Introduction. Let s learn to design digital circuits We ll start with a simple form of circuit:

Transcription

1 Digitl Design Chpter 2: Comintionl Logi Design Slies to ompn the tetook Digitl Design, irst Eition, rnk Vhi, John Wile n Sons Pulishers, Copright 27 rnk Vhi Instrutors of ourses requiring Vhi's Digitl Design tetook (pulishe John Wile n Sons) hve permission to moif n use these slies for ustomr ourse-relte tivities, sujet to keeping Digitl this opright Design notie in ple n unmoifie. These slies m e poste s unnimte pf versions on pulil-essile ourse wesites.. PowerPoint soure (or pf with nimtions) m not e poste to pulil-essile wesites, ut m e poste for stuents on internl protete sites or istriute iretl to stuents other eletroni mens. Copright 26 Instrutors m mke printouts of the slies ville to stuents for resonle photooping hrge, without inurring rolties. An other use requires epliit permission. Instrutors m otin PowerPoint rnk soure Vhi or otin speil use permissions from Wile see for informtion. Introution 2. Let s lern to esign igitl iruits We ll strt with simple form of iruit: Comintionl iruit A igitl iruit whose outputs epen solel on the present omintion of the iruit inputs vlues Digitl Design Copright 26 rnk Vhi Note: Slies with nimtion re enote with smll re "" ner the nimte items Digitl iruit Comintionl igitl iruit Sequentil igitl iruit 2?

2 4.5A 4.5A Swithes 2.2 Eletroni swithes re the sis of inr igitl iruits Eletril terminolog Voltge: Differene in eletri potentil etween two points Anlogous to wter pressure Current: low of hrge prtiles Anlogous to wter flow Resistne: Tenen of wire to resist urrent flow Anlogous to wter pipe imeter V = I * R (Ohm s Lw) V + 9V 2 ohms 4.5 A 9V Digitl Design Copright 26 rnk Vhi 3 Swithes A swith hs three prts Soure input, n output Current wnts to flow from soure input to output Control input Voltge tht ontrols whether tht urrent n flow The mzing shrinking swith 93s: Rels 94s: Vuum tues 95s: Disrete trnsistor 96s: Integrte iruits (ICs) Initill just few trnsistors on IC Then tens, hunres, thousns... rel soure input soure input ontrol input ontrol input () isrete trnsistor vuum tue off output on output IC Digitl Design Copright 26 rnk Vhi qurter (to see the reltive size) 4 2

3 Moore s Lw IC pit ouling out ever 8 months for severl ees Known s Moore s Lw fter Goron Moore, o-founer of Intel Preite in 965 preite tht omponents per IC woul oule roughl ever er or so Book over epits relte phenomen or prtiulr numer of trnsistors, the IC shrinks hlf ever 8 months Notie how muh shrinking ours in just out ers Enles inreil powerful omputtion in inreil tin evies To s ICs hol illions of trnsistors The first Pentium proessor (erl 99s) neee onl 3 million Digitl Design Copright 26 rnk Vhi An Intel Pentium proessor IC hving millions of trnsistors 5 The CMOS Trnsistor 2.3 CMOS trnsistor Bsi swith in moern ICs A positive voltge here......ttrts eletrons here, turning the hnnel etween soure n rin into onutor. nmos gte soure gte oie rin IC pkge onuts oes not onut pmos gte Digitl Design Copright 26 rnk Vhi () Silion -- not quite onutor or insultor: Semionutor IC oes not onut onuts 6 3

4 Boolen Logi Gtes Builing Bloks for Digitl Ciruits (Beuse Swithes re Hr to Work With) 2.4 Logi gtes re etter igitl iruit uiling loks thn swithes (trnsistors) Wh?... Digitl Design Copright 26 rnk Vhi 7 Boolen Alger n its Reltion to Digitl Ciruits To unerstn the enefits of logi gtes vs. swithes, we shoul first unerstn Boolen lger Tritionl lger Vrile represent rel numers Opertors operte on vriles, return rel numers Boolen Alger Vriles represent or onl Opertors return or onl Bsi opertors AND: AND returns onl when oth = n = OR: OR returns if either (or oth) = or = NOT: NOT returns the opposite of ( if =, if =) AND NOT OR Digitl Design Copright 26 rnk Vhi 8 4

5 Boolen Alger n its Reltion to Digitl Ciruits Develope mi-8 s George Boole to formlize humn thought E: I ll go to lunh if Mr goes OR John goes, AND Sll oes not go. Let represent m going to lunh ( mens I go, I on t go) Likewise, m for Mr going, j for John, n s for Sll Then = (m OR j) AND NOT(s) Nie fetures ormll evlute m=, j=, s= --> = ( OR ) AND NOT() = AND = ormll trnsform = (m n NOT(s)) OR (j n NOT(s))» Looks ifferent, ut sme funtion» We ll show trnsformtion tehniques soon AND OR NOT Digitl Design Copright 26 rnk Vhi 9 Evluting Boolen Equtions Evlute the Boolen eqution = ( AND ) OR ( AND ) for the given vlues of vriles,,, n : Q: =, =, =, =. Answer: = ( AND ) OR ( AND ) = OR =. Q2: =, =, =, =. Answer: = ( AND ) OR ( AND ) = OR =. Q3: =, =, =, =. Answer: = ( AND ) OR ( AND ) = OR =. AND OR NOT Digitl Design Copright 26 rnk Vhi 5

6 Converting to Boolen Equtions Convert the following English sttements to Boolen eqution Q. is n is. Answer: = AND Q2. either of or is. Answer: = OR Q3. oth n re not. Answer: () Option : = NOT() AND NOT() () Option 2: = OR Q4. is n is. Answer: = AND NOT() Digitl Design Copright 26 rnk Vhi Digitl Design Copright 26 rnk Vhi Converting to Boolen Equtions Q. A fire sprinkler sstem shoul spr wter if high het is sense n the sstem is set to enle. Answer: Let Boolen vrile h represent high het is sense, e represent enle, n represent spring wter. Then n eqution is: = h AND e. Q2. A r lrm shoul soun if the lrm is enle, n either the r is shken or the oor is opene. Answer: Let represent lrm is enle, s represent r is shken, represent oor is opene, n represent lrm souns. Then n eqution is: = AND (s OR ). () Alterntivel, ssuming tht our oor sensor represents oor is lose inste of open (mening = when the oor is lose, when open), we otin the following eqution: = AND (s OR NOT()). 2 6

7 Boolen lger (mi-8s) Relting Boolen Alger to Digitl Design Implement Boolen opertors using trnsistors Cll those implementtions logi gtes. Let s us uil iruits oing mth -- powerful onept Digitl Design Copright 26 rnk Vhi Boole s intent: formlize humn thought Swithes (93s) Shnnon (938) Digitl esign or telephone swithing n other eletroni uses Showe pplition of Boolen lger to esign of swithse iruits Smol Truth tle Trnsistor iruit NOT OR AND Note: These OR/AND implementtions re ineffiient; we ll show wh, n show etter ones lter. 3 NOT/OR/AND Logi Gte Timing Digrms time time time Digitl Design Copright 26 rnk Vhi 4 7

8 Builing Ciruits Using Gtes Rell Chpter motion-in-rk emple Turn on lmp (=) when motion sense (=) n no light (=) = AND NOT() Buil using logi gtes, AND n NOT, s shown We just uilt our first igitl iruit! Digitl Design Copright 26 rnk Vhi 5 Emple: Converting Boolen Eqution to Ciruit of Logi Gtes Q: Convert the following eqution to logi gtes: = AND NOT( OR NOT() ) () () Digitl Design Copright 26 rnk Vhi 6 8

9 Emple: Set Belt Wrning Light Sstem Design iruit for wrning light Sensors s=: set elt fstene k=: ke inserte p=: person in set Cpture Boolen eqution person in set, n set elt not fstene, n ke inserte Convert eqution to iruit Notie Boolen lger enles es pture s eqution n onversion to iruit How esign with swithes? Of ourse, logi gtes re uilt from swithes, ut we think t level of logi gtes, not swithes Digitl Design Copright 26 rnk Vhi w = p AND NOT(s) AND k k BeltWrn p w s 7 Some Ciruit Drwing Conventions no no es es ok not ok Digitl Design Copright 26 rnk Vhi 8 9

10 Boolen Alger 2.5 B efining logi gtes se on Boolen lger, we n use lgeri methos to mnipulte iruits So let s lern some Boolen lgeri methos Strt with nottion: Writing AND, OR, n NOT() is umersome Use smols: *, +, n (in ft, * n e just ). Originl: w = (p AND NOT(s) AND k) OR t New: w = ps k + t Spoken s w equls p n s prime n k, or t Or even just w equls p s prime k, or t s known s omplement of s While smols ome from regulr lger, on t s times or plus Digitl Design Copright 26 rnk Vhi Boolen lger preeene, highest preeene first. Smol Nme Desription ( ) Prentheses Evlute epressions neste in prentheses first NOT Evlute from left to right * AND Evlute from left to right + OR Evlute from left to right 9 Boolen Alger Opertor Preenene Evlute the following Boolen equtions, ssuming =, =, =, =. Q. = * +. Answer: * hs preeene over +, so we evlute the eqution s = ( *) + = () + = + =. Q2. = +. Answer: the prolem is ientil to the previous prolem, using the shorthn nottion for *. Q3. =. Answer: we first evlute euse NOT hs preeene over AND, resulting in = * ( ) = * () = * =. Q4. = (). Answer: we first evlute wht is insie the prentheses, then we NOT the result, ieling (*) = () = =. Q5. = ( + ) * +. Answer: Insie left prentheses: ( + ( )) = ( + ()) = ( + ) =. Net, * hs preeene over +, ieling ( * ) + = () +. The NOT hs preeene over the OR, giving () + ( ) = () + () = + =. Digitl Design Copright 26 rnk Vhi 2

11 Boolen Alger Terminolog Emple eqution: (,,) = Vrile Represents vlue ( or ) Three vriles:,, n Literl Apperne of vrile, in true or omplemente form Nine literls:,,,,,,,, n Prout term Prout of literls our prout terms:,,, Sum-of-prouts Eqution written s OR of prout terms onl Aove eqution is in sum-of-prouts form. = (+) + is not. Digitl Design Copright 26 rnk Vhi 2 Digitl Design Copright 26 rnk Vhi Boolen Alger Properties Commuttive + = + * = * Distriutive * ( + ) = * + * + ( * ) = ( + ) * ( + ) (this one is trik!) Assoitive ( + ) + = + ( + ) ( * ) * = * ( * ) Ientit + = + = * = * = Complement + = * = To prove, just evlute ll possiilities Emple uses of the properties Show equivlent to. Use ommuttive propert: ** = * * = ** = ** =. Show + =. Use first istriutive propert + = (+ ). Complement propert Reple + : (+ ) = (). Ientit propert () = * =. Show + z equivlent to + z. Seon istriutive propert Reple + z (+ )*(+z). Complement propert Reple (+ ), Ientit propert reple *(+z) +z. 22

12 Emple tht Applies Boolen Alger Properties Wnt utomti oor opener iruit (e.g., for groer store) Output: f= opens oor Inputs: p=: person etete h=: swith foring hol open =: ke foring lose Wnt open oor when h= n =, or h= n p= n = Eqution: f = h + h p h p DoorOpener f oun inepensive hip tht omputes: f = hp + hp + h p Cn we use it? Is it the sme s f = (p+h)? Use Boolen lger: f = hp + hp + h p f = h(p + p ) + h p ( the istriutive propert) f = h() + h p ( the omplement propert) f = h + h p ( the ientit propert) f = h + h p ( the ommuttive propert) Sme! Digitl Design Copright 26 rnk Vhi 23 Boolen Alger: Aitionl Properties Null elements + = * = Iempotent Lw + = * = Involution Lw ( ) = DeMorgn s Lw ( + ) = () = + Ver useful! To prove, just evlute ll possiilities Digitl Design Copright 26 rnk Vhi Airrft lvtor sign emple Behvior Three lvtories, eh with sensor (,, ), equls if oor loke Light Aville sign (S) if n lvtor ville Eqution n iruit S = + + Trnsform () = + + ( DeMorgn s Lw) S = () New eqution n iruit Ciruit S Ciruit Alterntive: Inste of lighting Aville, light Oupie Opposite of Aville funtion S = + + So S = ( + + ) S = ( ) * ( ) * ( ) ( DeMorgn s Lw) S = * * ( Involution Lw) Mkes intuitive sense Oupie if ll oors re loke S 24 2

13 3 25 Digitl Design Copright 26 rnk Vhi Representtions of Boolen untions A funtion n e represente in ifferent ws Aove shows seven representtions of the sme funtions (,), using four ifferent methos: English, Eqution, Ciruit, n Truth Tle 2.6 Ciruit Ciruit 2 () () English : outputs when is n is, or when is n is. English 2: outputs when is, regrless of s vlue () () The funtion Truth tle Eqution 2: (,) = Eqution : (,) = + 26 Digitl Design Copright 26 rnk Vhi Truth Tle Representtion of Boolen untions Define vlue of for eh possile omintion of input vlues 2-input funtion: 4 rows 3-input funtion: 8 rows 4-input funtion: 6 rows Q: Use truth tle to efine funtion (,,) tht is when is 5 or greter in inr () () ()

14 Converting mong Representtions Cn onvert from n representtion to n other Common onversions Eqution to iruit (we i this erlier) Truth tle to eqution (whih we n onvert to iruit) Es -- just OR eh input term tht shoul output Eqution to truth tle Es -- just evlute eqution for eh input omintion (row) Creting intermeite olumns helps Q: Convert to truth tle: = + Digitl Design Copright 26 rnk Vhi Inputs ' ' ' Output Inputs Outputs Term = sum of = + Q: Convert to eqution = Stnr Representtion: Truth Tle How n we etermine if two funtions re the sme? Rell utomti oor emple Sme s f = h + h p? Use lgeri methos But if we file, oes tht prove not equl? No. Solution: Convert to truth tles Onl ONE truth tle representtion of given funtion Stnr representtion -- for given funtion, onl one version in stnr form eists Digitl Design Copright 26 rnk Vhi f = hp + hp + h f = h(p + p ) + h p f = h() + h p f = h + h p (wht if we stoppe here?) f = h + h p Q: Determine if =+ is sme funtion s = + +, onverting eh to truth tle first = + ' = + + Sme 28 4

15 Digitl Design Copright 26 rnk Vhi Cnonil orm -- Sum of Minterms Truth tles too ig for numerous inputs Use stnr form of eqution inste Known s nonil form Regulr lger: group terms of polnomil power ( > ) Boolen lger: rete sum of minterms Minterm: prout term with ever funtion literl ppering etl one, in true or omplemente form Just multipl-out eqution until sum of prout terms Then epn eh term until ll terms re minterms Q: Determine if (,)=+ is sme funtion s (,)= + +, onverting first eqution to nonil form (seon lre in nonil form) = + (lre sum of prouts) = + (+ ) (epning term) = + + (SAME -- sme three terms s other eqution) 29 Multiple-Output Ciruits Mn iruits hve more thn one output Cn give eh seprte iruit, or n shre gtes E: = +, G = + G G () Option : Seprte iruits () Option 2: Shre gtes Digitl Design Copright 26 rnk Vhi 3 5

16 Multiple-Output Emple: BCD to 7-Segment Converter f g e efg = () = w z + w z + w z + w z + w z + w z + w z + w z = w z + w z + w z + w z + w z + w z + w z + w z () Digitl Design Copright 26 rnk Vhi 3 Comintionl Logi Design Proess 2.7 Step Step 2 Step 3 Step Cpture the funtion Convert to equtions Implement s gtese iruit Desription Crete truth tle or equtions, whihever is most nturl for the given prolem, to esrie the esire ehvior of the omintionl logi. This step is onl neessr if ou pture the funtion using truth tle inste of equtions. Crete n eqution for eh output ORing ll the minterms for tht output. Simplif the equtions if esire. or eh output, rete iruit orresponing to the output s eqution. (Shring gtes mong multiple outputs is OK optionll.) Digitl Design Copright 26 rnk Vhi 32 6

17 Emple: Three s Detetor Prolem: Detet three onseutive s in 8-it input: efgh Step : Cpture the funtion Truth tle or eqution? Truth tle too ig: 2^8=256 rows Eqution: rete terms for eh possile se of three onseutive s = + + e + ef + efg + fgh Step 2: Convert to eqution -- lre one Step 3: Implement s gte-se iruit e f e ef g efg fgh Digitl Design Copright 26 rnk Vhi h 33 Emple: Numer of s Count Prolem: Output in inr on two outputs z the numer of s on three inputs Step : Cpture the funtion Truth tle or eqution? Truth tle is strightforwr Step 2: Convert to eqution = z = Step 3: Implement s gtese iruit z Digitl Design Copright 26 rnk Vhi 34 7

18 More Gtes 2.8 NAND NOR XOR XNOR NAND NOR NAND: Opposite of AND ( NOT AND ) NOR: Opposite of OR ( NOT OR ) XOR: Etl input is, for 2-input XOR. (or more inputs -- o numer of s) XNOR: Opposite of XOR ( NOT XOR ) Digitl Design Copright 26 rnk Vhi NAND sme s AND with power & groun swithe Wh? nmos onuts s well, ut not s (resons eon our sope) -- so NAND more effiient Likewise, NOR sme s OR with power/groun swithe AND in CMOS: NAND with NOT OR in CMOS: NOR with NOT So NAND/NOR more ommon 35 More Gtes: Emple Uses Airrft lvtor sign emple S = () Deteting ll s Use NOR Deteting equlit Use XNOR Deteting o # of s Use XOR Useful for generting prit it ommon for eteting errors 2 2 Ciruit A=B S Digitl Design Copright 26 rnk Vhi 36 8

19 AND OR X OR NOR XNOR ' ' NAND Completeness of NAND An Boolen funtion n e implemente using just NAND gtes. Wh? Nee AND, OR, n NOT NOT: -input NAND (or 2-input NAND with inputs tie together) AND: NAND followe NOT OR: NAND preee NOTs Likewise for NOR Digitl Design Copright 26 rnk Vhi 37 Numer of Possile Boolen untions How mn possile funtions of 2 vriles? 2 2 rows in truth tle, 2 hoies for eh 2 (22 ) = 2 4 = 6 possile funtions N vriles 2 N rows 2 (2N ) possile funtions or 2 hoies or 2 hoies or 2 hoies or 2 hoies 2 4 = 6 possile funtions f f f2 f3 f4 f5 f6 f7 f8 f9 f f f2 f3 f4 f5 AND XOR OR NOR XNOR NAND Digitl Design Copright 26 rnk Vhi 38 9

20 esor o r op r i M Deoers n Mues 2.9 Deoer: Populr omintionl logi uiling lok, in ition to logi gtes Converts input inr numer to one high output 2-input eoer: four possile input inr numers So hs four outputs, one for eh possile input inr numer Internl esign AND gte for eh output to etet input omintion Deoer with enle e Outputs ll if e= Regulr ehvior if e= n-input eoer: 2 n outputs Digitl Design Copright 26 rnk Vhi i i i 2 3 i i i i i ii ii i i i i 2 3 i i 2 3 i i 2 e 3 i i e Deoer Emple New Yer s Eve Countown Displ Miroproessor ounts from 59 own to in inr on 6-it output Wnt illuminte one of 6 lights for eh inr numer Use 664 eoer 4 outputs unuse 2 2 i i i2 i3 i4 i5 e Hpp New Yer Digitl Design Copright 26 rnk Vhi 4 2

21 Multipleor (Mu) Mu: Another populr omintionl uiling lok Routes one of its N t inputs to its one output, se on inr vlue of selet inputs 4 input mu nees 2 selet inputs to inite whih input to route through 8 input mu 3 selet inputs N inputs log 2 (N) selets Like rilr swith Digitl Design Copright 26 rnk Vhi 4 Mu Internl Design 2 i i s 2 mu i i 2 s i i 2 s i i s i (*i=i) i (+i=i) 4 i i i2 i3 s s i i i2 i3 4 mu Digitl Design Copright 26 rnk Vhi s s 42 2

22 r P Mu Emple Cit mor n set four swithes up or own, representing his/her vote on eh of four proposls, numere,, 2, 3 Cit mnger n ispl n suh vote on lrge green/re LED (light) setting two swithes to represent inr,, 2, or 3 Mor s swithes Use 4 mu 2 3 i i i2 4 i3 s s on/off Green/ Re LED Digitl Design Copright 26 rnk Vhi 4 mnger's swithes 43 Mues Commonl Together -- N-it Mu s i i s 2 i i s 2 i i s i 2 i s A B 4 4 I I 4-it 2 D s s 4 C Simplifing nottion: 4 C is short for 3 2 E: Two 4-it inputs, A (3 2 ), n B (3 2 ) 4-it 2 mu (just four 2 mues shring selet line) n selet etween A or B Digitl Design Copright 26 rnk Vhi 44 22

23 N-it Mu Emple our possile ispl items Temperture (T), Averge miles-per-gllon (A), Instntneous mpg (I), n Miles remining (M) -- eh is 8-its wie Choose whih to ispl using two inputs n Use 8-it 4 mu Digitl Design Copright 26 rnk Vhi 45 Aitionl Consiertions Shemti Cpture n Simultion 2. Inputs Inputs i i i Outputs 3 Simulte i Outputs 3 Simulte 2 2 Shemti pture Computer tool for user to pture logi iruit grphill Simultor Computer tool to show wht iruit outputs woul e for given inputs Outputs ommonl isple s wveform Digitl Design Copright 26 rnk Vhi 46 23

24 Aitionl Consiertions Non-Iel Gte Behvior -- Del Rel gtes hve some el Outputs on t hnge immeitel fter inputs hnge Digitl Design Copright 26 rnk Vhi 47 Chpter Summr Comintionl iruits Ciruit whose outputs re funtion of present inputs No stte Swithes: Bsi omponent in igitl iruits Boolen logi gtes: AND, OR, NOT -- Better uiling lok thn swithes Enles use of Boolen lger to esign iruits Boolen lger: uses true/flse vriles/opertors Representtions of Boolen funtions: Cn trnslte mong Comintionl esign proess: Trnslte from eqution (or tle) to iruit through well-efine steps More gtes: NAND, NOR, XOR, XNOR lso useful Mues n eoers: Aitionl useful omintionl uiling loks Digitl Design Copright 26 rnk Vhi 48 24