DeMorgan s Theorem. The dual DeMorgan D + E = D E D+E D E 1 D NOR D E. Slide 29 Modified; January 3, 2006 John Knight Digital Circuits p.

Transcription

1 DeMorgn s Theorem DeMorgn s Theorems, Simple orms DeMorgn s Theorem DeMorgn s Theorems, Simple orms DeMorgn The ul DeMorgn A + B = A B (DeM) D + E = D E (DeM2) Inverse The ul inverse A + B = A B D E = D + E A B A B A+B 0 AB AB D E D+E D E D E Equivlent grphil forms: A B A B = K = A + B K A B K D E D + E = G = D E G D NOR E NOR G A B A B = C = A + B C A AND B AND AND C D E D + E = = D E D OR E OR Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 29 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 56 DeMorgn s Theorem DeMorgn s Theorems DeMorgn s Theorems A theorem relting s n NORs. An OR gte with inverte inputs is equivlent to n AND gte with n inverte output. An AND gte with inverte inputs is equivlent to n OR gte with n inverte output. Inverting inputs n outputs of n OR mkes it n AND. Inverting inputs n outputs of n AND mkes it n OR. EXAMPLE Convert 4. PROBLEM Reue 42. PROBLEM Reue Chnging everything into NOT n AND gtes to n expression with 3 letters n inversion rs only over single letters. to four letters with inversion rs over single letters only. to four letters with inversion rs over single letters only. It turns out tht ny logi iruit n e me from AND n NOT gtes. DeMorgn s lw n e use to trnsform the iruits. 43. PROBLEM ( + )( + ) ( + )( + ) = ( + ) + ( + ) (DeM) = ( ) + ( ) (DeM2) = + (Cler rkets) = ( + ) (D) xy + xz = x(y+z) (+) + (e) + (e)e Convert ((rw + t)u + r)t into funtion with only AND n NOT opertions. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 29 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 57

2 DeMorgn s Theorem Rel CMOS Digitl Gtes Rel CMOS Digitl Gtes Trnsistor NOT +5 V +V DD Two swithes linke together X X JOLT Smll REE Jolt JOLT Smll REE Jolt A Q +V DD + A A= => Q lose => =0 Q PMOS trnsistor ts like: lose swith when A is 0 open swith when A is NMOS trnsistor ts like: open swith when A is 0 lose swith when A is Trnsistor E VDD E E n = =>Output Groune Output Inverte A B Trnsistor NOR NOR VDD A+B B or A= =>Output Groune Output Inverte One nnot mke AND or OR gtes iretly. All CMOS Gtes Invert Rel Gtes re, NOR n NOT Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment Slie 30 on Slie Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 58 DeMorgn s Theorem DeMorgn s Theorems Construting CMOS gtes from trnsistors DeMorgn s Theorem CMOS stns for Complementry Symmetry Metl-Oxie-Semionutor gtes. They lwys hve omplementry trnsistors, whih mens PMOS (turn off with one input) ove the output, n NMOS (turn on with one input) elow the output. The orret one inputs turn the lower NMOS trnsistors on, whih pulls the output own to zero thus inverting the output. The Esily Construte CMOS Gtes s with 2, 3 or 4 inputs, NORs with 2, 3 or 4 inputs, n NOTs. Gtes onstrute from other gtes To voi ll the extr inverters (NOTs) rel iruits re esigne to use s n NORs inste of ANDs n ORs. Thinking Thinking in -NOR logi is iffiult. Just look t ny inustril shemti use extensively for mintenne. The mrgin will e full of s n 0 s penille in y users. Converting AND-OR into -NOR is strightforwr mehnil proess. Muh less error prone thn oing logi with inverte signls n -NOR logi. These Notes If the logi is importnt ANDs n ORs will e use. If the gte esign is importnt, s when we tlk out CMOS gtes, s n NORs will e use.. If flsh is signl nme, flsh is n inverte signl. How n OR is me How n AND is me One wy of mking n XOR See prolem 4. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 30 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 59

3 DeMorgn s Theorem Rel CMOS Digitl Gtes Using DeMorgn s Theorem Equivlent Gte Symols NOR AND OR AND AND NOR OR Rel Gtes re n NOR NOR One nnot mke AND n OR gtes iretly. Esy to Unerstn Gtes re AND n OR AND OR Ciruit with rel gtes = () () Ciruit with simple gtes AND AND = + OR Whih one is esier for you to unerstn? Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 3 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 60 DeMorgn s Theorem DeMorgn s Theorems DeMorgn s Theorem The two expressions for the iruit with rel gtes n the iruit with simple gtes, re equivlent + = = () () Use High-True An-Or Signls for Thinking Thinking Thinking in nn-nor logi is iffiult. Just look t ny inustril shemti use extensively for mintenne. The mrgin will e full of s n 0 s penille in y users. Converting etween n-or n nn-nor is strightforwr mehnil proess. Muh less error prone thn oing logi with sserte low signls n nn-nor logi. These Notes If the logi is importnt ANDs n ORs will e use. If the gte esign is importnt, s when we tlk out CMOS gtes, s n NORs will e use. 44. PROBLEM Prove, using DeMorgn s Theroem(s), tht + = () () Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 3 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 6

4 DeMorgn s Theorem Ciruits Using s n NORS re Ciruits Using s n NORS re Hr to ollow Exmple Don t o thinking with s n NORs. Confusing multiple logi inversions. Does wter stop or strt the trin? Is SW strt or stop swith? SW PWR AIR DOOROPN DOOROPN2 WATER U. U.2 U.3 U3. Strt_O-Trin SW PWR + AIR (DOOROPN DOOROPN2) + WATER Do Thinking Prt of Design with AND/OR; Convert After Thinking is Done The sme iruit me with ANDs n ORs. The finl eqution n e written from inspetion. (Is there logi error?) SW PWR AIR DOOROPN DOOROPN2 WATER Strt_O-Trin SW PWR AIR (DOOROPN + DOOROPN2) WATER Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 32 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 62 DeMorgn s Theorem Strting The O-Trin Strting The O-Trin The AND-OR logi is muh lerer. It tkes ir, power, wter to strt the trin. Also two oors must not e open. However it tkes 4 gtes n 6 inverters to implement the AND-OR iruit me y ing inverters to s n NORs. The less ler -NOR iruit oes the sme logi, with 4 gtes n inverter. -NOR Logi is Confusing to Humns Multiple negtives re onfusing If your teher si, Never will I not, not give you A in Digitl Ciruits, it woul tke you some reful reing to etermine your mrk. Digrms with rel gtes This is the kin of igrm one woul use to uil iruit. It hs esily mnufturle gtes, i.e. NOT, n NOR. The numers U. et. re gte numers tht woul physilly ientify the gte on lyout igrm showing where eh prt ws. Esy to re igrms You shoul e le to spot the logi error. DOOROPN + DOOROPN2 Shoul you e le to strt the trin with one oor open?. The O-Trin is n Ottw interurn trin, whih stops right outsie the Engineering Builing t Crleton University. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 32 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 63

5 DeMorgn s Theorem DeMorgn Trnsfers Rel Gtes Into DeMorgn Trnsfers Rel Gtes Into Simple Gtes Ciruit with rel gtes = () () Use DeMorgn s gte symol t output Inverting irles nel eh other Ciruit with simple gtes AND AND OR = + Ciruits ment for unerstning the logi use ANDs n ORs. - Drw your iruits with ANDs n ORs. Ciruits for onstrution re rwn with s n NORs. Drw onstrution igrms y: - trnsforming the unerstnle iruit into the rel iruit - using the DeMorgn lternte symols. Thus Compromise rwings hve S n NORs with the irles re rrnge to nel eh other. lternte symol Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 33 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 64 DeMorgn s Theorem Trnsforming -NOR Digrms into Trnsforming -NOR Digrms into AND-OR Digrms Digrms in this ourse will e rwn with ANDs n ORs s muh s possile. Digrms for onstrution or mintinne, tht wnt to show extly wht gtes were use, will e rwn with s n NORs. This is prtiulrly true of oler igrms. A ompromise metho, whih is lmost s esy to follow, ut shows the rel gtes s use, is to mke lternte gtes with the lternte symols, the ones with the inverting irles on the inputs. NOR NOTE: One output irle nels ll the input irles it fees. 45. PROBLEM Trnsform this iruit into simple gtes. ) Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 33 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 65

6 Don t o initil esign in s n DeMorgn Trnsfers Rel Gtes Into Don t o initil esign in s n NORs. Design in AND/OR; Convert After Thinking is Done Rtionl People think est with AND n OR. Multiple inversions re very onfusing There is selom logil reson to invert exept t iruit inputs. Conversion AND/OR /NOR is esy using DeMorgn s form of gtes. Do it fter the thinking is one. Some people still seem to prefer esigning with s n NORs ut not in this lss Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 34 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 66 Don t o initil esign in s n Design With Coneptully Esy Logi Design With Coneptully Esy Logi In the post 20 er, people usully think out the logi. The etils of onstrution re one utomtilly. This mens you will normlly o AND-OR type logi. Think with AND-OR Buil with -NOR DeMorgn s Lw Trnsforms Rel Gtes to Simple Gtes 46. PROBLEM Trnsform this iruit into simple gtes. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 34 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 67

7 Don t o initil esign in s n AND/OR to /NOR Conversion AND/OR to /NOR Conversion Exmple using grphil form. Strt with AND/OR iruit = ( + )( + ) + ) 2. Selet lternte onnetion lyers. (Every seon lyer of onneting wires etween lyers of gtes.) One en of wires my e inputs or outputs. skip skip 2) 3. Put k-to-k inverting irles on oth ens of the les. A NOT gtes when neessry ((), () n ()) () () 3) () 4. Moving inverters, or inverting ules my mke logi simpler. Sometimes inverters on two (or more) input les shoul e move to the output. (See next exmple) (h) 4) Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 35 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 68 Don t o initil esign in s n Converting AND/OR Designs to / Converting AND/OR Designs to /NOR input. Strt with n AND-OR iruit. If you re strting with n eqution. rw the iruit. The originl formul will usully hve inversions only on inputs. However there my e inversions nywhere. 2. The iruit is rwn with lyer of onnetions whih fees lyer of gtes, feeing lyer of onnetions (green), whih fees nother lyer of gtes, whih fees nother lyer of onnetions (green). Some iruits my hve more or fewer lyers. Selet lternte onnetion lyers. Some iruits re niely lyere with the first lyer feeing seon lyer whih fees thir lyer et. However this step is not ext. There re often severl legitimte wys to efine lyers. Note some wires, like () n () my pss through gte level (green) without gte. 3. Put k-to-k inverting irles t oth ens of wires going through onnetion lyer. On les like () n (), you will hve to n inverter, sine there is no gte on whih to ple the seon irle. 4. This step is less utomti. Look for les where two inverters n e reple y one. When oing this e reful not to invert other onnetions on the sme input, like (h). When moving inverting irles mke sure tht the numer of irles etween the signl input n ll the gte inputs it goes to, re the sme efore n fter moving. BEORE 2 3) Numer of inversions from input to this gte input 2 (h) 4) () () ATER Numer of inversions from input to this gte input Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 35 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 69

8 Don t o initil esign in s n AND/OR to /NOR Conversion AND/OR to /NOR Conversion 4. Copy of step 4 on lst slie: It hs k-to-k ules n onsolite inverters 4) 5. Selet the unonventionl gtes. The ones with input ules. This is ompromise solution whih is: 5) firly rele represents rel gtes NOR NOR 6. Use DeMorgn lws on the unonventionl gtes. Hr to unerstn the logi ut goo for onstrution. 6) Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 36 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 70 Don t o initil esign in s n Converting AND/OR Designs to / On previous slie You must lwys two inverting irles to le, or none. Never just one irle, not even if one exists lrey. On this slie 5. Look t the gtes. Conventionl ones hve irles on their outputs n re s, NORs or NOTs. The unonventionl ones hve irles on their inputs ut re still s, NORs or NOTs. This step gives iruit whih is firly esy to re euse one n mentlly nel the k-to-k irles. However it still represents iruit you n uil esily euse it oes not ontin ANDs n ORs. 6. If esire, one n reple the DeMorgn forms of NOR n, the ones with irles on the inputs, with the more stnr forms with irle on the output. Sine -NOR igrms represent rel gtes, o not ple single irle on the input of gte, s ws one in the theoretil AND-OR igrm. 47. PROBLEM:. or the iruit on the right, fin the -NOR iruit when the onnetion on the levels shown re selete. A B C G G D E H 48. PROBLEM in n AND-OR equivlent for the iruit on the left. This requires working kwrs. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 36 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 7

9 Don t o initil esign in s n 2n Exmple of AND/OR to /NOR 2n Exmple of AND/OR to /NOR SKIP ) AND/OR iruit; = ( + )( + ) + 2) Selet every other onnetion levels 3) A nelling k-to-k inverting irles inverters where neessry. 4) Input inverters pirs my eome single inverters if move to the gte output 5) Selet the unonventionl gtes. Compromise Answer 6) DeMorgn onventionl gtes (optionl). Conventionl (Dinosur) Answer Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 37 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 72 Don t o initil esign in s n AND/OR /NOR Conversion AND/OR /NOR Conversion (ont) This is the sme iruit s the previous exmple. The ifferene etween the exmples strts t step 2 2. the lternte loks of les where the inverters re e re tht ones mrke SKIP in the first exmple. Compre step 2 here with step 2 on Slie In this step, the two inverters feeing the OR with inverting inputs looks too omplex. Throwing out the oule irles leves us with n OR gte. This is not llowe. Go to the output of the OR n k-to-k inversions there. This gives NOR gte whih is llowe. Both the lower gtes on the right re equivlent to n OR gte, ut one is simpler to implement s rel gtes. Equivlent gtes 6. Chnging the NORs n s with irles on their inputs, to NORs n s with irles on their outputs, is the finl step. Do it if you like suh rwings etter, or if the rwing stnrs your ompny uses require it, or if your oss sys to. Compre the results with those in steps 5 n 6 of the previous exmple. This iruit is mrginlly simpler thn the previous result. One NOT gte is sve. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 37 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 73

10 DeMorgn s Theorems, Generl orm 2n Exmple of AND/OR to /NOR DeMorgn s Theorems, Generl orm Astrt orm (A,B,C,... +,,) = (A,B,C,...,, +,) Ation Exmple I ) Tke Boolen expression ) Brket ll groups of ANDs Tke ul ) Chnge AND OR n OR AND ) Clen rkets Chnge ul into inverse e) Invert ll vriles = A B C = {A B C} ul = {A + B + C} ul = A + B + C = {A + B + C} Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 38 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 74 DeMorgn s Theorems, Generl orm Generl orm of DeMorgn s Theorem Generl orm of DeMorgn s Theorem The slie shows simplifie version The true generlize DeMorgn (A,B,C,... +,,0,) = (A,B,C,...,, +,,0) (A,B,C,... +,,0,) = (A,B,C,...,, +,,0) The interhnge of 0 n ws left out on the slie, sine esigners prtilly never hve 0 or in the expressions they operte on with DeMorgn s lw. DeMorgn generl lw is very similr to Dulity The ovious ifferene is the inverting rs. Another importnt ifferene is the pplition. DeMorgn trnsforms n expression into its inverse. Dulity tkes vli ientity n genertes nother vli ientity. 49. PROBLEM in n expression for tht hs inverting rs only over single letters, = ( + )( + ) + Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 38 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 75

11 Exmples: Generlize DeMorgn s 2n Exmple of AND/OR to /NOR Exmples: Generlize DeMorgn s Theorems (A,B,C,... +,,) = (A,B,C,...,, +,) Exmple II ) Tke Boolen expression = A B C + A B ) Brket ll groups of ANDs Tke ul ) Chnge AND OR n OR AND e) Clen rkets Chnge ul into inverse e) Invert ll vriles {A B C} + {A B } ul = {A+B+C} {A+B } ul = (A+B+C) (A+B ) = (A+B+C) (A+B ) Exmple III ) Tke ny Boolen expression ) Brket ll groups of ANDs ) Chnge AND OR n OR AND Ignore ny existing overrs ) Clen rkets e) Invert ll vriles = [A B C + D (A B + C)] A = {[ {A B C } + {D ( {A B} + C)} ] A} ul = {[{A + B + C} {D + ({A + B} C)}] + A} ul = {A + B + C} {D + {A + B} C} + A = {A + B + C} {D + {A + B} C} + A Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 39 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 76 Exmples: Generlize DeMorgn s Exmples using the Generlize Exmples using the Generlize DeMorgn s Lw Be sure to put rkets roun the ANDs When you o lger, you utomtilly o the ANDs efore the ORs. The nottion is esigne tht wy. Putting rkets roun n OR shows tht the OR shoul e one efore the AND. When you use DeMorgn s lw, n you trnsform + into (+)(+), the rkets mke sure tht the vriles in the trnsforme expression re operte on in the sme orer. Tht is you on t try to o ( + ). By pling the rkets roun the ANDs first, you o not get onfuse uring the trnsformtions. 50. PROBLEM () Convert ( + )( + ) to n expression with 3 letters n inversion rs only over single letters, using the Generlize DeMorgn s lw. () Compre this metho with tht of the Exmple on Comment on Slie 29 n see if the lger is shorter. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 39 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 77

12 Exmples: Generlize DeMorgn s Exmples: Generlize DeMorgn s Exmples: Generlize DeMorgn s Theorems Exmple IV ) Tke Boolen expression = A B C + A B ) Brket ll groups of ANDs ) Chnge AND OR n OR AND Ignore existing overrs ) Clen rkets if onvenient {A B C} + {A B } ul = {A+B+C} {A+B } ul = (A+B+C) (A+B ) e) Invert ll vriles = (A+B+C) (A+B) Note: Ignore ny inverting overrs exept over single letters = A B (C + A B) ul = {A+B}+(C { A+B }) = {A+B}+(C { A+B }) Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 40 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 78 Exmples: Generlize DeMorgn s Exmples using the Generlize Exmples using the Generlize DeMorgn s Lw Common Worry, Intermeite Overrs When pplying the generlize lw, o not osier ny overrs exept those on top of single letters. If the overr is over two or more letters, just rry it through without hnge. 5. PROBLEM Use DeMorgn s generl lw to remove ll ut one of the inverting rs from the inosur iruit on Slie 37. The expression is f = (+) ( ) Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 40 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 79

13 Cnonil orms Sum of Prouts Cnonil orms Stnr templtes Every logil expression n e onverte to either of these forms. Sum of Prouts S of Π, OR of ANDs + e + + f + Vriles ANDe together into terms. These terms re ORe together. Inversions re only over iniviul vriles. No rkets. Prout of Sums. P of S, AND of ORs Single vriles ORe together into terms. These terms re ANDe together. Inversions re only over iniviul vriles. Brkets only roun vriles in OR terms Questions Is + + (+e) S of Π? Is ( +)(+e) P of S? (++)(++e)(+)(f)(+) Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 4 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 80 Cnonil orms Cnonil orms Cnonil orms Definition of Boolen untion A funtion is wy of getting the output from the inputs. A funtion n e efine in mny wys. The truth tle is very si efinition A Σ of Π expression. + A Π of Σ expression. ( + )( + ) (ftore form) Other forms tht will e introue shortly. Krnugh mps Binry eision igrms Why we sy S of P for Sum of Prout n Σ is use for repetitive ition, s in x i = x 0 + x + x 2 + x, n i = 0 n Π is use for repetitive prout, s in x i = x 0 x x 2 x n i = 0. Cnonil mens oring to the rule or lw, prtiulrly the hurh lw. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 4 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 8

14 Krnugh Mps Prout of Sums. Krnugh Mps A truth tle, orer is importnt Rerw the truth tle Put the input lels on the sies; leve the empty oxes for. Lel on Another wy Arevite the sies of lelling lelling Comine leling Arrnge the rows so tht only one input it hnges t time. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 42 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 82 Krnugh Mps Krnugh Mps Krnugh Mps The mp is like truth tle Eh squre on the mp represents ifferent input omintion. All possile input omintions re represente on the mp. The inputs re lelle roun the eges of the mp. Not insie the squres s shown on the right. Arrngement of the squres As one steps from one squre to the next, either up, own, left or right, only one it shoul hnge in single step. If one goes to the nerest igonl neighor, two its will hnge. one it hnge one it hnge Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 42 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 83

15 Krnugh Mps Krnugh Mps Krnugh Mps Different untions using AND = = () 0 0 All the squres where =0, =. All the squres where =. 0 = = = =? All the squres where Wrp roun =? =? Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 43 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 84 Krnugh Mps Representing AND Terms Representing AND Terms Any single squre (Top row, first two mps) On these mps, ny single squre represents speifi vlues for three vriles, this is the sme s three term AND like Any two jent squres We me the mps so tht only one vrile hnges t time if one moves vertilly or horizontlly. (This is not true for igonl movements). Thus two jent squres lwys hve one ommon vrile. In the top row, thir mp, the squres n re. We n sy + = ( + ) = This shows tht ny two jent squres n e represente y two term AND. Any three jent squres You n only irle if the numer of squres is power of 2. Any four jent squres (Top row, fourth mp) There re two wys to look t this. One is tht ll the squres where = hve in them, hene one n esrie them s. Alterntely one n note tht squres tht re re = ( ) =. (ottom row, first two mps) These represent n respetively. Any eight jent squres If ll the squres on mp re, the funtion is =. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 43 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 85

16 Krnugh Mps Krnugh Mps Krnugh Mps Joining AND terms with ORs = = = 3 = = + + OR together the terms, n ple them on one mp. The terms n overlp = = 5 = 6 = = + + Using the lrger terms ( 4, 5, n 6 ) gives smller expression for. Bigger irles give smller gtes. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 44 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 86 Krnugh Mps Krnugh Mps, Joining AND terms with Krnugh Mps, Joining AND terms with ORs Cirling mps in ifferent wys Tke the truth tle for Boolen funtion, written s mp. One n irle it in severl wys. irst one n irle iniviul s. This gives long expression for. Another wy is to rek it up s, 2 n 3 s shown on the top line on the slie. A thir wy is to rek it into 4, 5, n 6, s shown on the ottom line in the slie. This gives smller eqution = = = = = Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 44 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 87

17 Krnugh Mps Krnugh Mps Properties Krnugh Mps Properties Mps for ifferent numers of input vriles 0 One 0 Two 0 Three our ive e e Simplifying Logi Cirle jent squres Must irle,2,4,8,6... squres (ones) 6 squres Six Digonl squres not jent Cirling with wrp roun Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 45 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 88 Krnugh Mps Krnugh Mp Properties Krnugh Mp Properties Mps my hve ny numer of vriles, ut- Most hve 2, 3, 4, or 5 vrile. One vrile is simple, (2 squres). five veriles hs two 4x4 mps, one for when e=, n one for when e=0. Six vriles hve 64 squres n were use in pre-omputer ys. Seven vriles is pst the limit of snity. You will hve 8 loks of 6 squres eh. Rules for irling s Ones on the mps n e irle to simplify logi. Only jent squres n e irle. Digonlly jent squres re not onsiere jent. Cirles must surroun, 2, 4, 8, 6... squres. Not 3, 5,6,7,9,... The mps wrp roun. A squre on n ege re jent to the squre on the opposite ege in the sme olumn (row). Lrger irles give simpler logi, ut- One must oey the ove rules. There re exeptions, prtiulrly with multi-output mps. 5A. PROBLEM irle the s on the two mps shown. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 45 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 89

18 Krnugh Mps Simplifition With Krnugh Mps Simplifition With Krnugh Mps Simplify = + + The term is reunnt = The Consensus Theorem + + = + OR together the terms, n ple them on one mp. Simplify = Use igger irle in the mile = + + Using the lrger terms (irles) gives smller expression for. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 46 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 90 Krnugh Mps Simplifition of Boolen untion Simplifition of Boolen untion A Boolen funtion n e efine in mny wys A truth tle A Krnugh mp without irles A irle Krnugh mp. A Σ of Π expression. A Π of Σ expression. et. Any funtion hs only one truth tle n only one unirle mp. There re usully severl wys of irling the mp or writing the lgeri expression for the sme funtion. One tries to fin the est efinition for some ojetive. Possile Ojetives Smller iruitry. Lower powere iruitry. ster iruitry. Mking the iruit smller is usully goo strt towr mking the iruit fster n lower powere. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 46 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 9

19 Krnugh Mps Simplifition With 4-Input Mps Simplifition With 4-Input Mps Simplify the funtion efine y this mp John s Solution = + + Tom s Solution = + + Usully igger is etter Simplify the funtion efine y this mp John s Solution Tom s Solution = = + + Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 47 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 92 Krnugh Mps Simplifition Simplifition Some Things to Do Use the lrgest Cirle possile John use when he shoul hve use. Try to voi unneessry overlp John hs two overlpping irles. Tom voie oth. 5B. PROBLEM in the simplest S of P expression for the logi funtion efine y the Krnugh mp on the right. Get with 9 letters. If it tkes more, o Pro 5A Mp of Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 47 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 93

20 Krnugh Mps Simplifition With 4-Input Mps Simplifition With 4-Input Mps Simplify the funtion John s Solution efine y this mp = + + Tom s Solution = + + Don t forget 4 orners Simplify the funtion John s Solution efine y this mp = Your Solution essentil = + Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 48 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 94 Krnugh Mps Simplifition Simplifition Essentil Terms A term (irle) is essentil if it ontins t lest one tht nnot e irle y ny other irle, of the sme or lrger size.y Your solution, The term, is essentil in tht no other irle (exept smller ones) will over the squre =0., is essentil in tht no other irle (exept smller ones) will over =0., is essentil in tht no other irle (exept smller ones) will over =0 n. One must hve the irles for these essentil terms. Squres Not Covere y Essentil Terms All the squres exept n 0 re overe y the essentil terms. One must look for terms to over these squres. This is the only ple there is hoie. There re three terms tht over one or oth of these squres. Choose the ones to give the simplest expressions. 52. PROBLEM in the simplest Σ of Π expression for the mp ) on the right.. This is often lle n essentil prime implint ) ) ) ) Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 48 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 95

21 Don t Cres in Krnugh Mps Simplifition With 4-Input Mps Don t Cres in Krnugh Mps eiml igit Binry representtion Binry Coe Deimls (BCD) eiml igit Binry representtion 0 0 not use x x x x x x Binry representtion Representing 3-igit numer with 2 its. Digit Digit Digit Mp showing the vlues of for eh squre Mp showing the eiml equivlent of the input its x x x x x x Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 49 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 96 Don t Cres in Krnugh Mps Binry-Coe Deimls Binry-Coe Deimls These re use minly for sening numers to isplys whih people hve to re. Mny yers go they were use to o ommeril rithmeti. The story ws tht onverting eiml frtions to inry use smll errors whih oul umulte n throw off your nk ount. or exmple $0.70 (eiml) = $0.0,,,... (inry) Binry-oe eiml igits use 4 its. The tle shows tht four of the sixteen 4-it omintions re unuse. If one hs iruit whih hs inry-oe eiml inputs there will e four input omintions whih never hppen. If they never hppen, then one oes not hve to worry out the iruits output for these input omintions. These omintions re lle on t re inputs n n e use to simplify the iruit. 52A. PROBLEM Write the yer in inry oe eiml. In 23, you shoul hve 6 its. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 49 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 97

22 Don t Cres in Krnugh Mps Rouning BCD numers Rouning BCD numers Roun 3-igit BCD numers to 2-igit. 783 roun to roun to roun to 790 (ritrry hoie) 797 roun to 8 Prt () of Prolem Detet if BCD igit x x x x x x Mp lotions of BCD igits x x x x 0 x x =igit 5 Digit 5 Ciruit re s show igits 5 () Lest sig igit < 5 - Sen inrement sig to next ig Inrm 0 7 (2) If inrementing mkes ny igit >9 - Cler igit to 0 - Sen inrement sig to next ig 7 0 Inrm Inrm Inrm Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 50 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 98 Don t Cres in Krnugh Mps Rouning BCD numers. Rouning BCD numers. 53. PROBLEM Design the prt (2) of the rouning iruit. A iruit tht will hek if: - the present igit is 9 AND - there is rry in from the previous igit. If so, it will sen rry out to the igit on its left. Minimize the logi using on t res Inrm Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 50 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 99

23 Don t Cres in Krnugh Mps Don t Cres In Krnugh Mps Don t Cres In Krnugh Mps Detet if BCD igit is 5 or more. Use of Don t Cres. The BCD igits > 9 never hppen We on t re out output for them. Mke these outputs on the mp x x x x x x Mp lotions of BCD igits x x x x 0 x x = igit 5 Digit 5 Ciruit my e irle or not s esire. Cirle to minimize logi Here we irle 4 out of 6 s This mkes = Wht s with this? Inputs to use the 6 outputs never hppens, ut if they i: - the 4 irle outputs woul now e, n - the 2 unirle ones woul now e 0. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 5 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. Don t Cres in Krnugh Mps Don t Cres In Krnugh Mps Don t Cres In Krnugh Mps If one hs input omintions tht never hppen, one oes not re wht outputs they generte euse those outputs n never hppen. We put on t res on the mp squres for input omintions tht never hppen. One n irle these on t res or not s onvenient. There is ommon error on the slie. The irle oul e extene to inlue ll of. This woul give the finl eqution s = PROBLEM Tke the hex igits 0,,2,3,4,5,6,7,8,9,A,B,C,D,E,. Plot their lotion on Krnugh mp in the sme wy the BCD igits were plotte. Then esign logi iruit whih will use four its w,x,y,z (efining hex igit s input, n give high output if the igit is ivisle y 3, i.e. it is 3, 6, 9, C or. 55. PROBLEM Design iruit whih will use four its,,, efining BCD igit s input, n gives high output if the igit is ivisle y 3. Utilize the inputs tht nnot hppen, to give on t re outputs, n hene simplify the logi. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 5 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 0

24 Don t Cres in Krnugh Mps Don t Cres In Krnugh Mps Simplifition With Don t Cres Simplify the funtion efine y this mp John s Solution = + + Don t hve to irle ll the. Tom s Solution = + 4 orners not so goo Simplify the funtion efine y this mp John s Solution = Tom s Solution = + + Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 52 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 02 Don t Cres in Krnugh Mps 56. PROBLEM. In the l you esigne one lok of omprtor for two inry numers X=x 2 x x 0 n Y=y 2 y y 0. The lok ompre x i with y i n lso utilize input A i+, B i+, from the omprison one for higher orer its, to give outputs A i n B i A typil lok is shown. We rop the umersome susripts n write A -, B - to tell the A, B inputs from the outputs. The - - in the 5th line of the truth tle inputs mens tht if A,B =,0 then, no mtter wht x n y re, the output is A -,B - = 0. Do not onfuse these with the on t res in the outputs,, whih re the result of input omintions tht never hppen. Complete the Krnugh mp for A - inluing the on t res. Then eue the expression for A - whih shoul hve 4 letters. A 3 =0 B 3 =0 A B Don t Cres In Krnugh Mps x y x 2 y 2 A - B - A 2 B 2 A B x y A - B x y A B x 0 y 0 A0 B 0 A - = if x>y or AB=0 B - = if x<y or AB=0 AB= never hpens y xy AB A 0 x Mp of A - B Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 52 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 03

25 Don t Cres in Krnugh Mps Don t Cres Don t Cres Common mistkes. Cirling on t res when there is no nee. Rememer you irle them only if onvenient. 2. Over irling 3. orgetting wrp-roun. 4. Not enlosing on t res whih woul mke the irle lrger Printe; 03/0/06 Deprtment of Eletronis, Crleton University Slie 53 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 04 Don t Cres in Krnugh Mps Common Mistkes Common Mistkes Top: The four orners woul e etter. There is no nee to inlue the upper two s. Mile: The four orners re one twie. Also the top ovl oul e oule with wrp roun. Bottom: The re ovl oul e wrppe roun n oule. Printe; 03/0/06 Deprtment of Eletronis, Crleton University Comment on Slie 53 Moifie; Jnury 3, 26 John Knight Digitl Ciruits p. 05