电磁阀 资料.docx
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电磁阀 资料.docx
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电磁阀电磁阀资料资料CO-ENERGY(AGAIN)Inthelinearcase,energyandco-energyarenumericallyequal.thevalueofdistinguishingbetweenthemmaynotbeobvious.Whybotherwithco-energyatall?
EXAMPLE:
SOLENOIDWITHMAGNETICSATURATION.Previoussolenoidconstitutiveequationsassumedelectromagneticlinearity.arbitrarilylargemagneticfluxescouldbegenerated.Inrealityfluxcannotexceedsaturationflux.Forsufficientlyhighcurrentsbehaviorisstronglynonlinear.Solenoid&co-energypage1NevilleHoganMODELTHATPHENOMENON.Assumeanelectricalconstitutiveequationasfollows:
whereL(x)isposition-dependentinductanceasbefore.Forsufficientlysmallcurrentstherelationisapproximatelylinear.Forsufficientlylargecurrentsthefluxlinkagereachesalimitingvalue,s.Solenoid&co-energypage2NevilleHoganMECHANICALCONSTITUTIVEEQUATIONmaybefoundusingthestoredenergy.Findthestoredenergyatafixeddisplacement.Findforceasthegradientwithrespecttodisplacement.Thatyieldstherelationbetweenforceandfluxlinkage.F=F(,x)Butfluxcannotbespecifiedarbitrarily.Realisticboundaryconditionsrequirecurrentinput.Tofindtherelationbetweenforceandcurrent,substitute.=(i,x)Solenoid&co-energypage3NevilleHoganSTOREDELECTRICALENERGY(atafixeddisplacement)Needtoinverttherelationbetweenfluxlinkageandcurrent.Ingeneral,anythingbutstraightforward.Inthiscase,alittlealgebrayieldsthefollowing.USEWITHCAUTION!
Ifsthisexpressionyieldsanimaginarynumberforthecurrent.Solenoid&co-energypage4NevilleHoganALITTLECALCULUS(andsomemorealgebra)yieldsanexpressionforenergy.Partialderivativewithrespecttodisplacement:
ls2l2Substituteforfluxlinkage:
Stillmorealgebrasimplifiesthisexpression:
Solenoid&co-energypage5NevilleHoganCOMMENT:
Theassumedelectricalconstitutiveequationwaschosenprimarilyforpedagogicsimplicityamorerealisticrelationmaybe(far)lesstractableusuallynosimplealgebraicformexistsConsequentlyaprocedurerequiring(a)inversion(b)integrationwithrespecttofluxlinkageand(c)differentiationwithrespecttodisplacementmaybeimpractical.Solenoid&co-energypage6NevilleHoganANALTERNATIVEAPPROACH:
useco-energyinsteadofenergy.Totalstoredenergy:
Electricalco-energy:
(aLegendretransformationwithrespecttocurrent)Forceisthenegativegradientofthisco-energy.Solenoid&co-energypage7NevilleHoganElectricalco-energyatafixeddisplacement:
KEYPOINT:
Co-energymaybefoundwithoutinvertingtherelationbetweenfluxlinkageandcurrent.Partialdifferentiationwithrespecttodisplacement:
Nofurtheralgebrarequired.Solenoid&co-energypage8NevilleHoganREMARKSIngeneral,co-energyfunctionsareusefulformultiportand/ornonlinearenergystorageelements.Invertingconstitutiveequationsmaybeavoided.Inthelinearcase,energyandco-energyarenumericallyequal.thevalueofdistinguishingbetweenthemmaynotbeobvious.Inthenonlinearcase,thetwoarenotequal.Distinguishingbetweenthemisimportant.Inthisexample,energyisupper-boundedco-energyisnot.Ingeneral,energyisconservedco-energyneednotbe.Solenoid&co-energypage9NevilleHoganNOTE1:
*Integrationtofindenergy:
whereyisa“dummyvariable”.rearrange:
substitution:
defineSolenoid&co-energypage10NevilleHoganDoesthatmakesense?
InthelimitasapproachessSolenoid&co-energypage11NevilleHoganPlausible;impliesthatonlyafiniteamountofenergymaybestored.Inthelimitasapproaches0,binomialseriesexpansionoftherootyieldshigherordertermsasexpectedforalinearinductor.Solenoid&co-energypage12NevilleHoganNOTE2:
*Integrationtofindco-energy:
Doesthismakesense?
InthelimitasL(x)isE*siThisistheareaofarectangleofsidesiands.Notethatco-energymayincreasewithoutbound,whereasenergymaynot.InthelimitasSolenoid&co-energypage13NevilleHoganL(x)isseriesexpansionofthesquareroot+higherordertermsasexpectedforalinearinductor.Solenoid&co-energypage14NevilleHoganNOTE3:
*Partialdifferentiationwithrespecttodisplacement:
Solenoid&co-energypage15NevilleHoganNOTUSED*Alternativesaturationfunctions:
“polynomial”saturationwherehyperbolictangent(logisticfunction?
)*Completerevision:
new“saturation”functionmightlooklikeSolenoid&co-energypage16NevilleHogan
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