数字电路英文版第五单元

CHAPTER5BINATIONALLOGICCHAPTER5Dualsymbol(对偶符号)Node(节点)Signaltracing(信号跟踪)Universalgate(万能门)Combinationallogic(组合逻辑)Sequentiallogic(时序逻辑)KAYTERMDualsymbolsThenegative-ANDisthedualsymbolfortheNORgate,andthenegative-ORisthedualsymbolfottheNANDgate.NodeAmonconnectionpointinacircuitinwhichagateoutputisconnectedtooneormoregateinputs.SignaltracingAtroubleshootingtechniqueinwhichwaveformsareobservedinastep-by-stepmannerbeginningattheinputandworkingtowardtheoutputorviceversa.Ateachpointtheobservedwaveformisparedwiththecorrectsignalforthatpoint.UniversalgateEitheraNANDgateoraNORgate.Thetermuniversalreferstothepropertyofagatethatpermitsanylogicfunctiontobeimplementedbythatgateorbyabinationofgatesofthatkind.§5.1BASICBINATIONALLOGICCIRCUITSInChapter4,youlearnedthatSOPexpressionsareimplementedwithanANDgateforeachproducttermandoneORgateforsummingalloftheproductterms.ThisSOPimplementationiscalledAND—ORlogicandisthebasicformforrealizingstandardBooleanfunctions.2.Inthissection,theAND—ORandtheAND-OR-Invertareexamined;andtheexclusive-ORandexclusive-NORgates,whichareactuallyaformofAND-ORlogic,arecovered.3.AND-ORLogic(SOP)ABDCABXCD&&≥1ABCDXX=AB+CD4.Fig.5-1InputOutputABCDABCDX0000000000100000100000011011010000001010000110000011101110000001001000101000010110111100101110110111101011111111TABLE5-1TruthtablefortheAND-ORlogicinFig.5-15.X=AB+CDAND-OR-InvertLogic(POS)ABDCABXCD&&≥1ABCDXX=AB+CD=(A+B)(C+D)6.ABXExclusive-ORLogicABXX=AB+AB=A◯BABX=17.ABABInputOutputABX0000111011108.X=AB+AB=A◯BABXExclusive-NORLogicXORABXX=AB+ABX=AB+ABABAB9.ABXX=1X=AB+AB=AB+AB=A◯B10.§5.2IMPLEMENTINGBINATIONALLOGICInthissection,examplesareusedtoillustratehowtoimplementalogiccircuitfromaBooleanexpressionoratruthtable.MinimizationofalogiccircuitusingthemethodscoveredinChapter4isalsodiscussed.11.FromaBooleanExpressiontoaLogicCircuitLet’sexaminethefollowingBooleanexpression:X=AB+CDEANDORX=AB+CDEABCDE12.Asanotherexample,let’simplementthefollowingexpression:X=AB(CD+EF)ANDNOTORANDABCDEFCDEFDCD+EFX=AB(CD+EF)13.(a)X=AB(CD+EF)=ABCD+ABEFABCDABEFDABCEFX=ABCD+ABEF(Sum-of-productsimplementationofcircuitinpart(a).)14.FromaTruthTabletoaLogicCircuitInputOutputABCX00000010010001111001101011001110ABCABCX=ABC+ABC15.X=ABC+ABCXABCABCAABBCC16.§5.3THEUNIVERSALPROPERTYOFNANDANDNORGATESUptothispoint,binationalcircuitsimplementedwithANDgates,ORgates,andinvertershavebeenstudied.Inthissection,theuniversalpropertyoftheNANDgateandtheNORgateisdiscussed.17.TheuniversalityoftheNANDgatemeansthatitcanbeusedasaninverterandthatbinationsofNANDgatescanbeusedtoimplementtheAND,OR,andNORoperations.Similarly,theNORgatecanbeusedtoimplementtheinverter,AND,OR,andNANDoperation.18.TheNANDGateasaUniversalLogicElementBBAAAAAABAB=ABABA(a)ANAND

gateusedasaninverter(b)TwoNAND

gateusedasanANDgate19.AB=A+BAABBABA+B(c)ThreeNAND

gateusedasanORgateAABBABA+BA+BAB=A+B(d)FourNAND

gateusedasanNORgateG1G2G3G1G2G3G420.TheNORGateasaUniversalLogicElementBBAAAAAA+BA+BA+BA(a)ANOR

gateusedasaninverter(b)TwoNOR

gateusedasanORgate21.A+B=ABAABBAB(c)ThreeNOR

gateusedasanANDgateABABABAB(d)FourNOR

gateusedasanNANDgateABABABG1G2G3G1G2G3G422.§5.4BINATIONALLOGICUSINGNANDANDNORGATESInthissection,youwillseehowNANDandNORgatescanbeusedtoimplementalogicfunction.RecallfromChapter3thattheNANDgatealsoexhibitsanequivalentoperationcalledthenegative-ORandthattheNORgateexhibitsanequivalentoperationcalledthenegative-AND.23.Youwillseehowtheuseoftheappropriatesymbolstorepresenttheequivalentoperationsmakes“reading”alogicdiagrameasier.24.NANDLogicAB=A+BX=(AB)(CD)=(A+B)(C+D)=(A+B)+(C+D)=AB+CD=AB+CDBAABCDCDX=AB+CDNANDnegative-OR25.BACDX=AB+CDG2actsasANDG1actsasORG3actsasANDBACDBACDAB+CDAB+CDBubblescancelBubblescancel26.NANDLogicDiagramsAABBCCDDEFEF(AB+C)D+EFXX=ABCDEF=ABCD+EF=(AB+C)D+EF=(AB+C)D+EFABAB+C(AB+C)DEFABABCABCDEFBubblescancelsbarBubblesaddsbartoCBubblescancelsbarORANDORAND27.NORLogicA+B=ABX=A+B+C+D=(A+B)(C+D)=(A+B)(C+D)C+DNORnegative-ANDCDAB28.A+BBACDX=(A+B)(C+D)G2actsasORG1actsasANDG3actsasORBACDX=(A+B)(C+D)BubblescancelBubblescancel29.A+B=ABNORLogicDiagramsAABBCCDDEFEF[(A+B)C+D](E+F)X=A+B+C+D+E+F=(A+B+C+D)(E+F)=((A+B)C+D)(E+F)=((A+B)C+D)(E+F)A+B(A+B)C(A+B)C+DE+FA+BA+B+CA+B+C+DE+FBubblescancelsbarBubblesaddsbartoCBubblescancelsbarANDORANDOR30.§5.5OPERATIONWITHPULSEWAVEFORMSSeveralexamplesofgeneralbinationallogicalcircuitswithpulsewaveforminputsareexaminedinthissection.Keepinmindthatthelogicaloperationofeachgateisthesameforpulseinputsasforconstant—levelinputs.31.Theoutputofalogiccircuitatanygiventimedependsontheinputsatthatparticulartime,sotherelationshipofthetime-varyinginputsisofprimaryimportance.32.TheoutputofanANDgateisHIGHonlywhenallinputsareHIGHatthesametime.TheoutputofanORgateisHIGHonlywhenatleastoneofitsinputsisHIGH.TheoutputofaNANDgateisLOWonlywhenallinputsareHIGHatthesametime.TheoutputofanNORgateisLOWonlywhenatleastoneofitsinputsisHIGH.33.EXAMPLE5—11.ShowingtheoutputsofG1,G2,andG3withinputwaveforms,AandB,asindicated.ABG1G2G3X=AB+AB34.FIGURE5--33ABXG2output=ABG3output=AB35.X=AB+ABEXAMPLE5—12.DeterminetheoutputswaveformsXforthelogiccircuitasfollow:ABCDX=Y3+Y4=Y1C+Y2D=(A+B)C+CD=AC+BC+CDY1=A+BY3=Y1CY2=CY4=Y2D36.BACDX37.Y1=A+BY2=CY3=Y1CY4=Y2D=AC+BC+CDACACCDBCUsingK-mapstoEliminateTimingHazardsAL=AAAAALAALBCCACBCCBACBCLL=AC+BCABC00011110011111L=AC+BC+AB(111110)ACBACBCLAB00000AB0000010111111010CD0000(A+C)(B+C)(A+C+D)(A+B+D)ACBCACDACBCACDABDLLChapter5:CombinationalLogicTrue/FalseTheNANDgateisanexampleofbinationallogic.Fig.5-1isanexampleoftheimplementationofAND-OR-INVERTERlogic.ABCDX38.3.Theabbreviationforanexclusive-ORgateisXOR.4.X=ABC+BCDisintheformofasum-of–productsexpression.5.TheK-mapinFig.5-2showsthecorrectimplementationoftheexpression,X=ACD+AB(CD+BC).ABABABABCDCDCDCD000000000000111139.6.NANDgatescannotbeusedtoconstructNORgates.7.NORgatescanbeusedtoconstructANDgates.8.Theexpressions,ABandA+B,areequivalent.9.Theeffectofaninvertedoutputbeingconnectedtotheinvertinginputofanothergateistoeffectivelyeliminateoneoftheinversions,resultinginasingleinversion.40.10.ThelogiccircuitandassociatedwaveformsshowninFig.5-3arecorrect.ABCXABCX41.MultipleChoice11.WhichfigureinFig.5-4representsAND-ORlogic?BACDXABCXACXABCXBa.b.c.d.&&≥142.12.WhichofthefiguresinFig.5-5(a-d)isequivalenttoFig.5-5(e)?ABCDABCDABCDABCDABCDXXXXXa.b.c.d.e.43.13.WhichofthefiguresinFig.5-6(a-d)isequivalenttoFig.5-5(e)?ABABABXXXa.b.d.e.Xc.44.14.WhichofthefollowinglogicexpressionsrepresentsthelogicdiagraminFig.5-7?X=AB+ABb.X=AB+ABc.X=AB+ABd.X=AB+ABABX45.15.WhattypeoflogiccircuitisrepresentedbyFig.5-7?a.XORb.XNORc.XANDd.XNAND46.16.ThelogicexpressionsforinFig.5-8is?X=ABC+ACDb.X=(AB)(ACCD)c.X=(AB)(AC+CD)d.X=ABC(CBD)ABCDX47.17.TheexpendedexpressionsforinFig.5-8is?X=A(BC+CD)X=ACD+ABCD+ABCc.X=ABC+BCDd.X=AABCD+AABCD+ABCD48.18.UseBooleanalgebratodeterminethesimplestoutputequationforFig.5-8.(note:Theinstructormayrequireyoutoshowyourworkforthisproblem.)X=ABCDX=A(BC+CD)c.X=ABC+BCDd.X=A(BCD)49.19.UseK-maptodeterminethesimplestoutputequationforFig.5-8.(note:Theinstructormayrequireyoutoshowyourworkforthisproblem.)X=ABCDX=A(BC+CD)c.X=ABC+BCDd.X=A(BCD)50.20.WhichcircuitinFig.5-9implementstheequation,X=AB+AC+ABC?ABCABCABCABCABCABCBCAXXXXa.b.c.d.51.22.Howmanygates,includinginverters,arerequiredtoimplementtheequation,X=ACD+AB(CD+BC),beforesimplification?5b.9c.7d.323.Howmanygates,includinginverters,arerequiredtoimplementtheequation,X=ACD+AB(CD+BC),aftersimplificationwithalgebra?5b.9c.7d.353.24.Howmanygates,includinginverters,arerequiredtoimplementtheequation,X=ACD+AB(CD+BC),aftersimplificationwithaK-map?5b.9c.7d.354.25.TheNANDgateisreferredtoasa“universal”gate,becauseit.Canbefoundinalmostalldigitalcircuits.Canbeusedtobuildalltheothertypesofgates.Isusedinallthecountriesoftheworld.Wasthefirstgatetobeintegrated.55.26.WhichofthefiguresinFig.5-11(a-d)representstheNANDimplementationofaNORgate?a.c.b.d.56.27.WhichofthefiguresinFig.5-11(a-d)representstheNANDimplementationofanORgate?28.WhichofthefiguresinFig.5-11(a-d)representstheNANDimplementationofanINVERTERgate?57.29.TherelationshipbetweenaNANDgateandanegative-ORgateisexpressedby=.AB=A+Bb.A+B=A+Bc.AB=A+Bd.AB=A+B58.30.WhichofthefiguresinFig.5-12(a-d)isthelogicalequivalentofFig.5-12(e)?a.b.c.d.e.59.31.Whentheinvertedoutputofthegateisconnectedtotheinvertedinputofanothergate,.theinversionscancel.adoubleinversionoccursandthesignalisinverted.oneinversioncancelstheotherandonlyasingleinversionresults.Alloftheabovearecorrect.60.32.WhyaremultipleNANDgatesusedtoreplaceothersinglefunctiongates?Itise