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,EEETRANSACTIONSONCIRCUITTHEORY,VOL.CT-18,NO.5,SEPTEMBER1971507Memristor-TheMissingCircuitElementLEON0.CHUA,SENIORMEMBER,IEEEAbstract-Anewtwo-terminalcircuitelement-calledthememrirtor-characterizedbyarelationshipbetweenthechargeq(t)sSt%i(7Jd7andtheflux-linkage(p(t)=J-‘-mvfrjdTisintroducedosthefourthboriccircuitelement.Anelectromagneticfieldinterpretationofthisrelationshipintermsofaquasi-staticexpansionofMaxwell’sequationsispresented.Manycircuit~theoreticpropertiesofmemdstorrarederived.Itisshownthatthiselementexhibiissomepeculiarbehaviordifferentfromthatexhibitedbyresistors,inductors,orcapacitors.ThesepropertiesleadtoanumberofuniqueapplicationswhichcannotberealizedwithRLCnet-worksalone.I+”-3nlAlthoughaphysicalmemristordevicewithoutinternalpowersupplyhasnotyetbeendiscovered,operationallaboratorymodelshavebeenbuiltwiththehelpofactivecircuits.Experimentalresultsorepresentedtodemonstratethepropertiesandpotentialapplicationsofmemristors.(a)I.1NTR00~cnoNI+Y-3THISPAPERpresentsthelogicalandscientificbasisfortheexistenceofanewtwo-terminalcircuitelementcalledthememristor(acontractionformemory(b)resistor)whichhaseveryrighttobeasbasicasthethreeclassicalcircuitelementsalreadyinexistence,namely,theresistor,inductor,andcapacitor.Althoughtheexistenceofamemristorintheformofaphysicaldevicewithoutinternalpowersupplyhasnotyetbeendiscovered,itslaboratoryrealizationintheformofactivecircuitswillbepresentedinSectionII.’Manyinterestingcircuit-theoreticpropertiespossessedbythememristor,themostimportantofwhichisperhapsthepassivitypropertywhichprovidesthecircuit-theoreticbasisforitsphysicalrealizability,willbederivedinSectionIII.Anelectromagneticfieldin-terpretationofthememristorcharacterizationwillbepre-sentedinSectionIVwiththehelpofaquasi-staticexpansionofMaxwell’sequations.Finally,somenovelapplicationsofmemristorswillbepresentedinSectionV.1+”-3I+-3”--(clCd)II.MEMRISTOR-THEFOURTHBASICCIRCUITELEMENTFromthecircuit-theoreticpointofview,thethreebasictwo-terminalcircuitelementsaredefinedintermsofarelationshipbetweentwoofthefourfundamentalcircuitvariables,namely;thecurrenti,thevoltagev,thechargeq,Fig.1.Proposedsymbolformemristoranditsthreebasicrealizations.(a)Memristoranditsq-qcurve.(b)Memristorbasicrealization1:M-Rmutatorterminatedbynonlinearresistor&t.(c)Memristorbasicrealization2:M-LmutatorterminatedbynonlinearinductorC.(d)Memristorbasicrealization3:M-Cmutatorterminatedbynonlinearcapacitore.ManuscriptreceivedNovember25,1970;revisedFebruary12,197l.ThisresearchwassupportedinpartbytheNationalScienceFoundationunderGrantGK2988.TheauthorwaswiththeSchoolofElectricalEngineering,PurdueUniversity,Lafayette,Ind.HeisnowwiththeDepartmentofElectricalEngineeringandComputerSciences,UniversityofCalifornia,Berke-ley,Calif.94720.rInaprivatecommunicationshortlybeforethispaperwentintopress,theauthorlearnedfromProfessorP.Penfield,Jr.,thatheandhiscolleaguesatM.I.T.havealsobeenusingthememristorformodel-ingcertaincharacteristicsofthevaractordiodeandthepartialsuper-conductor.However,aphysicaldevicewhichcorrespondsexactlytoamemristorhasyettobediscovered.andtheflux-linkagecp.Outofthesixpossiblecombinationsofthesefourvariables,fivehaveledtowell-knownrela-tionships[l].Twooftheserelationshipsarealreadygivenbyq(t)=JLwi(T)d7andcp(t)=sf.mD(T)d7.Threeotherrela-tionshipsaregiven,respectively,.bytheaxiomaticdefinitionofthethreeclassicalcircuitelements,namely,theresistor(definedbyarelationshipbetweenvandi),theinductor(definedbyarelationshipbetweencpandi),andthecapacitor(definedbyarelationshipbetweenqandv).Onlyonerela-tionshipremainsundefined,therelationshipbetween9andq.Fromthelogicalaswellasaxiomaticpointsofview,itisnecessaryforthesakeofcompletenesstopostulatetheexistenceofafourthbasictwo-terminalcircuitelementwhichAuthorizedlicenseduselimitedto:CHONGQINGUNIVERSITY.DownloadedonJanuary15,2009at03:26fromIEEEXplore.Restrictionsapply.508IEEETRANSACTIONSONCIRCUITTHEORY,SEPTEMBER1971TABLEICHARACTERIZATIONANDREALIZATIONOFM-R,M-L,ANDM-CMUTATORS=‘PEII-2II-2II2-RANSMISSIONMATRIX::I=[T(P’][JBASH:REALIZATIONSUSINGCONTROLLEDSOURCESSYMBOLANDCHARACTERIZATION’:vi:Y!I.I,'1f)REALIZATION2REALIZATIONI‘2w+gf+R“2ilD---a+Ic(2)+i2+“2(li,dt)ilF--a-+(!%)-i2++“2Uqdt)-P0~R,b)=[10PdVp“I=dtdipiI=-7(q.Vl-RvR,iR)Y-RMUTATORREALIZATIONIREALIZATION2di2VI’-7REALIZATION2REALIZATIONIgq-:IdenticaltoTcR,(p)fcTypeIC-RMUTATOR1REALIZATtON4REALIZATION3REALIZATIONI“I=“2di2iI=-!TDq-yI+(!!k)i2++(VI)-“!M-LMUTATORL1REALIZATION2(q,#-WL,iL)mdi,v,=-dti,=v20PTML2fP”)o[1(IdenticaltoTLR2(p)mfcType2L-RMUTATOAREALIZATION2:f-pq*REALIZATlONIi,=-i2P0k,(P)o([1(I&tical10TLRltp)daTypeIL-RMUTATORlI;REALlZATlON3REALIZATlON4M-CMUTATORREALIZATIONIv,=-i2d”2il=r(tREALIZATION2r.,-ili2+(ipI-D---a“I;+xt/ildt)-I+3pq“2blcp=po1(Idtmticdlt0TCR2(p),faType2C-RWTATOFAuthorizedlicenseduselimitedto:CHONGQINGUNIVERSITY.DownloadedonJanuary15,2009at03:26fromIEEEXplore.Restrictionsapply.CHUA:MEMRISTOR-MISSINGCIRCUITELEMENT509+Ecc014l?&l50)R4(IOOK)RJ9lO)(2~4236)LIfI11h-b--&4-1AR,1SO-IOAJRLq-III’*-kck-+1“2--Port2Fig.2.Practicalactivecircuitrealizationoftype-l