Clock Skew and Signal Representation. Program. Timing Engineering
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1 lock Skew ad Sigal epreseaio h. 7 IBM Power 4 hip 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 /7/4 8 frequecy domai /7/4 8 frequecy domai Sequeial ircuis 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 lock skew ca occur because of differe wire leghs ad differe capaciive loads imig Egieerig urrely, sysem speed ad clock frequecy is super IMPOAN (Power is also impora, frequely a coflicig objecive) lock 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 urre subjec: basic priciples of clock circui aalysis Firs: IBM Power 4 chip example /7/4 8 frequecy domai /7/4 8 frequecy domai 4 IBM Power 4 hip Layou Pla lock 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 lock frequecy.4 Ghz rasisor size.8 µm 74.. rasisors 68 4 pis ( sigal) 5 W (@. GHz,.5 ) IBM J. es. Dev. ol 46 No. Ja (also olie) LOK Soluio: Hree lock Nework All blocks equidisa from clock source zero (relaive) skew Sub blocks should be small eough o igore irablock skew I pracice perfec Hree shape o realizable (ad load formed by blocks o ideical ad ad may more oidealiies) /7/4 8 frequecy domai 5 /7/4 8 frequecy domai 6
2 Power 4 lock Disribuio /7/4 8 frequecy domai 7 /7/4 8 frequecy domai 8 H ou i lock lie Model v () Liear driver models Oelump for iercoec Firs Order espose/emider F I = v() I = iiial value, a ime of swichig F = fial value, afer sufficie ime L ou v() as a fucio of ime / τ v( ) = F ( I F ) e lock geeraor H L Iercoec Flipflop v() ime as a fucio of v h F I = τ l F h Also rue whe I > F Shifs i ime whe swich /7/4 8 frequecy domai 9 /7/4 8 frequecy domai lock Skew Example Wha is he lock Skew bewee ideical clocks wih differe wire leghs (simple approach) s H = 5, L = L h = 4 s = 4 Ω L = pf s w W = Ω vs. Ω W = pf vs. pf w τ = (4)x() = s τ = (4)x() =.8s L F I = τ l =.6τ F kω x pf = s h =.6s =4.5s Skew (differece) =.9 s lock Skew Example Evaluaio τ = (4)x() = s τ = (4)x() =.8s =.6s =4.5s Is his a good compuaio? ompuaio is OK, bu model is iaccurae Skew =.9 s Model does o accou for Disribued aure of iercoec (see.5) Periodic aure of clock sigal (similar o siuaio i.7.)... Now we will cosider periodic aure of clock sigal /7/4 8 frequecy domai /7/4 8 frequecy domai
3 Periodic Sigals : period Symmeric square wave F I Firs Order espose / emider evisied = v() I = iiial value, a ime of swichig F = fial value, afer sufficie ime v() = v( ) = v( ) for arbirary ad = fudameal frequecy [Hz] (Herz) v() as a fucio of ime / τ v( ) = F ( I F ) e Afer = ατ sie wave Asymmeric square wave Sawooh v( ατ) = F ( I F ) e ( e ) F e I = Also rue whe I > F Shifs i ime whe swich /7/4 8 frequecy domai /7/4 8 frequecy domai 4 v H L Periodic Swichig ( ατ ) = = ( e ) H e v( ) v( )= ( e ) L e = ( e ) e v( ) H ad so o v() Assume / = ατ ad use v( ατ) = ( e ) F e I /7/4 8 frequecy domai 5 Periodic Swichig Numerical Example Assume L =, H =, v() =, α = e α =.6 e α =.7 ( e ) e v( ) = H =.6 H.7v ( ) =. 6 v( ) =.6L.7v =. =.6H.7v ( ) =. 76 v( ) =.6L.7v = =.6H.7v ( ) = v( ) =.6L.7v = =.6H.7v ( ) = H L v() oclusio: Afer iiial sarup behavior, high ad low swichig volages coverge o fixed values /7/4 8 frequecy domai 6 H v h.6 v l L A Sarup rasie oclusio: Afer iiial sarup behavior, high ad low swichig volages coverge o fixed values.69 A Seady Sae Waveforms ( 7.4.).7.69 H v h v l L A Seady Sae Pullup, < < ( ½) e τ e τ H L v = l e τ v h e τ e τ L H = e τ v ( ) = ( v l H ) e τ H Pulldow, ( ½) < < ( ) v ( ) = ( vh L ) e τ L /7/4 8 frequecy domai 7 /7/4 8 frequecy domai 8
4 lock Skew Example evisied s s =.6s =4.7s w w L Skew =.47 s L Same daa: H = 5 h = 4 s = 4 Ω L = pf W = Ω vs. Ω W = pf vs. pf Use formulas from previous slide (also see p. 6) Do his yourself!.9 s wih previous approach ime Domai Aalysis a be difficul, especially for periodic sigals whe ieresed i seadysae periodic behavior Should use righ ool for he job Frequecy domai mehods more appropriae for frequecy domai quesios his exra accuracy is defiiely worh he rouble! Bu i ca be doe much simpler: frequecy domai approach comig up soo /7/4 8 frequecy domai 9 /7/4 8 frequecy domai Priciple Frequecy Domai Aalysis. Decompose periodic ime domai sigal io sum of siusoidal sources. ompue sysem resposes o each idividual frequecy. Add decomposed oupu resposes Easier ha ime domai appr. if perfrequecy compoe compuaio is easy! 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 /7/4 8 frequecy domai /7/4 8 frequecy domai Fourier Series Sigal epreseaio Ay periodic sigal ca be represeed by a ifiie series of cosies = f () = A A cos( θ ) Usig cos( u) = cos cos u si si u his is equivale o: a = f ()d = f () = a ( a cos b si ) he coefficies a i ad b i are as follows a = f () cos d do eed o lear he proof b = f () si d Fourier Series for Square Wave a = v s () = f () = a ( a cos b si ). a = v s () d = d d = Average value see p.45 ad upcomig slide o eve/odd fucios. 5 = f= = π. b = v s () si d = si πd 5 = cos π. aleraes bewee ( ) π = cos π cos ad π = [ ( ) ] v s () = siπ π π,s =,,... /7/4 8 frequecy domai /7/4 8 frequecy domai 4 4
5 Fourier Series for Square Waveform, Eve vs. Odd Symmery Why are he cosiecoefficies of he square wave zero?,,8,6,4,,,,,,4,6,8, v s siπ =,,... π () = Usig six erms Usig four erms /7/4 8 frequecy domai 5 = Odd square wave = Eve square wave Eve symmery: f(x) = f(x), odd symmery: f(x) = f(x) Square wave fucio o he lef has odd symmery osie 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 /7/4 8 frequecy domai 6 omplex Form of he Fourier Series Fourier: = f () = a a cos b si see book Euler: e jϕ = cosϕ jsiϕ Prove! eve/odd properies of sie ad cosie cos = si = j j j ( e e ) j j ( e e ) f () f e = j = j f = f () e d omplex form of Fourier Series /7/4 8 frequecy domai 7 Expl. 7.4 omplex Fourier Series for Eve Symmeric Square Wave, < < 4 4 v() =, < < 4 4 π = j () ( ) e v = π = odd /7/4 8 frequecy domai 8 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 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 = / I = / = I I = / / I I /7/4 8 frequecy domai 9 /7/4 8 frequecy domai 5
6 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:. osider each idepede source i ur eplace all ohers by dead equivale (should keep depede sources) ompue respose o acive source. Add idividual resposes /7/4 8 frequecy domai i..k v 5 i v k v 4.7k i. i.k v i 5 i v 4.7k k v Superposiio Example 5. ' '' I = I I =. 69 ma ' '' I = I I =. 49 ma ' '' I = I I =. 6 ma 5. I ' = = // ' ' 4. 7 // I = I =. 5 ' ' 4. 7 // I = I = i 4.. I '.k v ' = = //. i '' //. '' i I = I =. 9. v k v 4.7k '' //. '' I = I =. 8 /7/4 8 frequecy domai 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 Seady Sae Siusoidal espose d v () v () = cos d v ( ) = Acos B si v v s () v () cos d Acos B si τ [ Acos B si] = cos d Acos B si Aτ si Bτ cos = cos A Bτ = A = τ B = B Aτ = τ () = cos si /7/4 8 frequecy domai /7/4 8 frequecy domai 4 cos si Siusoidal espose Summary v v τ () = cos si v () 7.7. Example 7. τ () = cos si cos Siusoidal espose oclusio v s () v () v τ () = cos si = a cos( ϕ ) (Usig cos( u) = cos cosu si si u), see ex chaper same frequecy bu differe phase ad ampliude he respose of imeivaria liear sysems drive by siusoidal sigals of a give frequecy is agai a siusoid wih same bu modified ampliude ad phase /7/4 8 frequecy domai 5 /7/4 8 frequecy domai 6 6
7 Overview ad Pla 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 I I I Fourier Series Superposiio A seady sae aalysis he respose of liear sysems o a siusoidal sigal is a siusoid of same frequecy Siusoidal Seady Sae Apply above ools for geeral periodic seady sae aalysis /7/4 8 frequecy domai 7 /7/4 8 frequecy domai 8 Fourier series for square wave Superposiio of Fourier Sources Deermie he respose of a load o a square wave, usig Fourier series, usig respose of si : v si π si π si π τ si cos si ( τ ) cos si ( τ ( τ ( τ ) cos si ( τ ) π ( τ ) π ( τ ) cos si ( τ ) π ( τ ) π ( τ ) v ( ) = cos si = ( τ,,... ( τ /7/4 8 frequecy domai 9,,,8,6,4,,,,,4,6,8,, L = H = τ =. Graph ( τ ) v ( ) = cos si = ( τ,,... ( τ. H. v h v l L compare o earlier resul L = H = τ =.5 h =.7 l =.69 /7/4 8 frequecy domai 4 Priciple Wrappig Up. Decompose periodic ime domai sigal io sum of siusoidal sources. ompue sysem resposes o each idividual frequecy. Add decomposed oupu resposes Easier ha ime domai appr. if perfrequecy compoe compuaio is easy! emark 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, bypassig 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. /7/4 8 frequecy domai 4 /7/4 8 frequecy domai 4 7
Clock Skew and Signal Representation
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