Exclusive OR Gate (XOR Gate) - Electronics Hub

Truth Table of 3 Input XOR Gate Home Logicgates ExclusiveORGate(XORGate) April15,2021 By RaviTeja WewilllearnaboutExORGateinthistutorial.ExORgateistheExclusiveORgate.ThisisnotfrequentlyusedasinclusiveORgate,whichisnothingbuttheregularORgate.XORgatehasitsownsignificance.Wewilllearnthesymbol,truthtableofXORGate,implementationusingothergates(AND,OR,NAND,NOR),popularXORICsandsomeimportantapplicationsofExclusiveORGate(XORGate). Outline IntroductiontoXORGateXORSymbolXORTruthTableXORBooleanExpressionXORGateEquivalentCircuitXORGateusingBasicLogicGatesWithNORGatesWithNANDGatesUsingAND,ORandNANDGatesPulsedOperationofXORGate3-InputEx-ORGate3-InputEx-ORGateLogicSymbolTruthTableof3InputXORGateCommonlyAvailableTTLandCMOSLogicEx-ORgateIC’s7486Quad2-InputExclusive-ORGateICApplicationsof Ex-ORGateUsedinAdder(Addition)Pseudo-RandomNumberGenerationCorrelationandSequenceDetectionConclusion IntroductiontoXORGate ExclusiveORGate,alsoknownasEXORGateorXORGate,isanimportantdigitallogicgate,whichimplementsanexclusiveorlogici.e.,theoutputisHIGHifandonlyifoneoftheinputsisHIGH.IfboththeinputsareLOWorHIGH,thentheoutputisLOW. XORSymbol Therearemultiplestandardsfordefininganelectroniccomponent.Generally,wefollowtheIEEE(InstituteofElectricalandElectronicsEngineers)andIEC(InternationalElectrotechnicalCommission)standards.TheXORlogicsymbolinIEEEandIECstandardsisshownbelow. TheBooleanexpressionforXORgatecannotdetermineddirectlylikeAND,ORgates.AsitisaHybridgate,theBooleanexpressionofoutputofXORgateisgivenbyacombiningofMultiplication,Additionandinvertingofinputs.WehavetousetheKarnaughMapsorK–MapsalongwiththetruthtabletoderivetheBooleanExpressionofXORgate. XORTruthTable ThetruthtableofXORgateisshowninthebelowtable.Fromthis,itisclearthatXORgateproducesalogicLOWi.e.,Logic‘0’atitsoutput,whenboththeinputsaresame(bothmaybeLOWorbothmaybeHIGH). lowlogicthatislogic‘0’,atitsoutput,whenWhenthetwoinputsaredifferentitproducesalogichighvaluei.e.,logic‘1’atitsoutput. Inputs Output A B Q 0 0 0 0 1 1 1 0 1 1 1 0 TheK-maprepresentationoftheabovetruthtableoftheXORGateisshownbelow. XORBooleanExpression UsingtheabovetruthtableandcorrespondingK-Map,wecannowderivetheBooleanExpressionforXORGate.IfAandBaretheinputsoftheXORgate,itsoutputisgivenas:  AB+AB  TheXORoutputisrepresentedas:  A⊕B  Itcanalsobewrittenas:  (A+B)(A+B)  ByapplyingDeMorgan’slaw,theaboveBooleanexpressioncanalsobewrittenas:  (A+B)(AB)  XORGateEquivalentCircuit TheEX-ORgateisdefinedasahybridlogicgatewith2inputstoperformtheExclusiveDisjunctionoperation.Fromtheabovecalculations,themainBooleanExpressionofXORgateis:  AB+AB  So,theXORcircuitwith2inputsisdesignedusingAND,ORandNOTgatesasshownbelow. Theoutputof2inputXORgateisHIGHonlywhenoneofitsinputsareHIGH.Ifboththeinputsaresame,thentheoutputisLOW. XORGateusingBasicLogicGates Ifaspecificgateisnotavailabledirectly,wecandesigntheXORGatebyusingmultiplegates.AnEX-ORgatecanbedesignedbyusingbasiclogicgateslikeNANDgateandNORgateastheyareuniversalgates. WithNORGates LetusnowseehowwecanimplementtheXORGateusingNORGates.Forthis,wehavetore-writetheaboveXORBooleanEquation.  Q=AB+AB   Q=AB+AB+AA+BB   Q=(A+B)(A+B)   Q=(A+B)(A+B)=(A’+B’)(A+B)  Takingcomplementonbothsides,weget:  Q=((A’+B’)(A+B))  UsingdeMorgan’sLaw,weget:  Q=(A’+B’)+(A+B)=(A’+B’)’+(A+B)’  Onceagaintakingcomplementonbothsides,weget:  Q=((A’+B’)’+(A+B)’)=((A’+B’)’+(A+B)’)’  ThisequationlookslikeitcanbeimplementedusingNORGates.WeneedtotallyfiveNORgates(twoforinvertingAandB,oneforNORofAandB,oneforNORofA’andB’andthefinalonetoobtaintheaboveequation).ThefollowingimageshowstheXORGateimplementedusingNORGates. WithNANDGates LetusnowseehowwecanimplementtheXORGateusingNANDGates.Forthis,wehavetore-writetheaboveXORBooleanEquation.  Q=AB+AB   Q=AB+AB+AA+BB   Q=(A+B)(A+B)   Q=(A+B)(A+B)=(A+B)(A’+B’)  ApplyingdeMorgan’sLawonthesecondtermintheaboveequation,weget:  Q=(A+B)(AB)  NowweneedtoimplementthiscircuitusingNANDgates.  Q=A(AB)+B(AB)=A(AB)’+B(AB)’  Takingcomplementonbothsides,weget:  Q=(A(AB)+B(AB))’=(A(AB)’+B(AB)’)   Q=(A(AB))’(B(AB))’=(A(AB)’)(B(AB)’)  Finally,onceagain,applycomplementonbothsides.  Q=(A(AB)’)’(B(AB)’)’=((A(AB)’)’(B(AB)’)’)’  ThisequationlookslikeitcanbeimplementedusingNANDGates.WeneedtotallyourNANDgates.ThefollowingimageshowstheXORGateimplementedusingNANDGates. UsingAND,ORandNANDGates LetusnowseehowwecanimplementtheXORGateusingNAND,ANDandORGates.Forthis,wehavetore-writetheaboveXORBooleanEquation.  Q=AB+AB   Q=AB+AB+AA+BB   Q=(A+B)(A+B)   Q=(A+B)(A+B)=(A+B)(A’+B’)  ApplyingdeMorgan’sLawonthesecondtermintheaboveequation,weget:  Q=(A+B)(AB)  ThefirsttermintheaboveequationrequiresanORGate,thesecondtermneedsaNANDgateandthefinalequationcanbeobtainedusinganANDgate. PulsedOperationofXORGate Thepulsedoperationof2inputXORgateisshownbelow. 3-InputEx-ORGate Insomecases,weneedtohaveXORGateswithmorethan2inputs.Morethan2inputXORfunctioniscalledas“Oddfunction”or“Modulo-2sum”.TheBooleanfunctionforthe3-inputXORgateis:  Q=A⊕B⊕C= ABC +ABC +ABC+ABC  Thetruthtableandlogicsymbolfor3-inputXORgateisgivenbelow. 3-InputEx-ORGateLogicSymbol TruthTableof3InputXORGate For3-inputXORgates,wecanhavetheHIGHinputwhenoddnumbersofinputsareatHIGHlevel.Sothe3-inputORgateiscalledas“OddfunctionedORgate”. Inputs Output A B C Q 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 CommonlyAvailableTTLandCMOSLogicEx-ORgateIC’s FollowingisalistofsomeofthecommonlyavailableXORICs. ICNumber Description 4030 Quad2-InputXORGates 4070 Quad2-InputXORGates 7486 Quad2-InputXORGates 74LS86 Quad2-InputXORGates 741G86 Single2-inputEx-ORGate 74136 Quad2-InputXORGateswithOpenCollectorOutputs 74386 Quad2-InputXORGates Ofthese,themostpopularTTLLogicbasedEX-ORGateICisthe74LS86,whichisaQuad2-inputXORIC.ComingtotheCMOSLogicbasedXORGateIC,theCD4030Quad2-inputXORICisapopularchoice. 7486Quad2-InputExclusive-ORGateIC IC7486isaquad2-inputXORgatei.e.,itcontainsfour2-inputXORGatesinasinglepackage.ThepindiagramandpindescriptionoftheICisshownbelow. PinNumber Description 1 Gate1InputA 2 Gate1InputB 3 Gate1OutputY 4 Gate2InputA 5 Gate2InputB 6 Gate2OutputY 7 Ground 8 Gate3OutputY 9 Gate3InputA 10 Gate3InputB 11 Gate4OutputY 12 Gate4InputA 13 Gate4InputB 14 PositiveSupply Applicationsof Ex-ORGate XORLogicGateisusedinmanyapplications.Someofthemareexplainedbelow. UsedinAdder(Addition) Wecandesignsinglebitadder(alsoknownasHalfAdders),whichwilladdtwobitsandproduceasinglebitoutput.ThesinglebitAdderdesignedbyusingXORgateisshownbelow. Forexample,ifweaddtwobits‘1’and‘1’inbinaryaddition,wegettheanswer’10’andindecimaladditionmethodweget2.ThemainprincipleofhalfaddersisthatthetrailingsumisachievedbytheoutputofXORgateandthecarrybitiscalculatedbyANDgate. Wecancascademanysinglebitaddercircuitstoformn-bitaddercircuit,tocalculatethesumoflongerbinarynumbers. Pseudo-RandomNumberGeneration LinearshiftregistersarealsocalledasPseudoRandomNumberGenerators(PNRs).Inordertogeneratetherandomnumbers,wearrangetheXORlogicgateinaspecificorderbyformingalinearfeedbackshiftregister. CorrelationandSequenceDetection XORgateiscapableofproducingalowlevelinputi.e.,0,whenallofitsinputsareHIGHorLOW.Whenwearesearchingforaparticularbitsequenceinalongdatasequence,weuseXORgatestofindtherequireddatabitsequence. Theaccuracyoffindingtherequiredstringofdatabitsinthetargetsequenceisdeterminedbycalculatingthenumberof0’sobtained.InmanycommunicationdevicessuchasdecodersandCDMAreceivers,weuseco-relaters,whichareusedtoextracttheparityofaspecificPseudoRandomNumbersequenceinagroupofPRNsequence. Conclusion AcompletetutorialonExclusiveORGate(XORGate).Youlearnedthesymbol,truthtableandBooleanExpressionofanXORGate,implementationofXORGateusingNORandNANDgates,a3-inputXORGate,itssymbol,truthtableandBooleanExpression,somecommonandpopularXORICsandalsosomeimportantapplicationsofXORGate. 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