Clock Skew and Signal Representation

Transcription

1 Clock Skew ad Sigal Represeaio Ch. 7 IBM Power 4 Chip 0/7/ frequecy domai

2 Program Iroducio ad moivaio Sequeial circuis, clock imig, Basic ools for frequecy domai aalysis Fourier series sigal represeaio Periodic sigals ca be represeed as a (ypically ifiie) series of sie ad/or cosie fucios Superposiio Liear sysems wih muliple sources ca be aalyzed by sudyig he effec of each source i ur Siusoidal Seady Sae he respose of liear sysems o a siusoidal sigal is a siusoid of same frequecy Apply above ools for geeral periodic seady sae aalysis 0/7/ frequecy domai

3 Sequeial Circuis From compuaio Sychroous circuis imig is uder corol of ceral clock sigal Works like a orchesra coducor Bu all gaes should receive (approximaely) he same clock sigal If o: circui will behave erroeously (Example 7.) imig differece bewee clock sigals a differe places is called clock skew Clock skew ca occur because of differe wire leghs ad differe capaciive loads 0/7/ frequecy domai 3

4 imig Egieerig Currely, sysem speed ad clock frequecy is super IMPORAN (Power is also impora, frequely a coflicig objecive) Clock archiecure has major ifluece o performace (speed, bu also power) Iel, IBM, ad so o, have large desig eams for opimizig clock desig Ad for weakig iercoec paers ec afer firs silico Curre subjec: basic priciples of clock circui aalysis Firs: IBM Power 4 chip example 0/7/ frequecy domai 4

5 IBM Power 4 Chip Layou Pla Clock frequecy.4 Ghz rasisor size 0.8 µm rasisors 6380 C4 pis (00 sigal) 5 W (@. GHz,.5 ) IBM J. Res. Dev. ol 46 No. Ja 00 (also olie) 0/7/ frequecy domai 5

6 Clock Nework Problem: how o equalize clock delay across whole chip Bu delay across chip (much) larger ha clock period Observe: Oly relaive skew is impora Soluio: H-ree Clock Nework All blocks equidisa from clock source zero (relaive) skew Sub blocks should be small eough o igore ira-block skew I pracice perfec H-ree shape o realizable (ad load formed by blocks o ideical ad ad may more o-idealiies) CLOCK 0/7/ frequecy domai 6

7 Power 4 Clock Disribuio 0/7/ frequecy domai 7

8 0/7/ frequecy domai 8

9 Clock lie Model H R ou R i v C () Liear driver models Oe-lump RC for iercoec L R ou C C Clock geeraor Iercoec Flip-flop v() H L R C 0/7/ frequecy domai 9

10 Firs Order RC Respose/Remider F I =0 R v() C I = iiial value, a ime of swichig F = fial value, afer sufficie ime v() as a fucio of ime v / τ () = F ( I F ) e ime as a fucio of v h F I = τ l F h Also rue whe I > F Shifs i ime whe swich 0 0/7/ frequecy domai 0

11 Clock Skew Example Wha is he Clock Skew bewee ideical clocks wih differe wire leghs (simple approach) R s R s R w C L H = 5, L = 0 h = 4 R s = 400 Ω C L = pf R W = 00 Ω vs. 300 Ω C W = pf vs. 3 pf C w C L F I = τ l F kω x pf = s h =.6τ τ = (40000)x() = s τ = (400300)x(3) =.8s =.6s =4.5s Skew (differece) =.9 s 0/7/ frequecy domai

12 Clock Skew Example Evaluaio τ = (40000)x() = s τ = (400300)x(3) =.8s =.6s =4.5s Skew =.9 s Is his a good compuaio? Compuaio is OK, bu model is iaccurae Model does o accou for Disribued aure of iercoec (see 3.5) Periodic aure of clock sigal (similar o siuaio i.7.)... Now we will cosider periodic aure of clock sigal 0/7/ frequecy domai

13 Periodic Sigals : period Symmeric square wave v() = v( ) = v( ) for arbirary ad = fudameal frequecy [Hz] (Herz) sie wave Asymmeric square wave Sawooh 0/7/ frequecy domai 3

14 F I Firs Order RC Respose / Remider Revisied =0 R v() C I = iiial value, a ime of swichig F = fial value, afer sufficie ime v() as a fucio of ime v / τ () = F ( I F ) e Afer = ατ v ( ατ) = F ( I F ) e α ( ) e α F e α I = Also rue whe I > F Shifs i ime whe swich 0 0/7/ frequecy domai 4

15 Periodic Swichig v() H L R C v v( )= Assume / = ατ ad use ( ) ( ) v ατ = e F e I ( ) ( α ) α ατ = v = e e v( 0) 3 v = H ( α ) α e e v L ( α ) α e e v( ) H α α ad so o 0/7/ frequecy domai 5

16 Periodic Swichig Numerical Example Assume L = 0, H =, v(0) = 0, α = e -α = 0.63 e -α = 0.37 ( α ) α e e v( 0) v = H = v 0 = v H 0.37v = ( ) ( ) = v = v v L = 0.63 H ( ) ( ) = v = v v = 0.63 L H 0.37v ( ) = ( 3 ) = v = v = 0.63 L H 0.37v ( 3 ) = H L R Coclusio: Afer iiial sarup behavior, high ad low swichig volages coverge o fixed values v() 0/7/ frequecy domai 6 C

17 AC Sar-up rasie H v h v l L Coclusio: Afer iiial sarup behavior, high ad low swichig volages coverge o fixed values 0/7/ frequecy domai 7

18 0/7/ frequecy domai 8 AC Seady Sae AC Seady Sae Waveforms ( 7.4.3) = τ τ τ H L e e e h v = τ τ τ L H e e e l v H L v l v h ( ) H H l C e v ) ( v = τ Pull-up, < < ( ½) ( ) L L h C e v ) ( v = τ Pull-dow, ( ½) < < ( )

19 Clock Skew Example Revisied R s R s R w C w C L C L Same daa: H = 5 h = 4 R s = 400 Ω C L = pf R W = 00 Ω vs. 300 Ω C W = pf vs. 3 pf Use formulas from previous slide (also see p. 336) =.60s =4.07s Skew =.47 s.9 s wih previous approach his exra accuracy is defiiely worh he rouble! Bu i ca be doe much simpler: frequecy domai approach comig up soo Do his yourself! 0/7/ frequecy domai 9

20 ime Domai Aalysis Ca be difficul, especially for periodic sigals whe ieresed i seady-sae periodic behavior Should use righ ool for he job Frequecy domai mehods more appropriae for frequecy domai quesios 0/7/ frequecy domai 0

21 Priciple Frequecy Domai Aalysis. Decompose periodic ime domai sigal io sum of siusoidal sources. Compue sysem resposes o each idividual frequecy 3. Add decomposed oupu resposes 3 Easier ha ime domai appr. if per-frequecy compoe compuaio is easy! 0/7/ frequecy domai

22 Program Iroducio ad moivaio Sequeial circuis, clock imig, Basic ools for frequecy domai aalysis Fourier series sigal represeaio Periodic sigals ca be represeed as a (ypically ifiie) series of sie ad/or cosie fucios Superposiio Liear sysems wih muliple sources ca be aalyzed by sudyig he effec of each source i ur Siusoidal Seady Sae he respose of liear sysems o a siusoidal sigal is a siusoid of same frequecy Apply above ools for geeral periodic seady sae aalysis 0/7/ frequecy domai

23 Fourier Series Sigal Represeaio Ay periodic sigal ca be represeed by a ifiie series of cosies f () = A A ( θ ) 0 cos = Usig cos( u) = cos cos u si si u his is equivale o: f () = a ( a cos b ) 0 si = he coefficies a i ad b i are as follows do eed o lear he proof 7.6. a 0 = f 0 ()d a = f () cos d 0 b = f () si d 0 0/7/ frequecy domai 3

24 Fourier Series for Square Wave v s () f () = a ( a cos b ) 0 si = ,s. a0 = v s () d = d 0d = Average value a = 0 see p.345 ad upcomig slide o eve/odd fucios 0. 5 = f= = π b = v () si d = si πd s cos π. 0 = = ( cosπ cos 0) π [ ( ) ] π = v s () = π =,3,... aleraes bewee 0 ad π siπ 0/7/ frequecy domai 4

25 Fourier Series for Square Waveform,0,00 0,80 0,60 0,40 0,0 v s () = =,3,... π siπ 0,00-0,0 0,00 0,0 0,40 0,60 0,80,00 Usig six erms Usig four erms 0/7/ frequecy domai 5

26 Eve vs. Odd Symmery Why are he cosie-coefficies of he square wave zero? =0 Odd square wave =0 Eve square wave Eve symmery: f(x) = f(-x), odd symmery: f(x) = -f(-x) Square wave fucio o he lef has odd symmery Cosie fucio has eve symmery ad sie has odd symmery. his is also rue for all harmoics Square wave o he lef oly o be represeed by sie waves, o cosie waves Square wave o he righ has eve symmery oly cosies Assymeric fucio boh sie ad cosie erms 0/7/ frequecy domai 6

27 Complex Form of he Fourier Series Prove! Euler: e jϕ = cosϕ jsiϕ eve/odd properies of sie ad cosie Fourier: f = () = a a cos b si 0 cos = si = j ( j j ) e e ( j j ) e e 7.8 see book 7.8 f f () = = f j e = j f 0 () e Complex form of Fourier Series d 0/7/ frequecy domai 7

28 Complex Fourier Series for Eve Symmeric Square Wave = π 0 0 v () =, 0, < < 4 4 < < Expl. 7.4 v = π j () ( ) e = odd 0/7/ frequecy domai 8

29 Program Iroducio ad moivaio Sequeial circuis, clock imig, Basic ools for frequecy domai aalysis Fourier series sigal represeaio Periodic sigals ca be represeed as a (ypically ifiie) series of sie ad/or cosie fucios Superposiio Liear sysems wih muliple sources ca be aalyzed by sudyig he effec of each source i ur Siusoidal Seady Sae he respose of liear sysems o a siusoidal sigal is a siusoid of same frequecy Apply above ools for geeral periodic seady sae aalysis 0/7/ frequecy domai 9

30 Superposiio Liear circuis wih muliple idepede sources ca be solved by addig he volages (or curres) caused by each source acig aloe, wih all oher sources replaced by heir dead equivales I I I R R R I = ( )/R I = /R I = /R = I I = /R /R 0/7/ frequecy domai 30

31 Superposiio Superposiio is a impora, powerful, geeral echique No oly for siusoidal sources, bu i geeral (see example) Bu i relies o he ework beig liear Algorihm:. Cosider each idepede source i ur Replace all ohers by dead equivale (should keep depede sources) Compue respose o acive source. Add idividual resposes 0/7/ frequecy domai 3

32 5-0. v 3 - i.k i k k - - v i v Superposiio Example 5. ' '' = I = I I. 69 ' '' = I = I I. 49 ' '' = ma ma I = I I ma 5 - i.k v i 3 - i v 3 4.7k k v I ' = = // ' ' 4. 7 // I = I =. 35 ' ' 4. 7 // I3 = I = v 3 - i -.k i 3 4.7k k v - i v - 4. ' 3 = = //. I ' '' //. '' I = I3 = '' //. '' I = I3 = /7/ frequecy domai 3

33 Program Iroducio ad moivaio Sequeial circuis, clock imig, Basic ools for frequecy domai aalysis Fourier series sigal represeaio Periodic sigals ca be represeed as a (ypically ifiie) series of sie ad/or cosie fucios Superposiio Liear sysems wih muliple sources ca be aalyzed by sudyig he effec of each source i ur Siusoidal Seady Sae he respose of liear sysems o a siusoidal sigal is a siusoid of same frequecy Apply above ools for geeral periodic seady sae aalysis 0/7/ frequecy domai 33

34 RC Seady Sae Siusoidal Respose v C v C d d () RC v () = cos C () = Acos B si cos v s () - R v C () C Acos d B si τ = d [ Acos B si] cos Acos B si Aτ si Bτ cos = cos A Bτ = B Aτ = 0 A = τ B = τ τ v C τ () = cos si τ τ 0/7/ frequecy domai 34

35 RC Siusoidal Respose Summary v C () = cos si τ 7.7. τ τ cos si - R C v C () Example 7. v C τ () = cos si τ τ 0/7/ frequecy domai 35

36 cos RC Siusoidal Respose Coclusio v s () - R v C () C v C τ () = cos si τ = a cos ( ϕ ) (Usig cos( u) = cos cos u si si u), see ex chaper τ same frequecy bu differe phase ad ampliude he respose of ime-ivaria liear sysems drive by siusoidal sigals of a give frequecy is agai a siusoid wih same bu modified ampliude ad phase 0/7/ frequecy domai 36

37 Program Iroducio ad moivaio Sequeial circuis, clock imig, Basic ools for frequecy domai aalysis Fourier series sigal represeaio Periodic sigals ca be represeed as a (ypically ifiie) series of sie ad/or cosie fucios Superposiio Liear sysems wih muliple sources ca be aalyzed by sudyig he effec of each source i ur Siusoidal Seady Sae he respose of liear sysems o a siusoidal sigal is a siusoid of same frequecy Apply above ools for geeral periodic seady sae aalysis 0/7/ frequecy domai 37

38 Overview ad Pla Fourier Series I R I R Superposiio AC seady sae aalysis I R Siusoidal Seady Sae 0/7/ frequecy domai 38

39 0/7/ frequecy domai 39 Superposiio of Fourier Sources τ τ τ si cos si si π si π 3 3 si π - R C si π si π 3 3 si π - R C v C ( ) ( ) ( ) π τ π τ τ si cos ( ) ( ) ( ) = = v C π τ π τ τ si cos ) (,3,... Fourier series for square wave Deermie he respose of a RC load o a square wave, usig Fourier series, usig respose of si : ( ) ( ) ( ) π τ π τ τ si3 3 3 cos ( ) ( ) ( ) π τ π τ τ si cos

40 Graph v C ( ) = =,3,... ( τ ) cos ( τ ) π ( τ ) si π,0,00 0,80 0,60 0, H v h h =0.73 0,0 0,00 0,00-0,0 0,0 0,40 0,60 0,80,00 L =0 H = τ = v l L compare o earlier resul L =0 H = τ = 0.5 l =0.69 0/7/ frequecy domai 40

41 Priciple Wrappig Up. Decompose periodic ime domai sigal io sum of siusoidal sources. Compue sysem resposes o each idividual frequecy 3. Add decomposed oupu resposes 3 Easier ha ime domai appr. if per-frequecy compoe compuaio is easy! 0/7/ frequecy domai 4

42 Remark he derivaive of a siusoid is a siusoid hus, liear sysems govered by differeial equaios bu drive by siusoidal exciaios (sources) ca be solved by algebraic equaios, by-passig differeial calculus his is why frequecy domai aalysis is ofe comparaively simple Bu oly for liear circuis Will be worked ou furher i ex chaper 9. 0/7/ frequecy domai 4