Different Approaches to Inductance Calculation of E-I Gapped Iron Core Inductor

Transcription

1 Ldja PETKOVSKA, Gga CVETKOVSKI Ss. Cyrl and Methdus Unversty f Skpje, Faculty f Electrcal Engneerng Dfferent Appraches t Inductance Calculatn f E-I Gapped Irn Cre Inductr Abstract. In the paper nductance calculatn by usng dfferent methds s presented. The bject f study s an Ε-Ι gapped rn cre nductr. The calculatn s started wth classcal analytcal apprach and by usng magnetc crcut thery (MCT); later n t s cntnued wth anther three numercal appraches, dne va Fnte Element Methd (FEM). The results f calculatns are cmpared wth the measured values f nductance. The evaluatn f dfferent methdlges s presented. Streszczene. Przedstawn różne metdy blczana ndukcyjnśc. Przedmtem badań jest szczelnwy dławk EI z rdzenem. Pkazan klasyczne pdejśce analtyczne za pmcą sec reluktancyjnych, a także trzy pdejśca numeryczne, wykrzystujące metdę elementów skńcznych. Wynk blczeń prównan z wynkam pmarów. (Różne pdejśca d blczena ndukcyjnśc szczelnwych dławków EI z rdzenem). Keywrds: Irn Cre Inductr, Magnetc Crcut Thery, Fnte Element Methd (FEM), Magnetsng Inductance, Leakage Inductance. Słwa kluczwe: dławk rdzenwy, tera bwdu magnetyczneg, metda elementów skńcznych, ndukcyjnść magneswana. Intrductn Irn cre cls, als called nductrs r reactrs, are used fr varus purpses n electrcal engneerng practce. Dependng n the pwer, nductrs are perfrmed as sngle phase, r three phases. Beng mre ecnmcally, they are usually made wth an rn cre. In practcal use, the mst frequently s fund a shell-type Ε-Ι shaped cre. The value f magnetzng nductance s ne f the mst mprtant features f rn cre nductr. In rder t adjust the requested value f the nductr nductance, a changeable ar-gap s always antcpated. Ether fr desgners r fr users, t s always an nterestng ssue hw t calculate the magnetzng nductance value, as accurate as pssble. Althugh the nductance can be easly estmated analytcally by usng smple magnetc crcut thery, ths apprach typcally s nt accurate as gnres flux leakage and frngng effects. T reslve these effects n mre detals, a numercal methd by usng the Fnte Element Analyss s appled. The gal f the present paper s t shw hw nductance s calculated by usng varus Fnte Element Methd (FEM) smulatns and t cmpare these results t the apprxmatn btaned thrugh a magnetc crcut apprach. Evaluatn f the dfferent appraches, wth respect t the accessblty and accuracy, s dne n the bass f the testng results. Example Gemetry The studed bject s a sngle phase varable nductance rn cre nductr, cnnected t street lghtng lamps; the apparent and actve pwer are varable, dependng n the wrkng and ladng cndtns []. The nductance varatn s btaned by peratng n the ar-gap length and n the number f turns f the exctatn wndng. Anther pssblty s t change the depth f the magnetc cre, nfluencng t the magnetsng and crcut parameters f the analysed structure. Dependng n the requested value f the nductr reactance, the peratng length f the ar gap can be adjusted n the range frm.4 mm up t. mm. The example gemetry s cmpsed f an E-I gapped lamnated cre wth a rectangular crss-sectn extendng fr 36 mm n the nt-the-page drectn. The rn cre s shell-type frmed by slcn-steel lamnatns f magnetc materal type EN 3-5, havng the relatve permeablty μ r 63. The rated wrkng value f the ar-gap length s.85 mm. The I-part s freely dsplacng, whle exctatn wndng les n the slts f the fxed E-part f the nductr and s cnssted f 68 cpper wre turns wth heavy buld nsulatn; the cpper fll fractn n the cl wndw s 49%. In accrdance wth the applcatn t whch s assgned the cnsdered nductr, the rated data are: vltage supply U n 5 Hz; current I n.5 A; pwer P n 5 W and pwer factr csϕ n.55. The startng vltage f frng the lamp s prescrbed t be U mn 6 V. The frnt vew f the gapped rn cre nductr, as well as ts gemetrcal crss-sectn s shwn n Fgure, (a) and (b) respectvely. (a) frnt vew Fg.. Studed E-I gapped rn cre nductr (b) crss-sectn Prblem Defntn The am f the study s determnatn the value f the magnetsng nductance f an E-I gapped rn cre nductr (ICI). The man gal s t present and t apply dfferent appraches t calculatn f ths mprtant electrmagnetc quantty and t dscuss bth ther accuracy and avalablty. In the paper, there wll be presented the methdlgy fr calculatn f the leakage nductance, as well; ths quantty cmbned wth the magnetsng nductance s enablng the pssblty t determne the man nductance f the rn cre nductr. The value f the man nductance s defnng the nductr's man reactance; cnsequently, t s determnng the behavur f the devce, and n partcular, the pwer factr and the reactve energy cnsumptn f the nductr. In the electrmagnetc devces where the varatn f nductance s nfluenced by the varatn f the ar-gap length, t s always very mprtant t nvestgate and study ther mutual dependence. It s bvus that magnetsng nductance f the gapped rn cre nductr s dependent n the ar-gap length. In accrdance t the methdlgy f each partcular apprach, t s cmprehended t carry ut several calculatns and/r analyses; thus, cllectng and cmbnng the results, nstead f calculatn nly ne value, the whle characterstcs wll be derved. 36 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 33-97, R. 8 NR 5/6

2 Furthermre, f Fnte Element Methd (FEM) s used, t s als pssble t assess the attractve frces between the mvable and fxed part f the analysed nductr. Leakage Inductance Calculatn The pwer factr f the nductr s dependent n the man reactance X, determned by the man nductance L. The nductance L f the rn cre nductr s cnssted f tw man parts: the leakage nductance L S f the cl (wndng), usually cnstant and nn-dependng n current value; the magnetsng nductance L m, usually dependng n the saturatn level f the rn cre, and changng nnlnear wth the exctatn current. () L L s + Fr calculatn f the leakage nductance, we are usng the exstng analytcal prcedures []. Intrducng the expressn fr detaled analytcal calculatn f the leakage nductance and havng avalable the exact gemetrcal dmensns f the studed Ε-Ι cre structure, the wndng leakage nductance s determned t be: () L S.98 [H] Ths value f the leakage nductance wll be used later n, when splttng the cmpnents f the man nductance. Magnetzng Inductance Calculatn Analytcal Apprach Calculatn f magnetzng nductance s always mre cmplcated nnlnear prblem, and dfferent appraches bth analytcal and numercal, can be used. Sme f them are mre exact, but mre cmplcated fr practcal use; anther are smple t apply, but nt gve enugh accurate results. The ntentn f ths paper s t present several dfferent appraches t nductance calculatn and t cmpare the btaned results. The apprasal f calculated results wll be gven n the bass f the measured values. The bref descrptn f the appled methds and the cmputatnal results are gven n cntnuatn. By usng Flux Lnkage Lnear case As a gudelne and frst step, a rugh apprxmatn f nductance value s dne at wrkng cndtn wth mnmum value f the vltage supply U mn 6V, and wth startng current n the cl I st.35a. Presumng the lnear magnetc cre, and adptng fr wndng vltage drp an average value f 5%, ne can calculate the wndng flux lnkage frm the expressn: E. 85U mn (3) Ψn NΦn L In fn fn where: N68 s number f turns f the wndng; The man nductance L s determned t be: (4) L.4538 [H] The magnetsng nductance s easly fund, by usng the value f the leakage nductance frm Eq. (): (5) L m.334 [H] By usng Flux Lnkage Nn Lnear case The prevus result s predcted by ntrducng several assumptns and ths smple apprach tends t rase the value f magnetzng nductance. Frst, the vltage drp f 5% s nly tentatve value. Secnd, even the vltage s wth a snusdal wave frm, the magnetsng current s sgnfcantly dstrted; applyng the fundamental harmnc fr current, we d nt take nt accunt the hgher level saturatn f the magnetc cre. Obvusly, ths value gven by Eq. (5) can be accepted nly as the frst apprxmatn quantty, and shuld be exacted. In Eq. (3) t s ntrduced a crrectn ceffcent fr the nn snusdal current wave. (6) Ψn NΦn L + Ls I νin where: IνI n s nn-snusdal magnetsng current, defned by a dstrtn ν and rated current I n. The current dstrtn ν s cmputed by mprvng the analytcal prcedure [3]. T take nt accunt saturatn f the cre, we suggest applyng an teratve prcedure. Ths apprach s recmmended t be used by the desgners, when develpng a new nductr []. The prcedure s carred ut teratvely n several steps; t takes 4-5 teratns t get equal the presumed (ntal) and the calculated (fnal) value f nduced vltage E, gvng the accurate value f the magnetc flux densty B n the cre. Afterwards, ne can fund fr ν.7 and the man nductance s calculated, (7) L.435 [H] whle the magnetsng nductance wll be: (8) L m.837 [H] By usng Magnetc Crcut Thery MCT A mre elabrated analytcal crcut mdel, based n the magnetc crcut thery f the E-I gapped rn cre nductr (ICI) s develped [,4]. It s pssble t take nt accunt the magnetc reluctance f the cre. At the begnnng as a gudelne, an analytcal expressn fr calculatn f the nductance s derved. In the frst apprxmatn we can easy assume that there s n frngng, but we d nt neglect the cntrbutn f the rn sectns, as usually the magnetc cre s saturated and the gap length s relatvely small. Frm the study f magnetc crcut, the fllwng expressn s dsplayed: (9) μn * g lfe,av + S g μr SFe where: g * g ttal length f the ar-gap alng a flux lne; S g crss sectn f the ar-gap; S Fe crss sectn f the rn cre; l Fe,av average flux lne length; μ r 65 relatve permeablty f the cre materal. If we cnsder nly the ar-gap magnetc feld, then we derve a smpler expressn: () S g N μ * g The nductance s essentally nversely prprtnal t the ar-gap length. Thus, cnstrants mpsed n the varatn f the ar-gap crrespnd t cnstrants n the varatn f the nductance tself. The varatn f peratng ar-gap length g f the ICI s between.4 and. mm. All gemetrcal dmensns f the analysed E-I gapped rn cre nductr, havng the tplgy dsplayed n Fg. (b), are knwn exactly. As the analyzed structure s PRZEGLĄD ELEKTROTECHNICZNY, ISSN 33-97, R. 8 NR 5/6 37

3 charactersed by a varable ar-gap, the calculatn f the magnetsng nductance s perfrmed at a wrkng length g av.85mm and the result s: () L m.73 [H] Magnetzng Inductance Calculatn Numercal FEM Apprach When nnlnear materals are nvlved n the analysed gemetry and the cre s saturated even at a presence f ar gaps, the calculatns get smewhat mre cmplcated. In rder t take nt cnsderatn all electrmagnetc phenmena, t s appled Fnte Element Methd (FEM) fr calculatn magnetc feld dstrbutn f rn cre nductr. It s als pssble t take nt accunt secndary effects: an extra flux cmpnent alng frngng paths crssng the gaps (uter and nner) and leakage n the cl wndws. FEM Calculatn f the Magnetc Feld Numercal magnetc feld slutn s perfrmed runnng FEMM [5]. The detaled FEM mdel f the E-I nductr s develped; the exact gemetry, bundary cndtns and all materal prpertes are carefully bult-n the FEM mdel f the studed structure. A farly fne mesh, especally n the ar-gap s antcpated; at maxmum length f. mm t s cnssted f 3,955 ndes and 47,648 elements. As an example, n Fg. s presented magnetc feld dstrbutn at rated current I n.5 A, fr the fllwng gaps: (a) g mn.4 mm; (b) g av.85 mm; (c) g max. mm. In rder t enable an evdent cmparsn f the feld strength, t s adpted the same scale f the flux densty ntensty and the same dfference f magnetc vectr ptental value A between tw adjacent flux lnes; cnsequently, each fgure s charactersed wth dfferent number f flux lnes. In addtn, t s analysed flux densty dstrbutn alng the mddle f the ar gap. The characterstcs at the rated exctatn current and three typcal ar-gap length are gven n Fg. 3 (a), (b) and (c), n the same way as prevusly (a) g mn.4 mm; A max.35 Vs/m; n 33 lnes 3 4 (a) g mn.4 mm; BBmd.854 T (b) g av.85 mm; A max.83 Vs/m; n lnes 3 4 (b) g av.85 mm; BBmd.486 T (c) g max. mm; BBmd.44 T (c) g max. mm; A max.73 Vs/m; n 9 lnes Fg.. Magnetc feld dstrbutn n E-I gapped rn cre nductr Fg.3. Magnetc flux densty alng the md-gap lne f the mddle leg 38 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 33-97, R. 8 NR 5/6

4 Fnte Element Magnetzng Inductance The fnte element slutns can be used drectly t fnd nductance [4,6]. There are a number f well establshed numercal methds f fndng nductance and generally speakng all wrk very well. The values f nductance fund frm FEM cmputatns generally agree well wth measured results. Results f nductance calculatn cnverge quckly wth decreasng mesh sze. Cnsequently, when analysng the gapped rn cre nductr by FEM, a partcular attentn s pad n the ar-gap regn, where the man change f the magnetc ccurrences takes place. In Fg. 4 s presented mesh f the sectn nfluencng the mst n the magnetzng nductance value. The shwn sectn s taken ver the mddle leg f the nductr; the argap mesh s pled between tw blue lnes; the straght black lne s the md-gap lne t whch the Fg. 3 s crrespndng. cmbnatn f the (,g) values, n the whle range f ther change. It s pssble t perfrm magnetstatc smulatn f the nductr, r t analyse tme dependent varable feld prblem, by ntrducng crrespndng values f exctatn current n the bth cases. As the feld slutn has been btaned, magnetc quanttes f nterest can be cmputed. Seres f cmputatnal results are well establshed bass fr accmplshng numercal Fnte Element Analyss (FEA). The magnetzng nductance f the nductr can be calculated n several ways, by usng dfferent numercal appraches based n the FEM results. A. Once the pstprcessr f the sftware FEMM has been run, the magnetsng nductance can be derved frm the magnetc energy equatn. It can be nted that anther dfferent expressn fr magnetc feld energy n a system wth just ne crcut, at a specfc wrkng pnt defned wth a par (,g ), can be derved n a frm: (5) W Settng these tw expressns fr energy equal t ne anther and slvng fr L m yelds: Fg. 4. A part f the ar-gap mesh at g max. mm Dependng f the analysed prblem, ne can cmpute ether apparent r dfferental magnetsng nductance [4]; bth are functns f the ar-gap length g. On the ther hand, due t the nnlnear characterstc f ferrmagnetc materals f the cre, the nductances are als functns f the currents flwng thrugh the exctatn cl. In general, the apparent nductance s defned at a partcular ar-gap length g, cnsderng the peratng pnt (,λ ) f the rn cre nductr. If the exctatn current s, whle the flux lnkage s λ, the apparent nductance s determned by the relatn: () L (,g ) app λ where: current n the exctatn wndng; λ flux lnkage f the cl, at the wrkng pnt. The rat between an nfntesmal varatn f flux lnkage dλ, and the crrespndng varatn f current d, arund the same peratng pnt (,λ ), defnes the dfferental nductance f the ICI, whch s: (3) L (,g ) df λ d d (, λ ) In lnear cndtns, snce the permeablty f the cre materal s cnstant, the tw values btaned frm the last tw equatns are equal: L app L df LL m. Alternatvely, the nductances can be cmputed numercally frm the varatn f magnetc energy. One way f expressng the ttal magnetc feld energy n the magnetc prblems s va the ntegral: (4) W A J dv W (, g) V The FEM calculatns have been perfrmed at varus (6) A JdV V Fr the example prblem, J s nnzer nly nsde the cls, s the A J ntegral need nly be taken ver the cls. The result f perfrmng the calculatn s: (7) L m.95 [H] B. An alternatve apprach s t btan stred energy va anther ntegral: (8) W B H dv V where ths ntegral s taken ver the entre prblem dman nstead f just ver the cls and s drectly cmputed frm energy ntegratn. The magnetzng nductance s derved frm Eq. (5): (9) W L m Fr the cnsdered prblem, the result f calculatn yelds t the value: () L m.95 [H] C. Fnally, usng pstprcessng results f the numercal FEM cmputatn, we calculate the flux lnkage: () λ A JdS S Drect mplementatn f the Flux Lnkage/Current f the wndng cl unambguusly s nterpreted as magnetzng nductance. It s nt lkely that nductance btaned frm Eq. () n a nnlnear prblem wll be the result sught; but, as we use nnlnear FEM slver at a specfed wrkng pnt (,λ ) the PRZEGLĄD ELEKTROTECHNICZNY, ISSN 33-97, R. 8 NR 5/6 39

5 calculated value s qute accurate, and the FEM result s: () L m.986 [H] Inductance characterstcs In cntnuatn, n the bass f the cmputatnal FEM results, the magnetsng nductance characterstcs L m f(g,) wll be derved. It s requested t run prgram FEMM by changng gap length g and wndng current. The energy cncept (A) frm the prevus headng, as an apprach t magnetsng nductance calculatn, s selected. On the bass f cmputatnal FEM results, the L m f(,g) characterstcs are derved; they are presented n Fg. 5. magnetsng nductance [H] exctatn current [A] Fg. 5. Magnetsng nductance characterstcs L m f(,g) g mm g. mm g.4 mm g.6mm g.8 mm g. mm Frm an analyss f the characterstcs, t s evdent that greater ar-gap length means lnear magnetc feld, yeldng t almst cnstant magnetsng nductance, ndependent n the exctatn current. Cmparatve analyss The exact value f the nductance can be reached nly by measurements. After testng f the cnsdered E-I nductr [,8] t s btaned the relevant result fr the man nductance L.43 [H]. Usng the calculated value f the leakage nductance L S, the magnetzng nductance s: (3) L mtest.85 [H] The cmputed results wth prpsed methdlges are cmpared wth the measured results at peratng cndtns wth same nput vltage and/r current. Cmparatve analyss presented n Table, s the best way fr evaluatn f dfferent appraches fr calculatn f nductance. Table Cmparatve results f magnetzng nductance L m [H] Analytcal results Numercal FEM results Lnear Measured case MCT A. B. C..334 Nnlnear case Obvusly the lnear calculatns f the magnetsng nductance are nt enugh relable; ther nly advantage s the smplcty. The result s errneus fr +8.3%. In Table s presented the nnlnear analytcal result btaned by teratve prcedure prpsed by the authrs. The mprvement f the accuracy s evdent and the cmputatnal errr s nly +.4%; but t shuld be emphassed that the prcedure s rather cmplcated. Applcatn f the magnetc crcut thery (MCT) gves satsfactry results; the calculated result dffers frm the measured ne fr 4.3%. Calculatns based n the fnte element results, by usng three dfferent methdlges A, B and C are qute crrect. The cmputatnal errr s +3.54%, +.83% and +4.7%, respectvely; as expected the flux lnkage/current methd s the least accurate. But n general, the numercal FEM apprach s the mst accurate and relable. Cnclusn The am f the present paper s t shw hw nductance s calculated by usng varus pssbltes f Fnte Element Methd (FEM) smulatns and t cmpare these results t the apprxmatn btaned thrugh analytcal and magnetc crcut apprach. All btaned results are cmpared wth the expermentally measured value. The studed structure s an rn cre nductr wth varable ar gap. Fr rugh estmatn f the nductance t s suggested t start wth analytcal apprach. The magnetsng nductance can be easly calculated by usng smple magnetc crcut thery, althugh ths apprach typcally s nt enugh accurate as gnres flux leakage and frngng effects. The best apprach s t apply Fnte Element Methd. Cnsequently, the magnetsng nductances btaned frm the FEM smulatns are always hgher than thse analytcally cmputed; ths s essentally due t the flux lnes external t the ar-gap, as can be seen n Fg.. The numercal results are always clser t the measured values, prvng the FEM calculatns as the mst accurate. References [] Petkvska L. et. all, Develpment f a Gapped Irn Cre Inductr fr Street Mercury Lamps, R&D Prject fnanced by the Mnstry f Scence and Educatn, Fnal Reprt, pp., Faculty f Electrcal Engneerng, Skpje, Macedna, (3). [] K a l a n t a r v P.L., C e t l n L.A., Inductance Calculatns, Bk, Publshed by Energatmzdat, Sank Petersburg, Russa, (986). [3] M l a k a r F., K l a r I., Small Transfrmers and Inductrs,, Manual fr Electrcal Engneers, Vl. 8, Publshed by Elektrtehnsk Vestnk, Ljubljana, Slvena, (97). [4] B a n c h N., Electrcal Machne Analyss Usng Fnte Elements", Bk, CRC Press, Taylr & Francs Grup, Bca Ratn, FL, USA (5), pp [5] M e e k e r D., "Fnte Element Methd Magnetcs", FEMM Versn 4.., User s Manual, Natck, MA, USA, (6). [6] S a l n S. J., Fnte Element Analyss f Electrcal Machnes, Bk, Kluwer Academc Publshers, Nrwell, MA, USA, (995). [7] Petkvska L., Cvetkvsk G., Sarac V., Fnte Element Methd Cupled wth Genetc Algrthm as a Desgn Optmsatn Tl f Electrmagnetc Devces, Shrt Papers Prceedngs f the XVIII Sympsum n Pwer Electrncs, Electrcal Drves, Autmatn and Mtn SPEEDAM'4, Vl. /, TD pp. 3-6, Full manuscrpt publshed n CD, pp. 66-6, Capr, Naples, Italy, (4). [8] Petkvska L., Cvetkvsk G., Dfferent Appraches t Inductance Calculatn f an E-I Gapped Irn Cre Reactr, Bk f Dgest f the Thrd Slvenan-Plsh Jnt Semnar n Cmputatnal and Appled Electrmagnetcs'5, pp , Marbr, Slvena, (5). Authrs: Prf. Dr. Ldja Petkvska, Ss. Cyrl and Methdus Unversty, Faculty f Electrcal Engneerng, P. O. Bx 574, Skpje, Macedna, E-mal:ldjap@etf.ukm.edu.mk,; Assst. Prf. Dr. Gga Cvetkvsk, Ss. Cyrl and Methdus Unversty, Faculty f Electrcal Engneerng, P. O. Bx 574, Skpje, Macedna, E- mal: ggacvet@etf.ukm.edu.mk; 4 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 33-97, R. 8 NR 5/6