MSA993726029

AppendixAFormulaeThisappendixprovidessomeoftheadvancedequationsusedintheelectronicindustryforfailureidentification,aswellasmiscella-neousinformationaboutblockdiagrams,howandwhentousetheappropriatereliabilitycalculations,andsoon.Specifically,thefollowingitemsarecovered:•TheArrheniusequationandtheEyringTypemodel•TheEyringandReich-Hakimmodel•Equationsforacceleratedfactors•Reliabilityblockdiagrams•Developmentofblockdiagramsfromsystemtounits•EquationsforthecalculationoftheMTBFinseries•EquationsforthecalculationoftheMTTFinparallel•Equationsforthereliabilityandmeanlifeofcomplexsystemswithcyclicalunits•ConversionfactorsforfailureratemeasurementunitsCD-1THEARRHENIUSEQUATIONwhereCisaconstant,EaisactivationenergyineV,Tistemperaturein0K,KisBoltzman’sconstant8.625
10-5eV/0KAccelerationequation:wherer1/r2istheaccelerationfactor.Activationenergyequation:Standarddeviationequation:wheret50isthemedianlifeortimeto50percentfailureandt16isthetimeto16percentfailure.THEEYRINGTYPEMODELWhereAandBareconstantsistheactivationenergyineVTisthetemperatein0K1150CARHBT
K
E
eAet
16501ntt1221111nTTrrKeEa

122111TTKEerra
KTEaeR
TCCD-2AppendixAKistheBoltzmanconstantineV/0KRHisrelativehumidityin%ECAistheelectricfieldinthecorrosionareaTHEEYRINGANDREICH-HAKIMMODELwhereGisafactorcombiningalltheconstantsEQUATIONSFORACCELERATEDFACTORSTheequationsforaccelerationfactorsassociatedwithtemperatureandvolt-ageacceleratedburn-insusethefollowing.TheArrheniusequationis:Accel.factorwhere:KBoltzman’sConstant,8.662E-5eV/0Kactivationenergy,eVTsstresstemperature,0KTd/operatingtemperature,0KThevoltageaccelerationequationis:Accel.factorwhere:CelectricfieldconstantToxthicknessofgateoxide,AVSstressvoltage,VVDdynamiclifeoperatingvoltage,Vexp
VDVS
TCoxTsTd
K11expKA,eB,e1GCAE
T
RHTRHe11501CATKEeBeRHeeAt
FormulaeCD-3Thepurposeofburn-inistocreateacceleratedtemperatureandvoltagestressconditionstoscreenoutearlylifefailures,especiallythoserelatedtoMOSoxidedefects.FAILURERATECONVERSIONExample:Ifyouknowthatthefailurerateis10fr/106hr,youcanfindthefailureratein%fr/1,000hrifyoumultiply10by10-1,or10
10-11;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 2k
1–211
(n)0exxn1dxisthegammafunction.Single,linearlyincreasing,failurerateunit.IntheRayleighpdfkIntheWeibullpdf2,k,and0.22.22
TkT2
TT1
CasenumberSystemstructureMTTFCommentTableA.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.1
T11T1111
2
T22T2221
211
T=1,2
T=2,...n
T=n,.12...nii1n1222G12122T111
eT222
dT,whereG
n,zexxn1dxistheincompletegammafunction.If12and12,thentheMTTFbecomes1
111112

1where
ngammafunction.0TableA.1EquationsforthecalculationoftheMTBFinseries.(Continued)FormulaeCD-9CasenumberSystemstructureMTTFComment67nseriesWeibullfailurerateunits,allwiththesameshapeparameterandlocationparameter.nrseriesunits,nwithaconstantfailurerateandrunitswithalinearlyincreasingfailurerate.1
T11T111
2
T22T22Tnn11
1
T1,2
T2,...,n
Tn12...ni,i1n,,n
T
....
111i
1i1nwhere
ngammafunctionn1
Tk1T,n2
Tk2T2,...,nr
TkrT.nrn2n1... k12

ek212k12,where12ki,i1rk12and
z1
2 122ex22dx.12...n22TableA.1EquationsforthecalculationoftheMTBFinseries.CD-10AppendixACasenumberSystemstructureMTTFComment12Twoparallel,constantfailurerateunits.Twoparallel,constantfailurerateunits.1
T,12
T,211112.12111112111

22G12,1222
1eT111
T222
where
ngammafunction,andG
n,z1exxn1dx,istheincompletegammafunction.If12and12,thentheMTTFbecomes
1111
12
111212121
T11T1111
,2
T22T2221
,21.TableA.2EquationsforthecalculationoftheMTTFinparallel.(Continued)FormulaeCD-11CasenumberSystemstructureMTTFComment34nparallel,constantfailurerateunits.nparallel,Weibullunitshavingthesameshapeparam