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AppendixAFormulaeThisappendixprovidessomeoftheadvancedequationsusedintheelectronicindustryforfailureidentification,aswellasmiscella-neousinformationaboutblockdiagrams,howandwhentousetheappropriatereliabilitycalculations,andsoon.Specifically,thefollowingitemsarecovered:•TheArrheniusequationandtheEyringTypemodel•TheEyringandReich-Hakimmodel•Equationsforacceleratedfactors•Reliabilityblockdiagrams•Developmentofblockdiagramsfromsystemtounits•EquationsforthecalculationoftheMTBFinseries•EquationsforthecalculationoftheMTTFinparallel•Equationsforthereliabilityandmeanlifeofcomplexsystemswithcyclicalunits•ConversionfactorsforfailureratemeasurementunitsCD-1THEARRHENIUSEQUATIONwhereCisaconstant,EaisactivationenergyineV,Tistemperaturein0K,KisBoltzman’sconstant8.62510-5eV/0KAccelerationequation:wherer1/r2istheaccelerationfactor.Activationenergyequation:Standarddeviationequation:wheret50isthemedianlifeortimeto50percentfailureandt16isthetimeto16percentfailure.THEEYRINGTYPEMODELWhereAandBareconstantsistheactivationenergyineVTisthetemperatein0K1150CARHBTKEeAet16501ntt1221111nTTrrKeEa122111TTKEerraKTEaeRTCCD-2AppendixAKistheBoltzmanconstantineV/0KRHisrelativehumidityin%ECAistheelectricfieldinthecorrosionareaTHEEYRINGANDREICH-HAKIMMODELwhereGisafactorcombiningalltheconstantsEQUATIONSFORACCELERATEDFACTORSTheequationsforaccelerationfactorsassociatedwithtemperatureandvolt-ageacceleratedburn-insusethefollowing.TheArrheniusequationis:Accel.factorwhere:KBoltzman’sConstant,8.662E-5eV/0Kactivationenergy,eVTsstresstemperature,0KTd/operatingtemperature,0KThevoltageaccelerationequationis:Accel.factorwhere:CelectricfieldconstantToxthicknessofgateoxide,AVSstressvoltage,VVDdynamiclifeoperatingvoltage,VexpVDVSTCoxTsTdK11expKA,eB,e1GCAETRHTRHe11501CATKEeBeRHeeAtFormulaeCD-3Thepurposeofburn-inistocreateacceleratedtemperatureandvoltagestressconditionstoscreenoutearlylifefailures,especiallythoserelatedtoMOSoxidedefects.FAILURERATECONVERSIONExample:Ifyouknowthatthefailurerateis10fr/106hr,youcanfindthefailureratein%fr/1,000hrifyoumultiply10by10-1,or1010-11;con-sequently,thefailurerateis1%fr/1,000hr.[1FIT1fr/109hr.;1BIT1fr/1010hr.].Thereadermaybenefitfromthefollowingconversionfactorsforfailureratemeasurementunitsfoundabundantlyintheliterature.CD-4AppendixAEFDABCOutputInputCDABOutputInputBCAOutputInputFigureA.1Reliabilityblockdiagrams.Whenyouknow:Youcanfind:Ifyoumultiplyby:FailuresperFailuresper106millionhours,hour,fr/hrfr/106hr%/1,00010-1%fr/1,000hr10-1Percentfailures10-1per1,000hoursFITS103BITS104PPMhours,parts1failingpermilliondevice-hours%/1,000fr/106hr10fr/hr10-5FITS104BITS105PPMhours10FITSfr/106hr10-3%/1,000hr10-4BITSfr/106hr10-4%/1,000hr10-5FormulaeCD-5CD-6AppendixADABCEF12345678abcdefghijklmnopqrstuvwxSystemSubsystemEUnit7FigureA.2Developmentofblockdiagramsfromsystemtounits.FormulaeCD-713Single,constantfailurerateunitSingle,Weibullfailurerateunit.1 2k1–2 11 (n)0exxn1dxisthegammafunction.Single,linearlyincreasing,failurerateunit.IntheRayleighpdfkIntheWeibullpdf2,k,and0.22.2 2TkT2T T 1CasenumberSystemstructureMTTFCommentTableA.1EquationsforthecalculationoftheMTBFinseries.ThecalculationsforMTBF,MTTF,complexsystems,andtheconversiontablehavebeenadaptedfrom:Kececioglu,D.1991.ReliabilityEngineeringHandbook.Vol.2.pp.5–7,24–25,195–197.ReprintedbypermissionofPrenticeHall,EngelwoodCliffs,NewJersey.(Continued)CD-8AppendixACasenumberSystemstructureMTTFComment45Twoseries,Weibullfailurerateunits.nseries,constantfailurerateunits.1T1 1T1 1112T2 2T2 221211T=1,2T=2,...nT=n,.12...nii1n12 22G1212 2T1 11eT2 22dT,whereGn,zexxn1dxistheincompletegammafunction.If12and12,thentheMTTFbecomes1 111 11 21where ngammafunction.0TableA.1EquationsforthecalculationoftheMTBFinseries.(Continued)FormulaeCD-9CasenumberSystemstructureMTTFComment67nseriesWeibullfailurerateunits,allwiththesameshapeparameterandlocationparameter.nrseriesunits,nwithaconstantfailurerateandrunitswithalinearlyincreasingfailurerate.1T1 1T1 112T2 2T2 2Tn n111T1,2T2,...,nTn12...ni,i1n,,nT.... 111 i1i1nwhere ngammafunctionn1Tk1T,n2Tk2T2,...,nrTkrT.nrn2n1... k12ek212k12,where12ki,i1rk12andz12 122ex22dx.12...n2 2TableA.1EquationsforthecalculationoftheMTBFinseries.CD-10AppendixACasenumberSystemstructureMTTFComment12Twoparallel,constantfailurerateunits.Twoparallel,constantfailurerateunits.1T,12T,211112.121 1 111 2 111 22G12,12 221eT1 11T2 22where ngammafunction,andGn,z1exxn1dx,istheincompletegammafunction.If12and12,thentheMTTFbecomes 111 11 211 1 212121T1 1T1 111,2T2 2T2 221,21.TableA.2EquationsforthecalculationoftheMTTFinparallel.(Continued)FormulaeCD-11CasenumberSystemstructureMTTFComment34nparallel,constantfailurerateunits.nparallel,Weibullunitshavingthesameshapeparam