Chapter 2: Combinational Logic Design. Instructor: Dr. Hyunyoung Lee. Based on the slides by Frank Vahid

Transcription

1 Chpter 2: Comintionl Logic Design Instructor: Dr. Hunoung Lee Bsed on the slides rnk Vhid Copright 2 rnk Vhid Instructors of courses requiring Vhid's Digitl Design tetook (pulished John Wile nd Sons) hve permission to modif nd use these slides for customr course-relted ctivities, suject to keeping this copright notice in plce nd unmodified. These slides m e posted s unnimted pdf versions on pulicl-ccessile course wesites.. PowerPoint source (or pdf with nimtions) Digitl m Design not e posted 2e to pulicl-ccessile wesites, ut m e posted for students on internl protected sites or distriuted directl to students other electronic mens. Instructors m Copright mke printouts 2 of the slides ville to students for resonle photocoping chrge, without incurring rolties. An other use requires eplicit permission. Instructors m otin PowerPoint rnk Vhid source or otin specil use permissions from Wile see for informtion.

2 Introduction 2. Let s lern to design digitl circuits, strting with simple form of circuit: Comintionl circuit Outputs depend solel on the present comintion of the circuit inputs vlues Vs. sequentil circuit: Hs memor tht impcts outputs too () = Digitl = Sstem = Digitl = Sstem if =, then = if =, then = Digitl Design 2e Copright 2 rnk Vhid () Motion sensor Light sensor Digitl Sstem Lmp Note: Slides with nimtion re denoted with smll red "" ner the nimted items if = nd =, then = if = nd =, then = if = nd =, then = if = nd =, then = (c) = Digitl = Sstem = Digitl = Sstem = Digitl = Sstem Cnnot determine vlue of solel from present input vlue 2

3 Switches 2.2 Electronic switches re the sis of inr digitl circuits Electricl terminolog Voltge: Difference in electric potentil etween two points (volts, V) Anlogous to wter pressure Current: low of chrged prticles (mps, A) Anlogous to wter flow Resistnce: Tendenc of wire to resist current flow (ohms, Ω) Anlogous to wter pipe dimeter V = I * R (Ohm s Lw) 9 V = I * 2 ohms I = 4.5 A 4.5 A V 9 V 2 ohms 4.5 A If 9V potentil difference is pplied cross 2 ohm resistor, then 4.5 A of current will flow. + 9V 4.5 A Digitl Design 2e Copright 2 rnk Vhid 3

4 Switches A switch hs three prts Source input, nd output control input off Current tries to flow from source input to output Control input Voltge tht controls whether tht current cn flow The mzing shrinking switch 93s: Rels 94s: Vcuum tues 95s: Discrete trnsistor 96s: Integrted circuits (ICs) Initill just few trnsistors on IC Then tens, hundreds, thousnds... rel source input source input control input () discrete trnsistor vcuum tue output on output IC Digitl Design 2e Copright 2 rnk Vhid qurter (to see the reltive size) 4

5 Moore s Lw IC cpcit douling out ever 8 months for severl decdes Known s Moore s Lw fter Gordon Moore, co-founder of Intel Predicted in 965 tht components per IC would doule roughl ever er or so Book cover depicts relted phenomen or prticulr numer of trnsistors, the IC re shrinks hlf ever 8 months Consider how much shrinking occurs in just ers (tr drwing it) Enles incredil powerful computtion in incredil tin devices Tod s ICs hold illions of trnsistors The first Pentium processor (erl 99s) needed onl 3 million Digitl Design 2e Copright 2 rnk Vhid An Intel Pentium processor IC hving millions of trnsistors 5

6 The CMOS Circuit 2.3 CMOS (Complementr metl oide semiconductor) circuit Bsic switch in modern ICs A positive voltge here... source gte oide drin...ttrcts electrons here, turning the chnnel etween the source nd drin into conductor IC pckge Silicon -- not quite conductor or insultor: Semiconductor IC Digitl Design 2e Copright 2 rnk Vhid 6

7 The CMOS Circuit 2.3 CMOS (Complementr metl oide semiconductor) circuit Bsic switch in modern ICs A positive voltge here......ttrcts electrons here, turning the chnnel etween the source nd drin into conductor nmos gte gte source oide drin IC pckge conducts does not conduct pmos () IC gte Silicon -- not quite conductor or insultor: Semiconductor does not conduct conducts Digitl Design 2e Copright 2 rnk Vhid 6

8 Boolen Logic Gtes Building Blocks for Digitl Circuits (Becuse Switches re Hrd to Work With) 2.4 Logic gtes re etter digitl circuit uilding locks thn switches (trnsistors) Wh?... Digitl Design 2e Copright 2 rnk Vhid 8

9 Boolen Alger nd its Reltion to Digitl Circuits To understnd the enefits of logic gtes vs. switches, we should first understnd Boolen lger Trditionl lger Vriles represent rel numers (, ) Opertors operte on vriles, return rel numers (2.5* + - 3) Boolen Alger Vriles represent or onl Opertors return or onl Bsic opertors AND: AND returns onl when oth = nd = OR: OR returns if either (or oth) = or = NOT: NOT returns the opposite of ( if =, if =) AND NOT OR Digitl Design 2e Copright 2 rnk Vhid 9

10 Boolen Alger nd its Reltion to Digitl Circuits Developed mid-8 s George Boole to formlize humn thought E: I ll go to lunch if Mr goes OR John goes, AND Sll does not go. Let represent m going to lunch ( mens I go, I don t go) Likewise, m for Mr going, j for John, nd s for Sll Then = (m OR j) AND NOT(s) Nice fetures ormll evlute m=, j=, s= --> = ( OR ) AND NOT() = AND = ormll trnsform = (m nd NOT(s)) OR (j nd NOT(s))» Looks different, ut sme function» We ll show trnsformtion techniques soon ormll prove Prove tht if Sll goes to lunch (s=), then I don t go (=) = (m OR j) AND NOT() = (m OR j) AND = AND OR NOT Digitl Design 2e Copright 2 rnk Vhid

11 Evluting Boolen Equtions Evlute the Boolen eqution = ( AND ) OR (c AND d) for the given vlues of vriles,, c, nd d: Q: =, =, c=, d=. Answer: = ( AND ) OR ( AND ) = OR =. Q2: =, =, c=, d=. Answer: = ( AND ) OR ( AND ) = OR =. Q3: =, =, c=, d=. Answer: = ( AND ) OR ( AND ) = OR =. AND OR NOT Digitl Design 2e Copright 2 rnk Vhid

12 Converting to Boolen Equtions Convert the following English sttements to Boolen eqution Q. is nd is. Answer: = AND Q2. either of or is. Answer: = OR Q3. is nd is. Answer: = AND NOT() Q4. is not. Answer: () Option : = NOT(NOT()) () Option 2: = Digitl Design 2e Copright 2 rnk Vhid 2

13 Converting to Boolen Equtions Q. A fire sprinkler sstem should spr wter if high het is sensed nd the sstem is set to enled. Answer: Let Boolen vrile h represent high het is sensed, e represent enled, nd represent spring wter. Then n eqution is: = h AND e. Q2. A cr lrm should sound if the lrm is enled, nd either the cr is shken or the door is opened. Answer: Let represent lrm is enled, s represent cr is shken, d represent door is opened, nd represent lrm sounds. Then n eqution is: = AND (s OR d). () Alterntivel, ssuming tht our door sensor d represents door is closed insted of open (mening d= when the door is closed, when open), we otin the following eqution: = AND (s OR NOT(d)). Digitl Design 2e Copright 2 rnk Vhid 3

14 Relting Boolen Alger to Digitl Design Boolen Boole s intent: formlize lger humn thought (mid-8s) Smol NOT OR AND Switches or telephone (93s) switching nd other electronic uses Truth tle Showed ppliction Shnnon (938) of Boolen lger to design of switchsed circuits Trnsistor circuit Digitl design Implement Boolen opertors using trnsistors Cll those implementtions logic gtes. Let us uild circuits doing mth - powerful concept Digitl Design 2e Copright 2 rnk Vhid.8 V.2 V.6 V V nd ech ctull corresponds to voltge rnge Net slides show how these circuits work. Note: The ove OR/AND implementtions re inefficient; we ll show wh, nd show etter ones, lter. 4

15 NOT gte (Inverter) () () When the input is When the input is time Digitl Design 2e Copright 2 rnk Vhid 5

16 OR gte () () time When n input is When oth inputs re Digitl Design 2e Copright 2 rnk Vhid 6

17 AND gte () () time When oth inputs re When n input is Digitl Design 2e Copright 2 rnk Vhid 7

18 Building Circuits Using Gtes Recll Chpter motion-in-drk emple Turn on lmp (=) when motion sensed (=) nd no light (=) = AND NOT() Build using logic gtes, AND nd NOT, s shown We just uilt our first digitl circuit! Digitl Design 2e Copright 2 rnk Vhid 8

19 Emple: Converting Boolen Eqution to Circuit of Logic Gtes Strt from the output, work ck towrds the inputs Q: Convert the following eqution to logic gtes: = AND NOT( OR NOT(c) ) c Digitl Design 2e Copright 2 rnk Vhid 9

20 More emples = AND (s OR d) 2 = ( AND NOT()) OR ( AND NOT(c)) 2 3 s d () c () Strt from the output, work ck towrds the inputs Digitl Design 2e Copright 2 rnk Vhid 2

21 Using gtes with more thn 2 inputs = AND AND c c c () () Cn think of s AND(,,c) Digitl Design 2e Copright 2 rnk Vhid 2

22 Emple: Set Belt Wrning Light Sstem Design circuit for wrning light Sensors s=: set elt fstened k=: ke inserted Cpture Boolen eqution set elt not fstened, nd ke inserted Convert eqution to circuit w = NOT(s) AND D k k s BeltWrn w Timing digrm illustrtes circuit ehvior We set inputs to n vlues Output set ccording to circuit Inputs k s Outputs w Setelt Digitl Design 2e Copright 2 rnk Vhid time 22

23 Gtes vs. switches Notice Boolen lger enles es cpture s eqution nd conversion to circuit How design with switches? Of course, logic gtes re uilt from switches, ut we think t level of logic gtes, not switches BeltWrn w = NOT(s) AND k s BeltWrn w k w k s Digitl Design 2e Copright 2 rnk Vhid Setelt 23

24 More emples: Set elt wrning light etensions Onl illuminte wrning light if person is in the set (p=), nd set elt not fstened nd ke inserted w = p AND NOT(s) AND k k p s Belt Wrn w Given t= for 5 seconds fter ke inserted. Turn on wrning light when t= (to check tht wrning lights re working) w = (p AND NOT(s) AND k) OR t k p s t BeltWrn w Digitl Design 2e Copright 2 rnk Vhid 24

25 Some Gte-Bsed Circuit Drwing Conventions no es no es ok not ok Digitl Design 2e Copright 2 rnk Vhid 25

26 Boolen Alger 2.5 B defining logic gtes sed on Boolen lger, we cn use lgeric methods to mnipulte circuits Nottion: Writing AND, OR, NOT() is cumersome Use smols: * (or just ), +, nd Originl: w = (p AND NOT(s) AND k) OR t New: w = ps k + t Spoken s w equls p nd s prime nd k, or t Or just w equls p s prime k, or t s known s complement of s While smols come from regulr lger, don t s times or plus "product" nd "sum" re OK nd commonl used Boolen lger precedence, highest precedence first. Smol Nme Description ( ) Prentheses Evlute epressions nested in prentheses first NOT Evlute from left to right Digitl Design 2e Copright 2 rnk Vhid * AND Evlute from left to right + OR Evlute from left to right 26

27 Boolen Alger Opertor Precedence Evlute the following Boolen equtions, ssuming =, =, c=, d=. Q. = * + c. Answer: * hs precedence over +, so we evlute the eqution s = ( *) + = () + = + =. Q2. = + c. Answer: the prolem is identicl to the previous prolem, using the shorthnd nottion for *. Q3. =. Answer: we first evlute ecuse NOT hs precedence over AND, resulting in = * ( ) = * () = * =. Q4. = (c). Answer: we first evlute wht is inside the prentheses, then we NOT the result, ielding (*) = () = =. Q5. = ( + ) * c + d. Answer: Inside left prentheses: ( + ( )) = ( + ()) = ( + ) =. Net, * hs precedence over +, ielding ( * ) + = () +. The NOT hs precedence over the OR, giving () + ( ) = () + () = + =. Boolen lger precedence, highest precedence first. Smol Nme Description ( ) Prentheses Evlute epressions nested in prentheses first Digitl Design 2e Copright 2 rnk Vhid NOT Evlute from left to right * AND Evlute from left to right + OR Evlute from left to right 27

28 Boolen Alger Terminolog Emple eqution: (,,c) = c + c + + c Vrile Represents vlue ( or ) Three vriles:,, nd c Literl Appernce of vrile, in true or complemented form Nine literls:,, c,,, c,,, nd c Product term Product of literls our product terms: c, c,, c Sum-of-products Eqution written s OR of product terms onl Aove eqution is in sum-of-products form. = (+)c + d is not. Digitl Design 2e Copright 2 rnk Vhid 28

29 Boolen Alger Properties Commuttive + = + * = * Distriutive * ( + c) = * + * c Cn write s: (+c) = + c + ( * c) = ( + ) * ( + c) (This second one is trick!) Cn write s: +(c) = (+)(+c) Associtive ( + ) + c = + ( + c) ( * ) * c = * ( * c) Identit + = + = * = * = Complement + = * = To prove, just evlute ll possiilities Emple uses of the properties Show c equivlent to c. Use commuttive propert: **c = *c * = c ** = c ** Show c + c =. Use first distriutive propert c + c = (c+c ). Complement propert Replce c+c : (c+c ) = (). Identit propert () = * =. Show + z equivlent to + z. Second distriutive propert Replce + z (+ )*(+z). Complement propert Replce (+ ), Identit propert replce *(+z) +z. Digitl Design 2e Copright 2 rnk Vhid 29

30 Emple tht Applies Boolen Alger Properties Wnt utomtic door opener circuit (e.g., for grocer store) Output: f= opens door Inputs: p=: person detected h=: switch forcing hold open c=: ke forcing closed Wnt open door when h= nd c=, or h= nd p= nd c= Eqution: f = hc + h pc h c p DoorOpener f Cn the circuit e simplified? f = hc' + h'pc f = c'h + c'h'p ( the commuttive propert) f = c'(h + h'p) ( the first distri. propert) f = c'((h+h')*(h+p)) (2nd distri. prop.; trick one) f = c'(()*(h + p)) ( the complement propert) f = c'(h+p) ( the identit propert) c h p DoorOpener f Simplified circuit Digitl Design 2e Copright 2 rnk Vhid Simplifiction of circuits is covered in Sec. 2. / Sec

31 Emple tht Applies Boolen Alger Properties Commuttive + = + * = * Distriutive * ( + c) = * + * c + ( * c) = ( + ) * ( + c) Associtive ( + ) + c = + ( + c) ( * ) * c = * ( * c) Identit + = + = * = * = Complement + = * = ound inepensive chip tht computes: f = c hp + c hp + c h p Cn we use it for the door opener? Is it the sme s f = hc + h pc? Appl Boolen lger: f = c hp + c hp + c h p f = c h(p + p ) + c h p ( the distriutive propert) f = c h() + c h p ( the complement propert) f = c h + c h p f = hc + h pc Sme! Yes, we cn use it. ( the identit propert) ( the commuttive propert) h c p DoorOpener f Digitl Design 2e Copright 2 rnk Vhid 3

32 Boolen Alger: Additionl Properties Null elements + = * = Idempotent Lw + = * = Involution Lw ( ) = DeMorgn s Lw ( + ) = () = + Ver useful! To prove, just evlute ll possiilities Digitl Design 2e Copright 2 rnk Vhid 32

33 Emple Appling DeMorgn s Lw ( + ) = () = + Aircrft lvtor sign emple c Circuit Behvior Three lvtories, ech with sensor (,, c), equls if door locked Light Aville sign (S) if n lvtor ville Eqution nd circuit S = + + c Trnsform (c) = + +c ( DeMorgn s Lw) S = (c) New circuit S c Circuit S Alterntive: Insted of lighting Aville, light Occupied Opposite of Aville function S = + + c So S = ( + + c ) S = ( ) * ( ) * (c ) ( DeMorgn s Lw) S = * * c ( Involution Lw) Mkes intuitive sense Occupied if ll doors re locked Digitl Design 2e Copright 2 rnk Vhid 33

34 Commuttive + = + * = * Distriutive * ( + c) = * + * c + ( * c) = ( + ) * ( + c) Associtive ( + ) + c = + ( + c) ( * ) * c = * ( * c) Identit + = + = * = * = Complement + = * = Digitl Design 2e Copright 2 rnk Vhid Emple Appling Properties Null elements + = * = Idempotent Lw + = * = Involution Lw ( ) = DeMorgn s Lw ( + ) = () = + or door opener f = c'(h+p), prove door sts closed (f=) when c= f = c'(h+p) Let c = (door forced closed) f = '(h+p) f = (h+p) f = h + p ( the distriutive propert) f = + ( the null elements propert) f = 34

35 Complement of unction Commonl wnt to find complement (inverse) of function when is ; when is Use DeMorgn s Lw repetedl Note: DeMorgn s Lw defined for more thn two vriles, e.g.: ( + + c)' = (c)' (c)' = (' + ' + c') Complement of f = w' + w''z' f ' = (w' + w''z')' f ' = (w')'(w''z')' ( DeMorgn s Lw) f ' = (w+'+')(w'+++z) ( DeMorgn s Lw) Cn then epnd into sum-of-products form Digitl Design 2e Copright 2 rnk Vhid 35

36 Representtions of Boolen unctions 2.6 English : outputs when is nd is, or when is nd is. () Eqution : (,) = + Eqution 2: (,) = () English 2: outputs when is, regrdless of s vlue (c) Circuit Truth tle (d) Circuit 2 The function A function cn e represented in different ws Aove shows seven representtions of the sme functions (,), using four different methods: English, Eqution, Circuit, nd Truth Tle Digitl Design 2e Copright 2 rnk Vhid 36

37 Digitl Design 2e Copright 2 rnk Vhid 37 Truth Tle Representtion of Boolen unctions Define vlue of for ech possile comintion of input vlues 2-input function: 4 rows 3-input function: 8 rows 4-input function: 6 rows Q: Use truth tle to define function (,,c) tht is when c is 5 or greter in inr () c () c d (c) c

38 Converting mong Representtions Cn convert from n representtion to nother Common conversions Eqution to circuit (we did this erlier) Circuit to eqution Strt t inputs, write epression of ech gte output Equtions Truth tles 6 Circuits 5 c c' h p h+p = c'(h+p) Digitl Design 2e Copright 2 rnk Vhid 38

39 Converting mong Representtions More common conversions Truth tle to eqution (which we cn then convert to circuit) Es just OR ech input term tht should output Eqution to truth tle Es just evlute eqution for ech input comintion (row) Creting intermedite columns helps Equtions 3 Inputs 4 2 Truth tles Outputs 6 Circuits Term 5 = sum of = + Q: Convert to truth tle: = + Digitl Design 2e Copright 2 rnk Vhid Inputs ' ' ' Output Q: Convert to eqution c c c c = c + c + c 39

40 Emple: Converting from Truth Tle to Eqution Prit it: Etr it dded to dt, intended to enle detection of error ( it chnged unintentionll) e.g., errors cn occur on wires due to electricl interference Even prit: Set prit it so totl numer of s (dt + prit) is even e.g., if dt is, prit it is hs even numer of s Wnt eqution, ut esiest to strt from truth tle for this emple c P Convert to eqn. P = ''c + 'c' + 'c' + c Digitl Design 2e Copright 2 rnk Vhid

41 Digitl Design 2e Copright 2 rnk Vhid 4 Emple: Converting from Circuit to Truth Tle irst convert circuit to eqution, then eqution to tle c c' ()' ()'c' c ()' c' Inputs Outputs

42 Stndrd Representtion: Truth Tle How cn we determine if two functions re the sme? Recll utomtic door emple Sme s f = hc + h pc? Used lgeric methods But if we filed, does tht prove not equl? No. Solution: Convert to truth tles Onl ONE truth tle representtion of given function Stndrd representtion for given function, onl one version in stndrd form eists Digitl Design 2e Copright 2 rnk Vhid f = c hp + c hp + c h f = c h(p + p ) + c h p f = c h() + c h p f = c h + c h p (wht if we stopped here?) f = hc + h pc Q: Determine if =+ is sme function s = + +, converting ech to truth tle first = + ' = + + Sme 42

43 Truth Tle Cnonicl orm Q: Determine vi truth tles whether +' nd (+)' re equivlent = + ' = (+)' Not equivlent Digitl Design 2e Copright 2 rnk Vhid 43

44 Cnonicl orm Sum of Minterms Truth tles too ig for numerous inputs Use stndrd form of eqution insted Known s cnonicl form Regulr lger: group terms of polnomil power c ( > ) Boolen lger: crete sum of minterms Minterm: product term with ever function vrile ppering ectl once, in true or complemented form Just multipl-out eqution until sum of product terms Then epnd ech term until ll terms re minterms Q: Determine if (,)=+ is equivlent to (,)= + +, converting first eqution to cnonicl form (second lred is) Digitl Design 2e Copright 2 rnk Vhid = + (lred sum of products) = + (+ ) (epnding term) = + + (Equivlent sme three terms s other eqution) 44

45 Cnonicl orm Sum of Minterms Q: Determine whether the functions G(,,c,d,e) = cd + 'cde nd H(,,c,d,e) = cde + cde' + 'cde + 'cde(' + c) re equivlent. G = cd + 'cde G = cd(e+e') + 'cde G = cde + cde' + 'cde G = 'cde + cde' + cde (sum of minterms form) Equivlent Digitl Design 2e Copright 2 rnk Vhid H = cde + cde' + 'cde + 'cde(' + c) H = cde + cde' + 'cde + 'cde' + 'cdec H = cde + cde' + 'cde + 'cde + 'cde H = cde + cde' + 'cde H = 'cde + cde' + cde 45

46 Compct Sum of Minterms Representtion List ech minterm s numer Numer determined from the inr representtion of its vriles vlues 'cde corresponds to, or 5 cde' corresponds to, or 3 cde corresponds to, or 3 Thus, H = 'cde + cde' + cde cn e written s: H = m(5,3,3) "H is the sum of minterms 5, 3, nd 3" Digitl Design 2e Copright 2 rnk Vhid 46

47 Multiple-Output Circuits Mn circuits hve more thn one output Cn give ech seprte circuit, or cn shre gtes E: = + c, G = + c c c G G () Option : Seprte circuits () Option 2: Shred gtes Digitl Design 2e Copright 2 rnk Vhid 47

48 Multiple-Output Emple: BCD to 7-Segment Converter w z Converter f g e c d f g e c d cdefg = () () Digitl Design 2e Copright 2 rnk Vhid 48

49 Multiple-Output Emple: BCD to 7-Segment Converter f g e c d = 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 2e Copright 2 rnk Vhid 49

50 Comintionl Logic Design Process 2.7 Step : Cpture ehvior Step Cpture the function Description Crete truth tle or equtions, whichever is most nturl for the given prolem, to descrie the desired ehvior of ech output of the comintionl logic. Step 2: Convert to circuit 2A: Crete equtions 2B: Implement s gtesed circuit This sustep is onl necessr if ou cptured the function using truth tle insted of equtions. Crete n eqution for ech output ORing ll the minterms for tht output. Simplif the equtions if desired. or ech output, crete circuit corresponding to the output s eqution. (Shring gtes mong multiple outputs is OK optionll.) Digitl Design 2e Copright 2 rnk Vhid 5

51 Emple: Three s Pttern Detector Prolem: Detect three consecutive s in 8-it input: cdefgh Step : Cpture the function Truth tle or eqution? Truth tle too ig: 2^8=256 rows Eqution: crete terms for ech possile cse of three consecutive s = c + cd + cde + def + efg + fgh c d c cd Step 2: Crete eqution -- lred done Step 2: Implement s gte-sed circuit e f g cde efg def fgh Digitl Design 2e Copright 2 rnk Vhid h 5

52 Emple: Numer of s Counter Prolem: Output in inr on two outputs z the # of s on three inputs Step : Cpture the function Truth tle or eqution? Truth tle is strightforwrd Step 2: Crete equtions = c + c + c + c z = c + c + c + c Optionl: Let's simplif : = 'c + 'c + (c' + c) = 'c + 'c + Step 2: Implement s gte-sed circuit Digitl Design 2e Copright 2 rnk Vhid c c c c c c 52 z

53 Used in previous circuit Simplifing Nottions c c c c () () ' c List inputs multiple times Less wiring in drwing Drw inversion ule rther thn inverter. Or list input s complemented. Digitl Design 2e Copright 2 rnk Vhid 53

54 Emple: Kepd Converter Kepd hs 7 outputs One per row One per column Ke press sets one row nd one column output to Press "5" r2=, c2= Gol: Convert kepd outputs into 4-it inr numer -9 to *, # nothing pressed: * # c c2 c3 r r2 r3 r4 Converter w z Digitl Design 2e Copright 2 rnk Vhid 54

55 Emple: Kepd Converter Step : Cpture ehvior Truth tle too ig (2^7 rows); equtions not cler either Informl tle cn help Digitl Design 2e Copright 2 rnk Vhid Step 2: Implement s circuit (note w = r3c2 + r3c3 + r4c + r4c3 + r'r2'r3'r4'c'c2'c3' shrle gtes)... = r2c + r2c2 + r2c3 + r3c + r'r2'r3'r4 c'c2'c3' = rc2 + rc3 + r2c3 + r3c + r4c + r4c3 + r'r2'r3'r4'c'c2'c3' z = rc + rc3 + r2c2 + r3c + r3c3 + r4c3 + r'r2'r3'r4'c'c2'c3' 55

56 Emple: Sprinkler Controller Microprocessor outputs which zone to wter (e.g., c= mens zone 6) nd enles wtering (e=) Decoder should set pproprite vlve to Microprocessor Digitl Design 2e Copright 2 rnk Vhid d d d2 d3 c d4 d5 decoder d6 e d7 zone 7 3 Equtions seem like nturl fit 5 zone Step : Cpture ehvior d = ''c'e d = ''ce d2 = 'c'e d3 = 'ce d4 = 'c'e d5 = 'ce d6 = c'e d7 = ce 56

57 Emple: Sprinkler Controller Step 2: Implement s circuit c d Microprocessor d d d2 d3 c d4 d5 decoder d6 e d7 zone zone d d2 Digitl Design 2e Copright 2 rnk Vhid d = ''c'e d = ''ce d2 = 'c'e d3 = 'ce d4 = 'c'e d5 = 'ce d6 = c'e d7 = ce e d3 d4 d5 d6 d7 57

58 More Gtes 2.8 NAND NOR XOR XNOR NAND NOR NAND: Opposite of AND ( NOT AND ) NOR: Opposite of OR ( NOT OR ) XOR: Ectl input is, for 2-input XOR. (or more inputs -- odd numer of s) XNOR: Opposite of XOR ( NOT XOR ) Digitl Design 2e Copright 2 rnk Vhid NAND sme s AND with power & ground switched nmos conducts s well, ut not s (resons eond our scope) so NAND is more efficient Likewise, NOR sme s OR with power/ground switched NAND/NOR more common AND in CMOS: NAND with NOT OR in CMOS: NOR with NOT 58

59 More Gtes: Emple Uses Aircrft lvtor sign emple S = (c) Detecting ll s Use NOR Detecting equlit Use XNOR Detecting odd # of s Use XOR Useful for generting prit it common for detecting errors 2 2 c Circuit A=B S Digitl Design 2e Copright 2 rnk Vhid 59

60 Completeness of NAND An Boolen function cn e implemented using just NAND gtes. Wh? Need AND, OR, nd NOT NOT: -input NAND (or 2-input NAND with inputs tied together) AND: NAND followed NOT OR: NAND preceded NOTs Thus, NAND is universl gte Cn implement n circuit using just NAND gtes Likewise for NOR Digitl Design 2e Copright 2 rnk Vhid 6

61 Numer of Possile Boolen unctions How mn possile functions of 2 vriles? 2 2 rows in truth tle, 2 choices for ech 2 (22 ) = 2 4 = 6 possile functions N vriles 2 N rows 2 (2N ) possile functions or 2 choices or 2 choices or 2 choices or 2 choices 2 4 = 6 possile functions f f f2 f3 f4 f5 f6 f7 f8 f9 f f f2 f3 f4 f5 AND XOR OR NOR XNOR NAND Digitl Design 2e Copright 2 rnk Vhid 6

62 Decoders nd Mues 2.9 Decoder: Populr comintionl logic uilding lock, in ddition to logic gtes Converts input inr numer to one high output 2-input decoder: four possile input inr numers So hs four outputs, one for ech possile input inr numer Internl design AND gte for ech output to detect input comintion Decoder with enle e Outputs ll if e= Regulr ehvior if e= n-input decoder: 2 n outputs Digitl Design 2e Copright 2 rnk Vhid d i d i d2 i i d3 i i i i ii ii i i d d d2 d3 d d d2 d3 i i d d d2 d3 i i i i d d d2 i i e d3 e d d d2 d3 62 d d d2 d3

63 New Yer s Eve Countdown Displ Microprocessor counts from 59 down to in inr on 6-it output Wnt illuminte one of 6 lights for ech inr numer Use 664 decoder 4 outputs unused Decoder Emple Processor 2 2 i i i2 i3 i4 i5 e 664 dcd d d d2 d3 d58 d59 d6 d6 d62 d Hpp New Yer Digitl Design 2e Copright 2 rnk Vhid 63

64 Multipleor (Mu) Mu: Another populr comintionl uilding lock Routes one of its N dt inputs to its one output, sed on inr vlue of select inputs 4 input mu needs 2 select inputs to indicte which input to route through 8 input mu 3 select inputs N inputs log 2 (N) selects Like ril rd switch Digitl Design 2e Copright 2 rnk Vhid 64

65 Mu Internl Design 2 i d i s 2 mu i i 2 s d i i 2 s d i i s i (*i=i) d i (+i=i) 4 i i d i2 i3 s s 4 mu i i i2 i3 d s s Digitl Design 2e Copright 2 rnk Vhid 65

66 Mu Emple Cit mor cn set four switches up or down, representing his/her vote on ech of four proposls, numered,, 2, 3 Cit mnger cn displ n such vote on lrge green/red LED (light) setting two switches to represent inr,, 2, or 3 Mor s switches Use 4 mu Proposl 2 3 i i i2 i3 4 s s d on/off Green/ Red LED Digitl Design 2e Copright 2 rnk Vhid 4 mnger's switches 66

67 Mues Commonl Together N-it Mu s i d i s 2 i d i s 2 i d i s i 2 d i s A B 4 4 I I 4-it 2 s s D 4 C Simplifing nottion: 4 C is short for c3 c2 c c E: Two 4-it inputs, A (3 2 ), nd B (3 2 ) 4-it 2 mu (just four 2 mues shring select line) cn select etween A or B Digitl Design 2e Copright 2 rnk Vhid 67

68 N-it Mu Emple rom the cr's centrl computer T A I M I I I2 8-it 4 D I3 s s 8 D To the ovemirror displ We'll design this lter utton our possile displ items Temperture (T), Averge miles-per-gllon (A), Instntneous mpg (I), nd Miles remining (M) ech is 8-its wide Choose which to displ on D using two inputs nd Pushing utton sequences to the net item Use 8-it 4 mu Digitl Design 2e Copright 2 rnk Vhid 68

69 Additionl Considertions Non-Idel Gte Behvior -- Del 2. (.8 V) ( V) () idel Rel gtes hve some del Outputs don t chnge immeditel fter inputs chnge time () more relistic time (c) time with del ut otherwise idel Digitl Design 2e Copright 2 rnk Vhid 69

70 Circuit Del nd Criticl Pth k BeltWrn p s ns t ns ns ns.5 ns ns ns ns ns ns w ++++ = 5 ns = 6.5 ns ++ = 3 ns Criticl pth del = 6.5 ns Hence, circuitʼs del is 6.5 ns Digitl Design 2e Copright 2 rnk Vhid Wires lso hve del Assume gtes nd wires hve dels s shown Pth del time for input to ffect output Criticl pth pth with longest pth del Circuit del del of criticl pth 7

71 Active Low Inputs Dt inputs: flow through component (e.g., mu dt input) Control input: influence component ehvior Normll ctive high cuses input to crr out its purpose Active low Insted, cuses input to crr out its purpose Emple: 24 decoder with ctive low enle disles decoder, enles Drwn using inversion ule d d i d i d i d2 i d2 e d3 e d3 Digitl Design 2e Copright 2 rnk Vhid () () 7

72 Schemtic Cpture nd Simultion Inputs Inputs i i i Outputs d3 Simulte i Outputs d3 Simulte d2 d2 d d d d Schemtic cpture Computer tool for user to cpture logic circuit grphicll Simultor Computer tool to show wht circuit outputs would e for given inputs Outputs commonl displed s wveform Digitl Design 2e Copright 2 rnk Vhid 72

73 Chpter Summr Comintionl circuits Circuit whose outputs re function of present inputs No stte Switches: Bsic component in digitl circuits Boolen logic gtes: AND, OR, NOT Better uilding lock thn switches Enles use of Boolen lger to design circuits Boolen lger: Uses true/flse vriles/opertors Representtions of Boolen functions: Cn trnslte mong Comintionl design process: Trnslte from eqution (or tle) to circuit through well-defined steps More gtes: NAND, NOR, XOR, XNOR lso useful Mues nd decoders: Additionl useful comintionl uilding locks Digitl Design 2e Copright 2 rnk Vhid 73