Karnaugh Maps (K-Map) | 1-6 Variables Simplification ...

3 variable K-map can be in both forms. Note the combination of two variables in either form is written in Gray code. So the min terms will not be in a decimal ... KarnaughMaps(K-Map),1-6VariablesSimplification&Examples TableofContents WhatisKarnaughMap(K-Map)?GreyCodeBCDtoGrayCodeusingK-Map RulesofMinimizationinK-Map 2VariableK-MapExampleof2VariableK-Map 3VariableK-MapExampleof3VariableK-Map 4-variableK-MapExampleof4VariableK-Map 5&6VariableKarnaughMaps5VariablesK-MapExampleof5VariablesK-Map 6-VariableKarnaughMapExampleof6VariableK-Mapmaping WhatisKarnaughMap(K-Map)? KarnaughmaporK-mapisamapofafunctionusedinatechniqueusedforminimizationorsimplificationofaBooleanexpression.Itresultsinlessnumberoflogicgatesandinputstobeusedduringthefabrication. BooleansexpressioncanbesimplifiedusingBooleanalgebraictheoremsbuttherearenospecificrulestomakethemostsimplifiedexpression.However,K-mapcaneasilyminimizethetermsofaBooleanfunction. Unlikeanalgebraicmethod,K-mapisapictorialmethodanditdoesnotneedanyBooleanalgebraictheorems. K-mapisbasicallyadiagrammadeupofsquares.Eachofthesesquaresrepresentsamin-termofthevariables.Ifn=numberofvariablesthenthenumberofsquaresinitsK-mapwillbe2n.K-mapismadeusingthetruthtable.Infact,itisaspecialformofthetruthtablethatisfoldeduponitselflikeasphere.Everytwoadjacentsquaresofthek-maphaveadifferenceof1-bitincludingthecorners. KarnaughmapcanproduceSumofproduct(SOP)orproductofSum(POS)expressionconsideringwhichofthetwo(0,1)outputsarebeinggroupedinit.Thegroupingof0’sresultinProductofSumexpression&thegroupingof1’sresultinSumofProductexpression.TheexpressionproducedbyK-mapmaybethemostsimplifiedexpressionbutnotunique.Therecanbemorethan1simplifiedexpressionforasinglefunctionbuttheyallperformthesame. GreyCode InGraycode,everytwoconsecutivenumberhasadifferenceof1-bit.AsthesquaresinK-mapalsodiffersfromitsadjacentsquareby1-bitwhichiswhythevariablesinK-maparewritteningreycode.ThegraycodeensuresthateachcellofK-mapisin1-bitdifferencewitheachother. Youmayalsoread:CounterandTypesofElectronicCounters BCDtoGrayCodeusingK-Map ThetableforBCDtoGraycodeisgivenbelow. RulesofMinimizationinK-Map Whilegrouping,youcanmakegroupsof2nnumberwheren=0,1,2,3… Youcaneithermakegroupsof1’sor0’sbutnotboth. Groupingof1’sleadtoSumofProductformandGroupingof0’sleadtoProductofSumform. Whilegrouping,thegroupsof1’sshouldnotcontainany0andthegroupof0’sshouldnotcontainany1. Thefunctionoutputfor0’sgroupingshouldbecomplementedasF’. Groupscanbemadeverticallyandhorizontallybutnotdiagonally. Groupsmadeshouldbeaslargeaspossibleeveniftheyoverlap. Alltheliketermshouldbeinagroupeveniftheyoverlap. Uppermost&lowermostsquarescanbemadeintoagrouptogetherastheyareadjacent(1-bitdifference).Samegoesforthecornersquares. EachgrouprepresentsatermintheBooleanexpression.Largerthegroup,smallerandsimpletheterm. Theproductofthoseliteralsthatremainsunchangedinasinglegroupmakesthetermoftheexpression. Don’tcare“x”shouldalsobeincludedwhilegroupingtomakealargerpossiblegroup. Karnaughmapof2to4variablesisveryeasy.However,5and6variableK-mapisalittlebitcomplex.Wewilldiscussonebyoneindetails. Youmayalsoread:DigitalFlip-Flops–SR,D,JKandTFlipFlops 2VariableK-Map 2variableshave2n=22=4minterms.Thereforethereare4cells(squares)in2variableK-mapforeachminterm. ConsidervariableA&Bastwovariables.TherowsofthecolumnswillberepresentedbyvariableB. Thesquarefacingthecombinationofthevariablerepresentsthatmintermasshowninfigbelow. Groupingin2variablesK-mapiseasyastherearefewsquares. Exampleof2VariableK-Map FunctionF(A,B) F=∑(m0,m1,m2)=A̅B̅+A̅B+AB̅ K-mapfromTruthtable Wemade2groupsof1’s.eachgroupcontains2minterms. Inthefirstgroup,variableAischanging&Bremainsunchanged.SothefirsttermoftheoutputexpressionwillbeB̅(becauseB=0inthisgroup). Inthe2ndgroup,VariableBischangingandvariableAremainsunchanged.SothesecondtermwillbeoftheoutputexpressionwillbeA̅(becauseA=0inthisgroup). Nowthesimplifiesexpressionwillbethesumofthesetwotermsasgivenbelow, F=A̅+B̅ Comparethisexpressionwiththeoriginalexpressionofthefunction,thisexpressiononlyusesonegateduringitsimplementation. Youmayalsoread:RippleCarryAndCarryLookAheadAdder 3VariableK-Map 3variablesmake2n=23=8minterms,sotheKarnaughmapof3variableswillhave8squares(cells)asshowninthefiguregivenbelow. 3variableK-mapcanbeinbothforms.NotethecombinationoftwovariablesineitherformiswritteninGraycode.Sothemintermswillnotbeinadecimalorder. Theuppermost&lowermostcellsareadjacentinthefirstformofK-map,theleftmostandrightmostcellsarealsoadjacentinthesecondformofK-map.Sotheycanbemadeintogroups. Someexamplesofgrouping: Youcanmakegroupsof2,4&8cellshavingsame1sor0s. Noticethegroupsoftheuppermost&lowermostcells.Theyareadjacentasthereisonlyone-bitdifference.Thatiswhytheycanbegroupedtogether.Don’tmakeunnecessarygroups.All1sor0sshouldbegrouped,notallpossiblegroupsof1sor0sshouldbemade. Exampleof3VariableK-Map F(A,B,C)=∑(m0,m1,m2,m4,m5,m6) Thisexampleshowsthatyoucanmakethegroupsoverlapeachothertomakethemaslargeaspossibleandcoverallthe1s. Inthisfirstgroup(m0,m2,m6,m4),A&Barechangingsowewilleliminateit.However,Cremainsunchangedinthisgroup.SothetermthisgroupproducewillbeC̅(becauseC=0inthisgroup). Inthe2ndgroup(m0,m1,m4,m5),AandCarechangingsoitwillbeeliminatedfromtheterm.However,Bremainsunchangedinthisgroup.SothetermthisgroupproducewillbeB̅(becauseB=0inthisgroup). Thesumofthesetwotermswillmakethesimplifiedexpressionofthefunctionasgivenbelow. F=B̅+C̅ Anotherexampleofgroupingof2isgivenbelow.Itshowshowthecornermintermsaregrouped. Inthefirstgroup(m0,m4),Aischanging.B&Cremainsunchanged.SothetermwillbeB̅C̅(B=0,C=0inthisgroup). In2ndgroup(m3,m7),Aischanging.B&Cremainsunchanged.BCwillbethetermbecauseB=1,C=1inthisgroup. SoThisK-mapleadstotheexpression F=B̅C̅+BC Thesetwoexamplesshowthatagroupof4cellsgiveatermof1literalandagroupof2cellsgivesatermof2literalsandagroupof1cellgivesatermof3literals.Sothelargerthegroup,thesmallerandsimplethetermgets. Youmayalsoread:RingCounter&JohnsonCounter–Construction&Operation 4-variableK-Map 4variableshave2n=24=16minterms.Soa4-variablek-mapwillhave16cellsasshowninthefiguregivenbelow. Eachcell(minterm)representthevariablesinfrontofthecorrespondingrow&column. Thevariablesareingraycode(1-bitchange).Thefourcellsofthecornerareadjacenttoeachotherasthereisa1-bitdifferenceeveniftheyarenottouchingphysically.Sotheycanbegroupedtogether. Someexampleofgroupingin4-variablek-mapisgivenbelow: Asyoucanseeintheexampleabovethe4cornercellsmakeagroup.Inthesecondexample,leftmostcolumnscanbegroupedwithrightmostcolumnanduppermostrowwiththelowermostrow. Thesegroupsshouldbeaslargeaspossiblecontaining1,2,4,8or16cells.Thetermsoftheexpressiondependonthesegroups.Ifthegroupcontains: Onesquare,thenitwillgiveatermof4literals Twosquares,thenitwillgiveatermof3literals Foursquares,thenitwillgiveatermof2literals Eightsquare,thenitwillgiveatermof1literal Sixteensquarewhichwillcoverthewhole4-variablek-mapwhichmeansconstant1output. Exampleof4VariableK-Map F(A,B,C,D)=∑(m0,m1,m2,m4,m5,m6,m8,m9,m12,m13,m14) Firstofall,trytomakethebiggestpossiblegroupsasshowninthisexample.Corner1scanalsobemadeintoagroupof4.Theremaininglast1shouldbecombinedwiththepre-madegrouptomakeabigoverlappinggroup. Groupof8willgiveatermof1literalthatremainsunchangedi.e.C̅ Cornergroupof4willgivetermwith2literalsthatremainunchangedi.e.B̅D̅ Thelastgroupof4willgiveA̅D̅becausetheyremainedunchangedinthegroup. Sotheexpressionwillbe F= C̅+B̅D̅+A̅D̅ Youmayalsoread:DigitalAsynchronousCounter(RippleCounter)–Types,Working&Application 5&6VariableKarnaughMaps K-MapisusedforminimizationorsimplificationofaBooleanexpression.2-4variableK-mapsareeasytohandle.However,therealchallengeis5and6variableK-maps. Visualizationof5&6variableK-mapisabitdifficult.Whenthenumberofvariablesincreases,thenumberofthesquare(cells)increases.AnddrawingtheK-mapbecomesabitcomplicatedbecauseofdrawingtheadjacentcells. 5VariablesK-Map 5variableshave32minterms,whichmean5variablekarnaughmaphas32squares(cells). A5-variableK-mapismadeusingtwo4-variableK-maps.Consider5variablesA,B,C,D,E.their5variableK-mapisgivenbelow. Theseboth4-variableKarnaughmaptogetherrepresentsa5-variableK-mapforvariableA,B,C,D,E.NoticevariableAoverthetopofeach4-variableK-map.ForA=0,theleftK-mapisselectedandrightmapforA=1. Eachcorrespondingsquares(cells)ofthese24-variableK-mapsareadjacent.VisualizethesebothK-mapsontopofeachother.m0isadjacenttom16,soism1tom17soonuntilthelastsquare. Therule(method)ofgroupingissameforeachofthe4-variablek-maps.However,youalsoneedtocheckthecorrespondingcellsinbothK-mapsaswell.Afewexampleofgroupingisgivenbelow. Intheseexamples,eachgroupisdifferentiatedusingdifferentcolors. Exampleof5VariablesK-Map F(A,B,C,D,E) = ∑ (m0,m2,m5,m7,m8,m10,m16,m21,m23,m24,m27,m31) Thisisthe5-variablek-mapforthefunctiongivenabove.TherearefourgroupsmadeinthisK-map.Eachgrouphasadifferentcolortodifferentiatebetweenthem. Theredcolorgroupisagroupof4mintermsmadebetweenboth4-variablek-mapsbecausetheyareadjacentcellsanditoverlapsthegreengroup. Theyellowgroupisalsoagroupof4mintermsmadebetweenadjacentcellsofthe4-variablek-maps. Thegreengroupisagroupof4mintermsmadeintheleft4-variablek-map.Thebluegroupisof2min-termsmadeintheright4-variablek-mapbecausetherearenocommonadjacentcellsintheotherk-map. Greencolorgroupof4mintermwillproducethetermA̅C̅E̅.Theindividual4-variableK-mapwillproduceC̅E̅astheyarenotchanginginthegroupbutvariableAshouldalsobetakenintoaccountbecausethisindividual4-variablek-mapisbeingrepresentedbyA̅. TheredcolorgroupwillproduceC̅D̅E̅.ThisgroupismadebetweenbothK-mapswhichmeansvariableAchangesandinindividualK-map,Bchangessothesebothvariableswillbeeliminatedfromtheterm.OnlyC̅D̅E̅remainsunchangedinthisgroup. TheyellowgroupwillproduceB̅CEbecausetheseliteralsarenotchanginginthisgroup. Bluegroupof2mintermswillproducethetermABDEastheyremainunchangedinthisgroup. Thesimplifiedexpressionwillbethesumofthese4terms,whichisgivenbelow: F=A̅C̅E̅+C̅D̅E̅+B̅CE+ABDE Youmayalsoread:DigitalSynchronousCounter–Types,Working&Applications 6-VariableKarnaughMap 6-variablek-mapisacomplexk-mapwhichcanbedrawn.Visualizing6-variablek-mapisalittlebittricky. 6variablesmake64minterms,thismeansthatthek-mapof6variableswillhave64cells.Itsgeometrybecomesdifficulttodrawasthesecellsareadjacenttoeachotherinalldirectionin3-dimensionsi.e.acellisadjacenttoupper,lower,left,right,frontandbackcellsatthesametime.wewilldrawitlike5-variablek-mapasshowninthefigurebelow. The6-variablek-mapismadefrom4-variable4k-maps.AsyoucanseevariableAontheleftsideselect2k-mapsrow-wisebetweenthese4k-maps.A=0fortheuppertwoK-mapsandA=1forthelowertwoK-maps.VariableBontopoftheseK-mapsselect2k-mapscolumn-wise.B=0forleft2K-mapsandB=1forright2K-maps. Imaginethese4-variableK-mapsasasinglesquare,thesek-mapsareadjacenttoeachotherhorizontallyandverticallybutnotdiagonallybecausethesecellshave1-bitdifference.Thegroupsbetweenthesek-mapsshouldbemadeasdonein5-variableK-mapbutyoucannotmakegroupsbetweendiagonalk-maps. Someexamplesofgroupingin6-variableK-maparegivenbelow. Groupof16min-termsbetween4k-mapsastheyarealladjacent.Visualizethesek-mapsontopofeachother. Inthisexample,thereare5groupsof4min-terms.Noticethemin-termsinthediagonalK-maps,theymakeaseparategroupbecausetheseK-mapsarenotadjacent. Exampleof6VariableK-Mapmaping F = ∑(m0,m2,m8,m9,m10,m12,m13,m16,m18,m24,m25,m26,m29,m31,m32,m34,m35,m39,m40,m42,m43,m47,m48,m50,m56,m58,m61,m63) Its6-variableK-mapisgivenbelow: Thereare5groupsinthisK-mapeachcoloreddifferent. Greengroupismadeof16min-termsbetweenall4individualK-maps.Inthisgroup,ABkeepschangingsotheywillbeeliminatedfromtheterm.C&Earealsochangingsotheywillbeeliminatedfromthetermtoo.SothetermwillbecomeD̅F̅becausetheyremainunchangedthroughoutthegroup. Redgroupismadeof4min-terms.Inthisgroup,Bischangingsoitwillbeeliminated.Disalsochanging.SotheonlyremainingunchangedliteralswillmakethetermwhichisA̅CE̅F. Bluegroupisalsomadeof4min-terms.TheonlychangingvariablesareDFthroughoutthisgroupsotheywillbeeliminatedfromtheterm.Thenon-changedliteralinthisgroupareA̅B̅CE̅whichwillbethetermproducedbythisgroup. TheYellowgroupisalsoagroupof4min-termsandthechangingvariablesinthisgroupareAE.TheliteralthatremainsunchangedareBCDFinthisgroup. Theblackgroupisof4min-termstoo.ThisgroupproducesthetermAB̅EFbecausetheyaretheunchangedliteralsinthisgroup. Thesimplifiedexpressionofthefunctionwillbethesumofthese5termsfromthesegroups.Theexpressionisgivenbelow: F=D̅F̅+A̅CE̅F+A̅B̅CE̅+BCDF+AB̅EF Youmayalsoread: DigitalLogicNOTGate–DigitalInverterLogicGate DigitalLogicORGate DigitalLogicANDGate Exclusive-NOR(XNOR)DigitalLogicGate DigitalLogicNORGate DigitalLogicNANDGate TagsK-MapKarnaughMaps ElectricalTechnology011minutesread ShowFullArticle Facebook Twitter LinkedIn Tumblr Pinterest Reddit VKontakte Skype Messenger Messenger WhatsApp Telegram ShareviaEmail Print ElectricalTechnology AllaboutElectrical&ElectronicsEngineering&Technology.Follow ElectricalTechnologyonFacebook,Twitter,Instagram,Pinterest,YouTube,&LinkedintogetthelatestupdatesorsubscribeHeretogetlatestEngineeringArticlesinyourmailbox. 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