Digital Circuit Engineering

Transcription

1 igitl Ciruit ngineering IGITL 2nd istributive (X + )(X + ) = X + Simplifition bsorption YX + X = X Y + XY = X + Y XY + XY = X emorgn Simple VLSI SIGN X + Y = XY X Y = X + Y emorgn Generl If (, b,... z,+,.,0,1) Then (, b,... z,.,+,1,0) Remember to brket N terms Crleton University 2006 ig Cir II p. 0 Revised; Jnury 24, 2007 Slide i emorgn s Theorem Simple two vrible forms s equtions nd s gtes with inverted inputs Why Rel Gtes re /, not N/ Two symbols for ; two for N- designs re esier to think bout - designs must be done to use rel gtes esign with N-; Implement with - Chnge between them using emorgn s Theorem N/ to / Conversions Generlized emorgn s Thorem Common rrors igitl Ciruits II p. 1, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide i

2 emorgn s Lw emorgn s Simple orm Used To ind the Inverse of xpressions emorgn s Lw ul emorgn s Lw = + (em) + = (em) Inverse The dul inverse + = = quivlent grphil forms: = K = + K K + = G = G G = C = + C N N N C + = = ig Cir II p. 2 Revised; Jnury 24, 2007 Slide 2 emorgn s Lw emorgn s Lws on Complementing emorgn s Lws on Complementing xpressions theorem relting s nd s. n gte with inverted inputs is equivlent to n N gte with n inverted output. n N gte with inverted inputs is equivlent to n gte with n inverted output. Inverting inputs nd outputs of n mkes it n N. Inverting inputs nd outputs of n N mkes it n. XMPL Convert ( + b)( + ) to n expression with 3 letters nd inversion brs only over single letters. ( + b)( + ) = ( + b) + ( + ) (em1) = ( b) + ( ) (em2) = b + (Cler brkets) = (b + ) (1) xy + xz = x(y+z) 41. PROLM Redue (+b) + b to four letters with inversion brs over single letters only. 42. PROLM Redue d(de) + (de)e to four letters with inversion brs over single letters only. Chnging everything into NOT nd N gtes It turns out tht ny logi iruit n be mde from N nd NOT gtes. emorgn s lw n be used to trnsform the iruits. 43. PROLM Convert ((rw + t)u + r)t into funtion with only N nd NOT opertions. igitl Ciruits II p. 3, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 2

3 Why Rel Gtes re /, not N/ Rel CMOS igitl Gtes Trnsistor NOT +5 V JOLT Smll R Jolt JOLT Smll R Jolt +V Two swithes, X hndles linked together with stiff wire +V + Q 1 Q 1 =1 => Q 1 losed => =0 X PMOS trnsistor ts like: losed swith when is 0 open swith when is 1 NMOS trnsistor ts like: open swith when is 0 losed swith when is 1 Trnsistor V nd =1 =>Output Grounded Output Inverted Trnsistor V + or =1 =>Output Grounded Output Inverted One nnot mke N or gtes diretly. ll CMOS Gtes Invert Rel Gtes re, nd NOT ig Cir II p. 4 Revised; Jnury 24, 2007 Slide 3 Why Rel Gtes re /, not Construting Gtes One Cn uild Construting Gtes One Cn uild Mking CMOS gtes from trnsistors CMOS stnds for Complementry Symmetry Metl-Oxide-Semiondutor gtes. They lwys hve omplementry trnsistors, whih mens PMOS (turn off with one input) bove the output, nd NMOS (turn on with one input) below the output. The orret one inputs turn the lower NMOS trnsistors on, whih pulls the output down to zero thus inverting the output. The sily Construted CMOS Gtes s with 2, 3 or 4 inputs, s with 2, 3 or 4 inputs, nd NOTs. Gtes onstruted from other gtes To void ll the extr inverters (NOTs) rel iruits re designed to use s nd s insted of Ns nd s. Thinking Thinking in - logi is diffiult. Just look t n industril shemti used extensively for mintenne. The mrgin will be full of 1 s nd 0 s penilled in by users. Converting N- into - is strightforwrd mehnil proess. Muh less error prone thn doing logi with inverted signls 1 nd - logi. These Notes If the logi is importnt Ns nd s will be used. How n is mde If the gte design is importnt, s when we tlk bout CMOS gtes, s nd s will be used. b How n N is mde One wy of mking n X See problem 41.. b 1. If flsh is signl nme, flsh is n inverted signl. igitl Ciruits II p. 5, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 3

4 emorgn; Two Symbols for, Two for emorgn s Lws Give lternte Symbols quivlent Gte Symbols N N N Rel Gtes re nd One nnot mke N nd gtes diretly. Ciruit with rel gtes b d = (b) (d) sy to Understnd Gtes re N nd N N N Ciruit with simple gtes b d = b + d Whih one is esier for you to understnd? ig Cir II p. 6 Revised; Jnury 24, 2007 Slide 4 emorgn; Two Symbols for, Two Construting Gtes One Cn uild emorgn s Theorem The two expressions for the iruit with rel gtes nd the iruit with simple gtes, re equivlent b + d = = (b) (d) Use High-True nd-or Signls for Thinking Thinking Thinking in nnd-nor logi is diffiult. Just look t ny industril shemti used extensively for mintenne. The mrgin will be full of 1 s nd 0 s penilled in by users. Converting between nd-or nd nnd-nor is strightforwrd mehnil proess. Muh less error prone thn doing logi with sserted low signls nd nnd-nor logi. These Notes If the logi is importnt Ns nd s will be used. If the gte design is importnt, s when we tlk bout CMOS gtes, s nd s will be used. 44. PROLM Prove, using emorgn s Theroem(s), tht b + d = (b) (d) igitl Ciruits II p. 7, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 4

5 Strt the O-Trin; Use - or N-? Ciruits Using s nd s re Hrd to ollow on t Think Using s nd s. Confusing multiple logi inversions. oes wter stop or strt the trin? Is SW1 strt or stop swith? SW1 PWR IR OOPN1 OOPN2 WTR U1.1 U1.2 U1.3 U3.1 Strt_O-Trin SW1 PWR + IR (OOPN1 OOPN2) + WTR Ciruits Using Ns nd s re sy to ollow o Thinking Prt of esign with N/; Convert fter Thinking is one The sme iruit mde with Ns nd s. The finl eqution n be written from inspetion. (Is there logi error?) SW1 PWR IR OOPN1 OOPN2 WTR Strt_O-Trin SW1 PWR IR (OOPN1 + OOPN2) WTR ig Cir II p. 8 Revised; Jnury 24, 2007 Slide 5 Strt the O-Trin; Use - or Strting The O-Trin Strting The O-Trin 1 The N- logi is muh lerer. It tkes ir, power, wter to strt the trin. lso two doors must not be open. However it tkes 4 gtes nd 6 inverters to implement the N- iruit mde by dding inverters to s nd s. The less ler - iruit does the sme logi, with 4 gtes nd 1 inverter. - Logi is Confusing to Humns Multiple negtives re onfusing If your teher sid, Never will I not, not give you in igitl Ciruits, it would tke you some reful reding to determine your mrk. igrms with rel gtes This is the kind of digrm one would use to build iruit. It hs esily mnufturble gtes, i.e. NOT, nd. The numbers U1.1 et. re gte numbers tht would physilly identify the gte on lyout digrm showing where eh prt ws. sy to red digrms You should be ble to spot the logi error. OOPN1 + OOPN2 Should you be ble to strt the trin with one door open? 1. The O-Trin is n Ottw interurbn trin, whih stops right outside the ngineering uilding t Crleton University. igitl Ciruits II p. 9, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 5

6 Chnging Rel Gtes into Simple Gtes emorgn s Lw Trnsfers / Gtes Into N/ Gtes Ciruit with rel gtes b d Ciruit with simple gtes b d N N = (b) (d) Use emorgn s gte symbol t output Inverting irles nel eh other = b + d Ciruits ment for understnding the logi use Ns nd s. - rw your iruits with Ns nd s. Ciruits for onstrution re drwn with s nd s. rw onstrution digrms by: - trnsforming the understndble iruit into the rel iruit - using the emorgn lternte symbols. Compromise drwings hve S nd s with the irles re rrnged to nel eh other. lternte symbol ig Cir II p. 10 Revised; Jnury 24, 2007 Slide 6 Chnging Rel Gtes into Simple Gtes Trnsforming - igrms into Trnsforming - igrms into N- igrms igrms in this ourse will be drwn with Ns nd s s muh s possible. igrms for onstrution or mintinne, tht wnt to show extly wht gtes were used, will be drwn with s nd s. This is prtiulrly true of older digrms. ompromise method, whih is lmost s esy to follow, but shows the rel gtes s used, is to mke lternte gtes with the lternte symbols, the ones with the inverting irles on the inputs. NOT: One output irle nels ll the input irles it feeds. 45. PROLM Trnsform this iruit into simple gtes. ) igitl Ciruits II p. 11, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 6

7 esign in N nd Gtes on t do initil design in s nd s. esign in N/; Convert fter Thinking is one Rtionl People think best with N nd. Multiple inversions re very onfusing There is seldom logil reson to invert exept t iruit inputs. Conversion N/ / is esy using emorgn s form of gtes. o it fter the thinking is done. Some people still seem to prefer designing with s nd s but not in this lss ig Cir II p. 12 Revised; Jnury 24, 2007 Slide 7 esign in N nd Gtes esign With Coneptully sy Logi esign With Coneptully sy Logi In the post 2000 er, people usully think bout the logi. The detils of onstrution re done utomtilly. This mens you will normlly do N- type logi. Think with N- uild with - emorgn s Lw Trnsforms Rel Gtes to Simple Gtes 46. PROLM Trnsform this iruit into simple gtes. igitl Ciruits II p. 13, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 7

8 N/ to / Conversions Using emorgn in Grphil orm xmple: b 1. Strt with N/ iruit = ( + )(b + ) + d d 1) 2. Selet lternte onnetion lyers. (very seond lyer of onneting wires between lyers of gtes.) One end of wires my be inputs or outputs. skip skip 2) 3. Put bk-to-bk inverting irles on both ends of the leds. dd NOT gtes when neessry ((), (b) nd ()) () (b) 3) () 4. Moving inverters, or inverting bubbles my mke logi simpler. Sometimes inverters on two (or more) input leds should be moved to the output. (See next exmple) (h) 4) ig Cir II p. 14 Revised; Jnury 24, 2007 Slide 8 N/ to / Conversions Grphil Conversion N/ to / Grphil Conversion N/ to / input 1. Strt with n N- iruit. If you re strting with n eqution. drw the iruit. The originl formul will usully hve inversions only on inputs. However there my be inversions nywhere. 2. The iruit is drwn with lyer of onnetions whih feeds lyer of gtes, feeding lyer of onnetions (green), whih feeds nother lyer of gtes, whih feeds nother lyer of onnetions (green). Some iruits my hve more or fewer lyers. Selet lternte onnetion lyers. Some iruits re niely lyered with the first lyer feeding seond lyer whih feeds third lyer et. However this step is not ext. There re often severl legitimte wys to define lyers. Note some wires, like () nd (b) my pss through gte level (green) without gte. 3. Put bk-to-bk inverting irles t both ends of wires going through onnetion lyer. On leds like () nd (b), you will hve to dd n inverter, sine there is no gte on whih to ple the seond irle. 4. This step is less utomti. Look for leds where two inverters n be repled by one. When doing this be reful not to invert other onnetions on the sme input, like (h). When moving inverting irles mke sure tht the number of irles between the signl input nd ll the gte inputs it goes to, re the sme before nd fter moving ) Number of inversions from input to this gte input 1 2 (h) TR 4) (b) () Number of inversions from input to this gte input igitl Ciruits II p. 15, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 8

9 N/ to / Conversions Grphil Conversion (ont.) 4. Copy of step 4 on lst slide: It hs bk-to-bk bubbles nd onsolidted inverters 4) 5. Selet the unonventionl gtes. The ones with input bubbles. This is ompromise solution whih is: firly redble represents rel gtes 5) 6. Use emorgn lws on the unonventionl gtes. Hrd to understnd the logi but good for onstrution. b d 6) ig Cir II p. 16 Revised; Jnury 24, 2007 Slide 9 N/ to / Conversions Grphil Conversion N/ to / On previous slide You must lwys dd two inverting irles to led, or none. Never dd just one irle, not even if one exists lredy. On this slide 5. Look t the gtes. Conventionl ones hve irles on their outputs nd re s, s or NOTs. The unonventionl ones hve irles on their inputs but re still s, s or NOTs. This step gives iruit whih is firly esy to red beuse one n mentlly nel the bk-to-bk irles. However it still represents iruit you n build esily beuse it does not ontin Ns nd s. 6. If desired, one n reple the emorgn forms of nd, the ones with irles on the inputs, with the more stndrd forms with irle on the output. Sine - digrms represent rel gtes, do not ple single irle on the input of gte, s ws done in the theoretil N- digrm. 47. PROLM:. or the iruit on the right, find the - iruit when the onnetion on the levels shown re seleted. C G G H 48. PROLM ind n N- equivlent for the iruit on the left. This requires working bkwrds. igitl Ciruits II p. 17, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 9

10 N/ to / Conversions, 2nd xmple of N/ to / Seleting the 1st nd 3rd levels, insted of the 2nd nd 4th b SKIP d 1) N/ iruit; = ( + )(b + ) + d 2) Selet every other onnetion levels 3) dd nelling bk-to-bk inverting irles dd inverters where neessry. 4) Sve n inverter by using bk-to-bk output inverters insted of input inverters. b d 5) Selet the unonventionl gtes. Compromise nswer 6) emorgn onventionl gtes (optionl). Conventionl (inosur) nswer ig Cir II p. 18 Revised; Jnury 24, 2007 Slide 10 N/ to / Conversions, N/ / Conversion N/ / Conversion (ont) This is the sme iruit s the previous exmple. The differene between the exmples strts t step 2 2. the lternte bloks of leds where the inverters re dded re tht ones mrked SKIP in the first exmple. Compre step 2 here with step 2 on Slide In this step, the two inverters feeding the with inverting inputs looks too omplex. Throwing out the double irles leves us with n gte. This is not llowed. Go to the output of the nd dd bk-to-bk inversions there. This gives gte whih is llowed. oth the lower gtes on the right re equivlent to n gte, but one is simpler to implement s rel gtes. quivlent gtes 6. Chnging the s nd s with irles on their inputs, to s nd s with irles on their outputs, is the finl step. o it if you like suh drwings better, or if the drwing stndrds your ompny uses require it, or if your boss sys to. Compre the results with those in steps 5 nd 6 of the previous exmple. This iruit is mrginlly simpler thn the previous result. One NOT gte is sved. igitl Ciruits II p. 19, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 10

11 emorgn s Lw, the Generl orm emorgn s Theorems, Generl orm bstrt Nottion (,,C,... +,,) = (,,C,...,, +,) tion xmple I ) Tke oolen expression b) rket ll groups of Ns Tke dul ) Chnge N nd N d) Clen brkets Chnge dul into inverse e) Invert ll vribles = C = { C} dul = { + + C} dul = + + C = { + + C} ig Cir II p. 20 Revised; Jnury 24, 2007 Slide 1 emorgn s Lw, the Generl orm Generl orm of emorgn s Theorem Generl orm of emorgn s Theorem The slide shows simplified version The true generlized emorgn (,,C,... +,,0,1) = (,,C,...,, +,1,0) (,,C,... +,,0,1) = (,,C,...,, +,1,0) The interhnge of 0 nd 1 ws left out on the slide, sine designers prtilly never hve 0 or 1 in the expressions they operte on with emorgn s lw. emorgn generl lw is very similr to ulity The obvious differene is the inverting brs. nother importnt differene is the pplition. emorgn trnsforms n expression into its inverse. ulity tkes vlid identity nd genertes nother vlid identity. 49. PROLM ind n expression for tht hs inverting brs only over single letters, = ( + )(b + ) + d igitl Ciruits II p. 21, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 11

12 Generlized emorgn s Lw xmples (,,C,... +,,) = (,,C,...,, +,) xmple II ) Tke oolen expression = C + b) rket ll groups of Ns Tke dul ) Chnge N nd N e) Clen brkets Chnge dul into inverse e) Invert ll vribles { C} + { } dul = {++C} {+ } dul = (++C) (+ ) = (++C) (+ ) xmple III ) Tke ny oolen expression b) rket ll groups of Ns ) Chnge N nd N Ignore ny existing overbrs d) Clen brkets e) Invert ll vribles = [ C + ( + C)] = {[ { C } + { ( { } + C)} ] } dul = {[{ + + C} { + ({ + } C)}] + } dul = { + + C} { + { + } C} + = { + + C} { + { + } C} + ig Cir II p. 22 Revised; Jnury 24, 2007 Slide 11 Generlized emorgn s Lw xmples using the Generlized xmples using the Generlized emorgn s Lw e sure to put brkets round the Ns When you do lgebr, you utomtilly do the Ns before the s. The nottion is designed tht wy. Putting brkets round n shows tht the should be done before the N. When you use emorgn s lw, nd you trnsform b + d into (+b)(+d), the brkets mke sure tht the vribles in the trnsformed expression re operted on in the sme order. Tht is you don t try to do (b + )d. y pling the brkets round the Ns first, you do not get onfused during the trnsformtions. 50. PROLM () Convert ( + b)( + ) to n expression with 3 letters nd inversion brs only over single letters, using the Generlized emorgn s lw. (b) Compre this method with tht of the xmple on Comment on Slide 10 nd see if the lgebr is shorter. 51. PROLM Given H = (b + )g + de( + b), find H igitl Ciruits II p. 23, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 12

13 Generlized emorgn s Lw xmples: Generlized emorgn s Theorems xmple IV ) Tke oolen expression = C + b) rket ll groups of Ns ) Chnge N nd N Ignore existing overbrs d) Clen brkets if onvenient { C} + { } dul = {++C} {+ } dul = (++C) (+ ) e) Invert ll vribles = (++C) (+) Note: Ignore ny inverting overbrs exept over single letters = (C + ) dul = {+}+(C {+}) = {+}+(C { + }) ig Cir II p. 24 Revised; Jnury 24, 2007 Slide 12 Generlized emorgn s Lw xmples using the Generlized xmples using the Generlized emorgn s Lw Common Worry, Intermedite Overbrs When pplying the generlized lw, do not osider ny overbrs exept those on top of single letters. If the overbr is over two or more letters, just rry it through without hnge. 52. PROLM Use emorgn s generl lw to remove ll but one of the inverting brs from the dinosur iruit on Slide 10. The expression is f = (b+) ( ) ( d) igitl Ciruits II p. 25, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 13

14 emorgn; Review emorgn s Theorems Two input forms Common rrors + = K K Not brketing ll Ns before exhnging + + = N N C N C H = C + H = ++C H = { ( C)} + H dul = {+(+C)} + = = + G G H = {+(+C)} Not onsidering the less error prone grphil method, before doing messy lgebr. Generl form: (,,C,... +,,) = (,,C,...,, +,) rket ll Ned terms before trnsforming Writing s though, dul nd were ll equl. =+b =(b+) tke dul =(b+) invert vribles ig Cir II p. 26 Revised; Jnury 24, 2007 Slide 13 emorgn; Review xmples using the Generlized, X, emorgn nd nglish dpted from Ronld. Stndler's humorous nd informtive site bout nglish, written by rel engineer nd lwyer. 1 The quoted text is diretly from his work. Most legl style mnuls devote t lest one pge of ditribe to the mening of nd, or, nd/or. I n only onlude tht the intensity of these ttorneys' rgumenttion must ompenste for their ignorne! In nglish, there re two menings for : 1) The inlusive whih mens or or both. It is better expressed s N/ 2) The exlusive whih mens or but not both. In nglish, it lso mens only one of,, C... or, s opposed to logi where it mens n odd number of the set must be true. It is my impression tht most physiists nd mthemtiins generlly use or in the inlusive sense, nd most ttorneys in the US generlly use or in the exlusive sense. The people who write legl style mnuls don't seem to notie tht their ssertion bout or not only ontrdits the mthemtil definition tht is used in omputer dt bses, but lso is nrrow-minded pproh to the nglish lnguge. His point is illustrted by line from the Crleton University Helth nd Sfety doumenttion. ood nd beverges re not permitted in the lb..." Logilly, if =ood, nd =everge, this sttement mens the expression ~(&) must be true. Thus offee with no food, or dry food only, is llowed. Mp of No ( nd ) If one sys no (food or beverge) is permitted in the lb, lwyers would men ~( ), however engineers would men ~(+), whih is wht is desired, provided the brkets re used to void mbiguity. Using emorgn s Theorem, one gets ~(+) = (~)&(~) ie. No food nd no beverges re permitted. lterntely, Sdler suggests lerer nglish version, Neither food nor beverges re permitted. 1. Sdler s rtile is muh funnier thn this dpttion, but he is opyright lwyer, nd I m not going to plgirize his work here. igitl Ciruits II p. 27, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 14

15 ig Cir II p. 28 Revised; Jnury 24, 2007 Slide 14 xmples using the Generlized Solution to #52. f = (b+) ( ) ( d) Let = f = (b+) ( ) ( d) Our nswer is the inverse of = (b+) ( ) ( d) dd brkets to be sure we know where the overbr ends = [(b+) ( )] ( d) Tke dul dul = [(b )+(+)]+(+d) o side lultions using emorgn dul = [(b+) (+)]+ d = f = [(b+) (+)]+ d lternte Solution to #52. Insert overbrs [(b )+(+)] = [(b ) (+)] = [(b+) (+)] (+d) = d f = (b+) ( ) ( d) f = (b+) ( )+( d) f = (b+) (+)+( d) f = b + +( d) em x z = x+z em x z = x+z Swp (x+)(z+) = x + z igitl Ciruits II p. 29, dig2emorgn.fm Revised; Jnury 24, 2007 Comment on Slide 15