ARTICLE Thermal profile shaping and loss impacts of strain annealing on magnetic ribbon cores

18 Pages·2017·1.08 MB·English

ARTICLE
Thermal profile shaping and loss impacts of strain annealing on
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1.8 magnetic ribbon cores
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j/7 Richard Beddingfielda) and Subhashish Bhattacharya
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51 NorthCarolina State University, Raleigh, NorthCarolina,USA
.0
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/g Kevin Byerly
ro.io NationalEnergyTechnologyLaboratory,Pittsburgh,Pennsylvania,USA;andContractortotheUSDepartmentof
d//:sp Energy,AECOM, Pittsburgh,Pennsylvania, USA
tth Satoru Simizu
.sm Material ScienceandEngineering, Carnegie MellonUniversity, Pittsburgh, Pennsylvania,USA
ret/e Alex Leary
ro Materials andStructuresDivision, NASAGlenn Research Center,Cleveland, Ohio, USA
c/gro Mike McHenryb)
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g Material ScienceandEngineering, Carnegie MellonUniversity, Pittsburgh, Pennsylvania,USA
d
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m Paul Ohodnicki
a
c.w National Energy TechnologyLaboratory andMaterialsScienceandEngineering, Carnegie Melon University,
ww Pittsburgh, Pennsylvania,USA
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tth (Received 13January2018; accepted 7May2018)
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alia The use of the advanced manufacturing technique of strain annealing for nanocomposite
va magnetic ribbons enables control of relative permeabilities and spatially dependent perme-
,esu ability profiles. Tuned permeability profiles enable enhanced control of the magnetic flux
fo sm throughout magnetic cores, including the concentration or dispersion of the magnetic flux over
re specific regions. Due to the correlation between local core losses and temperature rises with
t ero the local magnetic flux, these profiles can be tuned at the component level for improved losses
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eg and reduced steady-state temperatures. We present analytical models for a number of assumed
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irb permeability profiles. This work shows significant reductions in the peak temperature rise with
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aC overall core losses impacted to a lesser extent. Controlled strain annealing profiles can also
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ht o adjust the location of hotspots within a component for optimal cooling schemes. As a result,
t tce magnetic designs can have improved performance for a range of potential operating
jb conditions.
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ta I. INTRODUCTION converter will be the passive devices and specifically
910 the magnetic components.1–3 Furthermore, magnetic
2 Passivedevicedesignsmustmeetthestringentrequire-
na components will continue to play a critical role in power
J 0 ments brought about by the continued increase in
1 n capabilities and adoption rates of wide band gap power converters due to their energy storage and isolation
o
,65 semiconductors. Specifically, magnetic component capabilities. The high reliability and wide design enve-
.57.6 designs need to consider new operating spaces and to lope of magnetic devices also make them critically
12.5 take advantage of new soft magnetic materials and important elements of any power converter design. It is
9 :sserd manadtetroiaplolporgoiceesssairnegm. Goveninegratlolyw,aprodwveerrycohnigvherteefrficcoienntcroielss bdeecsiagunseteocfhtnhiequaefosreamndentthieonceodmfpacotnoernsttshautsebdothnemedagtnoetbice
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a P through soft switching techniques that greatly reduce the fully characterized and improved.
I .ero losses associatedwiththe switchingdevices.Thisin turn Analytical predictions will enable improved screening
c/g means that a dominant loss component of a power and optimization of magnetic designs to reduce the
ro.e number of designs that require careful characterization
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dirb and more detailed optimization through experimental
ma a)Address all correspondence tothis author. prototype efforts and/or more computationally intensive
c.w e-mail: [email protected] applied electromagnetic tools such as finite element-
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w//:sptth b)mdTehacinissuiasocunrtihpsottarsgwaeu.atsFhooarrnetdheedbiyJtoMredRoiftpotorhslii,scpyjloeouansrneraerlveidfeeuwrritnaognhdtthtppeu:/br/ewlivcwiaewtwio.mnanrosdf. bcaansebde mdeovdeelolipnegd. toFuirntchoerrpmoorraete, naenwalymtiactaelriaelsxparnedssnioenws
m material processing strategies. By determining estima-
o org/editor-manuscripts/.
rf d DOI: 10.1557/jmr.2018.157 tions for currently unavailable materials or magnetic
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R.Beddingfieldetal.:Thermalprofileshapingandlossimpactsofstrainannealingonmagneticribboncores
cores, this approach can also help to guide the research motivation to provide deeper insight into thermal mod-
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51 and development goals for new alloy development as eling and loss predictions of traditional materials. These
.8
10 well as advanced manufacturing process optimization loss predictions are strongly coupled to thermal perfor-
2
.rm techniques. That is, the magnetic core research and mance and can provide insight for the thermal profile of
j/75 development can be refocused on the realization of the magnetic ribbon cores. An observed gradient in
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1.0 a subset of candidate designs that have shown promise temperature is due to the flux density concentration and
1
/gro through analytical expressions and have been confirmed theanisotropiccore.Thisanisotropyisrepresentedinthe
.io byfiniteelementmodelingtechniques.Suchanapproach proposed thermal model.Finally, an example is explored
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//:sp is particularly useful in understanding the role and with a comparison of physical cores designed for high
tth performance potential of magnetic cores that are post- power inductor applications.
.sm processed with strain annealing techniques in which the
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t/e permeability can be readily tuned along the length of the
ro II. MOTIVATION
c/g ribbon. The utility of an analytical design approach is in
ro.e determining which strain annealing profiles are most Fully understanding the losses for a magnetic compo-
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d useful such that the material and process development nent is critical for optimal design in power electronics
irb
m can focus on optimized designs. Similarly, the analytical applications. It is also important when determining the
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c.w approach can better define what baselines provide necessary cooling systems for a particular design. Mul-
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w//:sp a mDeuaenitnogtfhuel cmoymripaadriosofnf.actors that influence the design tcioprleelaoustshodruseatondthpeuebxlicciatatitoionnswexapvleofroermth.1e2–v1a5rHiaotiwonesveirn,
tth
ta of a power electronics converter, it is important to all of these prior investigations base their loss models on
elb characterizeandprovidepredictionsformagneticdesigns assumptions of uniform flux density within the core.
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liav using similar methodologies to that of semiconductor These models either use the maximum of average flux
a
,e device characterizations. That is, magnetic devices need densityortheoverallpeakfluxdensity.Whilethiscanbe
su
fo to be characterized for all operational sensitivities, e.g., usefulforpredictingthelossesasafunctionofexcitation
sm temperature, loading, dV/dt, etc. in a particular solution waveforms enforced by the electrical circuit, it does not
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t e space.4 This research focuses on two of the more account for core geometry, flux concentration and pinch-
roC e influentialaspectsofmagneticsdesign,namelythedirect ing, and other material effects. Permeability-engineered
gdirbm rmelaagtnioentiszhiinpgbleotwsseeesn. tIhteislocnaoltetedmtpheartatfuorre rsiosemeandpocworeer caonrdesthweitrhessuigltninifigcaflnutxvadreiantsiiotnyspirnofirelelastivweilplebrmeefaubritlhiteyr
a
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e electronics converter designs leveraging amorphous and susceptible to errors in loss estimations when using the
h
t o metal amorphous nanocomposite (MANC) ribbon-based previously mentioned approaches and these modeling
t tce core materials, additional loss contributions have been approaches cannot provide guidance to the optimal
jb
us ,0 determined to be significant and even dominant in some magnetic core designs in cases where permeabilities can
5:8 cases as a result of eddy current losses generated within be readily adjusted throughout a given magnetic core.
1
:2 the ribbon plane due to flux leakage from the magnetic In Ref. 16, the author derives and highlights the differ-
2
ta 9 core.Whilewedonotspecificallydiscussthatthesource ences in estimated total losses using the local peak flux
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02 of losses in this work, similar techniques are relevant to density-associated losses against the rational approach
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aJ 0 address those challenges and are the subject of on-going using a single global flux density. The solution space
1
n research activities. It is also important to note that new explored is useful for small components but does not
o
,6 post-processing techniques such as strain annealing5–10 exploresomeofthedesignpossibilitiesthatareneededfor
5
.57 andfieldannealing,11enablenewmaterialcharacteristics. the high power applications of interest. Similarly, Ref. 17
.6
1
2 These materials could lead to improved designs and utilized a similar analysis to recommend variations to the
.5
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:sse alternative solutions. However, without full characteriza- magneticcorecrosssection.Herethemotivationwas,asis
rd tion and analytical screening of the post-processing trueinthisanalysis,minimizingthermalvariationoverthe
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a P dependent performance of these materials, their develop- core profile. Unlike this paper’s approach, utilizing varia-
I .e ment is not able to successfully transition into applica- tions in effective permeability, the author of Ref. 17
ro
c/g tions and components for which they can have the most recommended varying heights along the toroid radius.
ro.e significant impact on overall performance. The charac- As the author mentions, this is problematic for manufac-
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dirb terization of magnetic designs and materials under turability.Furthermore, the author fell shortof a complete
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a operationally relevant conditions is paramount to contin- analysis and did not fully analyze the variation in core
c.ww ued success and improvements in magnetics design. lossesduetotheimprovedfluxdensityprofile.Finally,the
w//:sptth powTheirsmpaagpneertwicsil,lchoimghploignhetntthleevieml,pproervfeodrmcaanpcaebielintiaebsleind aanuathlyosriswa(sFoEnAly).abTlehetroeexisploarmepsolelutriooonmsinffiornitceoenlteimnueendt
m bythestrainannealingpost-processingtechniques.Itwill analysis and analysis that considers newly proposed
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A. Flux density definitions layers. This drawing illustrates some of the physical and
7
51.8102 inFflourxcdlaernitsyit,yitreispriemspenotratatinotntso. fiTrhset dloecfianlefltuhxe vdaernisaittiyoniss cshoonwstrnucintiothnis dpraarwaminegtewrsi.ll eSniambilleartlhye, dethtaeileddidmeefinnsiitoionns
.rmj/7 the typical flux density that is derived using the relation- of various thermal impedances as described in Sec. III.
55 ship between current and magnetic field using Ampere’s
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ro.iod pmraotdeuricatloufntdheer cinovnesstatingtatrieolnat,ivle,paenrdmethaebiplietyrmoefabthileitcyooref Giventheabove-definedfluxdensityprofiles,aproper
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tth .sm enable an arb0itrarily variable profile of relative perme- nbaeturreeporfestehneteridbboasnwaonunedqumivaaglneentticcciorrceusi.tAomf otdheelrmcaanl
ret/eroc/gro.e aaoabsnfsdicluiomtcriieerisncr,ugamdittihfueiassrtebltneuhtftietailspaedsoritmraheecefrtauwiboniincsli.tetiySoenuqvcuaoharfliaeatshlpoeoennrgmtloyterhoaaeibsdiclaoirtrayfeudpnhiurceosti,igfiohlrnet, irtmievspeisleytda,annacsceesd,s,ecstcahrpeiabrcmeidtao.l1r8sc,Oapatnahcdeirtacrnuecrsereesanrtcahniednrjsehchetaaiotvnefls,ofowrleloswpweietchd-
gd a similar approach of modeling power magnetics by
irbm is readily produced using strain annealing techniques either using isotropic materials or averages of bulk
a through a monotonic variation in applied tension during
c.w materials, Refs. 19 and20. In Ref. 21, the authors model
w the strain annealing process followed by standard core
w anisotropic cores but do not consider variations in heat
//:sp winding and fabrication processing techniques. The flux injectionduetoafluxdensityprofile.Toaccountforthis
tth ta elba fldexeuncxistiadttyeinosnisitycsucirsarelvendatl,idbI,yirnethgtheaerdnaluepmspslbicoeafrbeloexfcriattuanrtginoesn,ofsNhr,apbaeen.tdwTethehines lnoeccaelsizsaerdy htoeahtavinejeactilooncaldizueed ttoherlomcaallimzeoddello.ssTehsi,siptrois-
liava theinnerradius,r,andtheouterradius,wherekisusedto posed model is shown below in Fig. 2. This model
,esu fo sm draedfiinues.Tthheeraavtieoragbieetflwuexendetnhseityouinteartroardoiiudsisasnhdowthneinin(n2e)r. dsTeihsvecerresautlibzcseocsrnictphetenntrsiipcsirtrahilnegodfliascyareerwtsizooautfniodonnmevaarrgiibanbbeolteinc. Etthaaipccehknrienisntsog.
ret eroC egd BðrÞ¼lð2rpÞrNI(cid:1)(cid:1)(cid:1)(cid:1)ri#r#kri ; ð1Þ eHhfaofsewcetaivveetrh,reitrnhmgisamlcaacspasapcaaintcadintamcnecaetne,ereiaCdlsc’,sosncpoleyrcrebifisepccohonendasitnidcgearpetaodciftioytsr.
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s ,05 top view and a cutout side view of one of the ribbon convection and radiation are also treated as impedances
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oftheformshownin(3).Thecomponentsof(3)forallof corethickness,t,andthesumoftheribbonthicknessand
751 thethermalimpedancesofFig.2aredescribedinTableI. the gap thickness, G. This ratio can be near five for
.810 typical fill factors, around 80–85%, ribbon thicknesses,
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d//:sp andresearchstudies,arepresentedfromRefs.22–24.The kt ¼ðkGGþk(cid:5)ctÞ Gt þkkG(cid:6)c
tth .sm cthoereceisnatesrseomfablreidbboofnNoLfltahyeenrsthsulacyhetrh.aTthRenGisrtehgeiroandiisusthaet kr ðG(cid:7)þtÞ kGtGþk(cid:8)G (cid:7) (cid:8)
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t/e gap between two ribbon layers and t is the thickness of ¼F2 2(cid:2) c G þF c G(cid:2)2 þ1 :
roc/g a ribbon layer. The dimension lG can also be used to kckG kckG
ro account high thermal impedance materials between rib- ð4Þ
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gd bon layers and between the ribbon and air. This can be
irbm used to represent thin oxidation layers or epoxies. The The effective thermal impedance for convection is
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c.w radial G effect can be estimated by using the thermal determinedusingNewton’slawofcooling.Thevariation
ww impedance of air, oxidation, or epoxy with the core fill in the k value, replacing the traditional symbol h, for
//:sp factor. High fill factors would have low G region the coef[fic/Gci]ent of convective heat transfer in this imped-
tth ta impedanceswhilelowfillfactorcoreswouldhaveahigh anceisduetothedifferentcoolingmethodsrangingfrom
elb anisotropy in the tangential and radial dimensions. The natural, unforced air, convection to higher performance
alia thermal anisotropy varies significantly with the bulk forced liquid cooling. The thermal impedance modeling
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a ,e magnetic core fill factor. The ratio of thermal conductiv- radiation utilizes Stefan’s law and reorganizes the law
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fo ity between the tangential and radial directions of toroid into the form of a typical thermal impedance. In these
sm isshownin(4),whereF,thefillfactor,istheratioofthe impedances, T is used to represent the core layer
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I .e Subscript Description Dimension Length(l) Area(a) Thermalconductivity(kt)
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c/gro.e CGrr CGoarpeccoonndduuccttiioonn RRaaddiiaall lGt/2/2 22ppRRnnhh 05.6––110.4ww/m/mkk
gdirbm CGzz CGoarpeccoonndduuccttiioonn HHeeiigghhtt lhG//22 22ppRRnntt 05.6––110.4ww/m/mkk
ac.www//:sptth m CRRCCaaoooddnnnzIzIO OInIEunndCtneegerorernccvrrooaaennddcvviitaeaieotctciinttooiionnonn HHHRReeeaaiiiddgggiihhhaallttt TT(cid:2)–111TTaammbb 22ppRR1222L(ppp(lGlRRRGzn1nttt11th)) eerrss555(cid:5)(cid:5)(cid:5)–––TT555(cid:2)(cid:2)000000TT111aa44mmbb(cid:6)(cid:6)(cid:6)
orf d RadO Outerradiation Height T(cid:2)Tamb 2pRL(lGz1h) ers T(cid:2)Ta4mb
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temperature and T is used to represent the ambient temperature nearer to ambient. This affects closely
7 amb
51 temperature. adjacent layers due to the larger temperature difference.
.8
10 In Fig. 2, the basic model most properly represents
2
.rm interior layers of the ribbon. For the innermost and the C. Heating impact
j/755 outermost layers, a slight modification is needed. Specif- The heating of the core has several impacts. The core
1.01 ically, the model region defined as core–air interface must remain below certain temperatures, e.g., the Curie
/gro connects to both the z dimension and the radial di- or impregnated epoxy breakdown temperatures, with
.iod mension. These augmentations to the radial thermal safety margins that would affect fundamental core prop-
//:sp impedance account for the convection and radiation erties. However, in most practical power magnetic
tth .smre aasnsdoociuatteerdmowsitthribthbeonelxapyoesresd.Tbhreosaedarseurrfeapcreesoefnttehdebiynntheer dareesigimnsp,otrhtaenret atoreaevvoeind lsouwcehr acsrittihcaelttheemrmpearlatluimresitsthoaft
t/e two-edge regions of core–air interface inner and outer, insulating epoxies or plastics.
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c/g respectively. Whiletheselimitsexist,onemaywishtohavedesigns
ro.e Thismodel is thenconnectedto otherradial models to that further limit the maximum core temperature. The
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dirb represent the different tangential zones of the magnetics. temperature impacts on the operating performance of
ma Three examples of zones are uncovered, covered with cores are generally material specific with25 providing
c.ww heat sink material, and covered with heat injection somepolynomialapproachestoaccountforthevariations
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ta bothaffectthezdimensionthermalresistancesandwhile Nearly all performance metrics have varying sensitivities
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lba a zone covered with the heat injection component could to operating conditions including permeability, core loss,
liav potentially add to the heat injection. That is, a heat sink and saturation flux density. While ferrites have shown
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,esu material adds its own thermal model between the edge, this variability, it is not always the case for nanocrystal-
fo smre icnonnenre,cotironoutoterthiempceodrea.ncAesltedrneapteinvdeliyn,gaonheiatts ipnhjeycstiicoanl lvianrei.a4tiIonnthinat mstueadsyu,recdorecsowreerleosssheoswonvteor haavreeamsoinniambalel
t ero component has several aspects to consider for complete temperature range. However, the effective relative per-
C e thermal modeling. A prime example of a potential meability of the core changed significantly. In operating
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dirbma einxtaermnaalghneeatitcicnojemctpioonnecnot.mOpnoeneonfttihsethsiemepxlecsittamtioondicfiocials- udprotpop8ed0°rCou,gthhelyef3fe0c0t%iv.eTrehlaistivreespueltremdeainbilaitymoufchthelocwoerer
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eh tions is modeling the use of a thermal insulator between transformer magnetizing inductance. In a practical con-
t ot tcejb tahffeecetexdcitraetgioionncdooielsannodt hcaovree.sigWniitfihcatnhtishesacteflnoarwiop,atthhes vcoerntdeur,cttihoinslcoossuelds breecsauultseinofuapthtoreea-ti9m-feoilndcrienacsreeaisnethine
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s ,0 toambientandthusallmodelconnectionstothesinkcan required magnetizing current. This increase in current
5:8 beremoved.However,ifathermalconductorisused,one would also have to be accounted for in determining the
1
:22 must consider the heat injection caused by the heating of system maximum current rating and the selection of
ta 9 theexcitationwinding.Thethermalpathfromthecoreto winding gauge. However, this increase in magnetizing
1
02 n the thermally conductive bobbin to the excitation coil current could be used advantageously as certain
aJ 0 does have a path to free space and thus the thermal converters utilize magnetizing current to increase the
1
no resistancestoambientofconvectionandradiationcanbe window for which semiconductor switching losses are
,65 included. minimized. The authors in Refs. 27 and 28 showed that
.5
7.6 With a complete thermal model of the transformer, we the how so-called soft switching of converters could
12.5 can see that the local loss profile has a significant impact utilizemagnetizingcurrentforimprovedoperation.Thus,
9 :sse on the temperature profile of the core. The thermal regardless of the design, it is important to have a firm
rd anisotropy between the radial and tangential dimensions grasp of the overall operating temperature and to
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a P furthercouplesthetemperatureprofiletothelocallosses. understand the implications that this temperature has on
I .ero Where the thermal body be isotropic, the temperature the magnetic core performance.
c/g profile would normalize somewhat but still be heavily
ro.e dependent on the heat injection profile. The thermal
gdirb impedances in the z dimension have some variations III. STRAIN ANNEALING PROFILES
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a between the layers due to varying surface areas but it is As mentioned above, applying mechanical strain as
c.ww minimal.Theexceptiontothistrendistheinnermostand a post-production process to the magnetic ribbon causes
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m radiative and convective impedances at these two layers a core with an arbitrary relative permeability between
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manufactured. While the ultimate profile achievable investigationhavenotshownsignificantvariations.Another
751 could be arbitrary, four fundamental profiles are pre- simplification is that the core height, h, or strain-annealed
.8
10 sentedherefortheirlogicalandpracticalutilityaswellas ribbon width is constant. While applying strain, the ribbon
2
.rm their manufacturability during the strain annealing pro- stretchingcauseselongationintheaxisoftheappliedstrain
j/75 cess without the need for highly sophisticated process and shrinkage in the other axes. This can result in a re-
5
1.0 controls.Theseprofilesareconstantstrain,rampedstrain, duction of the thickness of the ribbon and the width of the
1
/gro ramped strain with initial strain, and exponential strain. ribbon. The cross sectional area is consequently reduced.
.io These profiles will be discussed further in detail in the This could lead to a slightly higher flux density than
d
//:sp respectivesubsections.Inthisanalysis,thereareamyriad predicted flux density in the ribbon required due to the
tth of variables to explore and potentially many local in- reducedarea.Thethirdassumptionisthatthemagneticand
.sm flection points in relative performance. As such, this electricpropertiesofthecorematerialwillnotchangewith
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t/e analysis is meant to provide an initial overview and to temperature. While it has been shown to have significant
roc/g allow further in-depth study for specific cases. For all of effects,4 incorporating the full impact requires a specific
ro.e the presented design cases, the core permeance was held corethermalmodel.Furthermore,theinitialcoreparameters
gd to be the same. In this way, the comparison of strain will remain sufficiently unaffected while the core temper-
irb
m annealing approaches is for magnetic devices with the atures remain low. It is with the understanding of the
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c.w sameinductance,asinductanceisequaltothepermeance variationsandtheassumptionsthattheforthcominganalysis
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tth ta lossestimationbasedontheaveragefluxdensity,theheat For this analysis, the loss term and the ribbon width, core
elb injection, and finally the loss calculation based on the height, and thickness terms were assumed constant and
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liav local flux density. unvarying with temperature.
a
,e Theprofilesarepresentedintermsofthecoreradiusto In the forthcoming analysis, the relative permeability,
su
fo sm eflausxilydeanpspiltyy.thHeopwroepvoers,edthpiserrmadeiaablliylitdyetfiontehdedperofifinlietiodnooesf bscyalle.dTbhyetshuebspcerrimptedabeniloittyesotfhferetyepsepaocfet,hle0s,triasinreapnrenseeanlitnedg
re
t e not provide insight into the strain profile. For this, we profile.Thedifferentialvolumeofthetoroidreducestothe
ro
C e mustdeterminethelengthalongtheribbonforwhichthe circumference at that radius multiplied by the differential
g
dirbm satsraainnsAhorcuhldimbeedeaapnplisepdi.raWl,hailesitmhepltiofireodidaipspcroonxsimtruactitoend rraaddiiuussvisoludmeneo.tTedhewraitthiokb.etAwseesnhoinwnnerirnadRiuesf,.r5i,,anadsoetutoefr
a
C e yields adequate results. The length function is then Steinmetz parameters k, a, the frequency, f, and loss
h
t o presented in (5). The layer number, L , is determined component, b, can be used to fit the loss curves for an
t tce by (6). Once a desired permeability profinle is chosen, the arbitrary excitation using (8). The authors in Ref. 29
jb
u
s ,0 length for which the strain should be applied, lr, given showed how a map or surface of measured losses could
5:8 a desired radius location, r, is easily determined in (7). be fitted using (8) for several common excitation wave-
1
:2 forms. In the following analysis, accurate parameters are
2 ta 91 L ¼pL (cid:3)2r þ t ðL (cid:2)1Þ(cid:4) ; ð5Þ assumed to have been measured and fitted. The scaling
02 r n i F n factor, k, has a wide range of values and can be used to
n
aJ 0 modify the equation to handle parameters extracted for
1 n Fðr(cid:2)rÞ either 1000’s of Hz or kHz. In other words, k, could be
o ,65 Ln ¼ t i ; ð6Þ differentbyafactorof1000a dependingoniffrequency,f,
.57 is analyzed in Hz or kHz. The frequency factor, a, is
.612.5 (cid:7)Fr2 Fr2 (cid:8) typicallyintherangeof1–3whilethefluxdensityterm,b,
9 :sse lr ¼p t (cid:2) ti (cid:2)r(cid:2)ri : ð7Þ is between 2 and 3.
rd
d
a
PI .e P ¼kfaBb : ð8Þ
ro V
c/g
ro.e A. Assumptions used in analysis The Steimetz fittings are often further expanded to
g
dirb As with any numerical analysis and modeling of consider loss partitioning in terms of different operative
ma physical systems, certain assumptions must be made. mechanisms.30Analternatelossmodel,emphasizingloss
c.ww Therearethreeprimaryassumptionsusedinthisanalysis. separation, is the 3-term Steinmetz equation that is
w//:sptth Tsthraeinfirasntdisptehramttehaebilloitsystoerrmte,mbp,edroateusrne.otFcuhtuarnegewworikthwthilel euxsepdretsosepdarutistiinogn pscoawlaerrslokhsyss,deknedsdityy, panerdukneixtcvwolhuimche aforer
m verify this assumption or provide functions and curves (i) hysteresis, (ii) classical eddy current, and (iii) excess
o
rf d to account for the impact. The initial samples under loss components, respectively.
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In the product literature, it is common to lump the last density is (10), where N is the number of turns in the
7
51 two terms as total eddy current losses with a power law exciting coil and I is the exciting current. The resulting
.810 exponentadependentoninduction.Statichysteresisloss corepermeance,P^,foracoreofheight,h,isderivedfrom
2
.rm dominatesatlowfrequenciesandeddycurrentsdominate the flux and is shown in (11). These dimensions can be
j/75 at high frequencies. seen in Fig. 1.
5
1.01/gro squTahreetphaicrtkitnieosnsinogftchoenmstaantetr,iakledadnyd, iisnvperrospeloyrtpioronpalorttoionthael B ¼NI lclnðkÞ ; ð10Þ
.io the material’s resistivity. Therefore, high resistivity is avg 2prðk(cid:2)1Þ
d i
//:sptth .sm dfrriebesqbiuoreendnscf(ioeArsM.mRWasgh)n,ileeMticAthcNeoCmrsep,soiasntneidvnittscyryoosptfearlaalitmnineogrmpahatoteuhrsiigahlmseAatraCel P^ ¼ f ¼Z hZ kriBðrÞdrdz¼ h l lnðkÞ : ð11Þ
re NI 2p c
t/e tunable with chemistry, the resistivity of AMRs is typi- 0 ri
roc/g callyare2–3timeslargerwhichsignificantlyreduceseddy Byusingthefluxdensityaveragedovertheentirecore
ro.eg current losses. Hysteretic losses are diminished in AMR, and the total core volume in the Steinmetz loss equation,
d MANCs, and bulk amorphous materials through random
irbm magnetic anisotropy.31 Typically, magnetic induction is we can see that the losses based on the average flux
ac.w sacrificed for glass forming ability (and resistivity) in density are shown in (12).
w (cid:5) (cid:6)
w//:sptth ta e AcgrrMyosuRtnadlliannbedemtwbaeuteelrknialatshm,eeo.grhp.,ihgSohiuessstteeaillnslo.dyMuscAtiaoNsnCscsooomfffpeacrrraeydsmtaiwldlidintlhee P Bavg‐lc ¼¼kkffaaVhpBrba2v(cid:5)gk2(cid:2)1(cid:6)(cid:7)NI(cid:8)b(cid:7)lclnðkÞ(cid:8)b :
lbaliava ,e smTehacetteiorainabslislsiitgyannitdfiochacinagtshltyAirmeMspisaRtci’vtssiteyadnddoyfMcauAmrrNeonrCptshloosiunssestmhtoianrteecrdiruaolcsses. i 2p riðk(cid:2)1Þ ð12Þ
su
fo sm higThh-felaonsosmesa.3l1ouseddycurrent(AEC)losses,thirdtermin Forunderstandingtheheatinjectionprofile,itisuseful
re toinvestigatethelocalpowerloss(13).Thisisequivalent
t e the three-term Steinmetz equation is associated with
ro to the heat injection, Q, in the thermal model. Here the
C eg dynamicdomainwallmotion.AECcontributionsoriginate volume is the local slice of volume.
dirb atamesoscaleduetolocalheterogeneitiesinthemagnetic (cid:7) (cid:8)
maC eht o pddiionsmsniaipniangtiomsntrauogcfnteeutnriece.r3gd2yo,3m3ataJihunisgtwhaaflsrlesqlaiunseenSrciisescstre,ibeilnisndgutociislnimguiastendtihsoeitinr- Ploc ¼Q¼kfahCðrÞtBðrÞb ¼kfaht2pr1(cid:2)b l2cNpI b :
t tce ropyinAMRandMANCshavebeenusedtolowerlosses ð13Þ
jb
u
s ,05:8 aintduhcigehdbfyretqruanensvceierss.e(ItnotAheMARMs,Ralneinsgotthro)pfiyeldisantynpeaiclianllgy, Finally,anestimationofthelossesusingthelocalflux
1:2 rolling, and more recently strain annealing.34 density is presented in (14). This method of loss
2
ta calculation applies the Steinmetz equation to small slices
9
102 B. Constant permeability of the core. As such, the differential volume is needed.
na This volume is equivalent to the circumference of the
J 01 n This strain annealing profile has no profile variation infinitelythinsliceandisrepresentedbyC(r).Theheight
o from the permeability of traditional cores. However,
,6 ofthecoreisascalartothevolumeandisindependentof
5.57 reducing the permeability of the core by applying strain the radius.
.6 could eliminate the need of air gaps or achieve some
1
2.59 :sserdd aroeptshpuellirtcasitnpioecnai,ficacoanclsloytnasndttaensptierersmtdraerianeblaiilstiitvyaepfpoplireerdmtheteoabetnihltieitryer.ibrIbinbobntohtnios. P(cid:5)Bloc‐lc(cid:6)¼kfah2p(cid:7)N2pI(cid:8)blbcri2(cid:2)b2(cid:5)k(cid:2)2(cid:2)bb(cid:2)1(cid:6) : ð14Þ
a
PI .ero T(9h)u.sA,sthaewroaduinadllytordoeipdencodreen,tthpiesrmsteraaibnilaitnyneisaliangcopnrsotfianlet With two estimations for losses, one based on the
c/gro wouldresultinamagneticfluxdensityprofilethatislike average flux density and the other based on the summa-
.eg any other core of the same constant permeability. As tion of local losses, a comparison of differences between
dirb such, this approach will be used as the baseline for the two can provide insight into the design cases for
mac.ww comparison of other strain annealing profiles. (w1h5i)c.hItthiseinmupmoertraicnatltosimnoptleifithcaattioalnl coofnusstianngt pBearvmg eisabvialiltiyd
w
//:sptth lðrÞ¼lc : ð9Þ desetsiimgnatiocnas.eIst irsesounlltyiwnhleenssthteharnatioa b1e0tw%eevnariinantieornanind
m By averaging the radially dependent flux density over outerradii,k,ishighandthelossterm,b,islowthatthe
o
rf d theboundariesofthecore,itisclearthattheaverageflux estimationdifferenceisatitsgreatest.However,usingthe
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average flux density to predict magnetic core losses other words, a constant permeability design total loss
7
51 obfuscates the thermal impacts by considering the losses couldbedeterminedbymultiplying(16)byapointofthe
.810 as a whole and not as a profile. Furthermore, this loss curve corresponding to the chosen design.
2
.rm estimation will miss the loss impacts of different strain (cid:7) (cid:8)
j/7551 annealing profiles with the same permeance. kfah2p NI blb : ð16Þ
.01/gro.iod//:sp PP(cid:5)(cid:5)BBalovcg‐‐llcc(cid:6)(cid:6)¼ln2(cid:5)ðkkÞ2(cid:2)b(cid:5)bk(cid:2)2(cid:2)1(cid:6)1ðk(cid:6)ð(cid:2)2(cid:2)1ÞbbÞ : ð15Þ Acommonradiusratioand2pfluxd0ensitylossexponent,b,
tth .sm Using (14), loss contours for the general design space forpowermagneticsis2and2.4,respectively.Atthisdesign
re for power magnetics can be established. Specifically, the point, the loss scalars for the low permeability designs are
t/ero locallossescanbestudiedunderfourdifferentscenarios. 959 and 382 for low and high inner radius designs,
c/g This would be the combination of low and high inner respectively.Thehigherinnerradiushasalowerlossscalar
ro.e radiiandlowandhighpermeability.Inthiscase,thelow reflectingthelowerpeakfluxdensityforagivenexcitation.
g
dirb inner radius is 10 mm and the high inner radius is Thehighpermeabilitylossscalarforthelowinnerradiusis
ma 100 mm. The relative permeability is chosen with the 2.41 (cid:3) 105 and 9.59 (cid:3) 104 for the high inner radius.
c.w
w focus of power inductors where low permeability is
w//:sptth ta e gahreiegnhe1r0palellr0ymfoeidarebtaihllei.tylIoncwatshpei.esrTmchaeesaevb,ailltiuhtyeescpoaefsretmhaeenacdboi1nli0ttoi0eulsr0asfrtoeurdlotihesdes Ca.flTLahtiinsflepuaexrrmgdereanadsbiietlyidtypprieosrfimaleel.oagTbihicliaistlyiasssbuemcaputisoentthoeapcehrimevee-
lba scalars that reflect division of common variables. These abilityincreasesproportionallytotheradiallydependent
lia
va commonvariablesareshownbelowin(16).Forallofthe decrease in magnetizing force caused by the increasing
,esu strain annealed permeability profiles, the contours in magnetic path length at larger radii. Here, the slope of
fo sm Fig. 3 will be usedas the baselineto compare theimpact the permeability is chosen as ls. The relative perme-
re of a strain annealed induced permeability profile on the ability scales only as a function radius and has no
t ero toroid’s magnetization losses. These curves show the effective offset, by the profile definition, a relative
C
eg scalartermofaconstantpermeabilitydesignlocalloss.In permeability that would be zero at a radius of zero if
d
irb
m
a
C
e
h
t o
t tce
jb
u
s ,0
5
:8
1
:2
2
ta
9
1
0
2
n
a
J 0
1
n
o
,6
5
.5
7
.6
1
2
.5
9
:sse
rd
d
a
P
I .e
ro
c/g
ro
.e
g
d
irb
m
a
c.w
w
w
//:sp
tth
m
o
rf d FIG. 3. Contoursofbaselinevaluesforlossesofvarioustoroiddesigns.
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zero relative permeability could exist. The strain should (cid:7) (cid:8)
751.810 bsuecahppthliaetdeiancha slatayierrstoefpwfaosuhniodnriwbibtohninicnretahseintgorloeindgtihs P(cid:5)Bavg‐ls(cid:6)¼kfahpri2(cid:5)k2(cid:2)1(cid:6) N2pI blsb : ð20Þ
2
.rm subjected to the same strain and results in the same
j/75 permeability over the layer. This profile has the mini- Thelocallossesarecalculatedforthisprofilein(21).It
5
1.0 mum variation between maximum and minimum flux is interesting to note that despite the flat flux density
1
/gro densityastheyshould,withinmanufacturingtolerances, profile,theheatinjectionprofilealongthecoreisnotflat.
.io be the same. The permeability function is shown below Thisisbecausethevolumeofthecoreringshasincreased
d
//:sp in (17). in the outer radii.
tth .smret/ero The average flux dlenðrsÞity¼alnsdr pe:rmeance for a graðd1e7dÞ Ploc‐ls ¼kfaht2pr(cid:7)l2sNpI(cid:8)b : ð21Þ
c/gro.eg mpeernmtieoanbeilditaybporvoefi,lethaereflushxodwennsbiteylopwroifinle(1i8s)flaantds(u1c9h).thAast Finally, the total losses, calculated locally, are shown
dirbm itisequivalenttotheaveragefluxdensityalongtheentire iFnig(.242.).InThtheepseerdceensitgcnhcaonmgepainrisloosnsse,sthisesshloopwenfabcetloorw, lin,
ac.w core profile. Therefore, the flux density is no longer is chosen to ensure that the linearly graded permeabilitsy
w
w dependent on the core radius.
//:sptth ta e Bavg‐ls ¼N2pIls ; ð18Þ rpeesrumltesabiinlitythceorseampeermceo(cid:7)arnecep(cid:8).ermean(cid:5)ce as (cid:6)the constant
lbaliava ,esu fo P^ ¼2hplsriðk(cid:2)1Þ : ð19Þ P(cid:5)Bloc‐ls(cid:6)¼kfah2p N2pI blbsri2 k22(cid:2)1 : ð22Þ
sm
re Thepowerlossduetotheaveragefluxdensity,(18),is Interestingly, when designed to match the permeance
t ero shown below in (20). Again, the basic Stienmetz estima- of the constant permeability profile the linearly graded
C
eg tion is used for losses. permeability,flatfluxdensityprofile,designhasthesame
d
irb
m
a
C
e
h
t o
t tce
jb
u
s ,0
5
:8
1
:2
2
ta
9
1
0
2
n
a
J 0
1
n
o
,6
5
.5
7
.6
1
2
.5
9
:sse
rd
d
a
P
I .e
ro
c/g
ro
.e
g
d
irb
m
a
c.w
w
w
//:sp
tth
m
o
rf d FIG. 4. Percentchangeinmagnetizationlossesfortoroidsutilizinggradedpermeabilityprofiles.
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loss increase for all four design cases. In fact, it is easily note that the subscript for the offset permeability is
751 shown that the loss ratio reduces to (23) which is chosen as ‘1’ instead of the letter ‘o’, for ‘offset’, to
.8
10 equivalent to (15). Again, using the common radius ratio avoidconfusionbetweenthestrainannealedpermeability
2
.rm and loss term for power magnetics of 2 and 2.4, re- profile and the permeability of free space, l .
j/75 spectively,wecanobservethatthereisa2.8%additional 0
51.01 loss for this profile and design point. lðrÞ¼l1þlsr ¼l1ð1þgrÞ : ð24Þ
/gro.iod//:sptth PP(cid:3)(cid:5)BBlloocc‐‐llg(cid:6)(cid:4)¼ln2(cid:5)ðkkÞ2(cid:2)b(cid:5)bk(cid:2)2(cid:2)1(cid:6)1ðk(cid:6)ð(cid:2)2(cid:2)1ÞbbÞ : ð23Þ efuxntTrcaotciosteinmdalpflprifoeymrmtheetahbaeinlaistlylyospiisse,etphqeeurimovfaeflsaeebntitlpitteyormtsheuaecbhriliigtthyht,atlm1t,ohsiest
.sm c form of (24). The benefits of this are apparent in
ret/e Togeneralize,thechangeinlossesisindependentofthe investigating the average flux density, (25), where the
roc/g inner radius and the effective constant permeability that offset permeability is treated as scalar to two types of
ro the graded permeability design is replacing. The increase permeability functions, constant and linearly graded.
.egd inlossesislowestforhighfluxdensitylosstermsandlow Permeance is shown in (26).
irbm radiusratios.Thehighestincreaseinlossesisforlowloss (cid:7) (cid:8)
ac.www//:sp ttoehframninsacwr1ei0at%hsehdiingchlroersaassdeesi.uIsinnragtemionase;granhleo,twtihceevrleeorsi,sstehasismfiionnrcirmethaasilsepeissntrlaaelstinys Bavg‐l1 ¼N2pIl1 riðlknð(cid:2)kÞ1Þþg : ð25Þ
tth ta elb adnennesiatlyindgiffperroefinlcee.sTahloenbgetnheefictoorfe rgeedoumcientrgytmheaypebaekmflourxe P^ ¼2hpl1ðlnðkÞþgriðk(cid:2)1ÞÞ : ð26Þ
a
lia important than the costs of slightly increased losses for
v
a ,e many designs if they are observed in practice. The estimated losses using the average flux density is
su fo shown in the following equation:
sm
re
t e
ro
C
egdirbmaC e P(cid:5)Bavg‐l1(cid:6)¼kfahpri2(cid:5)k2(cid:2)1(cid:6)(cid:7)N2pI(cid:8)b(cid:7)l1(cid:7)riðlknð(cid:2)kÞ1Þþg(cid:8)(cid:8)b : ð27Þ
h
t o
t tce
jb
u
s ,05 D. Linear graded permeability with offset The local power, heat injection for the strain annealed
:81:2 Logically, the next straining profile to investigate is profile that is linearly graded with an offset is shown in
2
ta 9 a linearly graded permeability that has a nonzero y the following equation:
102 intercept.Forthispermeabilityfunction,thepermeability (cid:7) (cid:7) (cid:8)(cid:8)b
naJ 01 when the radius is chosen as zero is nonzero, in contrast Ploc‐l1 ¼kfaht2pr N2pIl1 1r þg : ð28Þ
n to the previously mentioned linear graded permeability
o
,6 profile. This profile could account for two common
5.5 Thetotalpowerlossduetothisstrainannealingprofile
7 scenarios. First, the magnetic ribbon does not have the
.612 mechanical strength to withstand the peak strain forces is shown in (29). It should be noted that Bz[a, b] is the
.59 incompletebetafunction.35Thederivationfromthebasic
:sse neededforthelinearlygradedpermeability.Withalower localpowerlossequationto(29)isprovidedinAppendix
rd permeability slope applied, the effective permeability at
d A. Again, the utility of extracting the offset permeability
a P the low end must be increased to ensure that the
I .e permeance matches the base case of constant permeabil- from the slope permeability is shown through the
ro simplification in the derivation of (29). Specifically, l
c/g ity.Similarly,themagneticpropertiesoftheribbonatthe 1
ro.e beginning and end permeabilities could be significantly is able to be removed from the integrand.
gd (cid:7) (cid:8)
irbmac.ww wpweoirnrfdsoeormwthaaonnfcenpsoemrimsinenaaolb.tilTittooieosencfsaoururelddtehbgaertadethenedfo,mrcaaengdn.oeTptiehcriasritbimobnionan-l (cid:5)P(cid:5)Bloc‐l1(cid:6)¼kfah2p N2pI bð(cid:2)gÞb(cid:2)2lb1 (cid:6)
w//:sptth idmefiunmearnibdbomnaxsitmarutmanadlleonwdabpleermpeerambeilaitbieilsitywwithoualdlitnheeanr B(cid:2)gkri½2(cid:2)b; 1þb(cid:4)(cid:2)B(cid:2)gri½2(cid:2)b; 1þb(cid:4) : ð29Þ
m slope applied between the two. Functionally, the perme- There are now at least two types of linearly graded
o
rf d ability profile is described below in (24). One point to permeabilitywithanoffsetthatcouldbecomparedtothe
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