Edge-Triggered D Flip-flop. Formal Analysis. Fundamental-Mode Sequential Circuits. D latch: How do flip-flops work?

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1 E-Trir D Flip-Flop Funamntal-Mo Squntial Ciruits PR A How o lip-lops work? How to analys aviour o lip-lops? R How to sin unamntal-mo iruits? Funamntal mo rstrition - only on input an an at a tim; iruit must stal or nxt input an an. Clk D B S How os it work? How was it sin? E-Trir D Flip-lop Intuitiv xplanation: lok=0 => R=S=1, CL=PR=1, an - ar l. itr D=1 => A=1, B=0 or D=0 => A=0, B=1 D an an ut an - ar l. Lt lok an 0->1 Eitr B=1, A=0 => R=1, S=0 => =0, -=1 or B=0, A=1 => R=0, S=1 => =1, -=0 Wat i D ans wil lok=1? a) D was 0 an ans to 1 A=0, B=1 => R=1, S=0 as S=0, an in D os not an B. ) D was 1 an ans to 0 A=1, B=0 => R=0, S=1 B ans rom 0 to 1, ut A os not an. Fallin lok as no t. Funamntal mo rstrition applis. D E Formal Analysis Y D lat: + Y To analys tis iruit, rak t ak loop. It is onvnint to prtn tat all t ats av 0 lays an tat tr is a init ut small lay in t ak loop.

2 Analysis o D lat Y + an tout o as t nxt stat. Y + = D.E + Y.(D + E) Transition Tal Y 0 1 DE Y + Stat Tal S 00 DE S0 S0 S0 S1 S0 S1 S1 S0 S1 S1 S* Stal stats - nxt stat is t sam as prsnt stat (irl). Stat & Output Tal Stat Tal DE S 00 S0 S0, S0, S1, S0, S1 S1, S0, S1, S1, S*, = D.E + Y.E + Y.D = D.E + Y S 00 DE S0 S0 S0 S1 S0 S1 S1 S0 S1 S1 D ans to 1 E ans to 1 S* Can tra wat appns wn inputs an rom DE = 00, S= S0. D os to 1 ollow y Eto1.

3 Stat Tal PR E-Trir D Flip-Flop S 00 DE Y1 Y1 + S0 S0 S0 S1 S0 S1 S1 S0 S1 S1 Clk Y3 Y3 + S* Wat appns i D&E now an almost simultanously? Exat squn is unprital. Final rsult is unprital. D Y2 Y2 + Multipl Fak Loops Assum PR an CL ar at 1 - n inor. + Y1 = Y2.D + Y1.Clk + Y2 = Y2.D + Y1.Clk + Clk + Y3 = Y1.Clk + Y3.(Y2.D + Y1.Clk + Clk) = Y1.Clk + Y3.(Y2.D + Y1.Clk + Clk) = Y3 + Y1. Y2.Clk + Y1.Clk. D Transition Tal Y1 Y2 Y3 00 ClkD Y2 000 tn Y Y3 tn Y2 Y1* Y2* Y3*

4 Ras Suppos startin stat is Y1,Y2,Y3 =1, Clk,D=00 an Clk ans to 1 Exat orr o stat ans pns on t orr o an o intrnal varials Ra - Multipl intrnal varials an stat as a rsult o ONE input anin stat I inal stat os not pn on orr o intrnal stat ans: non-ritial ra Suppos t transition tal look lik: Transition Tal Y1 Y2 Y3 00 ClkD Y2 000 tn Y Y1* Y2* Y3* Critial Ra to avoi! Stat & Output Tal S 00 ClkD S0 S2, S2, S0, S0, S1 S3, S3, S0, S0, S2 S2, S6, S6, S0, S3 S3, S7, S7, S0, S4 S2, S2, S7, S7, S5 S3, S3, S7, S7, S6 S2, S6, S7, S7, S7 S3, S7, S7, S7, S1,S4,S5 ar unstal S2,, S3,, S6, annot S*, - ra Flow Tal S 00 ClkD S0 S2, S6, S0, S0, S2 S2, S6, -,- S0, S3 S3, S7, -,- S0, S6 S2, S6, S7, -,- S7 S3, S7, S7, S7, S*, - Eliminat unstal stats, multipl ops an unraal stats

5 Flow Tal S 00 ClkD SM SM, S6, SM, SM, S3 S3, S7, -,- SM, S6 SM, S6, S7, -,- S7 S3, S7, S7, S7, S*, - S0 an S2 ar ompatil an may mr Funamntal Mo Dsin Stat Dsin Spiiations Driv a primitiv low tal Ru t low tal Mak a ra-r stat assinmnt Otain t transition tal an output map Otain azar-r stat quations Stat Minimisation Two stats an onsir quivalnt i tir outputs an nxt stats ar t sam (or all ominations o inputs). i.. ty ar inistinuisal rom outsi. Stats&karquivalnt (t two rows ar t sam). Rpla all instans o k y (only applis to row j). Nowstats&jarquivalnt. Stat Minimisation Prsnt stat s 00 xy a,0,0 a,0 a,0,0,0 a,0 a,0,0 i,0 a,0,1,1,0,0,1,1 i,0 a,0,1,0,0 a,0 a,0,1 j,0 a,0,1,0,0,0 a,0 i i,0 i,0 a,0 a,0 j,0 k,0 a,0 a,0 k,0 i,0 a,0,1 s*,z

6 Stat Minimisation Prsnt stat s 00 xy a,0,0 a,0 a,0,0,0 a,0 a,0,0 i,0 a,0,1,1,0,0,1,1 i,0 a,0,1,0,0 a,0 a,0,1,0 a,0,1,0,0,0 a,0 i i,0 i,0 a,0 a,0 Stat Minimisation Not tat tis is an inormal approa. W av ru t numr o stats rom to 9. W an ru t numr o stats to 6. Tis an on y onstrutin quivaln lasss. Can also on usin an impliation tal. Impliation tal an also us or inompltly spii systms. Impliation tal ontains all possil ominations o stats: s*,z i a -i -i Impliation Tal -i i- - a I outputs ar irnt, put X. I outputs ar sam, list nxt stats tat must quivalnt. (I nxt stats ar t sam as prsnt stats, put a tik.) i a -i -i Impliation Tal -i a Now o rit to lt pass trou t tal, puttin in Xs wr impliations av alray n isount...&i annot quivalnt as -i as an X. 1st pass sown. i- -

7 i a -i -i Impliation Tal -i 2n pass. i- - a Impliation Tal Tr ar now tr squars lt unross. In tis as, w an u tat stats (,i), (a,), (,) ar quivalnt. Tus t stats n to implmnt t systm ar (,i), (a,), (,), (), (), (). Can av mor omplx ass: i i -i - a- a i -i - a- a- a- Impliation Tal - -i - -i - a- - a- a- Atr all passs a Equivaln Classs Start at Rit o impliation tal. In olumn, t squar (,) is not ross out so w list t pair (,). In olumn, w in (,i). In olumn, w in ot (,i) an (,), so w an a to (,i). Similarly, w an a to (,) an a to (,,i). i (,) (,i) (,) (,i) (,) (,,i) (,) (,,i) (,,) a (a,,,i) (,,) Equivaln Classs: (a,,,i) (,,) () ()

8 Inompltly Spii Systm Suppos tat w o not ar wat t output is in rtain stats. Furtr, rtain nxt stat transitions may not in. Tis an xploit to minimis stats. W on't ar aout intrnal stats, only aout outputs! W now talk aout ompatil stats. Two stats ar ompatil i (an only i) tir outputs, i spii, ar t sam atr t sam squn o inputs... onsir a systm wit 4 inputs an 2 outputs spii y t ollowin stat an output tal. (N.B. w av not onsir all ominations o inputs - only on input is tru at any tim.) Inompltly spii systm s Input K L M N a a,- a,- a,-,00,-,-,- -,- -,-,-,- -,-,- -,-,- -,-,-,- -,- -,- -,- -,-,- a,,- -,- -,- a,,-,-,- a,00 s*,yz a- a- a- a- a- a Impliation Tal a Dtrmination o maximum ompatils Sam pross as or inin quivaln lasss. Start at rit o tal an list ompatils. - - (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) a (,) (,) (,) (,) (,) (,) (,) (a) Not tat, unlik a ully spii systm, (,) ar ompatil, (,) ar ompatil, ut (,) ar not ompatil. Tus w av two ompatil pairs, insta o on lass.

9 Maximal Compatils Not tat w now av 8 ompatil sts. W start wit 8 stats, so tis osn't appar to av ain us anytin! But w o not n all t sts. W ar tryin to in ompatil stats, tus o not n to us a st. W n t ollowin sts: (a) (,) ut all t otr stats our in two sts. Tis is analoous to runany in K-maps. Tror, w slt suiint sts to ovr all t stats.. (a) (,) (,) (,) (,) or (a) (,) (,) (,) (,) Stat Assinmnt m stats an r stat varials ivs 2r! /(2 r - m)! possil assinmnts. No way o trminin wi is st. Can assin to mak stat varials maninul. Can assin to minimis numr o its anin twn stats. Not rom t impliation tal, tat (,) implis (,) an (,) implis (,). Stat Dsin Exampl Nativ -trir T lip-lop Inputs T C Output Commnts a Initial stat Atr a Initial output Atr Atror Atraor 00 Nativ -trir T Flip-lop /0 a/0 /1 / /0 /0 /1 / Atror Atror

10 Primitiv Flow Tal Stat 00 TC Output a - a a 0-0 a a- Impliation Tal a Compatil stats From Impliation tal, ompatil stats ar: (a,) (,) (,) (,) (,) (,) (,) (,) Ts an mr - on tniqu is: a Ru Flow Tal Stat 00 TC Output (a,) A D A A B 0 (,,) B B B C B 1 (,) C B C C D 1 (,,) D D D A D 0

11 Stat Assinmnt Ru Flow Tal A=00 B= Y1 Y2 00 TC Output D= C= Y1* Y2* -v -trir T lip-lop K-maps (Entris orrsponin to stal stats in rn): TC Y1 Y TC Y1 Y2 00 Y2*: Y1*: Y1* T.C.Y2 C.Y1.Y2 T.Y1.Y2 C.Y1.Y2 T.C.Y2 T.Y1.Y2 T.C.Y1 Tr ar 8 prout trms in Y1*. Not tat a 1 to 0 an in T or C an aus a azar, vn twn stal stats. Tror all prout trms ar n, xpt T.C.Y1 aus tat ovrs two stal stats, wit t sam inputs. Altou Y2 is 1 in on stat an 0 in t otr, t transition an t appn Y2* T.Y2 C.Y2 Y1.Y2 T.C.Y1 W n n Y1.Y2 or Y2* aus it ovrs a transition twn stal stats wn T ans rom 1 to 0. T rul is: inlu all prout trms xpt tos tat ovr stal pairs in t sam olumn (i.. wit t sam inputs) an tos wr a transition rom a stal stat to an unstal stat (in t sam row) an only our as a rsult o a 0 to 1 transition.