi = LH =/3k; HL = G xx G x G G G 3 G n 3 se LHi +kihli I NNDi = LH =/3; HL =/k HL LH I HL LH I 3 G x3 HL3 LH3 I G x HLn LHn I (n+) NOR gae delay = I +3k i i NND gae delay = I k+3 i i Propagaion delay from o : n = I i (i+) HLi LHi
8 Propagaion Delay in uliple-levels of Logic wih Sage Loading and Overdrives Will now consider o propagaion for his circui as an example wih differen overdrives 50f 0f
9 Propagaion Delay in uliple-levels of Logic wih Sage Loading Equal rise-fall gaes, no overdrive 50f 0f =HL In 0.5u proc =0ps, C =f,r PD =.5K n i+ i= = I
10 Propagaion Delay in uliple-levels of Logic wih Sage Loading Equal rise-fall gaes, no overdrive Equal Rise/all C IN /C Inverer NOR NND Overdrive Inverer HL LH NOR HL LH NND HL LH 3k+ 3+k n / I(i+) 5 = I(i+) i= i=
11 Equal rise-fall gaes, no overdrive 3k+ I NOR= In 0.5u proc =0ps, C =f,r PD =.5K (Noe: This C OX is somewha larger han ha in he 0.5u ON process) 3+k I NND= 3k 0f C C 3 0f =5 f 3k 7 3k 0 3k I =0.5 I 3 =.5 I =.5 I 5 =.5 I 6 =.5 0 3k 7 = C C 3 k 5 50f =.5 f 5 i+ i= = I 50f =3.5
12 Propagaion Delay in uliple-levels of Logic wih Sage Loading Equal rise-fall gaes, wih symmeric overdrive f 0f 5 In 0.5u proc =0ps, C =f,r PD =.5K (Noe: This C OX is somewha larger han ha in he 0.5u ON process) = I n (i+) i= i
13 Propagaion Delay in uliple-levels of Logic wih Sage Loading Equal rise-fall gaes, wih overdrive C IN /C Equal Rise/all Equal Rise/all (wih ) Inverer NOR NND Overdrive Inverer HL LH NOR HL LH NND HL LH 3k+ 3+k 3k+ 3+k n I(i+) / I(i+) i= i= i n (i+) = i= i n I
14 Equal rise-fall gaes, wih overdrive 3k i+ I NORi=i 3+ki I NNDi=i 8 In 0.5u proc =0ps, C =f,r PD =.5K 5 (Noe: This C OX is somewha larger han ha in he 0.5u ON process) 3k 3 C C 0f 5 0f =5 f 3k 6 3k 0 I =.5 I 3 =3 I =.5 I 5 =5 I 6 =.5 3k 0 3 k 0 3k 7 C C = 50f =.5 f 50f = I n (i+) i= i =3.6
15 Propagaion Delay in uliple-levels of inimum-sized gaes Logic wih Sage Loading 50f 0f In 0.5u proc =0ps, C =f,r PD =.5K (Noe: This C OX is somewha larger han ha in he 0.5u ON process) =?
16 Propagaion Delay in uliple-levels of inimum-sized gaes Logic wih Sage Loading 50f 0f =? ll gae I=/ NOR: LH =/3k; HL = NND: LH =/3; HL =/k k: #inpus Observe ha a minimum-sized gae is simply a gae wih asymmeric overdrive
17 Propagaion Delay in uliple-levels of inimum-sized gaes Logic wih Sage Loading Equal Rise/all Equal Rise/all (wih ) inimum Sized C IN /C Inverer NOR NND Overdrive Inverer HL 3k+ 3+k 3k+ 3+k / / / ll gae I=/ NOR: LH =/3k; HL = LH NOR HL LH NND HL LH /3 /(3k) /k /3 NND: LH =/3; HL =/k k: #inpus I I / (i+) n I(i+) i= n n (i+) i= i i= HLi LHi
18 inimum-sized gaes LH=/3 HL= = k /3 / / / HL= LH= = 3k / 0f C C 0f =5 f / / / HL= LH= = 3k /6 I =3/ I 3 = I = I 5 =/ I 6 =.5 ll gae I=/ / = HL= LH= = 3k /9 / C C LH=/3 HL= = k / 50f =.5 f 5 = I(i+) i= HLi LHi 50f I C =C/C NOR: LH =/3k; HL = NND: LH =/3; HL =/k =63.5 k: #inpus
19 Propagaion Delay in uliple-levels of symmeric-sized gaes Logic wih Sage Loading 3k LH + I HL NOR= 3 LH +k I HL NND= / / 0f / 50f =?
20 Propagaion Delay in uliple-levels of Logic wih Sage Loading symmeric-sized gaes HL LH Equal Rise/all Equal Rise/all (wih ) inimum Sized symmeric ( HL, LH ) C IN /C Inverer NOR NND 3k+ 3+k 3k+ 3+k / / / HL +3 LH HL +3k LH k HL +3 LH Overdrive Inverer HL HL LH NOR HL /3 LH HL LH NND HL /(3k) /k LH HL LH /3 LH n I(i+) n I (i+) i= i= n n I(i+) I(i+) i i= HLi LHi i= HLi LHi / 5 = I(i+) i= HLi LHi
21 symmeric-sized gaes NOR: NND: HL= LH= 7/8 =? HL= LH=/ / / 0f C C 0f =5 f 7/ / I =63/8 I3 =9/ HL= LH= I =77/6 I5 =7/ I6 =.5 3/ = HL= LH=/ / 5/6 C C HL= LH= 50f =.5 f 5 = I(i+) i= HLi LHi 50f / =.6
22 Propagaion Delay in uliple-levels of Logic wih Sage Loading ixure of inimum-sized gaes, equal rise/fall gaes and f f =?
23 Driving Noaion Equal rise/fall (no overdrive) Equal rise/fall wih overdrive inimum Sized /3 symmeric Overdrive HL LH
24 Propagaion Delay in uliple-levels of Logic wih Sage Loading ixure of inimum-sized gaes, equal rise/fall gaes and f f /3 0f f 5 = I(i+) i= HLi LHi
25 Propagaion Delay in uliple-levels of Logic wih Sage Loading ixure of inimum-sized gaes, equal rise/fall gaes and / f 0f 3 5 = I(i+) i= HLi LHi
26 Propagaion Delay in uliple- Levels of Logic wih Sage Loading G x G xx G G G 3 G n I I 3 I I (n+) Equal rise/fall (no overdrive) G x3 G x n (i+) i= = I Equal rise/fall wih overdrive = I n (i+) i= i inimum Sized symmeric overdrive Combinaion of equal rise/fall, minimum size and overdrive n = I(i+) i= HLi LHi n = I(i+) i= HLi LHi n = I(i+) i= HLi LHi
27 Driving Large Capaciive Loads Example C L ssume C L =000C ssume driving by a reference inverer =? In 0.5u proc =0ps, C =f,r PD =.5K
28 Driving Large Capaciive Loads Example C L ssume C L =000C ssume driving by a reference inverer =000 is oo long! In 0.5u proc =0ps, C =f,r PD =.5K
29 Driving Large Capaciive Loads Example ssume C L =000C 000 ssume firs sage is a reference inverer C L =? = I (i+) i= i =
30 Driving Large Capaciive Loads Example ssume C L =000C 0 00 C L ssume firs sage is a reference inverer 3 (i+) = = i= i = 30 Dramaic reducion is propagaion delay (over a facor of 30!) Wha is he fases way o drive a large capaciive load? I
31 Opimal Driving of Capaciive Loads θ θ θ n- θ n- C L ssume firs sage is a reference inverer Need o deermine he number of sages, n, and he facors for each sage o minimize. n I n θi = (i+) = i= i θ i= i- where θ 0 =, θ n =C L /C This becomes an n-parameer opimizaion (minimizaion) problem! Unknown parameers: {θ, θ,...θ n-,n} n n-parameer nonlinear opimizaion problem is generally difficul!!!!
32 Opimal Driving of Capaciive Loads θ θ θn- θn- ssume firs sage is a reference inverer = Order reducion sraegy : ssume overdrive of sages increases by he same facor clear unil he load θ θ n i i= i- C L θ θ θ n- n θ C =C L C L This becomes a -parameer opimizaion (minimizaion) problem! Unknown parameers: One consrain : θ,n n θ C =C L One degree of freedom
33 Opimal Driving of Capaciive Loads θ θ θ n- C L = nθ = n θ C =C L θ θ n i i= i- θ n i i- i= θ = = nθ Unknown parameers: θ,n n θ C =C L n θ C =C L C n= ln lnθ C θ C = ln lnθ C L L Thus obain an expression for in erms of only θ
34 Opimal Driving of Capaciive Loads θ θ θ n- C L θ C = ln lnθ C L Is suffices o minimize he funcion lnθ-θ df θ = dθ ln θ θ ln 0 0 θ = e θ f θ= ln θ n θ C =C L C C n = ln n ln ln θ C C L L
35 Opimal Driving of Capaciive Loads θ θ θ n- C L θ OPT = e n OPT C ln L C n θ C =C L θ C = ln lnθ C L C L = eln nθ C
36 Opimal Driving of Capaciive Loads θ f= ln θ e e 3 θ minimum a θ=e bu shallow inflecion poin for <θ<3 pracically pick θ=, θ=.5, or θ=3 since opimizaion may provide non-ineger for n, mus pick close ineger
37 Opimal Driving of Capaciive Loads θ θ θ n- C L n-sage pad driver Ofen ermed a pad driver Ofen used o drive large inernal busses as well Generally included in sandard cells or in cell library Device sizes can become very large Odd number of sages will cause signal inversion bu usually no a problem
38 Opimal Driving of Capaciive Loads θ θ θ n- C L n-sage pad driver Example: Design a pad driver for driving a load capaciance of 0p, deermine for he pad driver, and compare his wih he propagaion delay for driving he pad wih a minimum-sized reference inverer. In 0.5u proc =0ps, C =f,r PD =.5K CL 0p nopt ln ln C f 7.8 Selec n=8, θ=.5 k- k- pk W =.5, W 3.5 nk
39 Opimal Driving of Capaciive Loads θ θ θ n- C L n-sage pad driver Example: Design a pad driver for driving a load capaciance of 0p, deermine for he pad driver, and compare his wih he propagaion delay for driving he pad wih a minimum-sized reference inverer. In 0.5u proc =0ps, k- k- C =f,r PD =.5K W nk=.5, Wpk 3.5 L n =L p =L IN Noe devices in las sage are very large!
40 Opimal Driving of Capaciive Loads θ θ θ n- C L n-sage pad driver Example: Design a pad driver for driving a load capaciance of 0p, deermine for he pad driver, and compare his wih he propagaion delay for driving he pad wih a minimum-sized reference inverer. In 0.5u proc =0ps, C =f,r PD =.5K k- k- pk W =.5, W 3.5 nk nθ =8.5 =0 ore accuraely: 7 C L = k= θ C 60
41 Opimal Driving of Capaciive Loads θ θ θ n- C L n-sage pad driver Example: Design a pad driver for driving a load capaciance of 0p, deermine for he pad driver, and compare his wih he propagaion delay for driving he pad wih a minimum-sized reference inverer. In 0.5u proc =0ps, C =f,r PD =.5K k- k- pk W =.5, W 3.5 If driven direcly wih he minimum-sized reference inverer C L = =500 C Noe an improvemen in speed by a facor of nk r
42 Opimal Driving of Capaciive Loads θ θ θ n- C L n-sage pad driver Example: Design a pad driver for driving a load capaciance of 0p, deermine for he pad driver, and compare his wih he propagaion delay for driving he pad wih a minimum-sized reference inverer. In 0.5u proc =0ps, k- k- C =f,r PD =.5K W nk=.5, Wpk 3.5 L n =L p =L IN Noe devices in las sage are very large!
43 Pad Driver Size Implicaions θ θ θ n- C L n-sage pad driver Consider a 7-sage pad driver and assume θ = 3 5 3
44 6 7
45 rea of Las Sage Larger han ha of all previous sages combined!
46 End of Lecure
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