ELECTRICAL ENGINEERING DEPARTMENT FINITE ELEMENTS FOR ELECTRICAL ENGINEERING

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1 ELECTICL ENGINEEING DEPTMENT FINITE ELEMENTS FO ELECTICL ENGINEEING mó gllo 6

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5 FINITE ELEMENTS FO ELECTICL ENGINEEING. FINITE ELEMENTS METHODS FO ELECTICL ENGINEEING PEFCE IMS To t to ctc th FE mthod to ls d dsg lctcl mchs d ts. To t to ctc th D fomlto. To t som commcl softw ogms fo modllg d lss of lctc d mgtc lmt o mch. SSESSMENT Nmcl solto of som lctos s std. If ossbl, com mcl slts wth ltcl o mtl slts. SPECILIZTION: Dvs d lctcl mchs. POGM OEIEW Hsto of FEM & FE FEM d FE t th EUETI lcto s Wht's th dffc btw FEM & FE?? Elctomgtcs & ltd lss INTODUCTION Mwll qtos Costttv ltos Elctosttc, Mgtosttc d Mgtodmc Flds Thml oblms od codtos dcto of D oblm to D oblm Mtls ots. L d No-l modls Pmt mgts PM modllg. NLYTICL SOLUTION Som mott thoms ltcl solto. Sto of vbls. Elctc mchs modllg Pmt Mgts Som mls NUMEICL SOLUTION. FUNDMENTLS ND SIC METHODS Gl cocts Clssfcto of lctomgtc EM oblms Ft Dffcs Mot Clo s mthod omto tchqs.gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC d 5

6 FINITE ELEMENTS FO ELECTICL ENGINEEING. PEFCE FINITE ELEMENT METHOD ONE DIMENSIONL FINITE ELEMENT NLYSIS Dsctzto d tolto Fomlto ssmbl of th qtos od codtos TWO DIMENSIONL FINITE ELEMENT NLYSIS Dom dsctzto Itolto tol fomlto ssmbl to fom th sstm of qtos Icooto of th bod codtos of th thd d Imosto of th Dchlt bod codto Nol oblms Pmt mgts PM modllg TNSIENT SOLUTION. oltg fd lctomgtc dvcs Colg of fld d lctcl cct qtos. Thc codctos. Th codctos. Eqtos fo th whol dom. MOEMENT MODELLING FO ELECTICL MCHINES INTEGL EQUTION METHOD Cocto btw dfftl d tgl qtos Th Momt MEthod od Elmt MEthod Comso of th FE d E Mthods COMPUTTION OF OTHE QUNTITIES. Post-ocssg. sc qtts. Dvd qtts. Eg Stod th Mgtc Fld. Ld Fl. Idctc c mf sstc. Jol losss ow. Cctc. Edd ct losss Foc d Toq. Mwll Stsss. tl Wo mthod. Co Losss GENEL CHITECTUE OF CD SYSTEM SED ON THE FINITE ELEMENT METHOD Th dt t modl Th solv Th ostocsso EmlsD: FEMM, MXWELL_S. PPLICTIONS D Solvd lctos. lctos to solv. sc blogh M... Ch, S.J. Slo. Nmcl mthods lctomgtsm. Ed. cdmc Pss.. J.P.. stos, N. Sdows. Elctomgtc modllg b ft lmt mthods. Ed. Mcl-Dc.. S.J. Slo. Ft lmt lss of lctcl mchs. Ed. lw gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC d 5

7 FINITE ELEMENTS FO ELECTICL ENGINEEING. PEFCE Comlmt blogh N. ch. Elctcl Mch lss sg Ft Elmts. CC. Tlo&Fcs. 5 G.. ch. Ft Elmt lss. Schm s Otls. Mc Gw-Hll P.P. Slvst,.L. F. Ft lmts fo lctcl gs d dto. Cmbdg Uvst Pss Hm, : lms. Nmcl modllg d dsg of lctcl mchs d dvcs. Ed. WIT Pss J. c, T.W. Psto. Ft lmt mthods lctcl ow gg. Ed. Ofod Uvst Pss.. M.N.O. Sd. Nmcl tchqs lctomgtcs. Ed. CC Pss. D. Pol, C.. bb. od Elmt Mthods fo Elctcl Egs. Wt Pss. 5.. Nogd. Élctodmq lqé. ss t cs hsqs d l élctotchq. Ed. Dod. 5. J. Sls. Comttol Mgtcs. Chm&Hll Ls Tchqs d l Ig. Softw s mls D. M. FEMM. s s ml. D. M. MIGE. s s ml. soft Co. Mwll D Stdt vso. GUCD. EFCD 6.. Ml..gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC d 5

8 FINITE ELEMENTS FO ELECTICL ENGINEEING. PEFCE Cos Ogzto Ths cos wll b dvlod tho d ctc sssos dctd th t tbl. Sssos wll b hld t : EUETI. 87 Ugll Stt. Mtg oom. floo. 5 to 7 h ch Thsd. dtld ogm s gv fl tbl. Ths cos s qvlt to ECTS ECTS s qvlt fom 5 to h of stdt ffot. Hlf of ths tm s dvotd to clsss both tho d ctc. Th stdt wll dvot th st of hs tm to sol std, solto of st css d dvlo th fl oct. Ech w I wll t docmtto ltd to m t clss o cos wb g. o I wold l o d d std ths bfo th clss. o Dts dctd th sllbs sbct to chg. Not: If docmtto cds th wb g qot, I wll CD-OM wth LL of th docmtto d th softw css fo ths cos wth ts Noml commcto mst b cd b -ml. I som css tlho o sol commcto wll b cctd. Ech w I wll st css. o Som css solvd. o Yo mst solv th solvd css d t ths to m bfo th fl dt dctd th docmtto. fl oct mst b comltd. o Yo mst tl to m bot ths wo. o If ossbl I f lctos tht tstg fo o. o If ossbl, o mst obt mtl slts to com wth mcl slts. o If ossbl, th oct mst b dvlod go to cog th go d tdscl wo. Gos m cld to to ol. Ectos mst b ovd b m. Th fl oct stto mst t lc blc ssso ttds wll b oth stdts of ths cos, oth ofssos, d, of cos, m Pocts mst cot ll of th followg tms: Poblm to solv. Mthmtcl modl of th oblm. Dwg of th modl. Smlfctos d smmts. od codtos d socs. Msh sd. Ghcl ms of fl, ottl, tc. M mcl slts obtd. Comso wth ltcl d/o mtl slts If ossbl Coclsos. Elctoc d wth ll of th smlto fls. ll wo mst b std lctoc fomt sch s MSWod o PDF fls. Pls t o wo. I do t cct hdwtt wo o wo whch hs b scd. Good lc.gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC d 5

9 FINITE ELEMENTS FO ELECTICL ENGINEEING. PEFCE SYLLUS W Clss Lsso //6 Tho INTODUCTION Mwll qtos, Elctosttcs, Mgtosttcs d Mgtodmcs Flds. Mtls ot. od codtos. 9//6 Tho INTODUCTION D d D modls. L d No-l modls. Pmt mgts. lctos. Elctc mchs modllg. Oth: Thml modllg. 6//6 Tho NUMEICL SOLUTION : Itodcto d bsc mthods. Ft dffcs. Motclo mthod. omto tchqs. Wghtd sdl mthods: ot collocto, sb dom collocto, lst sqs, Gl mthod. tol cls. //6 Tho FINITE ELEMENTS METHOD I ONE DIMENSIONL FINITE ELEMENT NLYSIS. Dsctzto d tolto. Fomlto. ssmbl of th qtos. od codtos. //6 Tho FINITE ELEMENTS METHOD II TWO DIMENSIONL FINITE ELEMENT NLYSIS. Dom dsctzto. Itolto. tol fomlto. ssmbl to fom th sstm of qtos. Icooto of th bod codtos. Nol oblms. Pmt mgts PM modllg. //6 Tho FINITE ELEMENTS METHOD III TNSIENT SOLUTION. oltg fd lctomgtc dvcs. Colg of fld d lctcl cct qtos. Thc codctos. Th codctos. Eqtos fo th whol dom. MOEMENT MODELLING FO ELECTICL MCHINES. 7//6 Tho INTEGL EQUTION METHOD Th Momt Mthod. od Elmt Mthod. /5/6 Tho COMPUTTION OF OTHE QUNTITIES Eg. Fl. Mgtc Losss. sstc. Idctc. Foc. Toq. Oth qtts. /5/6 Tho GENEL CHITECTUE OF CD SYSTEM SED ON THE FINITE ELEMENT METHOD Dt t modl. Solv. Postocsso. EmlsD: FEMM, MIGE, MXWELL_S, EFCD. 8/5/6 Pctc PPLICTIONS D Elctosttc flds: Dlctc mtls. Ecs: Hgh oltg solto lss. Stto cts: codctg mtl. Ecs: comtto o fs sstc. Noml ct. 5/5/6 Pctc PPLICTIONS D Mgtc flds: Elctc mchs. Pmt mgts. Ecs: D.C. mch lss. mt cto. b Dsg of fld col o ss DC moto. /6/6 Pctc PPLICTIONS D Edd cts. Ecs: Std to fqc vto d sh of th oto slot th otoc mdc schoos mch 8/6/6 Pctc PPLICTIONS D Comtto of losss. sstcs. Idctcs. Foc. Toq. Ecs: qvlt cct of dcto mch. b Toq clclto of dcto mch. c M dctc clclto o slt ol mch. 5/6/6 Pctc PPLICTIONS D Tsts. Ecs: Htg d Coolg of Slot of Elctc Mch..gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 d 5

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13 FINITE ELEMENTS FO ELECTICL ENGINEEING OEIEW OEIEW Th fld of Elctcl Egg o Elctomgtsm c b sbdvdd to th mo s: Thotcl Elctct Elctomgtsm ld Elctct Elctomgtsm Comtcol Elctomgtsm Thotcl lctct dls wth fdmtl lws d cls of lctomgtsm stdd fo th tsc sctfc vl. ld lctct tsfs ths thotcl owldg to sctfc d gg lctos, scll s gds th costcto of mthmtcl modls of hscl hom. Comttol lctomgtsm solvs scfc oblms b smlto thogh mcl mthods mlmtd o dgtl comts. Phsg old o bot mthmtcs, o m df comttol lctc s so who schs fo soltos to gv oblms, ld lctc s so who schs fo oblms tht ft gv soltos, d thotcl lctc s so who c ov th stc of oblms d soltos. lss d dsg of lctcl qmt s dffclt ts d to som scts: Coml gomt Md st of mtls volvd. Som of ths hv o-l chctstcs. Md hom st: o Elctomgtc fld. o Thml scts o Mchcl scts. Dmcl scts ddc o tm. Ol fw sml css c w fd o tm ltcl soltos. Nmcl solto s th ol vlbl mthod to fd sws to m oblms of lctcl qmt dsg. Th fst st of lss s th slcto of wht scts cosdd o cs. Oth scts glctd o god. Ths st s md MODELING. Ths sct s th coc of th Thotcl Eg. ltd to ths modllg s th dfto of th vs of lss; fo ml f w d to dtm th dstbto of tmt oom, w glct th vto of th to tmt ths s cosdd costt oth sct s th lvl of dtl th lss, fo ml: W c cosd ll of th codctos to slot stl o togth. W c cosd dsct dstbto of cts dsct mb of slots o cotos omto of ths sht of ct. Sttc o dmc lss cosdto o ot of th tm vtos Etc.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 8

14 FINITE ELEMENTS FO ELECTICL ENGINEEING OEIEW Th scod st of lss s th slcto of mcl mthod to solv th oblm. Ths ocss s md DISCETISTION of th oblm. Th som mthods fo ths ocss: Ft Elmt FEM od Elmt EM Ft Dffc FDM Momts mthod MM Motclo mthod MCM FDM s dqt fo l oblms wth gl gomt d tm-ddt oblms. E Ms sd som css wthot mshg th comlt gomt. Th mthmtcl scts of ths lmts ts lcto to l oblms. MM d MCM sd ol l css d sml gomts. FEM s th most sd mthod to l d ol oblms wthot stctos o th gomt. Ths sct s th coc of Comttol Eg. Wht Dos Dsctsto Loo L? Th coct of dsctsto wll b tl llsttd thogh tl ct oblm: fd th mt L of ccl of dmt d. Sc L π d, ths s qvlt to obtg mcl vl fo π. Dw ccl of ds d dmt d s t fg. Th scb gl olgo of sds, wh 8 Fg b. m olgo sds s lmts d vtcs s ods. Lbl ods wth tgs, Etct tcl lmt, s tht og ods 5, s show Fg c. Ths s stc of th gc lmt ctd Fg d. Th lmt lgth s L s π. Sc ll lmts hv th sm lgth, th olgo mt s th th omto to π s π L L L d π s. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 8

15 FINITE ELEMENTS FO ELECTICL ENGINEEING OEIEW ls of π obtd fo,,, lstd th scod colm of th followg tbl. s c b s th covgc to π s fl slow. Th s of th ow of s dqt to vod th s of ssodl fcto: π If w stt wth, cosdg tht s, d s α s α w c clclt of th vls s th followg tbl. Som ds bhd th FEM d oth dsctzto mthods c b dtfd ths ml. Th ccl, vwd s soc mthmtcl obct, s lcd b olgos. Ths dsct omtos of th ccl. Th sds, md s lmts, scfd b th d ods. Elmts c b std b dscoctg ods, ocss clld dsssmbl th FEM. Uo dsssmbl gc lmt c b dfd, ddtl of th ogl ccl, b th sgmt tht cocts two ods d. Th lvt lmt ot: sd lgth L, c b comtd th gc lmt ddtl of th oths, ot clld locl sot th FEM. Th tgt ot: th olgo mt, s obtd b coctg lmts d ddg th lgth; th cosodg sts th FEM bg ssmbl d solto, sctvl. Th thd d fl st of lss s comtto of ddtol slts d lss of th solto. If th slts t th dsd o wshd slts th lss mst b td.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 8

16 FINITE ELEMENTS FO ELECTICL ENGINEEING OEIEW Ths sct s th coc of ld Eg. Hsto of FEM & FE 85 Schllbch vodd th dfftl qtos b lcg thm wth omt st of lgbc qtos. 9. Cot sd cws cotos fcto dfd ov tgl. 956 T, Mt, To stblshd bod dfto of mcl lss. 96 Clogh sd fo th fst tm th tm of Ft Elmt Mthod. El 7 s FEM ws lmtd to sv mfm comts owd b th otcs, tomotv, dfs, cl dsts d, gl ss, hv dst. 7 s FEM fth hcd b Zwcz&Chg vtol och: Llc d Posso s qtos. Mthmtcs dvlod btt soltos: Gl, tz mgd s otmm soltos fo ct ctgos of gl oblms. Modlg d solto of o-l oblms. 995 mo th 8 blshd s bot Ft Elmt lss. Ft Elmt lss FE ws fst dvlod 9 b. Cot, who tlzd th tz mthod of mcl lss d mmzto of vtol clcls to obt omt soltos fo vbto sstms. Shotl thft, blshd 956 b T, Clogh, Mt, & To stblshd bod dfto of mcl lss. Ths ctd o th "stffss d dflcto of coml stcts".. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 8

17 FINITE ELEMENTS FO ELECTICL ENGINEEING OEIEW th l 7's, FE sd ol o sv mfm comts gll owd b th otcs, tomotv, dfs, d cl dsts, d th sco of lss ws cosdbl lmtd. Ft Elmt tcholog ws fth hcd dg th 7's b sch ol s Zwcz & Chg, wh th ld th tcholog to gl oblms dscbd b Llc & Posso's qtos. Mthmtcs w dvlog btt solto lgothms, th Gl, tz & lgh-tz mthods mgd s th otmm soltos fo ct ctgos of gl t oblms. Lt, cosdbl sch ws cd ot to th modlg & solto of o-l oblms, Hto & Csfld bg mo cotbtos. Whl cosdbl stds w md th dvlomt of th ft lmt mthod, oth s dd ot m sttc. owfl msh gto lgothms hv b dvlod. Commcl gtos hv th cblt of mshg ll bt th most dffclt gomt. So CE cocts hv lso mgd, t s ot sl to hv sgl CD modl fo odcg gg dwgs, cg ot mtc & ssmbl lss, s wll s bg sd fo ft lmt modllg. D to th d dcl th cost of comts d th homl cs comtg ow, st d dsto comts cbl of odcg cct slts fo ll ds of mts stdd PC's ov tms mo owfl th th bst scomts of th l 9's. Th ft lmt mthod ow hs t's oots m dscls, th d slt s tcholog tht s so dvcd tht t s lmost dgshbl fom mgc. Th vst ctlog of cblt tht comss FE, wll o dobt gow cosdbl lg th ft. CE s h to st, bt od to hss t's t ow, th s mst b fml wth m cocts, cldg th mchcs of th oblm bg modlld. ll lss q tm, c & most mottl, cfl lg. FEM d FE t th EUETI 98. Fst FEM cos, tght b J.C. Sbod ENSIEG, Gobl, Fc 98. Som ofssos vstd ENP d Gobl Fc. Softw FLUX-D t ts l lss Doctol dsstto of J. Llvís Ect dtmto of th ottls bg tsfomto s. th s of th momt s mthod. 99. Fst cotct wth C. Lmos ts Comb Uvst, Potgl. 99. Glzd Lst Sqs Mthod to modlzto of th stct twol modl 99.. gllo vstd th Comb Uvst fo td t CDdMG softw, dvlod b C.L. ts. Th Elctotchcl Dtmt bogth ths softw. 99. Elctotchcl Dtmt ogzd th fst cos bot Th FEM ld to Elctcl Egg. Ths cos ws tght b M. ot,. glló, J. Moó d wth th collboto of C.L. ts Comb Uvst 99-. CDdMG softw s sd th gl coss Dsg of Elctcl Mchs -5. FEMM softw s sd th gl coss Dsg of Elctcl Mchs, Eg Sstms d oth.. d ls sttd ts doctol dsstto bot hstss modllg d ts lcto to th lctcl mchs dsg.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 8

18 FINITE ELEMENTS FO ELECTICL ENGINEEING OEIEW lcto s I ssc, th ft lmt s mthmtcl mthod fo solvg od & tl dfftl qtos. cs t s mcl mthod, t hs th blt to solv coml oblms tht c b std dfftl qto fom. s ths ts of qtos occ tll vtll ll flds of th hscl sccs, th lctos of th ft lmt mthod lmtlss s gds to th solto of ctcl dsg oblms. D to th hgh cost of comtg ow of s go b, FE hs hsto of bg sd to solv coml & cost ctcl oblms. Clsscl mthods lo sll cot ovd dqt fomto to dtm th sf wog lmts of mo cvl gg costcto. If tll bldg, lg ssso bdg o cl cto fld ctstohcll, th coomc & socl costs wold b cctbl hgh. I ct s, FE hs b sd lmost vsll to solv stctl gg oblms. O dscl tht hs ld hvl o th tcholog s th osc dst. D to th tm dmds fo fst, stog, lght & mo ffct cft, mfcts hv to l o th tchq to st comttv. t mo mottl, d to sft, hgh mfctg costs of comots & th hgh md covg tht th dst s osd to, cft coms d to s tht o of th comots fl, tht s to cs ovdg th svc tht th dsg tdd. FE hs b sd otl hgh volm odcto & mfctg dsts fo m s, s to gt odct dsg wog wold b dtmtl. Fo ml, f lg mfct hd to cll o modl lo d to sto dsg flt, th wold d hvg to lc to mllo stos. Smll, f ol ltfom hd to sht dow d to o of th mo comots flg ltfom fm, tts, tc.., th cost of lost v s f gt th th cost of fg o lcg th comots, ot to mto th hg vomtl & sft costs tht sch cdt cold c. Th ft lmt mthod s v mott tool fo thos volvd gg dsg, t s ow sd otl to solv oblms th followg s: Stctl stgth dsg Stctl tto wth fld flows lss of Shoc dwt & mtls costcs Thml lss btos Csh smltos Fld flows Elctcl lss Mss dffso clg oblms Dmc lss Elctomgtc vltos Mtl fomg Cold lss Nowds, v th most sml of odcts l o th ft lmt mthod fo dsg vlto. Ths s bcs cotmo dsg oblms sll cot b solvd s cctl & chl sg oth mthod tht s ctl vlbl. Phscl tstg ws th om s go b, bt ow t s sml too sv.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 8

19 FINITE ELEMENTS FO ELECTICL ENGINEEING OEIEW Wht's th dffc btw FEM & FE?? Th tms 'ft lmt mthod' & 'ft lmt lss' sm to b sd tchgbl most docmtto, so th qsto ss s th dffc btw FEM & FE? Th sw s s, th s dffc, lbt sbtl o tht s ot ll mott ogh to los sl ov. Th ft lmt mthod s mthmtcl mthod fo solvg od & lltc tl dfftl qtos v cws oloml tolto schm. Pt sml, FEM vlts dfftl qto cv b sg mb of oloml cvs to follow th sh of th dlg & mo coml dfftl qto cv. Ech oloml th solto c b std b mb of ots d so FEM vlts th solto t th ots ol. l oloml qs ots, whl qdtc qs. Th ots ow s od ots o ods. Th sstll th mthmtcl ws tht FEM c vlt th vls t th ods, th s th ovtol mthod tz, th sdl mhod Gl & th vtol mthod lgh-tz. Th ft lmt lss s mlmtto of FEM to solv ct t of oblm. Fo ml, f w w tdg to solv D stss oblm. Fo th FEM mthmtcl solto, w wold obbl s th mmm ottl g cl, whch s vtol solto. s t of ths, w d to gt stbl lmt fo o lss. W m choos l stss, l st o smmtc t fomlto, wth l o hgh od olomls. Usg cws oloml solto to solv th dlg dfftl qto s FEM, whl lg th scfcs of lmt fomlto s FE,.g. l st tgl qdtc lmt. Elctomgtcs & ltd lss M ds of lctomgtc homo c b modld, fom th ogto of mcowvs to th toq lctc moto. lss of lctosttc d mgtc flds ssg thogh d od stct ovds sght to th sos, d hc ms fo gltg ths flds to tt scfc soss. FE c b sd to ls th l lctc o mgtc bhvo of dvcs. lss tcll volv th vlto of mgtc, lctc d thml flds. Fth lctos cld th lss of sh-mmo mtls & zolctc ffcts. lss c b sttc, hmoc o tst stt t. D to th comlt of th ctcl lctos of th tchq, t s ot sl to hv mgtc, dlctc d thml colgs sgl modl. Sch coml lss gll m lstc modllg doos ts. lcto s Th lcto s cld, bt ot lmtd to th dsg of:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 8

20 FINITE ELEMENTS FO ELECTICL ENGINEEING OEIEW ottg mchs DC motos, schoos mchs, dcto motos, st motos, colg dvcs, bshlss motos, swtchd lctc motos, PM motos, gtos Eg tsf d covso modls tsfoms, cbls, hgh voltg dvcs, sltos, coctos & fss. Elctcl cttos l motos, lctomgtc bs, cotctos, mgtc bgs, fl ctos, lctomgtc lchs. Ssos cctv d dctv, sd, dd cts o dstctv tstg, mgtosco, solvs, lctc mts. Fld gtos mss sctomts, mgtc codg, olsto flds, mgtsto dvcs. lss Ts Mgtosttc lss. Mgtc lss s sd to dsg o lz vt of dvcs sch s solods, lctc motos, mgtc shlds, mt mgts, mgtc ds dvs, d so foth. Gll th qtts of tst mgtosttc lss mgtc fl dst, fld tst, focs, toqs, dctc, d fl lg. Tst Elctomgtc lss. Tst mgtcs llows fomg tst o std stt C lss dsgg fo vt of DC o C dvcs sch s lctc motos, tsfoms, d so foth. Gll th qtts of tst tst mgtcs lss tm fctos of mgtc fl dst, fld tst, tl, dcd d totl ct dsts, focs, toqs, dctc, d fl lg. Tm-Hmoc Elctomgtc lss. Tm-hmoc lctomgtc lss s sd to lz mgtc flds csd b lttg cts d, vc vs, lctc cts dcd b lttg mgtc flds dd cts. Ths d of lss s sfl wth dfft dcto dvcs, solods, lctc motos, d so foth. Gll th qtts of tst hmoc mgtc lss lctc ct d ts soc d dcd comot, voltg, gtd Jol ht, mgtc fl dst, fld tst, focs, toqs, mdc d dctc. Elctosttc lss. Elctosttc lss s sd to dsg o lz vt of cctv sstms sch s fss, tsmsso ls d so foth. Gll th qtts of tst lctosttc lss voltgs, lctc flds, cctcs, d lctc focs. Ct Flow lss. Ct flow lss s sd to lz vt of codctv sstms. Gll th qtts of tst ct flow lss voltgs, ct dsts, lctc ow losss Jol ht. Thml lss. Thml lss ls mott ol th dsg of m dfft mchcl d lctcl sstms. Gll th qtts of tst thml lss tmt dstbto, thml gdts, d ht losss. Tst lss llows o to smlt tsto of ht dstbto btw two htg stts of sstm. Stss lss. Stss lss ls mott ol dsg of m dfft mchcl d lctcl comots. Gll th qtts of tst stss lss dslcmts, sts d dfft comots of stsss.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 8

21 INTODUCTION

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23 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM INTODUCTION Th oto of lctcl sstms dsgd to fom ct gg tss dds, l lst t, o lctcl, lctomchcl, o lctochmcl hom. Th lctcl scts of ths lctos dscbd b Mwll s qtos. Th tho of Elctomgtcs EM too log tm to b stblshd. It c b dstood b th fct tht th EM qtts c ot b s o tochd cotl to oths, sch s mchcl d thml qtts. ctll, th mot of th EM hom w stblshd b oth sctsts bfo Mwll 8 879: m Gss Fd Lz mog oths Colomb, Lotz, Llc. Mwll, todcg ddtol tm 86 to m s lw, cold sthsz th EM fo qtos. Th hscl ossblt of ths go of qtos log wth costttv ltos s so hgh tht v dfft hom.g. mcowvs d mt mgt flds c b csl dscbd b t. ddtoll, ths qtos svvd th fomlto of ltvt d w stmtl shg t; ths th lso svvd th todcto of qtm tho! Whl th fomlsm d th bsc cocts of th EM ltvl sml, lstc oblms c b v comlctd d dffclt to solv. Som mls of ths comlctos : Comlctd gomt. Mtls o-lt. No-sttc fld socs. It s mossbl to fd ltcl soltos fo m oblms d tht s th m so wh mcl mthods hv bcom wdl sd tools Elctcl Egg tod.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

24 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Mwll qtos Fd todcd w coct whch h vsod th sc btw tctg chgs to b flld wth flds. Fom Fd s ot of vw, lctc d mgtc flds dfd t ot v wh th s o chg st th. Th flds dfd tms of th foc tht wold b td o tst chg q f t w todcd t movg t sd v t th tm of tst. Th foc s smmzd tms of th lctc fld tst E d mgtc fl dst b th Lotz foc lw: Gss s lw E v f q Gss s lw dscbs how th lctc fld tst s ltd to ts soc. Th t chg wth bt volm tht s closd b sfc S s ltd to th t lctc fl thogh tht sfc b S S ε E ds D ds ρ d ρ d D ε E D s th lctc dslcmt fl dst d ρ s th chg dst. If dfft mtls st, Gss s tgl lw qs tht ε E ε b Eb σ s th tfc of dfft mtls. σ s s th sfc chg dst d dots th oml comot of E. m s tgl lw Th lw ltg th mgtc fld tst H to ts soc, th ct dst J, s: C H dl S J ds d dt S D ds sfc ct dst sfc S css dscott of th mgtc fld tst. m s lw qs tht H H b th tfc of sfc. s th sfc ct dst d dots th oml comot of H.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

25 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Chg cosvto lw Embddd th lws of Gss d m s ltosh btw th chg d ct dsts. l m s lw to closd sfc, If th coto C s gdd s th dwstg d S s th bg, th ths lmt s o whch th stg s dw tght so tht th coto shs to zo.ths th coto tgl vshs: C H dl S J ds d dt S D ds ft, th sfc tgl of th lctc dslcmt c b lcd b th totl chg closd: S d J ds ρ d dt Ths s th lw of chg cosvto. Ths qto shows tht th t ct ot of th volm qs tht th t chg closd b dcsg wth tm. Th cott codto ssoctd wth chg cosvto s dσ s J J b Imlct ths codto s th ssmto tht J s ft. Ths, th codto dos ot cld th ossblt of sfc ct. dt Fd s tgl lw. Th lws of Gss d m lt flds to socs. Th sttmt of chg cosvto mld b ths two lws lts ths socs. Nw tgl lws todcd tht do ot volv th chg d ct dsts. Fd s tgl lw stts tht th cclto of E od coto C s dtmd b th tm t of chg of th mgtc fl lg th sfc closd b tht coto:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

26 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM C E dl d dt S ds Th cott codto ssoctd wth Fd s lw s E E b Gss s tgl lw of mgtc fl Th t mgtc fl ot of go closd b sfc S mst b zo: S ds Th cott codto ssoctd wth Gss s tgl lw of mgtc fl s: SUMMY OF MXWELL S INTEGL LWS Nm Itgl lw Gss s lw ε E ds ρ d D ε E m s lw Fd s lw Mgtc fl cott Chg cosvto S C S b D ds H dl S C S ρ d E dl S J ds d dt S ds d dt S ds d J ds ρ d dt D ds SUMMY OF CONTINUITY CONDITIONS Nm Cott codto ε E ε E σ Gss s lw b b s m s lw H H b Fd s lw E E Mgtc fl cott 5 Chg cosvto s J J b b b dσ dt. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

27 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Dvgc thom Th dvgc of vcto s dfd tms of th lmt of sfc tgl: dv lm S ds I Cts coodts, th dvgc oto s dv z z Ths slts sggsts lttv otto. Th dlt oto s dfd s: z wh, d t vctos. So tht dv c b wtt s dot odct dv Th followg tbl shows th dvgc oto som commo coodt sstms. Cts coodts Cldcl coodts Shcl coodts z z φ z φ z s θ s θ θ θ φ s θ φ Dvgc thom shows mg to lc th tgto ov th volm to tgto od th coto: S ds d Th dv otto sggsts tht ths combto of dvtvs dscbs th otflow of fom th ghbohood of th ot of vlto. Cl oto Th cl of vcto s dfd tms of th lmt of coto tgl:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

28 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6 C S dl S cl lm I Cts coodts, th cl oto s z z cl z z So tht cl c b wtt s vcto odct cl Ths Cts coodts: z z Th followg tbl shows th cl oto som commo coodt sstms. Cts coodts z z z z Cldcl coodts z z z z z φ φ φ φ φ Shcl coodts φ θ θ φ φ φ θ φ θ φ θ φ θ θ s s s Stos s cl thom shows mg to lc th tgto ov th sfc to tgto od th coto: S C ds dl o-zo cl mls tht th cosodg vcto fld hs ottol ot. O w to loo fo cl s to mg tht th vcto fld cosods to th flow of wt. If w lc smll ddl whl th fld th th sc of o-zo cl sggsts tht th whl wll ott. If hs o dvgc, fld s sd to b solodl. If t hs o cl, t s ottol.

29 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Gdt oto Dfftl of scl fcto s d d d dz z I vcto otto ths c b showd how: d z d d dz Th gdt of scl fcto s dfd s Cts coodts gd z gd s vcto, dcl to sfc o whch s costt d otg th dcto of csg. W lso ot tht gd ots to th dcto of mmm chg. lg th bov thoms Stos d dvgc w c obt th dfftl foms of Mwll s qtos: Gss s lw S ε E ds ρ d S ε E ds ε E d ε E d ρ d ε E ρ m s lw C S d H dl J ds D ds dt S S H dl H ds C S H ds S J ds d dt S D ds H J dd dt Fd s lw C C d E dl ds dt S E dl E ds S S E ds d dt S ds E d dt. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 6

30 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Mgtc fl cott S S ds Chg cosvto S S J ds J ds ds d dt d ρ d J d d dt d dρ dt J d ρ d J SUMMY OF MXWELL S DIFFEENTIL LWS Nm Itgl lw Gss s lw ε E ρ m s lw dd H J dt Fd s lw d E dt Mgtc fl cott 5 Chg cosvto dρ J dt Dfftl foms btt th tgl foms. Itgls foms ddts o volm d th sfc of tgto. Dfftl foms ddt of ths. Costttv ltos Th fld vctos D d E d lso d H ltd b th ots of th mtls t ot th fld go. Ths oft fd to s th costttv ots of th mtl d gv b: b c D ε E µ H J σ E. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 6

31 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM ε s th mttvt of th mtl, µ s th mgtc mblt of th mtl d σ s th codctvt of th mtl. I som css ths vls c b dctd ltv fom: ε ε ε µ µ µ ε, µ th vls fo f sc d ε, µ th ltv vls of th mtl. Eqtos to d ltos to c th wll ow Mwll s qtos. Wth th ddto of th cott codtos w c solv lctomgtc oblm. Mwll s qtos do ot m dstcto btw low d hgh fqc lctos, bt fo ctcl lctos t s ossbl to dt thm to ths two sttos. W wll b tstd low-fqc hom. Wh dscbg low fqc oblms th Mwll s qtos c b dvdd to two gos: Elctosttcs d Mgtosttcs. d, mott ot: Ths c b ttd ddtl! Th followg gs wll b dvotd to ths och. Mwll s qtos Low fqc Hgh fqc Wvs Elctosttcs Mgtcs Mgtosttcs Mgtodmcs. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 6

32 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6 Scod od otos. Llc oto. It s ossbl to comb two vcto otos o scl fctos d vcto fctos. O of ths s th Llc oto clld shot, th Llc, ths s th dv of gd of scl fcto U: U gd z U U gd dv Followg tbl shows th Llc oto som commo coodt sstms. Cts coodts z U U U Cldcl coodts z U U U φ Shcl coodts s s s z U U U θ θ θ θ θ If s vcto fcto, w c dmostt tht: cl cl dv gd wh s clld th vcto Llc of. Ths s wtt s Cts coodts z wh, fo ml, th comot th O dcto s: z

33 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Elctosttc, Mgtosttc d Mgtodmc Flds I gl, th two clsss of lctomgtc flds c d dscbd: Th tm ddt sttc d Tm vg flds. Th c b scl d vcto flds. tcl scl fld fo ml s th lctosttc ottl dstbto,,z btw chgd lctods; d th mgtc fld tst H,,z sodg ct cg codcto s tcl vcto fld. W hv to dstgsh btw th slow d fst vg lctcl ct flow fld wth gd to th gomtcl dmsos of th ct cg codcto. Elctomgtc flds sttc Nosttc Elctc E Mgtc Ct flow J Slow vg qs-sttc fst vg Elctc E Mgtc Ct flow J Qsstto Elctomgtc wvs Ct flow J Th slow vg flds dstood to b flds ot ldg to ct dstbtos. Ths ms tht th o dd ct ffcts s th dmsos of th ct cg codcto smll th th tto dth of th fld. Th cts t thos fqcs dstbtd s th DC cs, foml ov th whol sfc of th codcto. Edd ct ffcts cosdd th flds wth fst vg tm ddc, d to th low fqc bg ttd s qs-stto. Hgh fqc flds, s focssd o t oblms d ldg to th lctomgtc wvs, ot cosdd ths cos.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

34 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Most of th hscl sss g gg c b dscbd b qssttc hom. Slowl vg d odc flds to Hz cosdd to b qs-stto. Elctcl g dvcs sch s motos, cttos, dcto fcs d hgh voltg tsmsso ls otd t low fqc. Tcl mls of qs-sttc flds th flds ctd b cols ottg lctcl mchs d tsfoms. Isd ths codctos th dslcmt ct s glgbl d th mgtc fld H otsd th col s clsvl ctd b th f ct dst J. Fo thos qs-sttc flds, m s lw s lcbl: H Dcdg whth th dslcmt ct c b glctd o ot, dds o th wvlgth λ of th oblm cosdd th fqc dom. If t s lg comd to th hscl dmsos of th oblm L, th dslcmt ct s glgbl. J I gl f λ 5...L th fld oblm c b cosdd s qs-sttc. Fo ths clss of oblm, th tstg flds v slowl d c b odc. So, th ctgos of oblms dstgshd: Sttc Slowl vg tst Tm-hmoc dd ct I tm-hmoc oblms ssodl vg fld qtts s ssmd. I tho, tm-hmoc solto s ol vld fo l sstms s ssodl ctto dos ot ld sgl fqc sos th o-l cs. Elctosttc flds Th two fdmtl lws govg ths lctosttc flds Gss s lw d Fd s lw, d th costttv lto btw D d E Gss s lw D ρ Fd s lw E Costttv lto D ε E I tms of th lctc scl ottl, E s ssd s o E E dl. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

35 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Combg, d th bov lto, gvs Posso s qto: o, f ε s costt: ε ρ ρ ε Wh ρ, ths qto bcoms Llc s qto: Llc s qto of th Elctc fld fo codctv md H w s th chg cosvto qto d costttv lto 5. Ths s sll fd to s ot fom of Ohm s lw. Chg cosvto Costttv lto J J σ E 5 lthogh ths sso coms fom qto ld to mgtc css, t dls wth lctosttc flds d tht s th so wh t s std h. Usg th costttv lto 5 d E w hv J σ o, f σ s costt: whch s Llc s qto. Mgtosttc flds Th bsc lws of mgtostàtc flds m s lw, d th lw of cosvto of mgtc fl m s lw Mgtc fl cott Costttv lto H J 6 7 µ H 8 I tms of th mgtc vcto ottl. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

36 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM lg th vcto dtt F F F to th qtos 6 d 8, sbstttg, d ssmg Colomb gg codto lds to Posso s qto fo mgtostàtc flds: µ J Wh J, ths qto bcoms Llc s qto: Th lctc vcto ottl I log to th mgtc vcto ottl, w c df th lctc vcto ottl T whch s ltd to th ct dst J b J T ssmg E to b tm ddt, E d, wth E J, w ow hv: σ T σ Comg ths wth th fomlto std th vos gh th followg qvlt ltoshs c b wtt: T J H E µ H J σ E Llc s qto fo ths oblm s sml to : T. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

37 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Mgtodmc flds low fqc qs-stto flds Th bsc lws of mgtodmc flds m s lw, wthot cosdto of tm vto of lctc dslcmt fl dst D D, Fd s lw, Mgtc fl cott d th costttv ltos. t m s lw H J Fd s lw d E dt Mgtc fl cott µ H J σ E 5 I tms of th mgtc vcto ottl d sbstttg Fd s lw, w c obt: E d dt t Now mlog Ohm s lw to clclt th dd cts J lds: J σ E σ t m s lw c ow b wtt, ldg th -fomlto fo th qsstto mgtc fld th tm dom: Sbstttg g σ J µ t d ssmg th Colomb gg codto lds to µ σ µ J t ssmg ssodl ctto cts wth gl fqc ω d ths sbstttg. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

38 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6 t ω lds th -fomlto th fqc dom to solv dd ct oblms: J µ σ µ ω W c dvlo oth qtos to, E, H, d J, sml w. Th followg tbl shows ths qtos. t H H σ µ t σ µ t E E σ µ t J J σ µ Ths qtos md s dffso qtos. Mgtodmc flds wvs Th bsc lw of mgtodmc flds m s lw, Fd s lw, Gss s lw d th costttv ltos. m s lw dt dd J H Fd s lw dt d E Gss s lw ρ D E D ε H µ 5 E J σ 6 µ ν Sbstttg m s lw wth s of ottl vcto :

39 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM E H ν J ε t E µ J µε t E µ J µε µ J t d ssmg th Lotz codto µε t µε t lds to µε t t t µε µ J µε µε µ J t D fo slow vg flds, d w obt: t H J µ J µε t Posso s qto fo mgtostàtc flds! Usg scl ottl dfd s: E t d sbstttg Gss qto gvs: ρ E ε t sg Lotz codto oc slts : ρ µ ε t ε. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 6

40 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Ths two qtos md o-homogos wv qtos. Th followg tbl shows ths. Ths two qtos wth th ddto of Lotz codtos qvlt to th fo Mwll qtos. Ths qtos vld wh. MXWELL EQUTIONS S FUNCTION OF POTENTIL µε µ J t ρ µ ε t ε µε t Th lctos of ths qtos ot wll b cosdd ths cos. Thml oblms Htg s v fqt homo o lctomgtc dvcs d, m sttos, th vlto of tmt s css to vod ovhtg stcts. I o, th dfft socs of ht sch s, Jol ffcts b dd d codctg cts, mgtc hstss d lso mchcl fcto. Now, w wll st bfl som tocs o ht tsmsso, bt fo mo dtld stto, sclzd fcs m b cosltd. Th th dfft ws of tht ht s tsmttd: Codcto dto Covcto Thml codcto Codcto s ocss wh th ht s tsmttd sd bod o btw dfft bods hvg hscl cotct. Th bsc qto dscbg thml codcto s Fo s qto wh: T c λ t T Q c s th thml cblt J / m C λ s th thml codctvt W / m C. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 6

41 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM T s th tmt C Q s th thml soc volmtc dst W / m If λ s ddt of th tmt o osto, th bov qto c b wtt s: c T T λ t Q λ Ths qto s sml to d c b solvd sml w. µ σ µ J t Fo stc, Q c b dfd s th Jol s ffct soc b: Q σ wh J s th ct dst fo both, dd o codctg cts, ddg o th stdd cs J Covcto tsmsso Covcto occs wh fld hs cotct wth htd sold bod. Th wll b costt movmt wh th htd tcls wll b lcd b cool os. Th m ffct, ht s tsmttd fom th bod to th fld b th followg qto Nwto s qto: wh λ dt ds h T h s th coffct of ht tsf b covto W / m C λ s th thml codctvt W / m C T s th tmt t th htd wll C T s th tmt of th fld t ot f fom th wll C Th qtt h dds o th fld ots, vloct d gomt. I ctcl lctos, h s dffclt to vlt d t s omll dtmd mtll. T. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 6

42 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM dto s s bfo, omll fo covcto d codcto, t lst two mtls mst b st th sstm. Ths s ot th cs fo dto. bod mts lctomgtc wvs. Ths dto c ch oth bod. Pt of ths wvs wll b flctd d t wll b bsobd b ths scod bod. Ths lst oto wll b tsfomd to thml g. bod t tmt T dts g to oth t tmt T, volvg t, ccodg to th followg sso wh T T dt λ ε γ ds γ s th Stf-oltzm costt ε s th mttvt of th bod Ths clss of ht tsmsso s somtms ot cosdd. od codtos O oblm cossts of fdg th ow fcto of PDE. I ddto to th fct tht stsfs qto L g wth scbd solto go, mst stsf ct codtos o S, th bod of. Th choc of th bod codtos ot ol flcs th fl solto, bt c fth dc th lss dom. Dfft fld soltos wth dfft bod codtos. Lft: Dchlt bod codto ght: Nm bod codto Th bod codtos tht c b mosd fom th m gos: Dchlt codto: ths codto s ssgg b fg dtmd vl of th ottl o gv bod cv. I ths w, ths cv s chctsd b costt vl of th ottl, th th qottl. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

43 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM ls slt tgtl to sch bod. I oth wods, o l cosss tht bod. Th Dchlt codto c b ssd s f o S. If f ths codto s md homogos codto; f f <> s ow s homogos codto. Nm codto: ths codto s ssgd b fg th oml dvtv of th ottl o gv bod cv. I oth wods, th ls cosss th bod ow w. Th Nm codto c b ssd s g o S If g ths codto s md homogos codto; f g <> s ow s homogos codto. Md bod codto h w o S wh h d w lct ow fctos o th bod S. dcto of th lss dom b ms of ssgmt of stbl Dchlt d Nm bod codtos. Ths bod codtos tcll sfl stcts chctsd b o o mo smmt s. Th lss s ccomlshd ol o t of th totl stct, mosg th Dchlt o th Nm codtos o th smmt s tslf. Cosd, fo ml, th th colm sgl-hs ctc. Th fl dst vctos tgt to th s. Th stct c b smlfd b mosg th Dchlt bod codto log th l.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

44 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM lso, th fl dst vctos omls to th s. Th stct c lso b smlfd b mosg th Nm bod codto log th l. H w hv dcd th comlt stct to ¼ of ts sz. Th std o th dcd stct wll b sml d fst.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

45 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM dcto of th dom b ms of th odc codtos I mlt-ol ottg mchs th fld lss c b dcd to v mb of ols b mlog odc bod codtos. Lt b th mb of ol s d sg th ol coodts, th odc bod codtos : π, θ, θ,,... Th lss c b dcd to odd mb of ols b mlog todc bod codtos: π, θ, θ,,.... glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

46 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM If th smmt o odct bods dvdtd, th std of coml stct s dcd to std ol o of ts ts. Ths lds th dobl dvtg: Th dcto of th dom to b lsd, wth th sltt dcto of clclto tm. Th ossblt of mo cct lss of th mg t. O od Poblms Tcll, ft lmt mthods bst std to oblms wth wll-dfd, closd solto gos. Howv, lg mb of oblms tht o mght l to ddss hv o tl ot bod. m ml s solod. Th bod codto tht o wold l to l s t. Howv, ft lmt mthods, b t, ml ft dom. Fottl, th mthods tht c b ld to gt soltos tht closl omt th o bod solto sg ft lmt mthods.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

47 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Tcto of Ot ods Th smlst, bt lst cct, w to ocd s to c bt bod f ogh w fom th of tst d dcl th o / o ths bod. l of thmb s tht th dstc fom th ct of th oblm to th ot bod shold b t lst fv tms th dstc fom th ct to th otsd of th obcts of tst. Tcto s th mthod mlod b most mgtc ft lmt ogms, bcs t qs o ddtol ffot to mlmt. Th dow sd to tcto s tht to gt cct solto th go of tst, volm of mch lg th th go of tst mst lso b modld. Usll, ths lg go to to th of tst c b modld wth ltvl cos msh to solto tms to mmm. Howv, som t tm d sc s stll qd to solv fo go whch o hs lttl tst. smtotc od Codtos Th sml w to omt o bod oth th tcto s to s smtotc bod codtos. Th slt s tht b cfll scfg th mts fo th md bod codto, d th lg ths bod codto to ccl ot bod, th bodd solto c b closl omtd. Cosd -D l oblm ol coodts. Th dom s ccl shll of ds o bodd go. s, vcto ottl gos to zo. O th sfc of th ccl, th vcto s scbd fcto of θ. Ths oblm hs ltcl solto, whch s:, θ m m m cos m θ α m wh th m d α m mts chos so tht th solto mtchs th scbd ottl o th sfc of th ccl. O cold th of ths solto s dscbg th solto to to ft lmt oblm wth ccl ot bod. Th solto s dscbd sd th ccl v ft lmt solto. Th tc s to t togth th ltcl solto otsd th ccl to th ft lmt solto sd th ccl. Fom sto, o c s tht th hgh-mbd hmoc, th fst th mgtd of th hmoc dcs wth sct to csg. ft ol shot dstc, th hgh mbd hmocs dc to th tt tht lmost ll of th o-sc solto s dscbd b ol th ldg hmoc. If s th mb of th ldg hmoc, th o-fld solto fo lg, bt ot ft, s closl dscbd b: m, θ cos m θ α m m. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

48 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Dffttg wth sct to lds: m cos m θ α m m m If th bov qto s solvd fo m d sbstttd to th comlt solto, th slt s: m Now, ths qto s v sfl slt. Ths s th sm fom s th md bod codto. If th ot dg of th solto dom s ccl, d th ot ft lmt bod s somwht movd fom th of m tst, th o dom solto c b closl omtd b lg th bov qto to th ccl bod. Som c mst b sd lg ths bod codto. Most of th tm, t s sffct to t. th obcts th solto go loo l dol wh vwd fom lg dstc. Howv, th oth css.g. -ol Hlbch mt mgt whch th ldg hmoc s somthg oth th. Yo d to s o sght to o scfc oblm to c th ot fo th ldg hmoc. Yo lso mst t th obcts of tst oghl th ct of th ccl ft lmt dom to mmz th mgtd of hgh-od fld comots t th ot bod. lthogh th lcto of ths bod codto qs som thoght o th t of th s, th slts c b qt good. Th followg fg sts th fld odcd b -cod col f sc. Th smtotc bod codto hs b ld to th ccl ot bod. Isctg th solto, fl ls to coss th ccl bod s f th solto dom w tl bodd. To l th smtotc od Codto, df w, md-t bod codto. h w o S Th, c th mts so tht: h µ w o ε wh o s th ot ds of th go mts gdlss of th wog lgth ts.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6

49 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM lv Tsfomto tcll good och to o bod oblms s th lv Tsfomto, tchq fst dscssd th cott of comttol mgtcs. Th stgths of ths tchq : th ffcts of th to go, tho, ctl modld b ths och; ss mt stto of th oblm s td l FEM- EM mthods, whch gv th sm ct solto bt dsl cols togth th bod ods. qs o scl fts th ft lmt solv to mlmt th tchq, oth th th blt to l odc bod codtos. Dvto I th f fld go, th mtl s tcll homogos.g. d f of socs I ths cs, th dfftl qto tht dscbs vcto ottl s th Llc qto: If w wt th bov qto ol otto, s dscbd b D lss: φ ssm tht th fld go of th oblm c b cotd ccl of ds o ctd t th og. Th f-fld go s th vthg otsd th ccl. O och to bodd oblms s to ttmt to m th bodd go oto bodd go, wh oblms c mo sl b solvd. Scfcll, w ds w to tsfom th bodd go otsd th ccl to bodd go. O sml w to m sch mg s to df oth vbl,, tht s ltd to b:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 6

50 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM sctg th bov tsfomto t c b s tht ths ltosh ms th to go oto ccl of ds o. Th t st s to tsfom th dfftl qto tht th fld mst stsf, to th md sc. Tht s, th Llc qto mst b wtt tms of d θ th th d θ. W c vlt dvtvs tms of std of b mlog th ch l: Now, w c ot tht t o, d ft som lgbc mlto w mst obt: φ Th tsfomd qto fo th ot go, hs ctl th sm fom s tht fo th go, ol tms of th th. Th mlcto s tht fo th -D l oblm, th to c b modlld sml b ctg oblm dom cosstg of two ccl gos: o ccl go cotg th tms of tst, d ddtol ccl go to st th f fld. Th, odc bod codtos mst b ld to cosodg dgs of th ccl to foc th cott of t th dgs of th two gos. Th s th cott of t th bod btw th to d to gos. Th scod ccl go ctl modls th ft sc solto, bt dos t o bodd dom o cold lws bc ot th fld fo ot sc b lg th vs of. s ml, cosd E-co lmto stc wth wdg od t. Sos tht th obctv s to dtm th fld od th E-co th bsc of fl t th.. wh th mgtc cct s o. I ths cs, th fl s ot costd to flow th tht s o wll dfd, bcs th lmtos tht comlt th fl th hv b movd.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 6

51 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Th gomt ws chos btl, th os h bg mo th ocd th th ctl oblm. Th mtl fo th co s l wth ltv mblt of 5. Th col cs bl ct dst of M/m. Th t gomt s show th followg fg. H th co s lcd wth ccl go, d scod ccl go s dw t to th go cotg th co. Podc bod codtos ld to th cs tht df th bods s show th fg. lso otc tht ot hs b dw th ct of th to go. ot ot hs b ld to ths ot tht scfs tht t ths fc ot. Th ct of th ccl ms to ft th logos o oblm, so t ms ss to df, ffct, t ft. If o fc ot s dfd, t s fl s to s tht th solto s ol q to wth costt. Th stto s logos to stto wh Nm bod codtos hv b dfd o ll bods, sltg o-q solto fo. Th sltg solto s show th followg fg. s s th tto, th fl ls to coss ot of th of th go cotg th co s f ffctd b th sc of th bod. Th fl ls th dom stg th to go, comltg th fl ths thogh th to go.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 6

52 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM dcto of D oblm to D oblm Th D oblm s dcd cosdg th smmt of th sstm. Th smmts c b of two ds: Pl smmt. Th lctomgtc hom sosd to b dtcl o ch l dcl to s, clld smmt s. Th fld s dtcl o ch scto of th lmt oml to th z-s, f w sos ft lgth of th sstm d glctg th d ffcts. Th lss s th gv o th, l. Th solto s v sml d sl obtd. Fo ml o th mgtostàtc oblm, w c obt th followg codtos: o Th ct dst vcto J hs th z s comot J,, J z, ol. Ths comot c b fcto of th d coodts,.. J z J z,. o Th mgtc vcto ottl hs comot lll to vcto J ol, tht s th z s comot,,, z. Ths comot s fcto of th d coodts,.. z z,. o Th fl dst vcto hs comots ol o th, l, s obtd fom: Tht s: z z l smmt. Th lctomgtc hom sosd s dtcl o ch sm-l obtd b otto thogh fd s. Z ϕ. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

53 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Th lss s th gv o th,z l. Th solto s v sml d s. Fo ml o th mgtostàtc oblm, w c obt th followg codtos: J θ, ol. Ths comot c b fcto of th d z coodts,.. Jθ Jθ, z. o Th mgtc vcto ottl hs comot lll to vcto J ol, tht s θ s comot,, θ,. Ths comot s fcto of th d θ coodts,.. θ θ, z. o Th fl dst vcto hs comots ol o th,θ l, s obtd fom: o Th ct dst vcto J hs θ s comot, J, Tht s: z θ θ Mtls ots. L d No-l modls It s ossbl to l Mwll s qtos vos sttos d combtos of dfft mtls. Fo ths os t s css to todc th coct of mgtc soto. Cosd mtl whos mgtc mblt s domt ct dcto. O sch mtl s sht of o wth g-otd stct o th lts md of sht mtl whch fom, fo ml, th co of tsfom, s th followg fg. µ >> µ It s sobl to ssm tht both css, th mgtc fl flows mo sl th dcto O. I th fst cs, ths s d to th otto of th gs d th scod d to th sc of smll gs btw th ls. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

54 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM of sht mtl. ssmg fld tst H whos comots H d H ql to H d f µ d µ th mblts th dcto O d O sctvl, w hv: µ H µ H W ot tht s lg th. W cocld tht th lto µ wh µ s scl, s ot gl sc t dos ot stsf th css bov mtod. cs of ths, w todc th coct of mblt tso dotd b µ. I gl fom, th lto btw d H c b wtt s: H µ µ z µ z µ H µ µ µ z µ z H µ z H µ z H z I gl lctos w c wt sml ltos to oth chctstcs, sch s th codctvt o mttvt: E ε D J σ E sds th coct of soto, whch comlcts th std of mgtc d lctc mtls, w todc oth homo, fqtl cotd lctomgtc dvcs. I ths dvcs, th mgtc mblt s ot costt bt dds o th tcl vl of H th mgtc mtl qsto. Ths homo s clld o-lt o stto. Th gl lto btw d H s ow: µ H H I som css th btt oto, fo mcl clclto, s to s th vs of µ. Ths s clld mgtc lctvt ν : ν ν µ To vod th s of gtv vls of, w c s modfd lto btw ν d. ν ν. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

55 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Fo mcl stblt ths fcto mst b cotos d dfftbl. mb of omtos ossbl fom olomls to otls. ol slcto s th cbc sl. Ths mthod gvs cws cbc olomls tht cotos wth cotos dvtvs. Th bov fg shows th hstss loo fo mgtc mtl. Th dscotos l shows th md oml mgtsto cv. Fo soft mgtc mtls, th hstss loo s ow d ths l s good omto. Fo hd mgtc mtls mt mgts th ttmt s slghtl dfft. Th t scto s ddctd to th modllg of mt mgts. Pmt mgts PM modllg. Th dvlomt of hgh g mt mgt mtls sch s SmCo d NdF hs ld to csd tst th s of mt mgt mtl lctcl mchs d cttos. s mtod th lst scto, fomgtc mtls chctsd b ow hstss loo. I cotst, hd mgtc mtls sch s PM hbt wd loos. It s oft cctbl to cosd th mgtc chctstc of PM b stght l th scod qdt of th hstss loo. Th tscto of th hstss loo wth th odt s clld th sdl o mc fl dst. Th tscto of th bscss d th loo s clld th coctv foc H c. Th two ossblts fo th modllg of PM mtl: Mgtsto modl Ct sht och. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

56 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM lthogh ths two mthods hv dfft sttg ot, th both slt th sm st of qtos. ssmg stght l s th chctstc of th PM mtl, th ol two mts qd to df th chctstc: Th slo of th l µ m Th -s tct Mgtc vcto modl Th dmgtsto chctstc s dfd b H M µ χ m wh χ m s th mgtc ssctblt, M th mgtsto vcto d H th fld stgth t th otg ot. I tms of th mt fl dst µ M Th cmtl mblt, th slo of th dmgtsto chctstc s H µ χ m χ m s v smll ostv mb so tht th t mblt of th mgt s ol slghtl lg th tht of th f sc. Th lctvt s dfd s ν µ d lg ths to th dmgtsto chctstc, lds χ m H µ M ν sg th Mwll qto fo mgtostàtc oblm: lds H ν J ν µ M Th scod tm, th mgtc vcto, o th ght-hd sd sts soc tm d c b dtfd s qvlt mgtc ct. J. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

57 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Ct sht och Th s of qvlt ct sht stg th PM mtl s s w to todc th mtls ots. N S Tg th mblt of th o co th bov fg to b ft, m s lw lds H m l m H δ Igog fgg d co ffcts, th fl dst s fom d: µ l m δ H Th tscto btw th g chctstc, th lod l, d th dmgtsto cv sts th otg ot of th sstm. Fo ll mgt w hv: m m m H H µ H µ m c χ m H Th PM c b std b ct sht wth totl m-ts N I H c l m d mtl of qvlt mblt µ H c g ssmg ft mblt of th o ts of th mtl co, H m l H δ H m c l m. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

58 FINITE ELEMENTS FO ELECTICL ENGINEEING INTODUCTION TO ELECTOMGNETISM Ths lds H c ν δ H l m c ll mgtc qtts otsd th PM m th sm s th cs of th mgtc vcto, bt shftd to th fst qdt of th mgtsto chctstc. Ths mthod s s to mlmt fo ctgl mgts wth mgtsto lll to two sds of th ctgl: lc th mgt b mtl of mblt µ H dd th ct sht log th two sds of th mgt to odc fld th dcto of th mgtzto. Th l ct dst /m mst b ql to H c, th coctv foc. Ths ds c b tsfd to PM wth bt sh. Th ct th sht s gv b: I b H c l wh l b s vcto otg th dcto of sht ct d whos mgtd s th lgth of th dg. cd s t vcto otg th dcto of mgtzto. b cd c. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6

59 NLYTICL SOLUTION

60

61 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL SOLUTION ltcl solto Som mott thoms Two thoms of fdmtl motc solvg EM oblms. Ths : Sosto cl. If ch mmb of st of fctos,,,..n, s solto to th PDE L g wth som scbd bod codtos, th l combto N lso stsfs L g. lso w c dvd th coml oblm to st of dcd oblms, whch s to solv th th ogl oblm. Th solto to th ogl oblm s gv b: Uqss thom. Ths thom gts tht th solto obtd fo PDE wth som scbd bod codtos s th ol o ossbl. I gl w, solto of U s ql dtmd b scfg th th vl of U o th oml comot of U th cs of scl ottl o th tgtl comot of U th cs of ottl vcto t ch ot of th bod sfc. N ltcl solto. Sto of vbls. Th mthod of sto of vbls somtms clld th Fo s mthod s covtol mthod fo solvg tl dfftl qto. scll, t tls sg solto whch bs dow to odct of fctos, ch of whch volvs ol o of th vbls. Fo ml, f w sg solto,, z, t to som PDE, w q tht t hs th odct fom:,, z, t X Y Z z T t solto of th fom th bov qto s sd to b sbl,, z, d t.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

62 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION W bg th lcto of sto of vbls b fdg th odct solto of th homogos scl wv qto c t Solto to Llc s qto c b dvd s scl cs of th wv qto. Dffso d ht qtos c b hdld th sm m s w wll tt wv qto. To solv ths qto, t s dt tht w fst st th tm ddc. W lt, t T t Sbstttg ths th bov qto: T U c U T '' dvdg b U T gvs: U U T '' c T th lft sd s ddt of T, whl th ght sd s ddt of ; th qlt c b t ol f ch sd s ddt of both vbls. If w lt bt costt b th commo vl of th both sds, th qto dcs to U U T '' c T Ths w hv b bl to st th sc vbl fom th tm vbl t. Th bt costt todcd th cos of th sto of vbls s clld th sto costt. W shll s tht gl th totl mb of ddt sto costts gv oblm s o lss th th mb of ddt vbls volvd. Th scod qto s od dfftl qto wth th gl solto T t ct ct Sc th tm ddc dos ot chg wth coodt sstm, th tm ddc ssd th bov qto s th sm fo ll coodt sstms. Thfo, w shll hcfoth stct o ffot to sg solto to th fst qto. Notc tht f, th tm ddc dss d ths qto bcoms Llc s qto. To otl th mthod cosd th Llc s qto Cts coodts. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

63 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of z Th solto c b wtt s th odct of th st soltos:,, z Z Y X z wh X s ol ddt o th vbl, Y o th vbl d Zz o th z vbl. Sbsttto of th gl solto to th ogl PDE d dvdg b z Z Y X gvs z z Z Y X I ths fom, ch tm dds o sgl vbl d, thfo, c b std. Fo th sto to b vld, ch tm mst b ql to costt to b dtmd. Th qto c b wtt s z z z Z Y X wh th th costts mst stsf z Th followg th dfftl qtos obtd: z Z z Y X z Now, th th qtos comltl ddt d c b solvd stl. combto of costts c b chos, f th bov qto s stsfd. Ths costts mst b slctd fo scfc lctos. Fo ml, w c ssm tht ; z d > >

64 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of Th gl soltos fo X d Y Y X Thfo, bcs < z th gl solto fo Zz s: z z z Z 6 5 wh th costts to 6 mst b vltd to obt tcl solto. Ths vltd fom th bod codtos of th oblm. of th costts c b zo, ddg o th bod codtos of th oblm. Fll, lthogh th solto s tms of ows to 6,,, z, ol s ows ddt. Fo ml, cosd two-dmsol bo dfd b two lll sfcs, both sm-ft tt d both t zo ottl. Th low sfc s t ottl. lso, th ottl t ft s zo. Clclt th ottl vwh th chl so dfd. Fo two dmsol lss, w c show tht: Th gl solto s: b b cos s, Fo th solto th dcto, w s th otl fom. Ths s bcs w tct sg vls of tht td to ft. Fo sch vls, otl foms mo covt th th hbolc foms. To stsf th bod codtos, w wt: t, b b t s s, b b

65 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of Ths gvs: m m π π wh m s tg, cldg zo. W wll, howv, cld m fom th solto bcs t lds to d l solto fo th fom b Smll, th gtv vls of m d ot b cosdd bcs gtv m wll ol chg th sg of th solto. Th gl solto t ths stg loos l: m m b b m π π π s, t s, b b m m π π Th solto t ths stg s: s, b C m C m π π t, ; to stsf ths codto, w cot sml sbsttt th gl solto. If w dd, th solto wold b ssodl th dcto d o costt C c stsf th bod codto. Howv, th solto m lso b wtt s sosto of soltos of th bov fom. W wt m C m m m m π π s, Now, w sbsttt : m C m m m π s, Th ltt fom s Fo s ss whch, ffct, omts th ls,,, b ft ss. I ths ss, C m, th mltd of th coffcts of th ss. To obt C m, w mltl both sds b

66 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of π s, wh s tg, d tgt both sds fom zo to. Ths s gl tchq w wll s g d ws dvlod b Fo hmslf: m m m m m m d m C d m C d s s s s s π π π π π wh th tgto d th sm tchgd. Ech sd of th lto s tgtd stl. Th lft hd sd gvs m fo m fo d s π π Fo th ght hd sd, w tgt ch tgl th sm. Fo vl of m, w gt m fo m fo C d m C m m s s π π To stsf both codtos bov, m mst b odd d m. oth vl lds zo. Ths:,,5,... ; m m C m π If w sbsttt ths th gl solto, w obt th gl solto sd th bo: m m m m m π π π s,,,5,...

67 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of Th followg tbls sms th gl soltos of Llc s qto dfft coodt sstms. Cts coodts z U U U X cos ' ' s Y cos ' ' b b b s b b b Zz cosh ' ' z z z z z z c c z c z sh c c c z z Cldcl coodts z U U U φ Zz cosh ' ' c c z c z sh c c c z z z z z z φ cos b b b s b φ l d c c d c c Y c J c

68 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of Shcl coodts s s s z U U U θ θ θ θ θ,...,, c c θ cos cos θ θ Q b P b Oth mls Cosd th s ffct o sold cldcl codcto. Th ct dst dstbto wth good codctg w obs th dffso qto t J J σ µ W wt to solv ths qto fo log codctg w of ds. ssm hmoc fld,.. t z J ω φ,,, ths J J σ µ Fo ftl log w, th bov qto dcs to o dmsol oblm cldcl coodts: J J σ µ Ths qto s modfd ssl qto of zo od. Hc th solto s c I c λ λ J wh µ σ ω δ δ µ σ ω λ δ s th s dth. Costt c mst vsh f J s to b ft t. t I c I c λ λ J J Ths I I λ λ J J

69 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION S ffct o l ft codcto W c sos l codcto sd of mgtc fld. Ths fld hs odc d ssodl. Ths codcto s ctl d s tds to ft coodt. Th Mwll s qtos fo ths cs : H J E t d wth th costttv ltos: µ H J σ E Th soltos : H, t H t J, t J t ωt ωt lg th cl oto: H J d sbstttg Ohm s lw: H σ E d lg th followg vcto dtt, d Gss s thom: H H H w c obt: µ H H ; µ. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of

70 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of t E H σ σ t H H µ σ I ths cs, ol th dmso s cosdd H d H d µ ω σ Th gl solto s: t H t H ω δ, o δ ω δ t H t H, δ, s ttto tm; Th fld dcss wth. Fo δ th fld s dcd b fcto -. δ ω t, s ogto fcto. W c obt sml sso fo th dst of ct th codcto J d J d µ ω σ δ ω δ t J t J, wth H J δ Th ct dst s gt o th sfc of codcto th sd th codcto. Fo ml co codcto 5 Hz fld th s dth s σ 56 6 Sm: m 5mm π π ω σ µ δ to cm.

71 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of O ctgl slot. Fll codcto Th followg fg shows o ctgl slot wth o c codcto. Ths slot hs hght of h m d wdth of b m. W cosd tht b << h. Th gl solto s ρ ω µ γ γ γ γ γ γ h sh ch b I J h sh sh b I H To clclt th lctc sstc d dctc w fst dtm th voltg do log th b. To bg wth, w clclt th dffc of lctc ottl log th b: h sh ch L b I dl J dl E L L γ γ γ ρ ρ Scod, w clclt th.m.f. d to th tm vto of ld fl. Th fl ld t th coodt s: h sh ch h ch L b I dz d h sh sh b I dz d H dz d Lh Lh Lh γ γ γ γ µ γ γ µ µ Th.m.f. s: t L L ω h sh ch h ch L b I L γ γ γ γ µ ω Th totl voltg do s:

72 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION d sg: b b L γ ch γ ρ I L b sh γ h I ch γ h ch γ ω µ L γ b sh γ h w c obt ω µ γ ω µ ρ µ ω ρ µ ω ρ γ ρ b γ ch γ h ρ I L b sh γ h Th mdc of slot-b s dtmd s Z b Th D.C. sstc s sml: I b γ ch γ h ρ L b sh γ h DC L ρ b h d Z b DC ch γ h γ h sh γ h Edg l d mg ts w c wt: X C C DC DC h h sh s h d d δ h h ch cos d d h h sh s h d d δ h h ch cos d d th followg fg shows th sstc, dctc d ctc vto s fcto of th dmsol mt δ h sstc gows wth th fqc. Idctc dcs wth th fqc.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

73 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of,5,5,5,5,5 5 c Xc Lc Som sfl omtos If 5 <. δ h w c wt: δ δ δ δ δ h h h X h h DC C DC C d b L h L L b L h h b L h h DC DC DC µ ω µ ω ρ ρ ω µ δ δ δ ω h h L X DC C b If >>> δ h DC DC C DC C L h h X h δ ω δ δ

74 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION Elctc mchs modllg Fld Dscto of Eg Flow: Potg's Thom Stt wth Fd s d m lws: E t H J mltlg th fst of ths b H d th scod b E d tg th dffc: H E E H H J E H H J E t O th lft of ths sso s th dvgc of lctomgtc g flow: S E H H, S s th clbtd Potg flow whch dscbs ow lctomgtc fld sstm. Th t of ths qtt s wtts sq mt th Ittol Sstm. O th ght hd sd two tms: H J E t Th fst tm s th t of chg of mgtc stod g. Th scod tm, sts l ow dssto. W wll dscss ch of ths mo dtl. Fo th momt, howv, ot tht th dvgc thom of vcto clcls lds: S d S d ol tht s, th volm tgl of th dvgc of th Potg g flow s th sm s th Potg g flow ov th sfc of th volm qsto. Ths tgl bcoms: S d ol H J E d t whch s sml lzto tht th totl g flow to go of sc s th sm s th volm tgl ov tht go of th t of chg of g stod ls th tm tht loos l dssto. fo w clos ths, ot tht, f th s moto of mtl wth th sstm, w c s th mcl. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

75 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION sso fo tsfomto of lctc fld btw obsvs movg wth sct to ch oth. H th 'md' fm s movg wth sct to th 'md' fm wth th vloct v: E ' E v Ths tsfomto dscbs, fo ml, th moto of chgd tcl sch s lcto d th flc of both lctc d mgtc flds. Now, f w ssm tht th s mtl moto th sstm w obsvg d f w ssg v to b th vloct of tht mtl, so tht E s msd fm whch th s o mtl moto tht s th fm of th mtl tslf, th odct of lctc fld d ct dst bcoms: E' v J E' J v J E' J v J J E I th lst st w sd th fct tht scl tl odct th od of th scl dot d vcto coss odcts c b tchgd d tht vsg th od of tms vcto coss odct sml chgs th sg of tht odct. Now w hv d ttto fo wht w hv clcltd: If th 'md' coodt sstm s ctll th fm of mtl moto, E' J J σ whch s sl s to b dssto d s ostvl dfd f mtl codctvt σ s ostv. Th lst tm s obvosl covso of g fom lctomgtc to mchcl fom: v J v F wh w hv ow dtfd foc dst to b: F J Ths s th Lotz Foc Lw, whch dscbs th tcto of ct wth mgtc fld to odc foc. It s ot, howv, th comlt sto of foc odcto lctomchcl sstms. s w ld l, chgs gomt whch ffct mgtc stod g c lso odc foc. Fottl, comlt dscto of lctomchcl foc s ossbl sg ol mgtc flds d tht s th toc of o t scto. Fld Dscto of Focs: Mwll Stss Tso Focs of lctomgtc og, bcs th tsfd b lctc d mgtc flds, th slt of thos flds d m b clcltd oc th flds ow. I fct, f sfc c b stblshd tht fll css. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of

76 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of mtl bod, th foc o tht bod c b show to b th tgl of foc dst, o tcto ov tht sfc. Th tcto τ dvd b tg th coss odct of sfc ct dst d fl dst o th -g sfc of mch bov ctll ms ss vw of th mcll dvd Lotz Foc Lw: Sbstttg th Mwll s qtos th Loz foc sso: J J H ν µ w obt: f ν Dvlog ths qto, w obt th followg qto w ol show th qto fo X comot: z f z z z ν If tm ν s ddd d sbtctd fom th bov qto, d th dtt s sd, th th foc comot bcoms: z z z f ν Som fth mltos gvs: z f z f z z ν ν Th mg sso m b cogzd s th dvgc of vcto f, whos comots :

77 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of z z f z T f T f T T ν ν ν sml dvlomt holds fo ch of th oth foc comots f d f z. Ths ths vctos c b combd to Tso T: T z z z z z ν Th foc dst c ow b wtt s th dvgc of ths tso: f T Th totl foc c b fod b tgto ov th volm: Td fd F Usg th dvgc thom, ths volm tgl m b dcd to sfc tgl: S TdS Td F W lmt th followg dvlomt to two dmsol gomt, so tht th sfc of tgto s l w cosd t dth. Th t oml d tgtl vctos to th sfc : t s s s s Th cmtl tgto th s th: dl ds wh dl s dfftl lgth log th tgto th. Th cmtl foc s ow:

78 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of dl s s f ds T f ν Th tgtl d oml comot : s t t s s s s dl f f s s s s dl f f ν ν Th tgtl d oml comots of th fl dst : t s s s s Sbstttg d som ft lgbc mltos, w c wt: dl f dl f t t t ν ν Th toq o c of ds s gv b: dl M t ν Eml: L Idcto Mch Th followg fg shows hghl smlfd ct of sgl sdd l dcto moto. Ths s ot how most l dcto mchs ctll blt, bt t s ossbl to show thogh smmt gmts tht th lss w c c ot h s ctll vld fo oth mchs of ths clss. Ths mch cossts of stto th sfc whch s std s sfc ct o th sfc of hghl mbl go. Th movg lmt cossts of th l of codctg mtl o th sfc of hghl mbl go.

79 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION Th movg lmt o 'shttl' hs vloct wth sct to th stto d tht moto s th dcto. Th stto sfc ct dst s ssmd to b: z ωt Not tht w gog som mott ffcts, sch s thos sg fom ft lgth of th stto d of th shttl. Sch ffcts c b qt mott, bt w wll lv thos tl lt, s th wht m l motos tstg. wd fom th shttl fo whch th dmso th dcto of moto s - - t, th ltv fqc s: z ω t ' ω t ' ω t s Now, sc th shttl sfc c sot sfc ct d s ctd b mgtc flds whch t ctd b th stto cts, t s sobl to ssm tht th fom of th oto ct s th sm s tht of th stto: m's Lw s, ths stto: whch s, coml mltds: s ωst' H g z s s H z s g Th - comot of Fd's Lw s s follows, ssmg th oblm s fom th z- dcto: ω s ωs E' µ H E'. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of

80 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of bt of lgbc mlto lds ssos fo th coml mltds of oto sfc ct d g mgtc fld: z s s z s s s s s g g H g g σ ω µ σ ω µ σ ω µ To fd sfc tcto, th Mwll Stss Tso c b vltd t sfc st blow th stto o ths sfc th - dctd mgtc fld s sml H z. Ths th tcto s H H T µ τ d th vg of ths s: * H H µ τ Tht s: Z s s s s g g g σ ω µ σ ω µ µ τ Now, f w cosd lctomgtc ow flow Potg's Thom: th - dcto: z H E S d sc th fm of th shttl s z H E ' µ ω s Z s s s s s g g g S τ ω σ ω µ σ ω µ µ ω ' Smll, vltd th fm of th stto:

81 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION S ω τ Ths shows wht w ld ssctd: th lctomgtc ow flow fom th stto s th foc dst o th shttl tms th wv vloct. Th lctomgtc ow flow to th shttl s th sm foc dst tms th 'sl' vloct. Th dffc btw ths two s th ow covtd to mchcl fom d t s th foc dst tms th shttl vloct. ottg Mchs Th s of ths fomlto ottg mchs s bt tc bcs, t lst fomll, dctol vctos mst hv costt dtt f tgl of focs s to bcom totl foc. I cldcl coodts, of cos, th dctol vctos ot of costt dtt. Howv, wth c d dstdg of th dcto of tcto d how t s tgtd w c m s of th Mwll Stss Tso MST och ottg lctc mchs. Now, f w go bc to th cs of ccl cld d tstd toq, t s tt cl tht w c comt th ccmftl foc b otg tht th oml vcto to th cld s st th dl t vcto, d th th ccmftl tcto mst sml b: τ φ µ H H φ ssmg tht th o fls sd th sfc of th oto, sml tgtg ths ov th sfc gvs zmthl foc. I cl ths s th sm s sodg th sfc of th oto b cotm of ftl smll bos, o sfc st otsd th oto d wth oml fcg otwd, th oth sfc st sd wth oml fcg wd. Of cos th MST s zo o ths sfc. Th mltlg b ds momt m gvs toq. Th lst st s to ot tht, f th oto s md of hghl mbl mtl, th zmthl mgtc fld st otsd th oto s ql to sfc ct dst. Glzto to Cotos Md Now, cosd sstm wth ot st mltlct of ccts bt cotm of ct-cg ths. I tht cs w cold dtf th co-g s: W ' λ dj d m wh tht s chos to ct ll of th ct cg codctos. Ths c b cd to b dcl to ch of th ct flmts sc th dvgc of ct s zo. Th fl λ s clcltd ov th tht cocds. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

82 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION wth ch ct flmt sch ths st sc ct hs zo dvgc. Th th fl s: λ d Now, f w s th vcto ottl, th fl ld b o of th ct flmts s: λ dl wh dl s th th od th ct flmt. Ths mls dctl tht th co-g s: W ' m J dl dj d Now: t s ossbl to m dl cocd wth d d b lll to th ct flmts, so tht: W ' m Fo l cotos md w c wt: vol dj d W m W m ' ol J d Pmt Mgts Pmt mgts bcomg v mo mott lmt lctc mch sstms. Oft sstms wth mt mgts ochd ltvl d-hoc w d md qvlt to ct tht odcs th sm MMF s th mgt tslf. Th costttv ltosh fo mt mgt lts th mgtc fl dst to mgtc fld H d th ot of th mgt tslf, th mgtzto M, H M µ Now, th ffct of th mgtzto s to ct s f th w ct wth dst: J * M Not tht ths ct cts" st l od ct mg mgtc fl dst. Mgtc co-g s:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

83 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION W ' m vol dm d Nt ot th vcto dtt: Now, C D D C C D W ' m vol dm d dm ds vol vol dmd dmd Th fst of ths tgls closd sfc vshs f t s t ov sfc st otsd th mgt, wh M s zo. Ths th mgtc co-g sstm wth ol mt mgt soc s W ' m vol dmd ddg ct cg cols to sch sstm s do th obvos w. Som mls lcto. Cosd th followg ct. Th mch cossts of cldcl oto d cldcl stto whch col d whch hv ssodl ct dstbtos o th sfcs: th ot sfc of th oto d th sfc of th stto. Th oto d stto bods md of hghl mbl mtl w omt ths s bg ft fo th tm bg, bt ths s somthg tht ds to b lood t cfll lt. W lso ssm tht th oto d stto hv ct dstbtos tht ll z dctd d ssodl: S Z Z S cos φ cos φ α. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

84 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of H, th gl φ s th hscl gl of th oto. D to th smmt, th z comot dos ot st. Posso s qto s fo ths cs s: Z S Z µ φ Th gl solto of ths qto s: m m m m m m m d c m m s b cos, φ φ Th bod codtos fo tgtl mgtc fld tst : J H H t t Th tgtl vl clcltd s: m m m m m m m t d c m m m s b H cos φ µ µ Fo ch bod: J d c m m m s b H d c m m m s b H m m m m m m m t m m m m m m m t cos cos φ µ φ µ wh J s th ct dst. If m w c wt: m m m m m m d c d c d f m J d c s b H t cos φ µ Th soltos of ths sstm of qtos :

85 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of m m J d J c b µ µ whch J m s th vl of ct dst. Th comlt soltos sltg fom both dst cts obtd b lg th sosto thom: cos cos,,, α φ µ φ µ φ S Z S Z Th mgtc fld dst s: φ φ cos cos,,, s s,,, α φ µ φ µ φ α φ µ φ µ φ φ S Z S Z S Z S Z th mgtc fld g s clcltd s: ol m d J W ths cs th dfftl volm lmt s:

86 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of dz d d d φ Th tgto z dcto s sml th logtd L of th mch: ol s css to tgt d φ coodts: [ ] [ ] φ φ φ π π π d L d L L d d J d J W S Z Z ol m cos cos cos cos α φ µ φ µ α φ µ φ µ S S Sbstttg d tgtg lds ll of th tgls of th t s cos θ θ d ths dfft to zo ol f : { } cos φ π µ L W s s m Th toq s obtd b dvto of th mgtc g: { } s φ π µ φ L W M s m If ε ε d glctg ε w c wt: { } s φ ε π µ L M s Th sm slt c b obtd b s of th Mwll Tso: π φ π φ φ φ φ φ µ φ τ µ µ τ d d M H H Fll w c clclt th toq b sg Llc s lw: df dm l d df th ct s:

87 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION d J dφ Fll th lmtl toq s: dm L J dφ Itgtg w c obt th sm vl clcltd b sg mgtc g. lcto. Pmt mgt fld lss Th ssmd gomt s show th followg fg. ssmd o bods t d d s. Th mt mgts, ssmd to b olzd dll d lttl, loctd btw d d. W ssm th ol s d tht ch mgt sbsms lctcl gl θ m. Th lctcl gl s st tms th hscl gl. If th mgts gd so tht th dll olzd mgts loctd od th zmthl og φ, th fdmtl hmoc of sc mgtzto s: M M cos φ wh th fdmtl mgtd s: M π µ m θ s m m π µ θ m s Sc th s o ct wh ths oblm, t s covt to tt th mgtc fld s th gdt of scl ottl: H ψ. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of

88 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION Th dvgc of ths s: ψ H Sc mgtc fl dst s dvgc f, w hv: o H ψ H M M cos φ M Now, f w lt th mgtc scl ottl b th sm of tcl d homogos ts: ψ ψ ψ ψ h h ψ M cos φ W c fd stbl solto to th tcl t of ths th go of mgtzto b sg: ψ C γ cos φ Cg ot th Llc clclto o ths: ψ C γ γ φ M cos φ cos whch wos f γ, whch cs: ψ M cos φ Of cos ths solto holds ol fo th go of th mgts < <, d s zo fo th gos otsd of th mgts. stbl homogos solto stsfs Llc s qto ψ h d s gl of th fom: ψ h cos φ cos φ th w m wt tl totl solto fo h fl dst s:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of

89 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of cos cos cos cos cos cos cos φ φ ψ φ φ φ ψ φ φ ψ M s < < < < < < Th bod codtos th followg: t th d ot bods t d s q tht th zmthl fld vshs, o φ ψ. t th mgt d ot d, φ H d mst b cotos: φ ψ φ H M ψ µ ft som lgbc tsfomtos, w c obt: M M S S S S cos cos S > < φ ψ φ ψ d: cos cos φ φ M H M H S S S S > < Th cs of s to b bt toblsom h, bt ths s sl hdld b otg tht:

90 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of l lm Scl css. Io f cs,,, ths bcoms: cos cos l cos φ φ φ M H M H fo M H > < Mchs wth o bods d wdgs slots. W tstd th flds t th bods. I sch cs, sll, th: o S. Th flds : cos cos φ φ M H M H S S S S S S d th mgtc fld dst s: S S g g S S S M M H cos cos φ µ φ µ µ g s gomtc fcto tht dscbs th gomt of th mgtc g. Th bov cs s vld fo mgts sd d. Fo mgts sd d, l S g

91 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of Fo th cs of mgts otsd d : S S g d fo mgts otsd d : l S S g Not tht fo th cs of smll g, whch both th hscl g g d th mgt thcss h m both mch lss th oto ds, t s stghtfowd to show tht ll of th bov ssos och wht o wold clclt sg sml, o dmsol modl fo th mt mgt: g h h m m g d ths s th whol sto fo th wdg slot, ow g, sfc mgt mch. Fl Th fl ld b sgl, fll tchd col whch ss gl fom zo to / π s: d L dl π φ If w cosd tht s ssodl dstbtd ths wll hv vl of: L Now f th ctl wdg hs N ts d sg th tch d bdth fctos, th totl fl ld s sml: w f N L λ

92 FINITE ELEMENTS FO ELECTICL ENGINEEING NLYTICL ESOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of q q q N b b w π γ γ γ π α α s s s Wh N s th col s slots, s th totl mb of slots th stto d q s th mb of slots b hs d ol.

93 NUMEICL SOLUTION: SIC METHODS

94

95 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION NUMEICL SOLUTION: FUNDMENTLS ND SIC METHODS Gl cocts Fqtl, dct ltc solto of PDE s ossbl ol fo sml css o d v stctv ssmtos. Fo th solto of dtld d lstc modls, mcl mthods oft th ol lttv vlbl. Th gol of ths cht s to todc mb of mthods d sml mls, to fmlz th d wth mcl d ctcl sss th solto of PDE's. Th m obctv of mcl mthod s to solv PDE o dsct st of ots of th solto dom, clld dsctzto. I od to do so, th solto dom s dvdd to sbdoms hvg th dsctzto ots s vtcs. Th dstc btw two dct vtcs s th msh sz. Tm s lso sbdvdd to dsct tvls, d w cll tmst th tvl btw two cosctv tms t whch th solto s obtd. Th PDE s th omtd, o dsctzd, to obt sstm of lgbc qtos, wh th ows th solto vls t th dsctzto ots. Ths sstm of lgbc qtos c th b solvd o comt b dct o ttv tchqs. It s mott to lz tht th dsctzto st lcs th ogl qto wth w o, d tht v ct solto of th dsctzd oblm wll ld omt solto of th ogl PDE, sc w todc dsctzto o. Clssfcto of lctomgtc EM oblms EM oblms clssfd tms of th qtos dscbg thm. Th qtos cold b dfftl o tgl o both. Most EM oblms c b sttd tms of oto qto L wh L s oto dfftl, tgl, o tgo-dfftl, g s th ow soc, d s th ow fcto to b dtmd. Fo ml th lctosttc oblm volvg Posso s qto: So tht g ρ ε. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

96 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION L ρ g ε s obsvd bfo, EM oblms volv l scod od tl dfftl qtos. I gl, scod od tl dfftl qtos PDE s gv b D fomlto: b c d f g Th coffcts, b, d c gl fctos of d ; th m lso dd o tslf, whch cs th PDE s sd to b ol. PDE whch g s qls to zo s tmd homogos; t s homogos f g. PDE gl c hv both bod vls d tl vls. PDEs whos bod codtos scfd clld std-stt qtos. If ol tl vls scfd, th clld tst qtos. l scod od PDE c b clssfd s lltc, hbolc, o bolc ddg o th coffcts, b, d c. Th bov qto s sd to b: lltc f b c < hbolc f b c > bolc f b c Elltc qtos ssoctd wth std-stt hom,.., bod vl oblms. Tcl mls of ths t of qto cld Llc s d Posso s qtos: µ J Hbolc PDE s ogto oblms. tcl ml s th wv qto: ρ µ ε t ε Pbolc PDEs gll ssoctd wth oblms whch th qtt of tst vs slowl comso wth th dom motos whch odc th vtos. Th most commo bolc PDE s th dffso o ht qto: µ σ µ J t c T Q T λ t λ oth t of oblm s clld gvl. Th stdd goblm s of th fom: L λ. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

97 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION mo gl vso s th glzd goblm hvg th fom: L λ M wh M, l L, s l oto fo EM oblms. I ths qto ol som tcl vls of λ clld gvls vld; ssoctd wth ths vls th cosodg fctos clld gfctos. Egoblms sll cotd vbto d wvgd oblms wh th gvls cosod to hscl qtts sch s soc d ct-off fqcs, sctvl. M qtos of ctcl motc m b of md t, o ot sl dtfbl ccodg to o of th bov ctgos. Nvthlss, th dstcto btw lltc, bolc, d hbolc qtos ovds v sfl gdl fo th slcto of solto ocds. Th m ochs whch sd fo th dsctzto of th ogl PDE to obt mcl oblm. Th most mott dsctzto ochs c b clssfd s Ft Dffcs Ft Elmts od lmts Momt s mthod Mot Clo mthod Oth mthods I th followg gs w l th fomlto of th Ft Dffcs mthod d ths w l th tol cls, tht commo to th Ft lmts, od lmts d Momt s mthods. fo ths commo todcto w l bot th fomlto of ch mthod. Fll w st th Motclo mthod tht s odtmstc mthod. ft dsctzto, t s css to chc f th omto s ot o f th dsctzd modl c odc solto t ll wh ogmmd to comt cod. Fo sccssfl solto, th mcl schm mst b stbl, covgt d cosstt. Th schm s stbl f th solto sts bodd dg th solto ocd. Th schm s covgt f th mcl solto tds to th l solto s th msh sz d th tm-st td to zo. Th schm s cosstt f th tcto o tds to zo s th msh sz d th tm-st td to zo. If mcl schm s cosstt, th stblt s css d sffct codto to chv covgc. schm whch s stbl bt ot cosstt m covg to solto of dfft qto wth whch t s cosstt. mb of os todcd wh PDE s dsctzd d solvd mcll. To smmz, w hv: Tcto o - th o todcd b th ft omto of th dvtvs. Dsctzto o - th o th solto d to th lcmt of th cotos qto wth dsctzd o. od-off o - th comttol o todcd b dgtl lgothms, d to th ft mb of dgts sd th mcl stto.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

98 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of Th od-off o s dom, d omll css wh th msh sz s dcsd. Covsl, th dsctzto o dcss wth th msh sz, sc mo msh ots.. mo solto todcd. Ft Dffcs Th ft dffc och s th most ol dsctzto tchq, owg to ts smlct. Ft dffc omtos of dvtvs obtd b sg tctd Tlo ss. Cosd th followg Tlo sos O O Th fst od dvtv s gv b th followg omtos: Fowd Dffc cwd Dffc Ctl Dffc omto fo th scod od dvtv s obtd b s of fowd d bcwd dffc: Ft Dffcs fo Llc s d Posso's qtos Lt s cosd Posso s qto fo homogos mtls two-dmsol Cts coodts:, f wh s mtl ot. Usg th bov omtos w obt:

99 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of,,,,,, Sbstttg to th Posso s qto, w obt: f,,,,,,, Ths qto s wtt fo ch od th oblm to obt st of smltos qtos. Ths qtos sgl. Ol wh th ottl of t lst o of th ods s scfd, th solto of ths ossbl. W s tht fo Posso s qto, th ght-hd sd s th soc fcto mltld b th of th ft dffc cll. Tht s th totl soc th cll. Itfcs btw mtls Cosd th followg fg, whch w hv fv-ot FD schm d bod btw two mtls. H w solv th cs of lctosttc oblm. If w ssm tht th tfc hs o chg, th th lcto of Gss s lw wll gv s qto fo th ottl t od tht clds th ffct of th mtl tfc. Usg th dottd l of th dffc cll s th Gss sfc fo smlct, w s h, w obt: h h h h h h h h h h h h h U h U ds d du d du E EdS ds D ε ε ε ε ε ε ε ε ε ε gg tms, w obt: ε ε ε ε ε ε ε ε

100 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of Th bov fg shows th scl cs of th co. Usg th sm ocd showd bov, w obt th followg qto: ε ε ε ε ε ε ε ε Nm od codtos Fo Posso s qtos to hv q solto, th th ottl o th oml dvtv of th ottl mst b scfd t v ot o th bod ol ot s css. If th ottl s scfd, ths s Dchlt codto d th ow odl ottl s lmtd. If th oml dvtv s scfd, th w c ocd s follows. Cosd th followg fg. Th ottl t ot, s ow. Th ot, s otsd to th dom of th solto. W ss th oml dvtv t, s: Solvg ths qto fo to ot:,, Sbstttg ths vl th gl ft dffc cll f,,,,,,, w obt:

101 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of f,,,,,, scl bt v commo cs s th homogos Nm bod codto wh. I ths cs th qottl ls dcl to th sfc. I ths cs:,, d th qto fo th cll bcoms: f,,,,,, Eqvlt cct stto W wsh show tht th ft dffc sso hs qvlt cct stto. I fct, cct smltos, sch s SPICE, hv b sd to solv fld oblms b th s of ths qvlt twos. Cosd th sm ft dffc cll showd th bov gs: f, Ths qto c b wtt s: f, Th qto fo th followg cct, c b wtt s: I Th two bov qtos hv th sm fom. Idtfg tms: f I, ; ; ; ; ; It s obvos tht th odl voltgs hv th sm mc vl s th odl ottl o th ft dffcs msh.

102 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION Ecs. Cosd sq wth th sd coctd to voltg soc d th oth sd coctd to god. Dtm th ottl sd th sq. I ths cs th govg qto s th Llc s qto,..:. lso t s covt to ssg gl msh,... Th qto fo od s dcd to:,,,,,,,,,,, Th ottl th ctd ot s th vg of th ottl th sodg ods. Th solto c b clcltd b sg of sdsht. Th bov fg shows th mlmtto of ths st of qtos. O th bods, w ssg dctl th ow vl d t th to ot w st th bov qto. Usg th tto mod th otos m of th sdsht w c solv th oblm b tto w mst slct covgc cto, fo ml dffc btw two cosctv ttos lss th. o th mb of ttos mst b lss th. Th solto s show th followg ct. 8 6 I ths cs th mb of ttos ws 6, d th o ws lss th.. It s obvos tht f o s hgh dst msh, o c obt mo cct slts.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of

103 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION Ecs to solv. Dtm th ottl sd ths bo. Ft Dffcs fo -D Pbolc Eqtos W cosd h th -D dffso qto t whch s dsctzd sc d tm wth fom msh tvls d tm-st sml och s to dsctz th tm dvtv wth fowd dffc s t,, t t. Th solto s ow t tm t d w solto mst b fod t tm t t t. Sttg fom th tl codto t t, th tm volto s costctd ft ch tmst th lctl, b dct vlto of sso obtd fom th dsctzd qto, o mlctl, wh solto of sstm of qtos s css. lct och s dl mlmtd b sbstttg th sc dvtv wth th - ot ft dffc vltd t th ct tm-st. Th lgothm, wtt fo gc ot of th dsctzto, s t,,,,, t Dfg α N ot msh:, w c wt th followg mt qto fo th whol sstm wth. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of

104 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of N N,,,,,, α α α α α α α It s s to show tht ths sstm of qtos s stbl f, d ol f: < < t α I gl, th tm st mst b v smll od to gt good slts fom th lct och. Fth, f w f th msh w mst dc th tm st, so od to chv hgh ccc, th comtto tm bcoms v sv. Imlct schms fl gl mlct schm s obtd b omto fo th sc dvtv wth wghtd vg of th ft dffc omto t t d t d th tm dvtv s omtd b fowd dffc: t t,,,,,,,, λ λ wtg th qto tms of t α :,,,,,, λ α λ α λ α λ α λ α α λ α Wh λ, th schm s fowd dffc qto Th clssc C-Ncholso schm s obtd wh 5. λ d wh λ th lct schm s covd bcwd dffc qto To dtm th stblt of th mthod w loo t th gvls of th mt. Fo th C-Ncholso schm, th mt qto s:

105 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of N N U S U C,,,,,, α α α α α α α α α α α α α α Th mt C s o-sgl d w c solv fo th ows t th d of th tm st s: U T U S C U W c fd th solto t tm st fom th tl codtos d ow of th mt T, so: U T U I od to s stblt th gvls of th T mt mst ll hv mgtd lss th. Ths gvls gv b: s s N N π α π α λ Wh N s th od of th mt. Ths sso ls btw - d fo vls of α, so th C-Ncholso mthod s stbl. Eqvlt cct stto Cosd th sm ft dffc cll showd th bov gs: t,,,,, Ths qto c b wtt s:,,,,,, t Th qto fo th cct showd, c b wtt s: dt d C d f w dsctz th bov qto wth sct to tm, w obt:

106 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION,,,,, C t, Th two bov qtos hv th sm fom. Idtfg tms:,, ;, ; C, ;,, It s obvos tht th odl voltgs hv th sm mc vl s th odl ottl o th ft dffcs msh. Ecs. Solv th followg oblm: Wth: d < t.sc du U dt U, d th followg bod codtos: U, t U, t. To solv ths oblm w c s gd wth d th C-Ncholso mthod. t. Th solto c b clcltd b th s of sdsht. S th followg cts. d tm Th followg fg shows th ottl t ch ot hozotl-s coodt d ch cv sts tvl of tm.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

107 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION 8 6 tm cs Th followg fg shows th volto of th ottl t ch od of th msh s fcto of tm hozotl s tm glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

108 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION Mot Clo mthods Th Mot Clo mthods obblstc mcl mthods mlod solvg mthmtcl d hscl oblms. Th mthod of omtl solvg oblms sg sqcs of dom mbs. Mot Clo mthods ld two ws: Smlto, fo ml to smlt to s moto to cto wll: ts zgzg th bg mttd b dom wl. Smlg, fs to mthods of ddcg ots of lg st of lmts b stdg ol smll, dom sbst. Fo ml, th vg vl of f ov <<b c b stmtd fom ts vg ov ft mb of ots slctd doml th tvl. Ths mots to Mot Clo mthod of mcl tgto. H, wo t dtl th gto of dom mbs d vbls. Yo c fd ths fomto boo o sttstcs. Nmcl Itgto Sos w wsh to vlt th tgl I f wh s -dmsol sc. Lt X X, X,..., X b dom vbl tht s foml dstbtd. Th fx s dom vbl whos m vl d vc s gv b f X f f X f f wh dx If w t N ddt smls of X,.., X, X,, X N, ll hvg th sm dstbto s X d fom th vg f X N f X... f X N f X N N w mght ct ths vg to b clos to th m of fx. Ths w c wt: I N N f X. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

109 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of Fo o-dmsol d two-dmsol tgls b lcto of th bov qto, w obt: N b d c N b X X f N c d b dd f I X f N b d f I, Fd dom Wl mthod to solv ottl oblms Sos tht th Mot Clo mthod s ld to solv Llc s qto go, sbct to Dchlt bod codtos:, t f f W bg b dvdg to msh d lcg b ts ft dffc qvlt qto. Th fl qto s wtt s:,,,,, h f h f h f h f f wh: / sq gd of msh sz s ssmd. Th qto m b gv obblstc ttto. If dom wlg tcl s sttosl t th ot,, t hs obblts,,, of movg fom, to, ;, ;, ;,, sctvl. ms of dtmg whch w th tcl shold mov s to gt dom mb, < < λ λ d stct th tcl to wl s follows:,, /,, / /,, / /,, / h h h h < < < < < < < < λ λ λ λ To clclt th ottl t,, dom wlg tcl s stctd to stt t tht ot. Th tcl ocds to wd fom od to od th gd tl t chs th bod. Wh t dos, th wl s tmtd d th scbd ottl t ths bod ot s codd. Ths vl s dotd b f. Th scod tcl s lsd fom, d llowd to wd tl t chs bod ot, wh th wl s tmtd

110 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of d th cosodg vl of th ottl s codd s f. Ths ocd s td fo N tms. Th ctd vl of ths st of ottls s th solto to th oblm,..: N f N f, wh N, th totl mb of wls, s lg. W c t ths ocss fo th whol st of gd ots to obt th solto t ch ot. If o d to solv th Posso s qto g, Th th FD stto s:,,,,, g h f h f h f h f f d th fl tm mst b codd t ch st of th dom wl. If m sts qd fo th th dom wl ogtg t, to ch th bod, th o cods, m g f Ths th solto c b ssd s N m N g N f N f,, Th followg fg shows Ecl mlmtto of ths mthodolog motc.ls. Yo mst dw th whol sstm. Th bod vls dctd b colos: Dchlt bods: t th clls d st th ow vl. Nm bods: t th cll g. ftwds, slct th cll wh o wsh to clclt th ottl d ss th stt btto ot show ths ct; ft th scbd mb of tsts 5 ths cs th clcltd vl s show. Not tht th mst b closd. If ths s o, th solto c t b chd.

111 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION omto tchqs Cosd th followg oblm: Th solto of sstm of dfftl o tgl qtos stblshd th gl fom: L f C { } wh L s l oto fo ml, f s ow fcto, d s th ow fcto to b dtmd. Th oto L s scfd go Ω sodd b th bod Γ. Po bod codtos C scfd o Γ. Th fcto wh th sbsct sts th ct solto of th oblm c b omtd b ssmg tht t vs ccodg to st of ow fctos, ch of thm mltld b ow coffct. Ths omt fcto coffcts, whos mb c b csd to cs th cso of th slts, c th b dtmd b solvg sstm of qtos. Ddg to th omto mthod w obt dfft qtos d mthods sch s: ft lmts, bod lmts, momt s mthod d oths cldg ft dffcs mthod! commo ft of ll th mthods ld s tht th omt solto s ssmd to b th ft sm ϕ ϕ wh ϕ th ow tl o bss fctos d th coffcts to b clcltd. Dffcs btw th mthods th ocss of dtmto of coffcts. Th tl fctos shold b ll ddt d fom comlt st go Ω. Fcto ϕ s sll scfd od to stsf Dchlt bod codtos o Γ. W todc h th so-clld sdl : L f Th sdl fcto vs th dom. Cosd ow how to dstbt th fcto. Th coct commo to ll of th mthods s tht th coffcts vltd b othogolzto of sdl to ct st of wghtg fctos w,.., b zog th odct:, w w dω,,,..., Ω Sbstttg, st of smltos l lgbc qtos s obtd. Th gl fom s: w w, f Lϕ, Lϕ If th odct of two fctos dfd bov Hs th followg ots:, v v Ω * dω. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of

112 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of,,,,, *,, > f v f v v v v v v W c s tht ths s smmtc d ostvl dfd. I ths cs th tol mthod lgh-tz mthod m b ld. Th omt solto of g L s dtmd s th mmm of th g fctol I : g L I,, Sbstttg th tl fctos d mmzg ths b tl dvto wth sct to ch coffct, I I I w obt th followg st of smltos qtos: N N N N N N N g g g L L L L L L ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ,...,,.,,, ,,, Fo Llc s d Posso s qtos d th vtos th followg tbl of fctols lcbl. PDE FUNCTIONL g [ ] dv g [ ] dv dt d dt dv dt d t dt d dt dv dt d t g [ ] dv g [ ] dv dz d d d d d

113 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION Ddg o th wghtg fcto w c obt dfft qtos to solv., w w dω,,,..., Ω Th followg tbl shows th wghtg fctos d th qtos to b solvd fo ch omto mthod fo th o-dmsol oblm. Momt mthod. Th wghtg fctos s th smlst:,,,.., fo o dmsol oblm. I ths w som hgh od momts of th sdl c b st to zo. Pot collocto. I ths cs N ots chos th dom d th sdl s st to zo o ths ots. Ths oto c b ttd s dfg th wghtg fctos tms of Dc mls. Sb-dom collocto. Th mthod s sml to th collocto mthod dscbd bov bt ow th sdl s qd to b zo ov ct go, th th ss of ots. Lst sqs. I ths cs w mmz th sq of th sdl. Gl. I ths cs th wghtg fcto s th sm s th omtg fctos. lgh- tz mthod fctol mmzto. Mthod Wghtg fctos Dtmto ocd of l lgbc qtos. Momts w,,,... Ω dω,,,...,. Pot collocto w δ b. Sb-dom b collocto δ Ω othws d. Lst sqs w 5. Gl w ϕ 6. lgh-tz b Ω d b I ϕ dω I I sll th sltt qtos th sm s obtd o th Gl mthod. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of

114 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION Eml. Solv th followg qto b lcto of ch omto mthod. : Ths qto hs th ct solto: W slct th followg tl fctos: s π sπ fo mthods fom to 6. ϕ ϕ ϕ ϕ s π ϕ sπ. Momt mthod. W slct th followg tl fcto, sml oloml, whch ft stsfcto of th bod codtos slts fo th s of smlct ol two tms wll b cosdd ol th fst two momts ql to zo: Th sdl s: 6 d th sstm of qtos to solv s: 6 d 6 d Th vls of th ow coffcts :. Pot collocto. W slct th followg mtchg ots: Th sdl s w δ / w δ / s π sπ π s π π sπ d th sstm of qtos to solv s: / / π s π / π sπ / π sπ / / π sπ / / Th vls of th ow coffcts :. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

115 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of sb-dom collocto. W slct th followg doms: / / δ δ Th sdl s dtmd s s s s s / / d d π π π π π π π π Th vls of th ow coffcts : Lst sqs. Th sdl d th qtos to solv : s s s s s s s ; s s s d d π π π π π π π π π π π π π π π π π π π π Th vls of th ow coffcts : d 6. Gl d lgh-tz mthods. Ths two mthods hv th sm qtos to solv. w ϕ Ω b d ϕ s s s s s s d d π π π π π π π π π π Th vls of th ow coffcts :.86.86

116 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL SOLUTION Th followg fg shows th ct d omt soltos. Th Gl d th momt s mthod hv th bst ccc. Not tht th momt s mthod omto fcto clds th ct solto ts fomlto ct ot sbdom lst sqs Gl momts ct momts ot sbdom lst sqs Gl Th Gl d th momt mthod th most cct. Not tht th momt mthod omto fcto clds th ct solto ts fomlto.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

117 NUMEICL SOLUTION: FINITE ELEMENT METHOD

118

119 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD FINITE ELEMENT METHOD ONE DIMENSIONL FINITE ELEMENT NLYSIS Th bod vl oblm to b cosdd s dfd b th ODE: d d d α β d f, L wh s th ow fcto, α d β ow mts d f s ow soc. Th bod codtos fo gv b: d α γ d L q Dsctzto d tolto Th fst st s to dvd th solto dom,l to smll sbdoms, whch ths cs wll b shot l sgmts. Lt l,,,.m dot th lgth of th th sgmt, wth M bg th totl mb of sgmts. Fth, lt,,,.n dot th osto of th th od wth d N L. oth lmts d ods mbd od fom lft to ght s show th t fg. To th fomlto cosstt wth two d th dmsol css, w dot locl mbg sstm th fomlto. I th followg, th ssct s sd to dot th qtt wth locl mb s sbsct, whl fo ll oth qtts th sbsct s globl mb.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

120 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 Th locl d globl sstms ltd b: M fo,,..., Th scod st s to slct th tolto fctos. W s l fctos. Wth th th lmt m b omtd b: b wh d b th costt to b dtmd. Fo l lmts, th two ods ssoctd wth ch lmt: o loctd t d th oth t. Scfg t ths two ods lds: b b wh dots th vl of t. Th solvg fo d b w obt: l l l b Sbstttg w obt: l N l N N l l

121 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 N fctos clld th tolto o bss fctos. Fomlto Th thd st s th fomlto of th sstm of qtos. W c s th tol och o th sdl wghtg fomlto Gl mthod. Th sdl fo th Gl mthod s: f d d d d β α d fo ch lmt th sdl c b wtt s:, d N W s N s wghtg fctos. Sbstttg w obt: fd N d d d d d N β α tgtg b ts fo th fst tm o th ght hd sd, w obt: d d N fd N d N d d d dn vd v dv d d dv N α β α α Th sbsttto of N lds to th qto:

122 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 d d N fd N d N N d dn d dn α β α d mt fom t bcoms: g b wh: d d N g fd N b d N N d dn d dn, α β α W ot tht s smmtc d, f α d β costts o c b omtd b costts wth ch lmt, mt lmts c b vltd ltcll; th slt s: 6 l f b b l l l l β α β α Wth th lmtl qto gv, w c ocd to fom th sstm of qtos b smmg t ov ll lmts: [ ] M M g b whch c b wtt s g b

123 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 7 ssmbl of th qtos To llstt th ssmbl of qtos, w cosd cs wth th lmts d fo ods. Usg th ltosh btw th globl d locl od mbs, w c d to mt d to colm. Ths, fo th fst lmt c b dd s: ; d th odct bcoms: Fo th scod d thd lmt w c wt sml w: ; ; Wh ths odcts ddd:

124 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 7 ccodg to th lto btw th globl d locl od mbs: mg ths chgs, w c wt: Smll w c d b to fd tht: ; g g g g g g g b b b b b b b Ect fo g d g, th oth lmts of g c b wtt s: g g g ft sbsttto of d d N d d N d d N g α α α ths bcoms:

125 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 7 d d d d g α α fom d w c wt: d sc d d α s cotos t, t s obvos tht g fo,. Thfo g hs two ol ozo lmts: d d d d g α α Fo gl oblm wth N ods w th hv: N fo g d d g d d g N N, ; α α od codtos Fst w cosd th Dchlt codto, tht s:. To foc ths codto w d to modf th sstm of qtos: w mos fo ml b sttg: ; b d fo,,,..n Th sltt sstm bcoms: b b b

126 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 7 Ths sstm s ot smmtc. Ths s ot dsbl, sc smmt s v mott ot whch c b lotd to dc comt mmo dmdd s wll s ocssg tm. To sto th smmt w m modf th bov qto s: b b b b b b s slt g shold b dscdd. Nt w cosd th bod codto q d d L γ α to b ld t L. If th bod codto s th homogos Nm codto q γ, th g N vshs. Othws, w hv: N L N q d d g γ α Thfo g N c b bsobd to d b: γ q b b b Eml. Dtm th ottl dstbto to th w whch s coctd to costt voltg. lso dtm th ct tht flows fo t. Th w hs costt scto d codctvt. Dvd th t dom lmts. Solto. Th lcbl qto s: J σ L E J σ f f d d d d β α β α W dvd th w to th lmts wth ql lgth: l L/.

127 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 7 dct lcto of th ssos fo d b, w c wt: 6 l f b b L l l L l l β α β α 6 6 L g b Ths s sgl sstm. lcto of C: ; w c wt: d fll, w obt: 6 6

128 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 S L S L S L ds L JdS I L L J L L J L L J d d E J / / / ρ σ σ σ σ σ σ σ σ σ σ σ σ W obt ltc solto to com wth mcl slts: L E J L d d E L L L L b b d d σ σ Ecs. Solv g f th w s md of two mtls: o fom to L/ d th oth fom L/ to L. I ths cs th lvt qto s: J σ f f d d d d β σ α β σ dct lcto of th ssos fo d b, w c wt:

129 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of l f b b L l l L l l L l l L l l σ β α σ β α σ β α σ β α 6 σ σ σ σ σ σ σ σ σ σ σ L g b Ths s sgl sstm. lcto of C: ; w c wt: σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ If, fo ml σ σ, w c obt:

130 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 / / / / / / / S L S L S L ds L JdS I S L S L S L ds L JdS I S L S L S L ds L ds J I L L L L J L L J L L J d d E J ρ σ σ σ ρ σ σ σ ρ σ σ σ σ σ σ σ σ σ σ σ σ σ σ Ecs to solv Solv th followg D oblm: < < f d d Clclt th solto wth th s of l tolto lmts d dsctzto ots:,.5,.7 d. Solv ssmg: f ; 5; 7. b f ; 5; 7. Solto: [ ] b [ ]

131 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD TWO DIMENSIONL FINITE ELEMENT NLYSIS Th bod vl oblm d cosdto s dfd b th scod od PDE: α α β f Th bod codtos to b cosdd gv b: o Γ d α α d d γ q d o Γ If th ots of th dom chctzd b α d α hv dscotts o bt chg, d fthmo, f th s o sfc soc of d t th dscott tfc, th stsfs th cott codtos: o Γ d α d d α d d α d d α d d o Γ Dom dsctzto Th fst st s to dvd th dom to mb of two dmsol lmts. W s tgl lmts h. To dtf ch lmt w c lbl th lmts wth st of tgs, d to dtf th ods tht th vtcs of th lmts, w c lbl thm wth oth st of tgs. Sc ch lmt s ltd to svl ods fo tgl od hs ts ow osto th ssoctd lmt ddto to ts osto th t sstm. Ths osto c lso b lblld wth tg mb fd to s th locl mb, cotst to th globl mb, whch dcts ts osto th t sstm.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

132 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD To lt ths th mbs- th globl od mb, th locl od mb, d th lmt mb w todc M, dotd b,, wh,, d,,,., M wh M dots th totl mb of lmts., s clld th coctvt ; s th locl mb of od, s th lmt mb, d th vl of, s th globl mb of th od. To llstt ths w cosd th ml show th followg fg. W hv lmts d 6 ods. Th, c b mbd s:,,, Th mto s ssgd cot clocws dcto lft ght; dow. I ddto to th dt dscbd bov, som oth dt lso css th FE fomlto: -,, whch ovd th coodts of th ods,. N, wh N dots th totl mb of ods. - Th vls fo α, α, β d f fo ch lmt. - Th vls of fo th ods sdg Γ. - Th vls of γ d q fo ch sgmt cocdt wth Γ. Itolto If l tgl lmts sd, th ow fcto wth ch lmt s omtd s:, b c wh, b d c costt coffcts to b dtmd d s th lmt mb. Fo l tgl lmts, th th ods loctd t. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

133 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 7 th vtcs of th tgl. Th vls of t ch od,, sctvl. Efocg t th th ods, w obt: c b c b c b Solvg d gg tms, w obt:,, N Wh, N th tolto o so fctos gv b:,,, c b N whch: lmt th th of E c b c b c b c b c b I th bov qtos,, dot th coodt vls of th th od th th lmt. It c b sl show tht th tolto fctos hv th ot: N, d, s slt, t od, dcs to ts odl vl., N vshs wh th obsvto ot, s o th lmt sd oost to th th od. Thfo, th vl of t th lmt sd s ot ltd to th vl of t th oost od, bt th t s dtmd b th vls t th two dots of ts ssoctd sd. Ths fts gts th cott of th solto coss th lmt sds. Th followg fg shows th tolto fcto N fo tgl lmt.

134 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 7 tol fomlto Th tol oblm qvlt to bod vl oblm s gv b: Γ o δf Wh: Ω Γ Ω Ω Γ Ω d f d q d F γ β α α s th fctol qto whch mst b mmzd. Fo smlct fst cosd th Homogos Nm od codto wth q γ fo whch th l tgl th fctol vshs. Ths th fctol c b wtt s: M F F Wh M dots th totl mb of lmts d F s th sb-fctol gv b: Ω Ω Ω Ω d f d F β α α wth Ω dotg th dom of th th lmt. Itodcg th sso,, N d dffttg F wth sct to lds:,, Ω Ω Ω Ω d fn d N N N N N N F β α α o:

135 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 7 [ ][ ] [ ] t F Wh: [ ] F F F F ; Th lmts of th mt [ ] gv b:,, Ω Ω d N N N N N N β α α d thos th vcto [t ] b:,, Ω Ω d fn t [ ] s smmtc mt. ssmg ow tht th coffcts f,,, β α α costt wth ch lmt d ql to f,,, β α α sctvl, th bov tgls c b vltd ltcll: f t c c b b δ δ β α α ssmbl to fom th sstm of qtos Wth th lmtl qto, w c ssmbl ll M lmts, th mos th stto qmt o F to fd th sstm of qtos: [ ][ ] [ ] [ ] M M t F F Th sstm of qtos c b wtt comctl s: [ ][ ] [ ] t

136 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD Wh [] s ssmbld fom [ ] d [t] s ssmbld fom [t ]. I sml w o th D ml, w c show tht th [] d [t] mt fo th fo lmt, 6 od ml sstm s ql to:,,, [ ] [] t t t t t t t t t t t t t Icooto of th bod codtos of th thd d Th sstm bov s dvd b th ssmto tht stsfs th homogos Nm bod codto o Γ. Now lt s cosd th gl cs wth o vshg γ d q. W dd to th fctol t tm:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 7

137 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 7 Γ Γ d q F b γ ssmg tht Γ s comsd b M s sds o sgmts lmts, th bov qto c th b wtt s: M s s S s s s s b ds q F γ Th ow fcto wth ch sgmt c b omtd s:,, s s s N wh: ζ ζ s s N N whch ζ s th omlzd dstc msd fom od to od th sgmt: ζ t od, d ζ t od, d btw th two ods t vs ll. Sbstttg d dffttg t wth sct to s lds: s s s s s s s s b d l qn d l N N F ζ ζ γ wh l s dots th lgth of th sgmt. I mt fom w c wt ths s: [ ][ ] [ ] s s s s s b t F wth th lmts [ s ] d [t s ] gv b:,,, d l qn t d l N N s s s s s s s s ζ ζ γ f γ d q costt wth ch sgmt d dotd b s γ d q s th bov tgls c b vltd ltcll d th slt s: q l t l s s s s s s 6 δ δ γ To cld F b to th sstm t shold b modfd:

138 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD F M F M s Fb s M s s s [ ][ ] [ t ] [ ][ ] [ t ] s M s s s F Wh b s, [ s s ], [ ] d [t s ] hv lso b gmtd. To do ths, w d tht lts th sgmts d th globl mb of th ssoctd ods. Ths, s,s,; s,..., M s ls sml ol s th coctvt,. I s,s w sto th globl mb of th th od of th sth sgmt. Cosd th bov ml, f Γ comss th sgmts th sds dfd b th ods 6,,, d, th s,s c b mbd s: s s s,s s,s 6 [ s ] c b ssmbld to [] b ddg ch c b ssmbld to [t] b ddg t s to t, s. s to d smll [t s ], s,, s s s s Imosto of th Dchlt bod codto W d to mos o th sstm th Dchlt bod codto whch ls to th ods o Γ. To llstt how tht s do, w cosd g th ml. ssm tht th ods, 5 d 6 o Γ d th hv th scbd vls, 5 d 6 sctvl. To mos th codto w c sml st: t fo,,,5,6 Ths dstos th smmt of th mt, d to sto ths ot w c m th followg modfctos: t t fo,,,5,6. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

139 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD I sml w, w c mos th oth codtos: mtcs d t bcom: , d fll, th [ ] ; [ t] t t t Wth th dlto of thd, ffth d sth qtos, th sstm bcoms: t t t oth och commol sd to mos Dchlt bod codtos wthot th modfcto of th sstm of qtos s to choos hg mb 7 d st: t 7 7 Fo ml to mos th qto ssoctd to bcoms: Povdd tht ll mt lmts d ows smll th 7, th bov qto s ffctvl qvlt to. Th w sstm of qtos th bcoms: t t t I whch h th smmt s td. Ths och s mch sml th th vos os s std. It qs ol two otos to mos bod codto. Mo mottl, ths tchq c sl b ld to. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

140 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 sstms whos mt s stod comct fom whch css fo dcg mmo dmd. Howv, th tchq dos ot mt lmto of th qtos ssoctd wth th Γ ods, whch s ts mo dsdvtg. Eml. Solv th mgtostàtc oblm dfd b th followg ods d chctstc lmts.,,, µ J /m J f J β µ α α µ Wth th bod codtos:. Itmdt clcltos: lmt th th of E c b c b c b c b c b b b b c c c Nod 5 X Y

141 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 f t c c b b δ δ β α α st Elmt ; ; t t t f µ µ µ µ µ µ µ µ µ µ µ β µ µ α α d Elmt ; ; t t t f µ µ µ µ µ µ µ µ µ µ µ µ β µ µ α α

142 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 t lmt ; ; ; t t t f µ µ µ µ µ µ µ µ µ µ µ β µ µ α α

143 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 7 5 ; ; t µ µ µ µ lcto of od codtos, w c obt th followg qto: m Wb / µ µ Eml. Dtm th lmts fo d T mt th cs of dffso qto:

144 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 7 J f J ωσ β µ α α ωσ µ s of th gl qto fo d t f t c c b b δ δ β α α w c wt fo ch lmt: J T c b c c b b c c b b c c b b c b c c b b c c b b c c b b c b ωσ µ bcoms coml mt!

145 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD Ecss to solv. Dtm th ottl th tmdt ods. b Dtm th ct dst, d th totl ct th fs showd th followg fg.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 7

146 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD Nol oblms I th lss of lctcl mchs d oth mgtc ts, th oblms bcoms ol d to th sc of fomgtc mtls. Good dsgs ot t, o, th stto ot. Th mblt µ / H s fcto of th locl mgtc fld, whch s ow t th stt of th oblm. Th mblt s ll of th lmts of mt, d w mst s ttv ocss d coctg th mblt tl t s cosstt wth th fld solto. Th most ol mthod of dlg wth ol oblms mgtcs s th Nwto-hso mthod. W l ths mthod th followg gs. Fo th Nwto-hso mthod, th mgtc lctvt, ν s fcto µ of mst b cotos d dfftbl. mb of omtos ossbl fom olomls to otls. Nwto-hso mthod Cosd fst th sgl o-l qto: g W wold l to fd th oots of g. W c d th bov qto od Tlo ss: g g dg d... g ol th fst od tms d gg, w c wt: g dg d Fom th tl stmto of,, w c fd oth omt vl: d so foth: g dg d. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 7

147 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD g dg d g dg d Eml. Solv th followg qto: f cos f df d cos s f df dlt E E E E o vss tto s mb Cosd th followg mltdmsol sstm of o-l qtos wth ddt vbls,,, :. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 7

148 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7,...,,,...,,,...,, f f f F Edg Tlo ss d tctg t ft th fst od tms gvs: { } F J Wh J s th Jcob mt gv b: f f f f f f f f f J Th omto c b clcltd b: F J F J F J lcto of th N- mthod to lmt W cosd h th two dmsol ol Posso qto fo th mgtc vcto ottl. W hv fod tht th qtos fo two dmsol fst od tgl : J J J ν Ths qto cold b wtt s: J J J f f f F ν To clclt th Jcob, w dfftt ths qtos wth sct to th odl vcto ottls. Fo f ths gvs:

149 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 f f f ν ν ν ν ν ν Th fst qto fo th N- tto s: [ ] [ ] J f f f f ν ν ν I mt otto ths bcoms: J ν ν ν w ow do th sm fo th scod d thd qtos to gt th whol lmt qto: J ν ν ν

150 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD ν Th vcto ottls h t fo th vos tto. W fd fom th stto cv stto. To vlt w ocd s follows: Fo tgl fst od lmts: b c b c b c b b b c c c b b b b c c c b b b c c c c W c smmz th ocss s follows: b b b c c c STEP POCESS ssm vl fo ν d fo ch lmt d od Evlt th mtcs sg ths vls d th mtl coffcts ssmbl th mt th oml w l bod codtos d solv fo th vcto 5 Fd th w b ddg to th vos vl of 6 l stog o covgc tst, sch s th ltv chg s smll th ε 7 If th tst fls, comtd th mt d t th ocss fom st Pmt mgts PM modllg Th dvlomt of hgh g mt mgt mtls sch s SmCo d NdF hs ld to csd tst th s of mt mgt mtl lctcl mchs d cttos. s mtod th lst scto, fomgtc mtls chctsd b ow hstss loo. I cotst, hd mgtc mtls sch s PM hbt wd loos. It s oft. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

151 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD cctbl to cosd th mgtc chctstc of PM b stght l th scod qdt of th hstss loo. Th tscto of th hstss loo wth th odt s clld th sdl o mc fl dst. Th tscto of th bscss d th loo s clld th coctv foc H c. Th two ossblts fo th modllg of PM mtl: Mgtsto modl Ct sht och lthogh ths two mthods hv dfft sttg ot, th both slt th sm st of qtos. ssmg stght l s th chctstc of th PM mtl, th ol two mts qd to df th chctstc: Th slo of th l µ m Th -s tct Mgtc vcto modl Th dmgtsto chctstc s dfd b H M µ χ m wh χ m s th mgtc ssctblt, M th mgtsto vcto d H th fld stgth t th otg ot. I tms of th mt fl dst µ M Th cmtl mblt, th slo of th dmgtsto chctstc s H µ χ m χ m s v smll ostv mb so tht th t mblt of th mgt s ol slghtl lg th tht of th f sc. Th lctvt s dfd s ν µ d lg ths to th dmgtsto chctstc, lds χ m H µ M ν sg th Mwll qto fo mgtostàtc oblm: H J. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

152 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD lds ν J ν µ M Th scod tm, th mgtc vcto, o th ght-hd sd sts soc tm d c b dtfd s qvlt mgtc ct. Th mt T mst b comtd to cld ths tm. Ths tm c b comtd wth: Ω νµ M N M N νµ dd M c c c M b b b Eml. Solv th mgtostàtc oblm dfd b th followg ods d chctstc lmts.,,, µ J M /m 5 5 µ.5 Dcto of mgtzto log s..8t Nod 5 X Y J µ ν µ M Wth th bod codtos:. Th d T mt th sm fo th st d d lmts, th thd lmt vls dvdd b th ltv mblt µ. 5 :. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

153 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of ; µ µ µ µ Th mo dffc s th ght hd sd wh th ct vcto s lcd b th mgtzto vcto sg th bov qto. I ths cs M d th w vcto T bcoms: b M b M b M t µ µ µ W c l th ot bod qtos to solv th oblm. Th vcto ottl solto s: m Wb /

154 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 7 Tst solto. W l th cs of th two dmsol tst mgtc oblm. Th PDE ths cs s: J t σ ν Th fctol fo ths qto s: d t J F Ω Ω ν σ Th fst d scod tms th sm s thos tht cotd th sttc cs. Th thd tm lds: Ω Ω Ω Ω d N N t t d N N d t 6 σ σ σ σ σ Th sstm of qtos mt vcto otto c b wtt s: T t Th solto s ol comtd t dsct ots tm, scd ft tvls t, th tm sts. Th Gl och c b ld th tm dom. Fst, w slct th sh fctos of tm, of cos s: T T t T t τ α ατ τ α ατ wth: t t t t t t t t t t τ τ τ

155 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD Th w l th sd mmzto wth ατ s wghtg fcto: ατ t ατ α τ ατt α τ T dτ ft som lgbc mlto, w obt: Commts: α t t α αt α T Ths sso s codtoll stbl fo α /. Fo α ths s lso f of osclltos. Fo α w obt th fowd dffc El mthod. Ths s lct mthod bcs th tm s vltd t th bgg of th tm tvl: t t T Fo α t gvs th bcwd dffc fll mlct mthod sc th tm s vltd t th d of th tm tvl: t t T Fo α / t gvs th C-Ncholso schm: t t α T α T oltg fd lctomgtc dvcs U to ths ot, w hv ssmd tht th soc of th mgtc fld s ct dst J. I m css, th sstm s voltg fd d th ct th col s ow. To solv ths oblm, both th fld qtos d th col voltg qto mst b solvd smltosl. Th voltg qto fo th col c b wtt s: d U dt. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 7

156 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD Wh U s th voltg o th col, d th col sstc d th mb of ts; s th mgtc fl Wb/m, w c stf ths t lt gtd th solto dom d ld b th col. Th ct cotbto of lmt of th msh s gv b: J Wh J s th ct dst d s th of th lmt. Now df w mt κ s th col t dst ts/m. If I s th ct of o t, w hv: J κ κ I I PI D C z I l d I E t cg DC ct I, ftl log z-dcto: ol o t z- dcto s std. Dstc btw codctos s d. l d Th - stto of th t. Lt s ow cosd th fl t lgth z-dcto of coss sfc tht hs th t s coto. It ds, obvosl:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 7

157 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD ϕ ds S wh S s th sfc of th t fo o t z-dcto, d s th otwd oml to S. sbstttg th bov qto th sso fo s fcto of : ϕ S S S ds lcto of Stos thom slts th followg: ϕ d S S S wh th l tgl th hs s dfd log th t. O th bss of th ssmtos vosl md, th fl of c b wtt s: C ϕ dz d S S S Th fst d thd tgl hv, sctvl, th followg vls: d - l th vcto ottl hs th sm vl wh th - d -coodt of th ot do ot chg d th z-coodt of th ot chgs. O th oth hd, th std ws fomd wth fc of o t z-dcto, d moov t shold b otd tht th tgto th movs oost dctos log th codctos. Th scod tgl s obvosl glgbl wth sct to th fst d thd sc th ws ftl log: w stdd st o t of th lgth z-dcto, bt th tso of th ctv lgth s ftl gt th th lgth of th coctos. W v t th followg foml: E D dz ϕ S l fo th fl b t of dth. If th lgth s ft, th totl fl s: l S s th zo mgtc vcto ottl vl s sll st th fld vlto s slt of th clclto o b ms of th Dchlt bod codto, o c clclt th mgtc fl wth sct to th ll vl d th wt th bov qto s l Wh s th ottl t ot th solto dom. Fth, ssmg tht s th vg ottl tgl lmt, w gt: l. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 7

158 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7 If s th mb of ts wth th lmt, w gt: [ ] [ ] l Q Q l l κ κ κ κ Usg El s schm to dsctz th tm dvtv, w obt: t Q t Q dt d Q dt d l κ κ [] t Q I U dt d U D PI t T t t D s tm ltd to th mt mgt o oth cts. [] U D I t Q t I t Q P t W c lso cld ddtol tm to t to ccot th tl sstc d dctc dt d L dt d U th fl qto s: [] [ ] [ ] [ ] U D I L t t Q t I L t t Q P t

159 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD Colg of fld d lctcl cct qtos. Two ts of codctos oft st lctcl mchs. Th c b th o thc codctos. I th fst cs th dd cts mst b glctd. Thc codctos Th followg fg shows thc codcto wth scto S t d lgth l. W c wt tht: z σ z Ths, w c df scl lctc ottl s: z Th voltg U t o th codcto s gv b: U l d dl l dz t Th ct dst th codcto s dtmd b: J d E σ J σ σ t t dz t σ U σ t l t d th totl ct th codcto s dtmd b: I t I t t JdSt σ dst St St t St U σ l t JdSt σ dst t l St St St U σ S t ds t U t σ dst t t gg ths qto, w c wt: U t t It t σ ds t Ths qto shows tht th voltg thc codcto s ql to th sm of th voltg do ov th DC sstc d voltg do d to dd cts. Ths qto, wth th ddto of th fld qto: S t t. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

160 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD solvs th oblm. t t t ν J σe σ ν σ σ U l Th codctos Th fg shows col md of N co ts of th codctos wth coss scto s, sl coctd. I ths codcto th ct dst s cosdd fom ov th coss-scto. W cll I f th ct codcto. U U t σ σ t l t It t σ dst t S t I t s s t ν t ν f ν σ σ ds St t t I σ σ t t t St l σ ds t t s th dcd ct dst σ s fom ov th coss scto S, w c t wt: s S t σ ds t t σ t d sbstttg th bov qto, ths s dcd to: I f ν s If th totl sfc of th col s S f Ncos, w gt: N S co ν I Th voltg U f t th tml of th wdg c b wtt s: f f. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

161 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD U N N o U U f co co f f N l I σs l I σs N c co f f f I U N f t l I σs N l s co S f N f N co co l σs I t s t l S co t N ds t σ ds N t l S f S f co f S f t s ds t σ ds t co l I σs f N co l σ s σs t Th fst tm s th voltg do ov th col sstc. Th scod tm s th voltg dcd th col. I th th codctos dom, th qtos bcom: U f I f f N co N S l S f ds t S f co ν I f f Eqtos fo th whol dom. ccodg to th bov ld, th st of qtos fo lctomgtc dvc stg mgtc mtls, mt mgts, thc d th codctos s: U U t f co t I σ σ ν µ M I t f t f N t St co f S f t ν f S f l I N l S σ ds t ds t t ft lg th tol o Gl mthod to th bov qtos, st of mt qtos s obtd s follows: U. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

162 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD PI f P' U t d Q' ' It U t dt d d Q '' I f L I f dt dt t D Mt s th d.c. sstc of th th codctos wdgs d L s th mt of ddtol dctcs. s mt cotg th DC sstc of thc codctos. Mtcs,, P, P, D, Q, d Q obtd b ssmblg th lmt mtcs: P S S S t N ν N σn N S co f th od blogs to th go of wdg, o P f od s lswh. f t N N ds ds U σ P' N ds l S f od s th go of wdg, o ql to lswh. ds N N D νµ M M dd Ω Ncol Q N ds S f od blogs th go of wdg, o ql to zo lswh. Q' S σ N S f od blogs to th thc codcto, o ql to zo lswh. Dfft combtos of th d thc codctos wdgs c b fod lctcl dvcs. W wll ot cosd ths h dtl: comlt fomlto mst cosd th coo btw th dfft cols d th stctos voltg d ct mst b cosdd. I gl fom, w c wt: t ds [ I f ] [ C][ I] ; [ It ] [ C ][ I ] [ U ] [ C ][ U ]; [ U ] [ C ][ U ] f t f. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 7

163 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD U, U, I d I th tl voltg d cts fo ml th hs smmtcl sstm d C, C, C d C th coo mtcs tht lt th tl vls to th tl vls. Movmt modllg fo lctcl mchs Th bsc qtos of moto s follows: dω J ρω M M c dt dθ ω dt Wh: J s th momt of t ρ s th dmg coffct ω s th ottol sd θ s th otto gl M s th lctomgtc toq M c s th tll ld mchcl toq. ll of th bov t to th movg t. Th followg fg shows lctc moto wh th oto s ottg. Th s mott qsto: If th sstm s movmt, th mt of th whol sstm tmddt? Th sos s Ys! tl w d to comt LL of th lmts o th sstm mtcs. If o costct gd wth movg bd th -g d ol th lmts th g o f sc mttd to dfom, th codctvt σ sch lmts s zo, whch llfs th cotbto of th, P d Q mtcs ddtoll th lmts of P d Q mtcs lso lls bcs th o codctos d ct th. Thfo, ol th tm ddc of th mt qs cosdto: w ol d to comt th vls of th lmts th movg bd.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 7

164 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD It s dqt to ct th o fo ls th g d l th movg bd to th tmdt ls. Ths ls f sc. Eml. Th followg sqc of cts shows th volto of th tmt slot of lctcl mch. Itll th mch s hot. t t s th mch s dscoctd fom th two d ms motolss. Th tmt flls to mbt vls. t s, t 5 s, t s, t 5 s, t s.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 7

165 FINITE ELEMENTS FO ELECTICL ENGINEEING NUMEICL ESOLUTION: FE METHOD. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 7

166

167 NUMEICL SOLUTION: INTEGL EQUTION METHOD

168

169 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM INTEGL EQUTION METHOD tgl qto IE s qto volvg ow fcto d th tgl sg. Th L tgl qtos tht most fqtl stdd fll to two ctgos md Fdholm d olt. Th followg tbl shows th fst, scod d thd ds of ths fo th o-dmsol cs d Fhdholm olt b f, t t dt f, t t dt b f λ, t t dt f λ, t t dt b f λ, t t dt f λ, t t dt Wh λ s scl l o coml mt. Fctos,t d f d th lmts d b ow, whl s ow. Th mt λ s somtms ql to t. Th fcto s ow s th l of th tgl qto. Notc tht th cs of olt Itgl qtos th lmt of tgto s vbl. If f, th tgl qtos bcom homogos. Cocto btw dfftl d tgl qtos Most od dfftl qtos c b ssd s tgl qtos, bt th vs s ot t. Whl bod codtos mosd tll dfftl qtos, th cootd wth tgl qtos. Fo ml, cosd th followg dfftl qto: d F,, d c b Itgtg w obt olt qto of th scod d: oth ml: F t, t dt c d F,, b d. glló. Elctcl Egg Dtmt. EUETI-UPC d 9

170 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM Itgtg oc lds: d d F, t dt ' Itgtg g b ts w obt: ' t F, dt G s fctos mo sstmtc ms to obt IE fom PDE s b costctg l fcto ow s th G s fcto fo tht oblm. Th G s fcto, lso ow s th soc fcto o flc fcto, s th l fcto obtd fom l bod vl oblm d foms th sstl l btw th dfftl d tgl fomltos. To obt th fld csd b dstbtd soc b th G s fcto tchq, w fd th ffcts of ch lmt oto of soc d sm thm. If G, s th fld t th obsvto ot o fld ot csd b t ot soc t th soc ot, th th fld t b soc dstbto g s th tgl of g G, ov th g of occd b th soc. Th fcto G s th G s fcto. Ths, hscll, th G s fcto sts th ottl t d to t ot chg t. Fo ml th solto to th oblm: g f o Is gv b: g ' G, ' dv' f G ds Wh dots th otwd oml to th bod of th solto go. It s obvos tht th solto c b dtmd f th G s fcto G s ow. So th l oblm s ot tht of fdg th solto bt tht of costctg th G s fcto fo th oblm. W ow llstt how to costct th G s fcto G cosodg to PDE. It s sll covt to lt G b th sm of tcl tgl of th homogos qto LG g d th solto of th ssoctd homogos qto LG. W lt G, ' F, ' U, ' Wh F, ow s th f-sc G s fcto o fdmtl solto, stsfs:. glló. Elctcl Egg Dtmt. EUETI-UPC d 9

171 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM d U stsfs LF δ, ' LU So tht b sosto G FU stsfs LG δ, '. lso G f o th bod qs tht: U F f o Notc tht F d ot stsf th bod codto. Fo solto F s ql to L th fdmtl F l π fo th two-dmsol cs d ' ' F π fo th th-dmsol cs ' ' z z' Th G s fcto of ch cs s ql to: G F U G F U F F l U π U π U s chos so tht G stsfs scbd bod codtos. Th followg tbl shows som G s fctos tht commol sd th solto of lctomgtc ltd oblms. Oto Llc s Std-Stt Modfd stdstt Eqto Eqto Hlmholtz s Hlmholtz s qto qto G G δ, ' G G δ, ' Solto s G δ, ' go -dmsol No solto ' ' -dmsol l H π -dmsol ' ' π π π Wth th d of th G s fcto, w c costct th tgl qto cosodg to th Posso s qto:. glló. Elctcl Egg Dtmt. EUETI-UPC d 9

172 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM s ρ ε ρ G, ' d ε I th dmsos ths qto slts : ρ d ε π THE MOMENT METHOD Th momt mthod hs b ld to so m EM oblms. W wll cosd tcl oblm: th dtmto of th cct of st tsmsso l o godd sstm. Cosd th st tsmsso l of th followg fg. If th l s ssmd to b ftl log, th oblm s dcd to two-dmsol oblm l. Lt th ottl dffc of th sts b U so tht st s mtd t U whl st s t U. O obctv s to fd th sfc chg dst ρ, o th sts so tht th totl chg t lgth o o st c b fod s Q l ρ dl Q l s th chg t lgth s dstct fom th totl chg o th st bcs w ttg th-dmsol oblm s twodmsol o. Oc Q s ow, th cctc t lgth C l c b fod fom Ql Cl To fd ρ, sg th momt mthod, w dvd ch st to sb-s of ql wdth so tht sb-s st mbd,,.,, whl thos st mbd,,..,. Th ottl t bt fld ot s:, ', ' l π ε ρ ' ' d d. glló. Elctcl Egg Dtmt. EUETI-UPC d 9

173 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM. glló. Elctcl Egg Dtmt. EUETI-UPC 5 d 9 Sc th tgl m b gdd s ctgl sb-s, th ottl t th ct of tcl sb- S s ' l d S ρ ε π O ρ Wh S d ' l ε π s th dstc btw th th d th sb-s, d ρ sts th ottl t ot I d to th sb-. I th bov qtos w ssm tht th chg dst s costt wth ch sb- ths s t f th s smll comd wth th totl dmsos of th st. Fo ll th sb-s w hv: U U U... ρ ρ ρ U U... ρ ρ Th w obt st of smltos qtos wth ow chg dsts. I mt fom U U U ,,,,, ρ ρ ρ U U ρ ρ Oc ρ s ow, w dtm C l : U U dl Q C l l ρ ρ It s s to show tht th lmts of mt c b dcd to:.5 l l ε π ε π

174 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM o s costt scl fcto commol t s t. Th followg tbl sts th comtd vl of th cctc lgth t fo dfft mb of sgmts st,. H m; W 5m; ε ε C F Eml. Cosd sq codctg lt mts o sd d lg o th z l, wth ct t th og, s show th followg fg. Lt ρ b th sfc chg dst o th lt, ssmd to hv zo thcss. Th ottl t ot sc s: ρ ', ', d' d' πε ' ' z Th bod codto s U costt o th lt. Th ow to b dtmd s th chg dst d th cctc of th lt. Cosd th lt dvdd to N sq sbsctos, s show th bov fg. Usg th slts fom l, w c wt: Wh: m d' N m m ρ πε ' ' m m d' To smlf th solto w df th followg fcto: N f o S w c wt: ρ, f d fll w obt th followg sso: N m fo m,,... N ot ths cs I ths cs th cctc c b clcltd s Q C ρds U ρ S U U N S N m S. glló. Elctcl Egg Dtmt. EUETI-UPC 6 d 9

175 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM Fo mcl slts th m lmts mst b vltd. Lt sd lgth of ch S : m d' πε ' ' m m m d' m S πε m b dot th N b l πε πε m b m Cosd th followg cs ths cosods to godg sstm of lctc ctl t: m ; b b m ; N Ths vls ld: 8.8 πε πε.5 πε m m πε πε d th cctc s: C C ε N S N m S F / m 78.ε F Th followg tbl shows cctc t lgth clcltd b sg th bov qto fo vos sb s: Nmb of sb s N C F/m ε good stmt of th t cctc s F/m.. glló. Elctcl Egg Dtmt. EUETI-UPC 7 d 9

176 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM OUNDY ELEMENT METHOD EM Th bsc d of th EM s to dsctz th tgl qto sg bod lmts. Th wll-ow momt mthod s qvlt to EM wh sg sbsctol bss d th dlt fcto s wghtg fctos. Ths, EM c b gdd s combto of th clsscl bod tgl qto mthod d th dsctzto cocts ogtd fom FEM. Cosd th cs of th Llc s qto,.. Ω Γ q q Γ Γ Γ Γ lcto of th wghtd sdl mthod w obt: Usg th followg dtts: Ω W dω W W W W W W d lg th Gss thom, w c wt: Ω W dω Γ W dγ Γ W dγ Ω W dω Th wghtg fcto W s chos to b th fdmtl solto, dtmd l: W δ ' Ths th dom tgl th bov qto c b wtt s: Ω W dω Ω δ ' dω Fo ot sd th dom. Combg th bov qtos, th followg tgl lto s obtd: W W dγ dγ Γ Γ. glló. Elctcl Egg Dtmt. EUETI-UPC 8 d 9

177 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM Pfomg sml mthmtcl mltos w c obt th followg tgl f : lto fo th Posso qto W W dγ dγ f W dω Γ Γ Ω Nots: Nm bod codtos t to ccot b th fst tgl Dchlt bod codtos t to ccot b th scod tgl. soc ot f sd th dom Ω s t to ccot b th dom tgl th bov qto. Fom th Posso s qto, th sdl to b mmzd c b wtt s: Ω f W dω Wh th obsvto ot s loctd o th bod Γ, th bod tgl bcoms sgl s ochs zo. To tct th sglt o th bod, w wt th qto W W dγ dγ Γ Γ I th followg fom: W W W dγ W dγ dγ dγ Γ Γ Γ Γ Γ Γ W st dtl th two-dmsol cs. I ths cs: d: W l π θ W dγ dγ d l lm lε ε θ π π ε Γ Γ θ θ W θ θ dγ dγ d lm ε θ π π ε Γ Γ ε π θ Sbstttg th l qto w obt:. glló. Elctcl Egg Dtmt. EUETI-UPC 9 d 9

178 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM c Γ W dγ Γ W dγ Wh: θ θ c π Ω Γ Ω If w cosd Posso s qto: Ω Ω f W dω Ω Ω f W dω f W dω π Ω Ω f W dω f l dω lm π ε θ θ lε f ε dθ dε Th fl qto s: c Γ W dγ Γ W dγ Ω f W dω I gl th o o th bod mst b ow. Dtmg ll vls of solto d ts oml dvtvs o th bod th solto t bt ot of th dom c b clcltd. Th lctosttc d mgtostàtc oblms th dfd b th followg qtos: c c Γ Γ W dγ W dγ Γ Γ W dγ W dγ Ω Ω ρ W dω ε µ J W dω Wh dots th lctosttc ottl d s th mgtc ottl vcto. od lmt dsctzto Th bsc d of th EM s to dsctz th bod of th dom d cosdto to st of lmts. Th ow solto ov ch lmt s omtd b tolto fcto, whch s ssoctd wth th vls of th fctos t th lmt ods, so tht th tgl qto c b covtd to sstm of lgbc qtos. Th bod gomt c b dsctzd ss of lmts.. glló. Elctcl Egg Dtmt. EUETI-UPC d 9

179 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM. glló. Elctcl Egg Dtmt. EUETI-UPC d 9 Th smlst solto s to s st of costt bod lmts oth soltos vlbl, sch s l o qdtc lmts. Th gomt of th costt bod lmt fo th two-dmsol cs s show th followg fg. Usg th costt bod lmt omto th tgl qto fomlto of th oblm dfd b th Posso s qto bcoms: Ω Γ Γ Ω Γ Γ s d W f d W d W c M Wh f s th costt vl of th soc o sgmt dom cotg socs. Itodcg th followg otto Ω Γ Γ Ω Γ Γ s d W f P d W H d W G U Q Th bov qto c b wtt fo ch s [ ] M P U H Q G c Ths lgbc qto sstm c lso b wtt mt fom s follows: P Q G U H

180 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM. glló. Elctcl Egg Dtmt. EUETI-UPC d 9 Eml. Solv Llc s qto ctgl bo. t th hozotl bod ls th ottl U scbd. t th vtcl bod ls th oml dvtv q s gv. W s costt lmts fo th solto. Fo smlct w s cos dsctzto wth lmts. I ths cs ol th bod vls,, q d q ow. Fo ch lmt w d to clclt th followg sso: Ω Γ Γ Ω Γ Γ s d W f d W d W c M I o cs: f. Ω Γ Ω c π π π θ θ W c wt: Γ Γ Γ Γ M d W d W ths cs fo qtos fo th fo ow bod vls obtd f ts th vls to. I mt otto ths qtos : q q q q G H ; Γ Γ Γ Γ d W H d W G ; l d dw W π π Clclto of mt lmts H d G 75. t l l. t / G d G d H π π π π D to th stl soto of th oblm, w c wt: G G G G H H H H Ths smmts do NOT hold gl fo EM.

181 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM H d G H G π π l π d l t π d.5 t. D to th stl soto of th oblm, w c wt: H H H H G G G G H d G H G π l π l π d b smmt: d t π d t H G H G.78.9 H d G H G π π l π d l t π d.5 t.. glló. Elctcl Egg Dtmt. EUETI-UPC d 9

182 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM. glló. Elctcl Egg Dtmt. EUETI-UPC d 9 Dgol tms H H H H. cs th vcto s dcl to wh th lod ot d fld ot loctd t th sm lmt. Dgol tms G to G. Mg chg of coodts sch tht L ξ Wh L s th lmt lgth. ft som lgbc mltos w c wt: l l t t L L d G Po Po π π Th mcl vls : L G G L G G ssmblg th bov qtos lds: q q q q q q q q G H Sbstttg th ow vls: ; q q q q q q W c solv th sltt sstm of qtos d obt th followg slt:

183 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM 5 ; 5 q q q q Nmcl Itgto Th bov ml shows th dffclts of mg ltcl solto fo th tgls volvd ths mthod. It s covt to s mcl mthod to comt thos tgls. Th most sd mthod s th Gss tgto mthod. I th followg gs w show th ctcl lcto of ths mthod. Th tgl c b omtd s: b N b z b b I f d w f E Wh N s th mb of tgto ots, z s th coodt of th th tgto ot, w s th ssoctd wghtg fcto d E s th o. ls of z d w lstd th followg tbl. ls fo mo ots c b obtd mcl mthods fcs. N ± z w I o cs, th tgls fo G d H c b clcltd b lcto to th followg qtos:. glló. Elctcl Egg Dtmt. EUETI-UPC 5 d 9

184 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM. glló. Elctcl Egg Dtmt. EUETI-UPC 6 d 9 N N w N N L H w L G Z Z Z Z z Z z Z L N L N L l ; ; ; ; ; ; π π Wh:, th coodts of th collocto ot. L s th lmt lgth N, N th comots of th t oml. s th dstc fom th collocto ot to th Gss tgto ots o th bod lmt., th ds dvtvs. I o cs d s slt to th lcto of th bov lgothm w obt th sm vls fo th mt coffcts. Comtto of solto to th dom Usg th gl sso w c wt: Γ Γ Γ Γ M d W d W c I o cs: Ω Γ Ω c π π π θ θ w obt th followg qto, vld fo ch to ot: Γ Γ Γ Γ M d W d W Fo ml, dtm th ottl t ot.5;. lcto of th bov lgothm w obt th followg vls:

185 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM.68 q G q G q G q.9 q.97 G q H.68 q.76 Comso wth th ltcl solto Th ltcl solto s th followg: 5 H.9 q.97 H H It s s to vf tht ths solto s comtbl wth ll of bod codtos. Th fls d dcto : d q 5 d q Th followg tbl smmzs th ltcl d mcl slts fo v ot Pot Coodts U ltcl Umcl Qltcl Qmcl.5; ; 5 5.5; ; 5 5 Ito.5; Th comso btw ltcl d mcl solto shows th followg: Fo th ottl Dchlt vbl v cos dsctzto lds to good ccc. Th lg o fo th fl Nm vbl q c b ld b th fct tht th dffttd qtt qs f dsctzto bcs tgto smooths whl dfftto cts oghss. If o s f bod msh wth, fo ml, s costt lmts of lgth, o cold obt th followg mt qto:. glló. Elctcl Egg Dtmt. EUETI-UPC 7 d 9

186 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM ; q q q q q q q q q.6 q.55 q.78 q.7 q.695 q Th solto of th bov sstm of qtos s: q q Th followg tbl smmzs th ltcl d mcl slts fo v ot Pot Coodts U Umcl Qltcl Qmcl ltcl.5; ; ; ; ; ; I ths cs th solto s mo cct s th l d cos solto.. glló. Elctcl Egg Dtmt. EUETI-UPC 8 d 9

187 FINITE ELEMENT METHODS FO ELECTICL ENGINEEING IEM Comso of th FE d E Mthods W commt h o som of th mo dffcs btw th two mthods. Ddg o th lcto som of ths dffcs c th b cosdd s dvtgos o dsdvtgos to tcl schm. FEM EM OS. t dom msh s qd. msh of th bod ol s qd. Et dom solto s clcltd s t of th solto. Solto o th bod s clcltd fst, d th th solto t dom ots f qd fod s ctos o th bod tcll lss cct th th ddt vbls. Dfftl Eqto s bg omtd. Ss smmtc mt s st st. oth d q of th sm ccc. Ol bod codtos bg omtd. Fll oltd o smmtc gtd. mtcs gtd. Elmt tgls s to Itgls mo dffclt to vlt, 5 vlt. d som cot tgds tht bcom sgl. Wdl lcbl. Hdls Cot v hdl ll l 6 ol oblms wll. oblms. ltvl s to mlmt. Mch mo dffclt to mlmt. 7 cs of th dcto sz of th msh, o oft hs of ol sg tht th oblm sz hs b dcd b o dmso. Ths s o of th mo lss of th EM - costcto of mshs fo comlctd obcts, tcll D, s v tm cosmg cs. Th m oblms wh th dtls of tst occ o th bod, o loclsd to tcl t of th dom, d hc t dom solto s ot qd. Th s of th G-Gss thom d fdmtl solto th fomlto ms tht th EM volvs o omtos of th dfftl Eqto th dom - ol ts omtos of th bod codtos. Th mtcs gll of dfft szs d to th dffcs sz of th dom msh comd to th sfc msh. Th oblms wh th mthod c gv s to th smll sstm d qcst solto - t dds tl o th volm to sfc to. Fo oblms volvg ft o sm-ft doms, EM s to b fvod. 5 EM tgls f hd to vlt. lso th tgls tht th most dffclt thos cotg sgl tgds hv sgfct ffct o th ccc of th solto, so ths tgls d to b vltd cctl. 6 fdmtl solto mst b fod o t lst omt o bfo th EM c b ld. Th m l oblms.g., vtll o homogos qto fo whch fdmtl soltos ot ow. Th ct s whch th EM s cll so, bt t c b th stctv ts lcblt. 7 Th d to vlt tgls volvg sgl tgds ms th EM t lst od of mgtd mo dffclt to mlmt th cosodg ft lmt ocd glló. Elctcl Egg Dtmt. EUETI-UPC 9 d 9

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189 COMPUTTION OF OTHE QUNTITIES. Post-ocssg

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191 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES COMPUTTION OF OTHE QUNTITIES. Post-ocssg Th os of ost-ocssg s to dv t fomto b th s of dct solto sll ottl ods Th fst slt s, sll, ghcl fomto. Ths fomto s sfl to cotol, fst omto, th vldt of solto: smmtcl dvc, sch s lctcl mch, th slt mst b smmtc. Coto lot of costt-vl ls of ottl,.. o. Th bov fg shows lctosttc cs. Th ls shows th costt ottl ls th sstm. Th ow ls dct th lctc fld tst E. Ths dwg s sfl to show th omt dcto of th fld. mmb tht lso E s dcl to. E Th followg fg shows th ls of costt vcto ottl schoos mch. Ths ls ls of costt dcto: d d. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

192 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES d d Fld ms. Th s of colod bds o shdg to dsl lmts sbct to stsss scfc gs,.g.. to. T. T bds, s of cosdbl hl wh vltg dsg. It s lso sll ossbl to c ot to obt th ctl stss t ths ot. Th followg fg shows th m of Elctc fld tst E dvc.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

193 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Th followg fg s m of dcto mch. Th stto tth hghl sttd s dctd b th colod fld m. Th followg fg shows th dst ct dstbto. Th followg fg shows tcl ot vl tbl.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

194 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Dstbto log l. Ths s sfl to show stl dstbto of som tstg vbls, sch s dcto, voltg, Elctc fld o Mgtc fld tst. Th followg fg shows th dstbto of log th g fo th bov schoos mch Ths dt c sll b otd to oth ogms, sch s Ecl tbl fom.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

195 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Eg Stod th Mgtc Fld Th g stod to th mgtc fld s qtt of m motc fo good dstdg of wht follows. Th smlst w to vlt th mgtc g stod to gv mtl cossts of th msmt of th g flow btw lctcl soc d th mtl tslf: ths s bcs w v good t vltg lctc g. Lt s cosd tos of mgtc mtl o whch th s wdg of N ts s wod. Th gomt of th tos s sch tht th dffc btw th tl ds d th tl ds s glgbl /. I th followg, th sgfct dmsos of th tos wll b th m ds m d th of th coss scto S. T m v - mtl dvc fo th vlto of th g stod mgtc mtl. Th mgtsto chctstc of th mtl wll b cosdd to b comltl gl. t ot o th mgtsto cv t s ossbl to df mblt µ sch tht: µ H I l mtls th mblt µ s costt, o mtt th vl of o H; o l mtls, th mblt s th comlctd fcto of d H. Lt th wdg b fd b dc voltg soc v. If th b T s closd, ct flows to th wdg. todcg qvlt sstc tht ts to ccot th Jol losss th wdgs, th Ohm lw fo ths cct ds: vt t N d ϕ t dt. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 9

196 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Sc th fl of th mgtc fl dst s costt coss ch coss scto: Mltlg b dt both lhs d hs: v t t NS d t dt vdt dt NSd d b lg th mé lw to th m ccmfc of th tos: vdt dt S Hd Th lhs th bov qto sts th g sld b th voltg gto tm tvl dt. I th hs of th sm qto, th fst tm sts th g losss tm tvl dt th qvlt sstc, d th scod tm s th vto of th g stod th mgtc fld btw tm stts t d t dt. Th tm S m s th volm of th tos. tgtg btw th tm stts t d t : vdt dt m H b db I th lhs th s th dffc btw th g sld b th voltg gto d th g dsstd b th Jol ffct th qvlt sstc. Th dffc,.. th hs, s th mg g st th sstm d cosdto: th g stod to th mgtc mtl. If th stts of o-l mtl s sch tht t wos ot H *, * th scfc g s th btw th cv d th -s fom to *. Fo l mtls, th tgl th hs of th bov qto bcoms: H b db µ H µ H Ths qtt hs th hscl dmsos of g t volm [J/m ] d s commol clld scfc mgtc g. Oc th vcto ottl th tolto ods hs b fod, th mcl vlto of th mgtc g s th sml. Pocdg lmt b lmt o hs: W l µ l Ωl d l µ l Ωl d l Th tgl c b ssd tms of th mcl vls of th tolto ods d tms of th sfc tgls of th sh fctos:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 9

197 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES W l L µ l Ω α α dω l l oth fom fo ths qto s: W W W l l l µ H H H Jd H d Ω Ω l l l Ω l Jd d l l l Ω l Ω l Hd HdS l l Ω l Hd l H J If w lt th sfc ov whch w t th scod tgl go to ft, th,, H S th sfc tgl gos to zo d W l Jd l Ω l Ld Fl mgtc fld s sd to b two-dmsol D wh th sts smmt tht ms ossbl to fd ttosss of th hscl hom l b l, fom h to ft. Ths s th stto wh th cts lws hv dcto lll to z-s, d th tsvs sctos to ths s lws st th sm gomt d th sm mtls, ot b ot. Ths s qvlt to sg tht,,z,,z, whtv z d z. I ths cs, sgfct smlfcto c b chvd. I fct, tl dvtv sct to z vshs, d t s fod tht th vcto ottl s lws lll to z-s, d thfo z z I th followg, fo th s of smlct, z wll b fd to sml s. Ths wll ot gt mbgt, sc, fo th sd ot, -D sttc flds th vcto ottl s fct scl ottl; mo csl, t s vcto whch hbts ol o comot. dvlog th vcto odct comots of : t s sl fod fo th. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 9

198 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES d d Wht bot th hscl mg of th vcto ottl? w wll cot to s ths m. Lt s cosd D fld, whch th two ftl log codctos lll to z-s, of glgbl coss scto, cg dc ct I d -I, sctvl. Th mgtc fld s th sm ot hvg th sm d coodts, o mtt th z coodt of th ot: th sm s t fo th vcto ottl, whch s ssmd to hv th vl th ght codcto of th t d th vl l th lft codcto. Th followg fg shows ths hscl stto, d th t fg shows th D gomt. D C z I l d I E t cg dc ct I, ftl log z-dcto: ol o t z- dcto s std. Dstc btw codctos s d. l d Th - stto of th t Fg Lt s ow cosd th fl t lgth z-dcto of coss sfc tht hs th t s coto: ϕ S ds S. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 9

199 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES wh S s th sfc of th t fo o t z-dcto, d s th otwd oml to S. sbstttg th sso fo s fcto of : ϕ S ds S lcto of Stos' thom tsfoms th bov qto to th followg: ϕs d S S wh th l tgl th hs s dfd log th t. O th bss of th ssmtos vosl md, th fl of c b wtt s: C E ϕs dz d S dz S D Th fst d thd tgl hv, sctvl, th followg vls: d - l th vcto ottl hs th sm vl wh th - d -coodt of th ot do ot chg d th z-coodt of th ot chgs. O th oth hd, th clclto ws fomd wth fc of o t z-dcto, d moov t shold b otd tht th tgto th movs oost dctos log th codctos. Th scod tgl s obvosl glgbl wth sct to fst d thd sc th ws ftl log: w stdd st o t of th lgth z-dcto, bt th tso of th ctv lgth s ftl gt th th lgth of th coctos. W v t th followg foml: ϕ S l tht sts v tstg ot of th vcto ottl: D mgtosttc oblms, th dffc btw vcto ottl two ots, d, sts th fl of thogh sfc hvg t lgth z-dcto d hvg th tcs of th sds lll to th z-s t th ots, d,. Ths hscl sct of th vcto ottl D mgtosttc oblms s v tstg fo th lss of mgtc flds, sc ths oblms th fl of s gl of gt tst th tslf. Th ot th bov qto vls lso tht w ot gl tstd th vl of th vcto ottl: th, w tstd th dffcs btw vcto ottl two ots. Moov, ls wh s costt lso fl ls: sc th fl of s ltd ol to th dffc btw vcto ottls, o mtt th ots wh th dffc s clcltd, f ths dffc s clcltd log two fl ls.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 9

200 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES W ld ddcd sfl ltosh btw th mgtc ottl d th fl of. Now w wt to gv dfto fo v mott qtt: th fl ld wth col. Lt s cosd two ts lg th sm l, wh th s ow dstbto of vcto ottl,. W show tht th fl of thogh th sfcs of th ts t lgth s: ϕ b, ϕ cd, If th ts ss-coctd fo stc b coctg wth shot-cct ots b d c, th stto ms ctcll chgd: fct th s w sfc sgl-bodd, two-fcs w th fl of c b vltd: ϕ d, ϕ b, ϕ cd, c d b z Two ts tht c b ss- o lll-coctd. Ths s mtl tst smlto fo th mcl vlto of th fl ld wth col. z cost D - stto of th stct Th stto chgs comltl f th ts lll-coctd, f ots, c d b, d shot-cctd. I ths cs t s fct mossbl to df. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

201 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES two-fcd, sgl-bodd sfc. I oth wods, t s mossbl to df th fl of thogh sfc, sc ths sfc dos ot st. t w ot tstd ol owldg of th fl; w lso tstd som ltd qtts, sch s th bc mf dcd sstm of ts. Fo stc, wht s th mcl vl of th voltg btw ots c d b d? Th sw c b fod b cosdg th followg fg, wh ch t s std hscl schmtsto s voltg gto ss wth sstc. sϕ s sϕ s wh s s th Llc oto. c v b d Th cct qvlt of th stct. Noto s thom c b ld to both sds of th cct obtg th cct th followg fg c v b d Noto tsfomto of th cct. oltg v s dl fod to b: v It shold b otd tht th bov qto sstcs ot lctl st: th ol costt o thm s th fct tht th ql. I sc of N ts lll coctd:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

202 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES N N v s N N ϕ Fomll, ths qto s th sso of th Fd s lw, sc voltg s s th dvtv of qtt tht hs th dmsos of mgtc fl. It s sobl to df ths qtt s fl: th fl ld wth th col ψ ψ N N ϕ Lt s ow cosd mssv t, md b codcto of ft dmsos - d -dctos. Ths t c b s s th lll cocto of ft sb-codctos, ch o hvg th sm sstc bt dfft fl to. Th fl ld wth th mssv t s: ψ lm N ϕ lm N N N N N l wh sffs l d dot th lft d th ght t of th t sctvl wth fc to th bov cs of two lmt ts, l sts th t of th wdg wh sb-ts hv odd mbs, th t wh sb-ts hv v mbs. Th followg fg sts D vw of th t. l z cost - stto of mssv t: th ct s z-dctd. mltlg both mto d domto of bov qto b S, th coss-scto of th t: ψ lm N S S N N lm l S N N wh S S / N s th sfc of th coss-scto of th lmt t. Th lmt of th sm ths qto s st th dfto of tgl. Thfo w hv fo th ld fl: l S. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

203 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES ψ ds ds S S l S wh th tgls hv to b vltd o th lft d o th ght t of th coss-scto of th t. Nmcl vlto of th tgls s th sml; sc gl tgls smmbl fctos, t s ossbl to ocd lmt-b-lmt. S N l s th mb of tgl lmts tht cov wthot hols d/o s ostos th go S wh th tgls hv to b vltd. O hs: Nl Nl ds dω S Ω Ω α dω wh d α dot th -th od ottl d sh fcto -th tgl lmt. Idctc Th dfto tslf of dctc s dctl ld wth th dftos of mgtc g d of ld fl. I gl, th dctc of wdg s gv b th to btw th fl ld wth th wdg d th ct to th wdg: L ψ If th sstm d cosdto s l, t s sl s tht qto s qvlt to: ψ NS NS HlS H W L I th bov gs, th mcl vlto of th ld fl d of th g stod th mgtc fld hv ld b dscssd; th mcl vlto of th dctc s thfo mmdt wh th ld fl o th g ow s: - th to btw th ld fl d th ct; - th to btw twc th mgtc g d th ct sqd. Fo o-l oblms, th stto s comltl dfft. Evlto of dctc fom th bov qtos lds gl to comltl dfft slts, thos comg fom L ψ hvg gt vls th thos fom. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

204 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES W L. Lt s sos tht th toodl dvc show l s md b o-l mtl, d tht th ct th wdg s sch tht th fl dst ch ot of th mtl s. T. If o lg fl s st, th s ootoll btw th fl ld wth th col d th fl dst, s wll s btw th ct d th mgtc fld: ψ NS Hl N thfo, f th dfto L ψ s dotd, th dctc s ootol to th W odct H, whl, followg dfto L, th dctc s ootol to th tgl of th mgtc fld. Ths stto s show th followg fg: th qtt H / s show; b s ghd th tgl of H vs.. fl dst fl dst mgtc fld H mgtc fld H b Th dfto of dctc fo o-l mtl: fom ld fl, b fom g stod mgtc fld. Fom ths fg t s vdt tht: H b db H th qlt bg vld ol fo l mtls. todcg qlt w c obt:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

205 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES W H Sl H H b db Hl / N N S Hl ψ It hs b show tht th dctc clclto fom stod g lds to mcl vls smll th tht fom ld fl, bt th qsto s th followg: wht dfto of dctc shold I s o-l oblms? Th sw s sml: t dds o wht I m loog fo, d o how I dfd th dctc tslf. Lt s cosd g th dvc. Th wdg s ow fd b ms of tmvg voltg ct. If w df th dctc s th to btw th ld fl d th ct, w hv: d d t dlψ v t t Lψ t Lψ t dt dt dt sc ow th dctc ot: th to btw ld fl d ct s ot mo costt, bt t vs wth th ct. Itgto of th bov qto s ot v sml, sc th s tm tht mls th owldg of th vto of th dctc lws th to btw ld fl d ct wth tm. smlfcto of ths qto c b chvd b wtg t s: d t dlψ d t v t t Lψ t dt d dt It wll b s th followg tht ths qto s th to solvg tst oblms volvg o-l mtl. It shold b vdt b ow tht th clclto of dl / d s ot so dffclt t lst cl, sc t wll sffc to dtm th bhvo of th dctc g, th to btw th ld fl d th ct vs. th ct, d to dv t wth sct to th ct tslf. Lt s sos tht th ct hs th vl *. Is th g stod th mgtc fld ql to Lψ * /? Ctl ot. If w wt to dtm th scfc g b ms of th dctc, w hv to clclt th qtt L W tht, obvosl, dds o th ct d sccssvl w hv to vlt L W * /. So, lso fo o-l oblms, both dftos of th dctc hold: th ol dffc wth l oblms s tht th do ot cocd. Thfo, ch dfto of dctc c b sd, ddg o whch d of slt w wt to obt. Slf d Mtl Idctc I ths gh th ttto wll b focsd o l oblms. Nmcl vlto of dctc fom th fl lg s dfd bov s th sml. Sml to solv th FEM oblm, to dtm th ld fl d to dvd t b th ct: th dctc hs b fod, o, lttvl, to vlt th totl mgtc g d to dvd t b o hlf of th ct sqd.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 9

206 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Ptcl ttto shold b d to th fct tht, f w tls smmt ots, w dtm ld fls d gs tht smll th th comlt gomt cs. To dtm th mtl dctc btw two wdgs, th g two ossblts: - fom mgtc g; - fom ld fl. I th fst cs, two FEM lss dd. I th fst o, th ct both dctcs th sm; th scod o, th ct o of th dctcs hs to b vsd. Th mgtc g th fst FEM lss s: W L L M m I th scod FEM lss th mgtc g s: W L L M m Fom th dffc btw W m d W m, th mcl vl of th mtl dctc fom mgtc g s sl vltd. Th vlto of th sm mt fom th ld fl s v sml: sffc to fd wdg wth gv ct, to fom FEM lss, to vlt th fl ld wth wdg, d to dvd ths qtt b th ct wdg. Slf d Mtl Idctc Coml Wdg Sstms Th oblm of th mcl vlto of th dctc, howv, s ot so sml s t fst sght. It m b sd tht th lst sts to vlt ths qtt sml f th och to th oblm ws ochd coctl sc th fst sts of th FEM lss. s ml of ths sttmt, lt s cosd th co dcto th followg fg, tht s md b th sb-wdgs. I th coctos of th th wdgs show: wdgs d coctd lll, d th coml s coctd ss wth wdg ; b th coss-scto of ths co s show. Fom th tl two, ths gmt of wdgs s s s sgl wdg, o mtt th tl coctos. Th qsto s th followg: wht s th mcl vl of th dctc fom tmls d? to b host, ths gmt s v stg, d t wold b th dffclt to fd ths dcto ctc.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 9

207 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES ' ' ' o co b th-wdgs co. th tl coctos: dottd ls st tl coctos, d slm ls st tl coctos; b th coss scto of th co md mbs dot th t codctos: cts gos d ' th sm wth oost dctos. Fo ths d of lss, sstv hom glctd s wll s dd ct ffcts. To vlt th tl dctc, th coml of th th wdgs m b fd wth t ct; oc fld solto s ow, both ld fl d g c b clcltd, d fom ths qtts th dctc. O of th most commo os cossts fdg th th wdgs wth th sm ct. t th dffcs btw th ct dsts, ths s mo o, bcs th totl ct th th wdgs s dfft. Oft, th cts wdgs of th t d sosd to b ql: thfo, ths cs, oth o cold b to st I.5 d I.5. Fom th cct schmtsto th bov fg t s vdt tht, f th s t ct, wdg cs th sm t ct. t wht bot wdgs d? Th sm of th cts th sd wdgs s sl o, bt th s o fomto tht llows to vlt -o th mcl vls of cts I d I. Th fct s tht o cct fomto s cldd FEM schmtsto. Thfo w d to m ths schmtsto stl. Fo o cs, th qvlt cct s show th followg fg I L M M L I I L M Th qvlt cct of th dcto.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 9

208 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES W loog fo th dctc s fom th tl cct,.. fom ots -. Sc w do ot ow th vls of I d I, o qvltl th to, th ol solto s to vlt stl th s dctv mts, tht c b do th FEM lss. I ch o of ths, ol o wdg s fd, th mg two wdgs cg zo ct. t th d of ch FEM lss, th slf dctc of th fd wdg s ow, s wll s th mtl dctcs btw ths wdg d th two o-fd wdgs. I ths w, th slf dctcs c b dtmd, d s mtl dctcs. Sos tht w fd wdg, d tht w fod th mcl vl of M. I cl, wh fdg wdg w shold ot dtm th vl of M, svg ths th comt tm to dtm ths dctc vl. Obvosl, M M, bt mcl lbotos ffctd b odg d tcto os. Ths ms tht, wth ctcll zo comttol ffot, w c com th mcl vls of M d M. Ths s good stmto of th ccc of th clcltos. Fo ctcl oss, two mtl dctcs c b cosdd ql f th dffc s lss th. %, th bss bg th smll. Oc ths ts hs b ccomlshd, th tl dctc c b fod b ms of sml cct lss. c mf tw th mch mts, bc mf s gl th fst st fo th dtmto of th ovll fomcs of dsg, d th followg t wll b show how t s ossbl to ddc cctl bc mf t th mch tmls sttg fom FEM lss. Th logcl sqc of ths tchq stts fom th owldg of th vcto ottl: bc mf s comtd b ms of smll mb of tgl dt closl ltd to th fl thogh sfc. It wll b vdt tht mcl dvto of sch tgl dt s css thoghot th wo, d thfo gt ccc s qd FEM comttos. Ths ms ot ol to f sccssv FEM soltos, bt lso to cotol th volto of sch mcl mts fom o FEM lss to oth. Ths s ot ol sgl FEM lss, bt lso comtvl fom o solto to oth, f svl FEM soltos qd fo vos gomtcl cofgtos of th sm dsg wth fc to ottg mch, fo dfft stto-oto ltv ostos. Lt ow s cosd D scto of ottg mch. Th followg hothss dotd: - lctomgtc flds c b cosdd D; - mgtc mtls l, d hstss s ot st fomgtc mtls o d PMs; - losss ctv mtls ot t to ccot; - th oto sd ω s costt.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 9

209 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES FEM solto tms of vcto ottl s cosdd blow. dfto, th fl ld wth sfc tht hs th, l ts tcs th ots d, s φ -. Th fl ld wth col ψ s: ψ, ds S wh S s th of th tsvs scto of th col d, s th vcto ottl h d th followg th mb of codctos ss slot wll b t s o; th sm holds fo th mb of ol s. Ths s th vl of th fl ld wth tst col t tm t ; vls fo dfft tms c b obtd wth w ltv ostos btw stto d oto. Howv, th fl ld wth th tst col t tm t s th sm fl ld t t wth th col whch s t gl t - t ω fom th ct col, wh ω s th ottg sd of th oto. Ths llows to lmt th mb of FEM clcltos to o, f th mgtc stct s sotoc. Wh th gomt d cosdto sts smll mb of cols ol d hs, th lvt mgtc stcts st chctstcs ot costt sc. Thfo, th mb of FEM clcltos to fom s t lst two: th fst o wh th s of th mgts smosd to th s of slot, th scod o wh th s of th mgts smosd to th s of tooth. Wh th fl ld wth col o wth hs s ow, th comtto of th lvt bc mf s cl sml ts, b lg th Fd lw: dϕ t dϕ θ dθ dϕ θ t ω dt dθ dt dθ wh θ s th gl osto, fc fm gdl coctd to th ottg fld, of th s of th col, d ω s th gl vloct of th ottg fld. Fom th bov qto t s vdt tht mcl dvto of th ld fl s th bss fo th dtmto of th bc mf. O ossblt s to clclt th bc mf t b ms of dct mcl dvto of th fl. ccc chbl ths w s oo, sc th ld fl s ow smll mb of ots of th tvl; o mtt wht d of mcl dvto lgothm, dvtvs of dg gt th o ot t to ccot. If th ld fl vs sddl th ot d cosdto, th mcl vls of hgh dvtvs ot glgbl, d ths slts gt mcl os. Ld fl c b omtd b ms of ltcl fctos, sch s: Fo sos, Lgg olomls, Tchbshv olomls, cbc oloml sls. If ltcl omto fo th ld fl s dtmd, th bc mf c b fod b ms of ltcl dvto, tht c b obtd wthot mcl os d wth o sgfct comttol. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 9

210 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES ffots. Ths s gl tolto oblm, wh mb N of s of θ,ψ vls ow closd tvl of th θ-s. Howv, dqt choc of th tvl log θ-s llows th -o owldg of mo costts o th dvtv of th ld fl: fo stc, th bc mf th tol s th bc mf s mmm o th ol s th bc mf s zo. Fo och to th tolto dos ot s to b sfl, sc th cso th mcl vlto of th Fo coffcts dcss s th od of th hmoc css. I cl ths s ot gt oblm, sc th mcl vl of th sd mt dcss wh th od of th hmocs css, d th o th mcl vl of th fl s oft ot cbl. O th oth hd, bc mf s clcltd b ms of ltcl dvto s th followg Eq: N dϕ θ θ ϕ s θ dθ It c b s fom th bov qto tht th hgh th od of th hmoc, th gt th wght. Nmcl tsts show tht N shold b t smll: ddto of two o th hmocs to qlttvl good θ cv oft slts wld ms of th w cv tht, t lst cl, shold b mo cs. sstc. Jol losss ow. I fd stt th ct dst vcto J s ow ot of th volm of th lzd sstm. Th sttos Jol losss J ρ J d Wth lttv cts, vg ssodll wth th tm, th vg Jol losss : J ρ J d Th qvlt lctcl sstc c b comtd s: P I J. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

211 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Cctc d sstc. lcto of th bsc cocts fo th lctosttc fld d codcto md, w c dtm th cctc d sstc fo th coml gomtc cofgtos: C S S ε E ds L L E dl E dl σ E ds Not th smmt th two fomltos. Fo l mtls w c s th lctosttc g och to clclt th cct of codcto: U C ; U D E dol ol ol ε E dol U s th lctosttc g d th ottl dffc. Edd ct losss Th dd ct dst s: J σω If cosods to th vl of th vcto ottl, th th loss lmt wll b J EL JJ * dd σ S Sc vs ll ov th lmt, th b th bov qto so dos J: { b c J b c J b c J } J Sbstttg d tgtg ov lmt, w obt: * * * { J J J EL J J J J J J } P σ. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

212 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Foc d Toq Loz s Foc lw s stghtfowd d sml to l to obt th foc ld to codcto. Gv th locl vl of fl dst, d ct dst w fd th locl foc vcto s df f J Ths qto s sfl fo fdg th foc o codctos. Fo ml cosds DC moto showd th followg fgs.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9

213 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9 lcto of Loz Foc sso w obt th foc cts ov th codcto. Th toq s gv b: F M Mwll Stsss Sbstttg th Mwll s qtos th Loz foc sso: J J H ν µ w obt: f ν Dvlog ths qto, w obt th followg qto w ol show th qto fo X comot: z f z z z ν If tm ν s ddd d sbtctd fom th bov qto, d th dtt s sd, th th foc comot bcoms: z z z f ν Som fth mltos gvs: z f z f z z ν ν Th mg sso m b cogzd s th dvgc of vcto f, whos comots :

214 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 9 z z f z T f T f T T ν ν ν sml dvlomt holds fo ch of th oth foc comots f d f z. Ths ths vctos c b combd to Tso T: T z z z z z ν Th foc dst c ow b wtt s th dvgc of ths tso: f T Th totl foc c b fod b tgto ov th volm: Td fd F Usg th dvgc thom, ths volm tgl m b dcd to sfc tgl: S TdS Td F W lmt th followg dvlomt to two dmsol gomt, so tht th sfc of tgto s l w cosd t dth. Th t oml d tgtl vctos to th sfc : t s s s s Th cmtl tgto th s th: dl ds wh dl s dfftl lgth log th tgto th. Th cmtl foc s ow:

215 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 9 dl s s f ds T f ν Th tgtl d oml comot : s t t s s s s dl f f s s s s dl f f ν ν Th tgtl d oml comots of th fl dst : t s s s s Sbstttg d som ft lgbc mltos, w c wt: dl f dl f t t t ν ν Th toq o c of ds s gv b: dl M t ν tl Wo mthod Th cl of th vtl wo tchq to stmt th foc s sd cosvtv g sstms. It s bsd o th comso of th g blc btw two dfft ostos, sosg vtl movmt log th dcto wh th foc s comtd. I lctomgtc oblms, th foc s ssd s th tl dvtv of mgtc cog wth sct to th coodts, log whch th foc comot mst b comtd: W W W F ' ' ' W c lso s th g:

216 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES F W W W Som commts: - If th vto s smll, th mcl solto s ot cbl. - If t s too bg cssv chg of th mgtc g s chvd - Th slt m b ffctd b th vto of th msh th two ostos. - Ths tchq qs t lst two fld soltos. Co Losss I gl, th mgtc o losss c b clcltd b th comosto of th losss: Hstss losss. Edd ct losss. Ecss losss lso clld omlos losss low w tt ch of ths th losss stl. Hstss Losss Hstss losss owg to th dscotos chct t th mcoscoc scl of th mgtzto ocss, tht s, d to th g lost b ch hs m. Th clssc ocd to dtm hstss losss s th Stmtz qto: Ph ηf α m O btt choc s to l th fct tht th volmtc dst of th g lost owg to th hs ms s th of th qssttc hstss ccl, d hc th g lost s th of th qssttc ccl h mltld b th volm of th sml. Th hstss losss c b clcltd wth th followg sso: P h f h wh f s th fqc of th tl ld fld.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 9

217 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Edd ct o clsscl Losss W sw l sctos tht codctg mtl sbmttd to tmvg fld, loos of dcd cts Edd cts ctd. I slb wth thcss d d lctc codctvt σ, d odc fld wth od T, ths losss c b dtmd b: T σ d t Pcl t dt T t Fo o-ssodl fls ths losss c b clcltd b sg Fo so of th dcto d smmg th ffcts of ll of th hmocs: P cl σπ f d 6 wh s th mltd of th th hmoc. EXCESS LOSSES Th css losss d to th stc of mgtc doms, whch hc dd cts th omts of dom wlls. Th clsscl modl bsd o Mwll s qtos sd th clsscl losss ddcto dos ot cosd th sc of doms d soss mgtzto ocss fctl homogos o sc, tht s, glctg th mgtc s ffct. t owg to th ffct of doms, lt dmc losss gt th clsscl losss; ths dffc of losss s clld css losss. P t P t P t d cl Th tott [7] tho stblshs tht th bhvo of doms o lg scl c b dscbd b th dmcs of mgtc obcts. Ths mgtc obcts gos of dom wlls coltd. I mtls wth lg doms, s th cs of otd g, o mgtc obct cosods to o dom wll. Th chg of mgtztos s codd, o gv momt, b ms of th smltos tvto of mgtc obcts. Th. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 9

218 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES mgtzto ocss s std s dstbto of cocv flds locll ssoctd wth mgtc obcts. Th s dmc blc mog th fld css to mt ct vto of d th cot flds d to dd cts. Tm vto of mls vto th blc d hc modfcto of dd cts. If w df css fld H t s th t of th tl fld ld to comst th fld ctd b th movmt of th mgtc obcts, css losss c b clcltd b ms of th followg sso [7]: T σ o T,5 d t P G dt dt wh G s th fcto coffct, whch sts th costt of ootolt btw th css fld d th t of chg of fl: H dφ Gσ dt t Th mt o s ltd to th mb of mgtc obcts d th css fld. Emtll t hs b dtmd tht th ltosh btw thm s l [8]: t H t Pctcl mlmtto of co loss clclto I gl, th sls of mgtc mtls sl two chctstcs of ts mtls: o Stto chctstc: fh Totl losss P FE f m, f. Usll ths ltosh s omtd b th followg qto: m β χ m f β > ; γ.6... To clclt th mgtc losss w c ocd s show: Fst, w dtm th ottl vcto d th dcto t ot of th modlld dom. Ths w c dtm th vg vl of dcto ch lmt: vlmt I th bov sso w cosd tht th lmts smll. If th lmts bg, th btt omto s to cosd th vg vl s th dcto t th ctod of th lmt.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 9

219 FINITE ELEMENTS FO ELECTICL ENGINEEING COMPUTTION OF OTHE QUNTITIES Usg ths vls w dtm th mgtc losss ch lmt: P FElmt f, f vlmt Fll b smmto of ths vls w obt th totl mgtc losss. P FE PFElmt Th bov ocss s vld ol th cs of ssodl fld dstbto. Fo oth css sll cotd th ctcl lctos ths vls wll b dtd. ffctv solto to ths oblm stll hs t b fod. fcs [] d ls., glló., d l Hoz J., Ph, P., Lmos ts C., Dsollo d modlo d hstss dámco to-psch s lccó l cálclo d máqs léccs ssh, Cogsso d Métodos Comtcos m Egh, CMCE, M J, Lsbo, Potgl [] Mgoz, I.D., Mthmtcl Modls of Hstss d th lctos, Elsv, [] d ls,., glló., d l Hoz J., Modlzdo dl cclo d hstéss mdt l modlo d Psch ssh vst Itcol Ivstgcó Tcológc, ol. º, M-J [] tott, G., Dmc glzto of th Scl Psch modl of Hstss, IEEE Tsctos o Mgtcs, ol.8, º5, , Stmb 99 [5] Follo F., Novov., movd och to ow losss mgtc lmtos d o-ssodl dcto wvfom, IEEE Tsctos o mgtcs, ol.6, º5,. 9-9, Stmb 99 [6] tott G., Gl ots of ow losss soft fomgtc mtls, IEEE Tsctos o mgtcs, ol., º,. 6-69, J 988 [7] tott G., Hstss mgtsm, cdmc Pss, 998. [8] tott G., Psql M., Phscl ttto of dcto d fqc ddc of ow losss soft mgtc mtls, IEEE Tsctos o mgtcs. ol.8, º5, , Stmb 99.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 9

220

221 GENEL LCHITECTUE OD CD SYSTEM

222

223 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE GENEL CHITECTUE OF CD SYSTEM SED ON THE FINITE ELEMENT METHOD I ctc th lss of dvc volvs th sts: Th dscto of th gomt, th hscl chctstcs d th msh Th lcto of th FEM Th vslzto d ttto of th slts of th smlto Ths th sts dfft d cosod to th th dfft modls: Th dt t modl: P-ocsso. Th modl to fom th lss: Solv. Th modl to ls th slts: Postocsso.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

224 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE Th dt t modl Th dt t modl s sd fo tg ll th fomto css fo th lss of th oblm. Ths modl ccomlshs th followg th fctos: Dscto of th gomt of th obct Msh gto Dfto of th gos d th bods Th msh gto cossts of fdg collcto of ods d collcto of ft lmts whch fom cctbl dsctzto of th dom. Sch dsctzto mst sct th bods of th dom d th tfcs btw dfft gos. lso, th sh of th FE mst t b too gl. Th followg fgs show coct d movd msh fo tcl oblm. Th followg fgs show th solto of th mgtc fld dct ct mch wth th s of two mshs. Th fst s cos d th slts oo. Th scod s fd msh; th solto s f.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

225 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE Th followg fgs show cs of schoos mch d th dtl of msh th g. To obt hgh solto w d to msh th g wth v f msh.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

226 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE Th followg fgs show th gto of th-dmsol msh. Th ods dfd b th coodts whl th lmts chctzd b th t d lst of th ods. I som css w cld th fomto bot th chctstcs of mtls d socs ths fl.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

227 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE Th solv Th solv comts th ows th FE oblm,. t solvs th l o o-l sstms of qtos. Its t s th dom dsctzto, som css th hscl chctstcs of th mtls, th socs d th bod codtos. ddtol fomto sch s t of oblm, tvl of tm clclto d mmm o s som css comlso. fo th solto of lgbc qtos, w mst : Ct th sbmt d sbvctos cosodg to ch lmt. ssmbl ths lmt mtcs d vctos to bld th sstm mt. l od codtos d smmts f ths st. Th solto of l lgbc sstms c b do svl ws: Dct mthods: Gss, Chols Sm dct mthods: ICCG loc ttv mthods: Gss-Sdl Wh th sstm of qtos s ot l, ths otos td ttv schm: Gss-Sdl, Nwto-hso,tc. Th ostocsso Th ostocsso fom two tss: Etcto of sgfct fomto. Sthtc stto of th mcl dt v ghc fclts. I most css th -ocsso d th ostocsso hv th sm tfc. Th dffc s th fomto vlbl t ch tm. Fo th ost ocsss w d to ow th vl of ottl t v od d th hscl chctstcs of v lmt. I th -ocsso t s ol css to ow th coodt ots.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of

228 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE EmlsD: FEMM, MXWELL_S. W st h som chctstcs of two ogms: FEMM, MXWELL_S. FEMM MXWELL_S Elctosttc Ys Ys DC codcto No Ys C codcto Ys Ys Mgtosttc Ys Ys Tst No No Tst oltg soc No No Thml No No Movmt No No Nmb of ods No lmtd No lmtd No-lt Ys Ys Eql m Ys Ys Colo ms Ys Ys cto ms Ys Ys Nmcl slts. Pot vls Ys Ys Ghcl stto of Ys Ys dvd qtts Msh tomtc tomtc Comlmt clclto Ys. Som tgls of l d Ys. T clclto wth hdd of otos. Otg sstm Wdows Wdows Th followg fgs show th c of som of ths ogms. Th -ocsso of FEMM. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of

229 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE Th ost ocsso of FEMM Th ost ocsso of FEMM. Comlmt clcltos. Th ost ocsso of FEMM. Ghcl stto of dvd qtts. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of

230 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE Th -ocsso of MXWELL. MXWELL hs tsv mgmt ogm. Th bov fg shows th mgmt wdow. I t w c s dfft otos to ct oct: Df modl. To dw th oblm modl St mtls. To ssg mtls to v. St bod d socs. To ssg both: bod codtos d socs of ctto voltg d ct St ctv mts. To dscb wht qtts mst b clcltd tomtcll sll foc, toq, dctc d oths St solto otos. To slct th otos of clclto: ml/tomtc msh, tc. Solv. To solv th sstm Post ocss. To t th ost-ocss ogm.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of

231 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE Th dwg m MXWELL. Th St otos MXWELL. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of

232 FINITE ELEMENTS FO ELECTICL ENGINEEING GENEL CHITECTUE Th ost-ocssg m MXWELL Th clclto th ost ocss m MXWELL. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of

233 EXMPLES

234

235 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Ths mls solvd b Qcfld softw. Yo c fd th dctd fls th cos CD-OM. Ecs N. Slot mbddd codcto. Modls lb.z d lb_.z Ts Dw lot of ct dstbto wth th codcto lcd th slot of lctc moto. Emt Ct dst s obtd b ottl msg log th t of codcto do b voltmt. Phs s dtmd b dgtl hsmt. Poblm t Pl-lll tm-hmoc l mgtc oblm. Gomt Poblm s to th ght of smmt s l b, cosodg bod codtos st. Gv dt ltv mblt of stl µ. ltv mblt of & co µ. Codctvt of co σ 57 S/m. Th stl s lmtd, so ts codctvt log th b s σ S/m. Fqc f 5 Hz. Th totl ct th b s 6, o f th two bs th slot. od codtos t th vtcl s of smmt l b H t. Th fld fds wth fomgtcs, ths t oth bods th fld s zo. tcl dstbto of ct dst log th sgl co b l t th bod of b. Poblm fl lb.z.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

236 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES tcl dstbto of ct dst log two co bs l t th bod of b. Poblm fl lb_.z.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

237 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Ecs N. Foc of tcto of two cldcl col cols. S followg modls Ts Fd th foc ld to th cols wth ct wth d wthot th shld btw thm. Emt Focs msd b s of dgtl blc o whch o of cols s stlld s th ght ct blow. Tst fomd wth cts vg, d dfft codctos d fomgtcs lcd btw cols. Gomt D to smmt of th fomlto ol hlf of th oblm bov b l s dfd, d t th s of smmt l b th bod codtos st. Gv dt Ct dst th col /m. ltv mgtc mblt of, lmm d co cols µ. ltv mgtc mblt of th stl shld µ. Elctcl codctvt of stl σ Sm/m. Elctcl codctvt of lmm σ 7 Sm/m. Cols wod b sltd w, so coss-scto codctvt cols σ Sm/m. od codtos log th hozotl smmt s l b. Eqto ot cldcl coodt sstm lds to cost t th s b. Th fld fds t th ft, so d to th codto of cott of th ottl t th l b.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

238 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Tm-hmoc lctomgtc fld, f 5 Hz. Shld t M foc of tcto, mn Modl No shld lb.bm Stl.96 lb_f.bm lmm.7 lb_l.bm DC mgtc fld. Shld t M foc of tcto, mn Modl No shld lbc.bm Stl. lbc_f.bm Ecs N. Pomt ffct. Modls lb_c.z d lb_f.z Ts Fd th ct dst dstbto log th coss scto of log lll codctos. Two ts of codctos lzd: two co ods d two stl tbs. Emt Ct dst s dfd b msg th voltgs o th codcto sgmts, th hs of ct s msd stl b th dgtl hsmt. Poblm t L l-lll oblm of tm-hmoc lctomgtc fld. Gomt D to oblm smmt ol -ght qt Ob s dfd, d t th s of smmt th bod codtos st. Co ods Stl tbs. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

239 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Gv dt ltv mgtc mblt of co d µ. ltv mgtc mblt of stl µ. Totl ct J. Fqc f 5 Hz. Elctc codctvt of co σ 57 Sm/m. Elctc codctvt of stl σ Sm/m. od codtos t th hozotl s of smmt l O H t. t th vtcl s of smmt Ob. Eqto ot th cldcl coodt sstm lds to cost t th s Ob. Fld fds t th ft so t th oth bods. Ct dst dstbto log th l O fo co ods. Modl lb_c.z Ct dst dstbto log th stl tb mt fom th ot to ot f clocws. Modl lb_f.z. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

240 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Ecs N. Elctomgtc shldg. S th followg modls. Ts Fd th lvl of mgtc fld dcto sd th shld. Shlds md of stl d co of th sm gomt lzd. Emt Ufoml tl mgtc fld s odcd b th lctc mgt. Th shld wth th msg col sd s lcd btw ts ols. EMF th col s msd: cs of DC ct th col - b bllstc glvomt th momt of swtchg o, cs of C ct - b s of mllvoltmt. Gomt Th shld cossts of two hlvs. Possbl ostos of th slot th shld s show b dottd l. Th slot cold b lgd to mm b shts of o-mgtc mtls. D to smmt ol ght- qt Ob s lzd, d t th s of smmt th bod codtos st. Shcl shld Cldcl shld Gv dt ltv mgtc mblt of d co µ. ltv mgtc mblt of stl µ. Mgtc fld s foml,.9 T. od codtos D to smmt t th l Ob H t. t th l O. Eqto ot th cldcl coodt sstm lds to cost.695 t th s O. Fld fds t th ft, so d to cott t th l O lso. Th fld s foml, d th ght bod hs th sm codto s th lft o H t. Shldg coffct - lto of mgtc fl dsts otsd d sd th shld.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6

241 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Tm-hmoc lctomgtc fld, f 5 Hz. Shld t Stl sh wthot slot Stl sh wth slot Stl cld wthot slot Stl cld wth slot Co sh wthot slot Co sh wth slot Co cld wthot slot Co cld wth slot Mgtc fl dst sd th shld mt, whl tl fom fld s 9 mt Shldg coffct Modl.8 69 lb_f.bm lbf.bm.6.69 lbc_f.bm..5 lbc_f.bm lb_c.bm.9 lb_c.bm lbc_c.bm lbc_c.bm DC mgtc fld. Shld t Stl sh wthot slot Stl sh wth slot Stl cld wthot slot Stl cld wth slot Mgtc fl dst sd th shld mt, whl tl fom fld s 9 mt Shldg coffct Modl lb_f.bm..7 lb_f.bm lbc_f.bm lbc_f.bm Ecs N 5. Mgtc fld of th cldcl col. S modls lb5.z d lb5_f.z Ts M lot of mgtc fl dst t th s of th col wth d wthot stl co. Emt Mgtc fl dst s msd lboto b mcowbmt. Poblm t L smmtcl oblm of mgtosttcs. Gomt D to oblm smmt ol -ght qt Ob s dfd, d t th s of smmt bod. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 6

242 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES codtos st. Gv dt Ct dst th col /m. ltv mgtc mblt of d co µ. ltv mgtc mblt of stl of th co µ 5. od codtos t th vtcl s of smmt l Ob H t. t th hozotl s of smmt O. Fom ot th cldcl coodt sstm w hv t th s O cost. Fld fds t th ft, so t th l O d to cott of. Ddc of th mgtc fl dst o th dstc to th col ct. No co. Modl lb5.z Ddc of th mgtc fl dst o th dstc to th col ct. Stl co. Modl lb5_f.z. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 6

243 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Ecs N 6. Mtl dctc of cols. S modl lb6.z Ts Fd th ddc of mtl dctc of col cldcl cols o th dstc btw thm. Emt EMF th ght col s msd b bllstc glvomt t th swtchg o. Poblm t L smmtcl oblm of mgtosttcs. Gomt Th fld soc s th lst col. D to th fld smmt ol -ght qt Ob s dfd. t th s of smmt th bod codtos st.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 6

244 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Gv dt ltv mgtc mblt of d co cols µ. Ct dst th lft col /m. Th s o ct th ght col, ths t hs o ffcto to th fld sh. od codtos t th vtcl s of smmt l Ob H t. t th hozotl s of smmt O. Fom ot th cldcl coodt sstm w hv t th s O cost. Fld fds t th ft, so t th l O d to cott of. Mtl dctc M - lto of th fl coctd wth ll ts of th ght col Ψ to th ct th lft col J whch s th og of th fl. L Ψ / J Ψ w H w s mb of ts of th ght col, - fl coss th ght col. Totl ct J * S scto * S modl lb6.z X, mm Fl coss th ght col, µwb. Mtl dctc M, µh *w *w.85.*w Ecs N 7. Mgtc fld smlto th g of DC lctc moto. S modls lb7.z d lb7.z Ts Dw th ls of mgtc fld DC lctc moto th o-fomgtc go. Clclt ol dssto coffcts fo two fomltos. Fst - th col s dfd b ct dstbto, th mgtc mblt of stl s ft. Scod - smlfd fomlto: stl ssmd to b wth ft mblt d th col s modld b ct l. Poblm t L l-lll mgtosttc oblm.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

245 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Gomt go s dfd b coto bcdfg. Dmsos: b mm, g o, cd 68 mm, bc 5 o, d o, f 55 mm. cto of oth ts of th moto s modld b bod codtos. Gv dt ltv mgtc mblt of stl of oto d stto µ. ltv mgtc mblt of µ. Ct dst th col /m. Elctc ow of moto 5 W. Smlfd modl lboto t Col s lcd b th ct l f, mgtc mblt of stl ssmd to b ft. Poblm fld s lcd b th sht of lctocodctg of o sh bcdfg. Obtg th ls of mgtc fld s lcd b th ls of ql lctc ottl th codctg sht. Ths lcmt s coct, s th ct of th fld H th d fld U th sht loos th sm. Ct flows t l f d gos ot fom l b. Ls of ql lctc ottl msd b dgtl voltmt. od codtos Modl of l moto: d to smmt t l Od H t, t l O. Smlfd modl lboto t: H t t th stl sfc ls dfg d bc d s of smmt Od, t th l O. Coffct of ol dssto - lto of th fll fl to th ffctv fl gog to oto of th ol. Modl t Dsstv fl, mwb Effctv fl, mwb Fll fl, mwb Coffct of ol dssto Modl l moto lb7.z Smlfd modl lboto t lb7.z. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

246 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Elctc moto bshlss DC moto wth mt mgts d th hs col ctto. Poblm T: ol l-lll oblm of mgtosttcs. Gomt: l lgth of th moto s mm. Th fo mgts md of Smm-Coblt wth ltv mblt of.5 d cocv foc of 55 /m. Th ct dsts fo th col slots s follows:,, /m o, -,, /m o -,,, /m o S, -,, /m o S -, d zo o T d T -. Th d ot fms md of Coblt-Ncl-Co-Io llo. Th -H cv fo th Coblt-Ncl-Co-Io llo: H / m T Th -H cv fo th stl: H /m T glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

247 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Mg: Nol mt mgt mt mgt d stl th Poblm T: ol l-lll oblm of mgtosttcs. Gomt: Gv: ll dmsos ctmts. Th mt mgts md of LNICO, cocv foc s 78 /m. Th olztos of th mgts log vtcl s oost to ch oth. Th dmgtzto cv fo LNICO: H, /m , T Th -H cv fo th stl: H, /m 6 8 6, T Poblm: Fd mmm fl dst Y-dcto Solto: To vod th flc of th bods whl modllg th bodd oblm, w'll clos th mgt ctgl go of d scf zo Dchlt bod codto o ts sds.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

248 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Comso of slts Mmm fl dst Y-dcto: z T NSYS. COSMOS/M. QcFld.7 Mg: Solod ctto solod ctto cossts of col closd fomgtc co wth lg. Clclt th mgtc fld d foc ld to th lg. Poblm t: ol smmtc oblm of mgtcs. Gomt: ll dmsos ctmts. Gv: ltv mblt of d col µ ; Ct dst th col,, /m ; Th -H cv fo th co d th lg:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

249 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES H /m T Poblm: Obt th mgtc fld th solod d foc ld to th lg. Solto: Ths mgtc sstm s lmost closd, thfo otwd bod of th modl c b t ltvl clos to th solod co. thc l of th otsd s cldd to th modl go t th lg sd, sc th mgtc fld ths cot b glctd. Msh dst s chos b dflt, bt to mov th msh dstbto, th ddtol vtcs ddd to th modl. W t o of ths vtcs t th col sfc t to th lg co, d two oths t to th co of th co t th both sds of th lg. coto fo th foc clclto closs th lg. It s t th mddl of th g btw th lg d th co. Whl dfg th coto of tgto, s stog zoom- mod to vod stcg th coto to stg dgs. Th clcltd foc ld to th lg F 7. N. Comso of slts Mmm fl dst Z-dcto th lg: z T fc.9 QcFld.8 fc D. F. Ostgd, "Mgtcs fo sttc flds", NSYS vso., Ttols, 987. Mg: Fomgtc C-mgt mt C-mgt th. Th ml dmostts how to modl cvd mt mgt sg th qvlt sfc cts. Poblm T: Pl oblm of mgtcs. Gomt of th mgt:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

250 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Gv: ltv mblt of th µ ; ltv mblt of th mgt µ ; Cocv foc of th mgt H c /m. Th olzto of th mgt s log ts cvt. Solto: To vod th flc of th bods whl modllg th bodd oblm, w'll clos th mgt ctgl go of d scf zo Dchlt bod codto o ts sds. Mgtzto of stght ts of th mgt s scfd tms of cocv foc vcto. Effctv sfc cts smlt mgtzto th mddl cvd t of th mgt. HMg: Slot mbddd codcto Poblm T: l oblm of tm-hmoc mgtc fld. Gomt:. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6

251 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES sold co codcto mbddd th slot of lctc mch cs ct I t fqc f. Gv: Mgtc mblt of µ ; Mgtc mblt of co µ ; Codctvt of co σ 58,5, S/m; Ct th codcto I ; Fqc f 5 Hz. Poblm: Dtm ct dstbto wth th codcto d coml mdc of th codcto. Solto: W ssm tht th stl slot s ftl mbl d m b lcd wth Nm bod codto. W lso ssm tht th fl s cotd wth th slot, so w c t Dchlt bod codto log th to of th slot. S HMg.bm oblm th Emls fold fo th comlt modl. Th coml mdc t lgth of th codcto c b obtd fom th qto Z / I wh s voltg do t lgth. Ths voltg do o th codcto c b obtd Locl ls mod of th ostocssg wdow, clcg bt ot wth th codcto. Comso of slts Z Ohm/m Im Z Ohm/m fc QcFld fc. od, "Itgodfftl Ft Elmt Fomlto of Two-Dmsol Std-Stt S Effct Poblms", IEEE Ts. Mgtcs, ol MG-8, No., J 98.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 6

252 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES HMg: Smmtc dobl l of codctos Poblm T: l oblm of tm-hmoc mgtc fld. Gomt: Two co sq coss-scto codctos wth ql bt oost cts cotd sd ctgl fomgtc cotg. ll dmsos mllmts. Gv: Mgtc mblt of µ ; Mgtc mblt of co µ ; Codctvt of co σ 56,, S/m; Mgtc mblt of cotg µ ; Codctvt of cotg σ,, S/m; Ct th codctos I ; Fqc f Hz. Poblm: Dtm ct dstbto wth th codctos d th cotg, coml mdc of th l, d ow losss th cotg. Solto: W ssm tht th fl s cotd wth th cotg, so w c t Dchlt bod codto o th ot sfc of th cotg. Th coml mdc t lgth of th l c b obtd fom th qto Z - / I wh d voltg dos t lgth ch codcto. Ths voltg dos ql wth oost sgs d to th smmt of th modl. To obt voltg do, swtch to Locl ls mod ostocssg wdow, d th c bt ot wth codcto. Th mdc of th l Z.9.7 Ohm/m. To obt ow losss th cotg:. I th ostocssg mod, choos Pc Elmts d c th cotg bloc to ct th coto.. Choos Itgl ls d slct Jol ht fom th lst of tgl qtts d choos Clclt. Th ow losss th cotg P.7 W/m.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 6

253 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Elc: Mcost tsmsso l shldd mcost tsmsso l cossts of sbstt, mcost, d shld. Poblm T: Pl-lll oblm of lctosttcs. Gomt: Th tsmsso l s dctd log z-s, ts coss scto s show o th stch. Th ctgl CD s scto of th shld, th l EF sts codcto st. Gv: ltv mttvt of ε ; ltv mttvt of sbstt ε. Poblm: Dtm th cctc of tsmsso l. Solto: Th svl dfft ochs to clclt th cctc of th l: To l som dstct ottls to th shld d th st d to clclt th chg tht ss o th st; To l zo ottl to th shld d to dscb th st s hvg costt bt ow ottl d cg th chg, d th to ms th ottl tht ss o th st. oth ths ochs m s of th qto fo cctc: C q / U. Oth ossbl ochs bsd o clclto of stod g of lctc fld. Wh th voltg s ow: C W / U,. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 6

254 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES d wh th chg s ow: C q / W Emt wth ths ml shows tht g-bsd ochs gv lttl bt lss ccc th ochs bsd o chg d voltg ol. Th fst och ds to gt th chg s vl of tgl log som coto, d th scod o ss ol locl vl of ottl, ths och s th smlst d m css th most lbl. S th Elc_.bm d Elc_.bm oblms Elc fo th cosodg th, ochs d th, ochs sctvl. slts: Thotcl slt C 78. F/m. och C 77.8 F/m 99.8% och C 78.7 F/m.% och C 77. F/m 99.6% och C 79.6 F/m.8% Elc: Two codcto tsmsso l Poblm T: l oblm of lctosttcs. Gomt: Th oblm's go s bodd b god fom th bottom sd d tdd to ft o oth th sds. Gv: ltv mttvt of ε ; ltv mttvt of dlctc ε. Poblm: Dtm slf d mtl cctc of codctos.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

255 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Solto: To vod th flc of ot bods, w'll df th go s ctgl lg ogh to glct sd ffcts. To clclt th cctc mt w st th voltg U o o codcto d U o th oth o. Slf cctc: C C Q / U, Mtl cctc: C C Q / U, wh chg Q d Q vltd o ctgl cotos od codcto d w fom th dgs. W chos th cotos fo th C d C clclto to b ctgls -6, d 6, sctvl. Comso of slts C F/m C F/m fc QcFld fc. hb,.. o, d. Mtt, " bsobg od Codto fo Qs-TEM lss of Mcowv Tsmsso Ls v th Ft Elmt Mthod", Jol of Elctomgtc Wvs d lctos, 99. Elc: Cldcl Dflcto lz Poblm T: Pl-lll lctosttc oblm Gomt: Cldcl dflcto lz CD s t of cldcl ccto wth gl scto of 7 7'. CD hs two slts md fo th tcls to t d t th CD fld. I ths ml th bm of lctos ts th CD dcl to th cld's ds wth tl tc g E 5 d gl dsso of 6 Gv: ds of tl cld. m ds of tl cld.7 m. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

256 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES CD voltg U Itl tc g of lctos E 5 ltv mttvt of ε ; Poblm: Df th bm focs ot. Solto: t th bgg w solv th lctosttc oblm clcltg th CD fld. ft tht w o th Pot Soc Emtt dlog sg th Ptcl Tcto commd w m. Usg th Emtt dlog g w osto th ot tcl mtt t th ct of th CD's tc slt -.76 m,.7 m d scf th g fo th sttg gls btw 6 d 68 dgs. Usg th Ptcl dlog g w choos th dsd tcl t - lcto - fom th lst, d df th vl of tl tc g E 5. To obt th slt, w clc l d vw th tcl tctos o sc. slts Th bm focs ot:.8,.7. Th focsg gl o.: Tho ss tht wth som vl of CD voltg ddg o th g of lctos, th bm wll b focsd t th t slt. I dl cs th voltg fo o ml wold b U 7. Th focsg gl d th CD voltg o ml slghtl dfft bcs of th CD fgg ffcts. G. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

257 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Ht: Slot of lctc mch Tmt fld th stto tooth zo of ow schoos lctc mch. Poblm T: Th l-lll oblm of ht tsf wth covcto. Gomt: ll dmsos mllmts. Stto ot dmt s 69 mm. Dom s -dg sgmt of stto tsvs scto. Two mt bs lg th slot ls ohmc loss. Coolg s ovdd b covcto to th l coolg dct d both sfcs of th co. Gv: Scfc co loss: 6 W/m ; Ht codctvt of stl: 5 J/ m; Ht codctvt of co: 8 J/ m; Ht codctvt of slto:.5 J/ m; Ht codctvt of wdg:.5 J/ m; I stto sfc: Covcto coffct: 5 W/ m ; Tmt of cotctg : C. Ot stto sfc: Covcto coffct: 7 W/ m ; Tmt of cotctg : C. Coolg dct: Covcto coffct: 5 W/ m ; Tmt of cotctg : C.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

258 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Ht: Cld wth tmt ddt codctvt v log cld ft lgth s mtd t tmt T log ts tl sfc d T o log ts tl sfc. Th thml codctvt of th cld s ow to v wth tmt ccodg to th l fcto λt C C T. Poblm T: smmtc oblm of ol ht tsf. Gomt: Gv: 5 mm, mm; T C, T o C; C 5 W/ m, C.5 W/ m. Poblm: Dtm th tmt dstbto th cld. Solto: Th l lgth of th modl s btl chos to b 5 mm. Comso of slts ds cm Tmt C QcFld Tho glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

259 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES THt: Htg d Coolg of Slot of Elctc Mch Chgg tmt fld th stto tooth zo of ow schoos lctc moto dg lodglodg ccl. Poblm T: Th l-lll oblm of ht tsf wth covcto. Gomt: ll dmsos mllmts. Stto ot dmt s 69 mm. Dom s -dg sgmt of stto tsvs scto. Two mt bs lg th slot ls ohmc loss. Coolg s ovdd b covcto to th l coolg dct d both sfcs of th co. Gv:. Wog ccl W ssm th foml dstbtd tmt bfo th moto ws sddl lodd. Th coolg codtos sosd to b costt dg th htg ocss. W tc of th tmt dstbto tl t gts lmost std stt. Th w stt to solv th scod oblm - gttg cold of th sddl stod moto. Th tl tmt fld s motd fom th vos solto. Th coolg codto sosd costt, bt dfft fom thos whl th moto ws bg lodd.. Mtl Pots Th thml codctvt vls th sm s th Ht ml. Fo tst lss th vls of scfc ht C d volm dst lso qd: Ht Codctvt J/ m Scfc Ht J/ m Mss Dst g/m Stl Co Co Islto.5 8 Wdg.5 5. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

260 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES. Ht socs d coolg codtos Dg th lodg hs th slot s htd b th ow losss co bs. Th scfc ow loss s 6 W/m. Wh lodd, th ow loss zo. W sos th tmt of cotctg to b th sm fo both hss of wog ccl. I t, th covcto coffcts dfft, bcs th coolg f s sosd to b stod wh th moto s lodd. I stto sfc Ot stto sfc Covcto coffct W/ m Lodg Tmt of cotctg C Covcto coffct W/ m Stod Tmt of cotctg C Coolg dct 5 Solto Ech hs of th lodg ccl s modld b st QcFld oblm. Fo th lodg hs th tl tmt s st to zo, d fo th coolg hs th tl thml dstbto s motd fom th fl tm momt of th vos solto. Moov, w dcd to b th coolg hs to two st hss. Fo th fst hs w choos tm st s smll s s, bcs th t of tmt chg s ltvl hgh. Ths llows s to s tht th tmt t th slot bottom fst css b omtl gd fo scods, d th bgs dcsg. Th scod stg of coolg, ft s, s chctzd b ltvl low t of tmt chgg. So, w choos fo ths hs th tm st to b 6 s. Fo htg ocss th tm st of s s chos. Pls s followg oblms th Emls fold: slts THtLd.bm fo lodg hs, d THtS.bm fo th bgg of stod hs, d THtS.bm fo th d of stod hs Tmt vs. tm ddc t th bottom of th slot wh tmt sso sll s lcd.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6

261 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES THt: Tmt sos of Sddl Coold W Dtm th tmt sos of co w of dmt d, ogll t tmt T, wh sddl mmsd t tmt T. Th covcto coffct btw th w d th s α. Poblm T: smmtc oblm of ol ht tsf. Gomt: Gv: d.565 ; T 7.77 C, T 8.88 C; C 8.6 J/g, ρ g/m ; α.7 W/ m. Poblm: Dtm th tmt th w. Solto: Th fl tm of 8 s s sffct fo th thotcl sos comso. tm st of.5 s s sd. Comso of slts Tmt, C Tm QcFld NSYS fc 5 s s s S th THt.bm m d THt_.bm l oblms th Emls fold. fc f F., "Pcls of Ht Tsf", Ittol Ttboo Co., Scto, Pslv, d Ptg, 959, Pg, Eml -.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 6

262 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES THt: Tst Tmt Dstbto Othotoc Mtl log mtl b of ctgl coss-scto s tll t tmt T d s th sddl qchd lg volm of fld t tmt T. Th mtl codctvt s othotoc, hvg dfft X d Y dctol ots. Th sfc covcto coffct btw th w d th s α. Poblm T: smmtc oblm of ol ht tsf. Gomt: Gv:, b λ.67 W/ m, λ 6.69 W/ m; T 7.78 C, T 6 C; 6.7 W/ m; C J/g, ρ 67. g/m. Poblm: Dtm th tmt dstbto th slb ft scods t th ct, co dg d fc cts of th b. Solto: To st th o-zo tl tmt w hv to solv l std stt oblm, whos solto s fom dstbto of th tmt T. tm st of. sc s sd. Comso of slts Tmt, C Pot QcFld NSYS fc, , ,..6 7., S th THt.bm m d THt_.bm l oblms th Emls fold. fc Schd P.J., "Codcto Ht Tsf", ddso-wsl Pblshg Co., Ic, dg, Mss., d Ptg, 957, Pg 6, Eml -7.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 6

263 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Sts: Pfotd lt th ctgl sht wth ctl hol sbct to tsl lodg. Poblm T: Pl oblm of stss lss l stss fomlto. Gomt of th lt: Lgth: mm; Wdth: 8 mm; ds of ctl og: mm; Thcss: 5 mm. Gv: Yog's modls E 7 N/mm Posso's to ν.. Th fom tsl lodg N/mm s ld to th bottom dg of th stct. Poblm: Dtm th coctto fcto d to sc of th ctl og. Solto: D to mo smmt o qt of th stct s std, d tl bods std X d Y dctos sctvl. Th coctto fcto m b obtd fom th lodg stss N/mm d th mmm comtd stss 6 N/mm s 6 /.65.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 6

264 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Col: Stss dstbto log solod v log, thc solod hs fom dstbto of ccmftl ct. Th mgtc fl dst d stss dstbto th solod hs to b clcltd. Poblm T: smmtc oblm of mgto-stctl colg. Gomt: Gv: Dmsos cm, cm; ltv mblt of d col µ ; Ct dst 5 /m ; Yog's modls E.75 N/m ; Posso's to ν.. Poblm: Clclt th mgtc fl dst d stss dstbto. Solto: Sc o of hscl qtts vs log z-s, th slc of th solod cold b modld. Th l lgth of th modl s btl chos to b. cm. dl comot of th fl dst s st ql to zo t th otwd sfc of th solod. l dslcmt s st ql to zo t th sd dgs of th modl to flct th ft lgth of th solod. Comso of slts Mgtc fl dst d ccmftl stss t. cm: z T σ θ N/m fc QcFld fc F.. Moo, "Mgto-Sold Mchcs", Joh Wl & Sos, N.Y., 98, Cht. S th ColMS.bm d ColS.bm oblms th Col.z fo mgtc d stctl ts of ths oblm sctvl.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

265 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Col: Cld sbct to tmt d ss v log, thc-wlld cld s sbctd to tl ss d std stt tmt dstbto wth T d T o tmts t d ot sfcs sctvl. Clclt th stss dstbto th cld. Poblm T: smmtc oblm of thml-stctl colg. Gomt: Gv: Dmsos cm, cm; I sfc tmt T C; Ot sfc tmt T o C; Coffct of thml so α -6 /; Itl ss P 6 N/m ; Yog's modls E N/m ; Posso's to ν.. Poblm: Clclt th stss dstbto. Solto: Sc o of hscl qtts vs log z-s, th slc of th cld c b modld. Th l lgth of th modl s btl chos to b. cm. l dslcmt s st ql to zo t th sd dgs of th modl to flct th ft lgth of th cld. Comso of slts dl d ccmftl stss t.875 cm: σ N/m σ θ N/m Tho QcFld fc S. P. Tmosho d Good, "Tho of Elstct", McGw-Hll oo Co., N.Y., 96, S th ColHT.bm d ColS.bm oblms Col.z fo th cosodg ht tsf d stctl ts of ths oblm. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

266 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Col: Tmt dstbto lctc w Clclt th tmt dstbto log ct cg w. Poblm T: smmtc oblm of lcto-thml colg. Gomt: Gv: W dmt d mm; sstc - Ω/m; Elctc ct I ; Thml codctvt λ W/ m; Covcto coffct α 8 W/ m ; mbt tmt T o C. Poblm: Clclt th tmt dstbto th w. Solto: W bt chos mm c of w to b std b th modl. Fo dt t w d th w ds 5 mm, d th sstvt of mtl: ρ π d / Ω m, d voltg do fo o mm c of th w: U I l -. Fo th ct flow oblm w scf two dfft voltgs t two sctos of th w, d zo ct codto t ts sfc. Fo ht tsf oblm w scf zo fl codtos t th sctos of th w d covcto bod codto t ts sfc. Comso of slts Ct l tmt: T C Tho. QcFld. fc W. ohsow d H. Y. Cho, "Ht, Mss, d Momtm Tsf", Ptc-Hll, N.J., 96. S th ColCF.bm d ColHT.bm oblms Col.z fo th cosodg ct flow d ht tsf ts of ths oblm.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

267 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES SM Elctc Moto Lst s swtchd lctc motos SM bcom commo solto th lcto wh d glto s qd. Som ol v ss tht SM os w lctcl dvs. Swtchd lctc moto s combto of sml bshlss moto d cotollg lctoc t. Th moto tslf cossts of slt ol stl oto wth o wdg, d slt ol stto wth col o ch ol. Dds o lgothm of cotol sstm, th moto c bhv dfft w ccodg to cstom ds. Dsg of SM moto d cotol sstm l o owg of th toq d fl lg s fcto of th ct d oto osto. Th ol w to obt sch ddc s mcl modlg of ol mgtc fld wth vos oto osto d ct th stto col. Tht wo c b s do b QcFld Wobch. Th SM sml llows to gt T f I, h d F f I, h ddcs fo SM moto, o gds th mb of slots d ols d gomtcl sh. To fd toq d fl fo gv oto osto d col ct w shold solv oblm of mgtosttc to fd mgtc fld dstbto. Th ol bod codto dd s th zo Dchlt codto o th ot sfc of th stto. Th fld socs gv ct th fowd d bc col hlvs, md wth bl d d o th stch bov. W t to ccot stl stto b dfg -H cvs fo stto d oto co. To chv hgh dg of glt, w bld th SM.dll l o th gomtc modl d dvc b tctv QcFld modl dto d stod to th..\sm_fls\sm_sc.mod fl. lso th hscl dt bod codtos, -H cvs fo oto d stto co, mgtc mblt of d co lso d dvc d stod to th..\sm_fls\sm_sc.dms fl. Thfo, wtg th SM.dll cod w c coctt o gttg ott mts. Th ol gomtcl mlto w hv to do ogmmtcll s otto of th oto to th dsd osto. Wh dg to modl oth moto, s wth dfft ol mb, w ol hv to lc th gomtcl modl../sm_fls/sm_sc.mod to oth o wth th sm m. low w fst dscss how to s th SM modl sld wth QcFld th Wobch vomt, d th bfl dscb ctg of sch modl b Wobch lcto wzd d ts ogmmg. Usg th SM modl To t th SM modl sld wth QcFld, do th followg sts:. Wobch Stt->Pogms->T lss->tools->wobch. Ol oc: gst th SM s log obct f o do ot do tht bfo. To gst th w obct, ss th gst Nw ElogObct btto d fd th m d locto of th SM.dll. QcFld stllto ogm ts t to th..\ctvfld\emls\sm fold.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

268 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES. O w modlg ssso b ssg th Nw Ssso btto. If th SM.dll ws ol gstd, ts co s th lst of gstd log obcts. Wh o clc t, th ll Pmts gs s wth SM stch d lst of mts. ccodg to th SM dsg th lst cots mts: oto Posto Ph; Stto Col Ct I G Dmt D Th fst two mts tdd to b vbl, d th lst o s sd b ostocsso wh bldg coto fo clclto toq. Clc o th dsd tm th bl colm to dcl th mt to b vbl o ot.. Swtch to th bl Pmts g. If o hv chos both oto osto d stto col ct to b vbl o vos st, o s th followg: Fo ch mt o c choos th Itto t fom th lst. Now two ttos vlbl: th l tto dvds gv g to ql ts whs th dom o llows to dstbt ch ot dvdll. 5. Wh o d wth t mt d tto dt, swtch to th slt tb d ss th Solv btto. Th slts wll th tbl wh clcltd.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

269 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Dg clclto o cm show d hd QcFld b Show QF btto, slct th ott qtt to dsl th tbl, dsl -lot st wdow. lso, th clclto cold b botd t tm b th Sto btto. 6. Wh clclto fshs t s good tm to sv ct modlg ssso to b bl vw t d clcltd dt t tm. If o hv Mcosoft Ecl stlld, o c ot th slt tbl to Ecl sdsht. 7. Wh o d to modl oth SM mch, ts gomt modl wth QcFld modl dto tctvl, d t ts co to th SM_Fls dcto d o fold wh th SM.dll lvs. Ev SM modl shold hv bloc lbld s "oto" tht wll b ottd o ch tto d of bloc lbld s "Wdg" d "Wdg- ". Th lst st foth d bc sds of stto col o ch ol. ll ths cols cosdd s coctd lll. How th SM modl ws ctd I th st of ths toc w dscss how to ct log obct fo Wobch sch s SM.dll. To do tht o shold b fml wth ogmmg sl sc d wth QcFld obct modl cls. Yo lso hv Mcosoft sl sc 6. stlld o o comt bfo stllg QcFld. Ctg of w log obct cossts of two m sts: ctg th slto of cod wth th QcFld Elog Obct Wzd d cstomzto th tomtcll gtd cod.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

270 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Ctg slto of cod sg QcFld Elog Obct Wzd. Tht ocss s dscbd fll dtls th Lsso of th ctvfld ttol. low w dscb ol th thgs, tht scfc fo th SM modl. Th t mts : Dscto Notto Ut l M M Dflt oto Posto Ph dg 6 Stto Col Ct I 5 G Mddl Dmt D mm Th fst mt - oto osto - s of gomtcl t. Tht ms tht th gomt modl shold b bld d mshd ch tm th mt vl chgs. Th scod mt s of hscl t. Wh ts vl chgs, th oblm s solvd g sg th sm gomt. Th lst mt - g mddl dmt - s of th ostocsso t. Wh t chgs b Wobch f w wt tht, th QcFld wll th gt w modl o solv oth oblm. Sch mt s sg ol fo lzg slts. W lso dcl two ott mts: Nm Notto Ut l Mchcl Toq M N*m Toq of th moto Pol Fl F Wb Th mgtc fl togh o ol Cstomzto th cod wtt b th wzd. I th cod modls ctd b wzd th sbots tht most ll cddts to modfcto std scl modl md Cstom. I o cs t s stod th SM.bs fl. Th fo ocds dfd: ModfModl - th lc fo cod modfg th gomt modl ccodg to ct vls of t mts; StLbl - clld b fmwo ch tm o of th hscl mt chgs ts vl. Its m gol s sttg vls fo lbls of blocs, dgs, d vtcs. Clcltos - good lc fo wtg cod clcltg ott mts. ldcoto - l sbot tht c b sd fo ctg coto, f th tgl vls sd fo clclto of ott mts. Th m ts of o ModfModl sbot s slctg th oto d otto t o th dsd osto. Th followg cod dos tht: ''' Locl vbls fo ll t mts Dm Ph s Dobl Ph thpmts"oto Posto".l ''' ''' TODO: Modf QcFld modl Mdl h Cost PI s Dobl.596 Dm oto s Shg St oto Mdl.Shs.Lbldsloc:"oto" If Not oto Is Nothg Th. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6

271 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES oto.mov qfotto, PotXY,, Ph / 8 * PI Ed If Pls ot tht w do ot d bld th msh d sv modfd modl - th fmwo dos t fo s tomtcll. Th StLbl fcto s clld ch tm wh o of th "hscl" mt chgs. Ths sbot hv ot to st ll ots to ll lbls. Istd, t chgs ol th vl of totl ct fo bloc lbls "Wdg" d "Wdg-" Cs "Stto Col Ct" ''' Dm I s Dobl I thpmts"stto Col Ct".l ''' TODO: Edt QcFld lbl cosodg 'Stto Col Ct' mt ''' Pobbl cod fo loc lbl: ''' - commt cod blow ''' - fo t d Edgs s 'qft' o 'qfedg' costts ''' - lc '' wth o QcFld lbl m ''' St Lbs Pb.DtDoc.Lblsqfloc 'Slct bloc lbls collcto St Lb Lbs"Wdg" 'Gt lbl St LbCt Lb.Cott 'Gt lbl cott LbCt.Lodg I LbCt.TotlCt T Lb.Cott LbCt 'Udt lbl cott St Lb Lbs"Wdg-" 'Gt lbl St LbCt Lb.Cott 'Gt lbl cott LbCt.Lodg -I LbCt.TotlCt T Lb.Cott LbCt 'Udt lbl cott ''' Th two sbots bov ogz modl cto, modfcto d solvg. Now w modf th ots tctg ott mts. Th sbot dog tht s Clcltos. s dstcto fom th cod gtd b wzd, w d th dfft cotos fo clcltg th tgl vls: th toq d th fl lgs wth lft d ght hlvs of stto col. Lt s t th cod fo ctg ch coto st sbot: ldcototoq,.ldcotowls, ldcotowms. Th Clclto ot ''' ''' TODO: ''' - f o hv ot ott mts: ''' lc 'PotXY, ' b Pot o d ccodgl t mts ''' - f o hv o ot ott mts: ''' mov ths cod lsdm Ct s QcFld.Coto ' Mchcl toq St Ct ldcototoqs Pt.l s.gtitglqfit_mwlltoq, Ct.bs ' Pol Fl St Ct ldcotowlss Pt.l s.gtitglqfit_fllg, Ct.bs. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 of 6

272 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES St Ct ldcotowmss Pt.l bspt.l - s.gtitglqfit_fllg, Ct.bs ''' Ech sbot tht blds coto s v sml:. Th coto sodg th oto fo clclto of th toq: Pvt Fcto ldcototoqs s QcFld.slt Dm Wd s QcFld.FldWdow 'slt fld wdow obct ' Gt FldWdow obct St Wd s.wdows ' Gt w Coto obct Dm Ct s QcFld.Coto St Ct Wd.Coto ''' ''' TODO: ld Coto h Dm s Dobl thpmts" G Dmt".l / # Ct.ddLTo PotXY,, Ct.ddLTo PotXY-,,.596 Ct.ddLTo PotXY,,.596 ''' St ldcototoq Ct Ed Fcto. Th t two fctos blds cotos fo fl clclto: Pvt Fcto ldcotowlss s QcFld.slt s QcFld.Coto Dm Wd s QcFld.FldWdow 'slt fld wdow obct ' Gt FldWdow obct St Wd s.wdows ' Gt w Coto obct Dm Ct s QcFld.Coto St Ct Wd.Coto Ct.Dlt T Ct.ddloc "Wdg" St ldcotowls Ct Ed Fcto Pvt Fcto ldcotowmss s QcFld.slt Dm Wd s QcFld.FldWdow 'slt fld wdow obct ' Gt FldWdow obct St Wd s.wdows ' Gt w Coto obct Dm Ct s QcFld.Coto St Ct Wd.Coto Ct.Dlt T Ct.ddloc "Wdg-" St ldcotowms Ct Ed Fcto Now w c coml o SM oct s SM.dll fl Fl->M SM.dll.. commd d s t fo clclto.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 8 of 6

273 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Tooth-Slot d G Modl I ths sml w vstgt mgtc fld d ct dstbto th smlfd modl of slot of lctc mch. Th sold ctgl codcto md fom coo ls th o ctgl slot. Th oost mt co s cosdd to b toothlss. O th m sc o c t th gomtc dmsos of th tooth, slot, g d codcto, s wll s fw hscl mts. Yo c chg vls o cct dflt os. Oc o d wth dt dtg, ss th Clclt btto. Th QcFld wdow s o ot ddg o th Show QcFld flg. Th clclto cossts of followg stgs:. Fst th gomt modl Slot.mod d th dt fo DC mgtc oblm ctd.. Th th m modl stts solvg of DC mgtc oblm d lz t slt. It clclt th t fl ld wth th codcto, th mgtc fl cossg th g "lod" fl d th fl lg. It lso clclt th Ohmc sstc d dctc of th codcto mt of l lgth.. Th th gomtc modl fo C mgtc oblm s ctd. To t th tooth stto to ccot, w dvd th tooth to 7 sb-gos. Th mblt vl chvd th ct of ch sb go th DC oblm s st to th C mgtc oblm.. Th th sml ogm solvs th C mgtc oblm d dtmts sstc d dctc of th codcto b C ct of scfd fqc. ll th clclto slts dsld th ot wdow. Yo c sv th ot to tt fl sg th Sv btto. Th Tooth ml ogm mlos th COMDLG comot of MS sl sc tht c b o c b ot stlld o o comt. To b s tht ll modls dd stlld, s th St. ogm fo stllg th Tooth sml o o comt.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 9 of 6

274 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Soc cods of th Tooth smls wtt sl sc. Th ogm s dsgd s st of obcts d clsss to solt th clclto d stto ls. Th cod tht commcts wth QcFld s loctd th Solv clss Solv.cls fl. Cbl: ctvfld Eml QcFld, focd b ctvfld tcholog m b ffctvl sd fo mlt-hscs lss of vos gg tss. Ths lss cold b hghl tomtd. It v c b mlmtd s Mcosoft Wod docmt qd b th st of mcos fo tomtc cto of QcFld oblm, solvg, ostocssg d ot gto. th comlctd ml - lss of ttco cbl - s vlbl s ctvfld Cbl Eml. If o hv wog owldg of sl sc, d dstdg of QcFld Obct modl - o wlcom to lz soc cod of ths mcos. Ths docmt dsls th slts of cbl lss bsd o scfc modlg mts. Pcts, tbls d ghs blow hv b tomtcll clcltd b Pofssol vso of QcFld, cotolld b cod mlmtd s MS Wod mcos. Cosodg QcFld oblms c b lzd b th Stdts vso.. Modl dscto. It mts. Clcltd cbl mts. Fld cts. Modl dscto. Fg. Cbl stch. Ths hgh-voltg tt-co cbl hs th tgl sctos wth hs codctos d od tl codcto th lss of th coss-scto bov. ll th codctos md of lmm. Ech codcto s sltd d th cbl s whol hs th-ld slto. Th cbl slto cossts of d ot sltos d otctv bdg stl t. Th sh cos of th hs codctos chmfd to dc th fld cow. Th cos of th codctos odd. Emt sc btw codctos s flld wth som slto, ossbl wth. It s oft qd to dsg cbl ccodg to mts of th codcto scto s. Codcto scto s dfd th Tbl. Th tbls to 7 dscb oth t mts.. It mts. Tbl. Codctos' gomtc mts. Phs codcto Mm Ntl codcto 5 Mm Thd odg ds Mm. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

275 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Tbl. Islto gomtc mts. Cbl-co slto thcss Mm I cbl slto thcss Mm Potctv stl bdg thcss Mm Ot cbl solto thcss Mm Tbl. Th cso. s clclto sobl o. Mm. Tbl. Codctos' lodg. Ct mltd. oltg mltd lctosttcs 65. Fqc 5 Hz. Ct hs fo sttc oblms Dg. Tbl 5. Codctos' hscl ots. ltv mblt Codctvt 6 S/m Thml codctvt W/ m Yog's modlo 6.9 N/m Posso's to. Coffct of thml so.-5 / Scfc dst 7 g/m Tbl 6. Stl bdg hscl ots. ltv mblt Codctvt 6 S/m. Thml codctvt 85 W/ m Yog's modlo N/m Posso's to. Coffct of thml so. / Scfc dst 787 g/m Tbl 7. Islto hscl ots. Co I Ot ltv mblt Codctvt S/m ltv lctc mtvt Thml codctvt... W/ m Yog's modlo N/m Posso's to... Coffct of thml so... / Scfc dst g/m. Clcltd cbl mts. Cbl hscl mts std th t tbl.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

276 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Cbl ot dmt s clcltd sg codcto d slto gomtcl mts t to Tbl d Tbl. Cbl l wght mt s clcltd fom gomtcl mts d scfc dsts of th cbl comots. Th whol cbl scfc dst s totl dst clcltd b tg to ccot ll cbl comots. Tbl 8. Cbl hscl mts Cbl ot dmt.8 Mm Wght mt.7 g Cbl scfc dst.9 g/m "Codctos' cctc" tbl holds slf- d mtl-cctcs of th cbl codctos. Ths vls clcltd th QcFld lctosttcs oblm sg th chg och. Fo ch tbl ow th st QcFld oblm s solvd. O of two codctos' sfcs cs t chg d th oth codcto's ottl s vltd. Th mtl cctc s ql to: C U / Q. Slfcctc s clcltd b th msmt of th ottl t th sm chgd codcto. Tbl 9. Codctos' cctc, F Codcto Codcto Codcto Nll-cod Codcto Codcto Codcto Ntl cod Codctos' dctcs std th Tbl. ls th colms 5 clcltd th mgtosttc oblm t th hs dfd th Tbl. ls th colms 6 9 clcltd C mgtc oblm. ll vls got sg th fl lg och b th foml: L / I. Th tbl dgol lmts st th slf-dctc vls. Tbl. Codctos' dctc I mgtosttc oblm I C mgtc oblm C- C- C- -cod C- C- C- -cod Codcto Codcto Codcto Ntl cod Tbl clds th mdc d mdc-l vls. I th mgtosttcs oblm th codcto's mdc ql to th sstc mt s clcltd b th foml: l / ρ S Jol ht mt mgtosttcs oblm s clcltd b th foml: P I, wh I s th oot-m-sq ct d s th codcto mdc. Th codctos' mdcs C mgtcs oblm clcltd sg th Ohm's lw s coml to of th codcto's vg ottl dvdd b th codcto totl ct dst. Th l t of ths to sts th sstc, mg t ctc d th modls mdc. Th Jol ht th C mgtc oblm s clcltd sg th cosodg QcFld tgl. Tbl. Codctos' mdc. I lctosttcs oblm I C mgtc oblm Codctos Nll cod Codcto Codcto Codcto Imdc, Ω sstc, Ω ctc, Ω Jol ht, W glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

277 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES Th gtd ht fld s otd fom th C mgtcs oblm to th ht tsf oblm. s slt of QcFld smlto o c s th cbl to sfc vg tmt, ht flow fom th cbl sfc d th vg tmts of ll codctos. vg tmts ltv mbs std Clss ssmd tht mbt sc tmt s C. Tbl. Cbl ht mts Eto sfc vg tmt.5 C Ht flow. W Codctos vg tmt, C Codcto Codcto Codcto Nll-cod Stss lss oblm s th tmost o, tht mots th tmt fld fom th ht tsf oblm d th mgtc focs fom th C mgtc oblm. D to ths mgtc d thml lodg th cbl comots bcom dfomd. Th mcl vls of ths dfomtos std th t tbl. Tbl. Stss lss oblm slts. Mml dslcmt 5.- Mm Mml Moh ct vl 8.67 N/m Th stgth vl s mott fo th cbl flt lss. Tbl. Th stgth. Mml stgth vl 8.78 /m Th "Stgth" fld s show o fg blow s wll s th "Totl ct dst", "Eg dst", "Momt fl dst", "Tmt" d "Dslcmt" fld cts. Scto 5. Fld cts.. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

278 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC of 6

279 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 of 6

280 FINITE ELEMENTS FO ELECTICL ENGINEEING EXMPLES. glló. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 of 6

281 FINITE ELEMENTS FO ELECTICL ENGINEEING PPLICTIONS M dctc dtmto ottg mchs. ltcl d Nmcl clclto: ddctcl och bstct. gllo,., Llvs, J., D ls,., Mtí, H., Pqé,. Elctcl Egg Dtmt. Elctoc Egg Dtmt EUETI, Poltchcl Uvst of Ctlo UPC c/ugll, 87, 86 clo S Tl. 97, f: 97, [email protected] Th cct dtmto of sttd mgtzg dctcs hs b th sbct of mch sch ov log tm. Ths slts css fo th ot dstmt of cotol glto loos d fo th movmt of tst sos d stblt of lctc dvs. Tdtoll th ltcl clclto volvs th dtmto of som mcl fctos, sch s th d-s d q-s ctc fctos. I fcs [] to [] th m ssos fo slt d o-slt ol mchs, bt ths vld ol fo th cosdd ol sh. If ossbl w shold s sso, o mthod, ddt of th ol sh. ltcl fomlto s ot dqt fo ths so. Now w c s th FE mthod to clclt ths d oth mts. I ddto th tm dvotd tod to th dsg of lctcl mchs hs b dcd d ths ms t mossbl to s lot of mcl o ghcl mthods. Th s of FEM ovds w to qcl d cctl clclt th sz of lctcl mch d ts mts. Ths hs b wtt to dscb ths mthodolog dctol vomt. wods. M dctc dtmto, FE mthod, cldcl d slt ol mchs.. ltcl clclto of mgtzg dctcs. I th followg ghs w dscb how obt ltcl sso fo th m dctcs. Ths mthodolog shows how ths fcto of th ol sh d how l ths dctol vomt.. Ufom g mch Th mgtzg dctc of fom g mch c clcltd ccodg to th followg ocd: Clclto of:. MMF ctd to th -hs qlbt ct sstm. g dcto ol cosds th fdmtl comot C. Totl fl hs D. M dctc dtmto: I L m Th followg sso s th slt of ths ocss. L m µ D L N ξ m π g q wth: m mb of hss, D g dmt, L lgth of th mch, g q qvlt g wth Ct s d stto cocto, N mb of ts hs, ol s.. Slt ol mchs Th clclto s sml, bt w fod som dffcs: - W clclt th MMF octo ov two s: dct d qdt s. - Ths w dtm th dcto ct fo ths two comots d dtm th fdmtl comot. - Ths w c clclt th fl d th m dctc fo v comot:. gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC d 7

282 FINITE ELEMENTS FO ELECTICL ENGINEEING PPLICTIONS. gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC d 7 I L I L q mq d md ; Ths ocss lds to th followg ssos: m q mq m d md L L L L ; Wh L m s th mgtzg dctc clclt sosg tht th g s fom d d d q coffcts tht ddd o ol sh. Th followg tbl shows ths coffcts fo dfft ol sh cofgto; th fst ow s fo clsscl slt ol schoos mch d th oths fo mt mgt mchs. Tbl I. dct d qdt cocto fctos. Slt ol schoos mch cos π ψ π ψ π ψ π π ψ π ψ π s s q d PMSM. Sfc mgts ; q d PMSM. Ist mgts [ ] g h c s s c s c s g g q g d π ψ π ψ π π ψ π ψ π π ψ π ψ π π ψ π ψ π PMSM. d mgts cos π ψ π ψ π π ψ ψ ψ π s q d D h D τ β ψ Wth: τ - ol tch, ψ ol c/ ol tch, h - mt mgt hght.. Nmcl dtmto of mgtzg dctc. Th mcl dtmto of mgtzg dctc volvs th lzto of Ft Elmt lss FE d th dtmto of th mgtc g stod th g. Th followg ghs dscb th ltos btw ths d th mgtc dctc. I ddto w dscb two ws fo th clclto of stod g; th fst s b tgto of dst of g d th scod s b cct modllg.. Mgtc g stos th g fom g mch. If w cosd dl mch wth ssodl dstbto of th dcto log th g, tht s, s τ π ˆ d w clclt th mgtc g stod th g, w obt: m q I g L D N m W ξ π µ 5

283 FINITE ELEMENTS FO ELECTICL ENGINEEING PPLICTIONS If w comb 5 wth w c wt: W m L I m m 6 sll m. Yo c obt th sm sso f o cosd th lctcl cct modl wth cold cols. Fo ml fo th dcto mch modl wth cols th stto d cols th oto, tht s: L L s s [ L] ;[ L ] L L L ; [ L ] t [ L ] [ L ] s s s L L s L s bs L cs L s cos θ L bs bs L cbs π L bs cos θ π L cs cos θ L cs bcs L cs L sb L csb cos θ π cos θ L bsb π cos θ L L b L c L b L b L cb L c L bc L c Lcsc cos θ π L sc cos θ π L bsc cos θ 7 Fg. smlfd mch. W obt: W L I I L s s Lbs bs Lcs cs L Lb b Lc c Lbs s bs Lbcs bs cs Lcs cs s Lb b Lbc b c Lc c W 8 L s s Lsb s b Lsc s c Lbs bs Lbsb bs b Lbsc bs c L cs cs Lcsb cs b Lcsc cs c W omttd th tms wth cos to smlf th sso. If w cosd th followg vls, cosodg to stt wth: s Iˆ; b bs c cs Iˆ 9 w obt:. gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC d 7

284 FINITE ELEMENTS FO ELECTICL ENGINEEING PPLICTIONS W L I L m σ s I Ect fo th lst tm, ths s th sm sso 6. Ths tm s slt to th dsso ffct d wll b ot cosdd fo th m dctc clclto.. Mgtc g stos th slt ol mch. W c obt sso fo th mgtc g stod th cs of th slt ol mch, bt ths ts log to dtm. W dvlo sso bsd o cct modl omto. I th slt ol mch w cosd th fst hmoc omto fo th dctc vto,.., L L L cosθ fo th -hs schoos mch w c wt: L f L L bb L cc L c b L L L L Fg. Slt ol mch σs σs L L sf L bc L σs L L L / / cosθ, L L cf L L / L L L sf L L bf cos cos cos cos θ θ π / θ π / cos θ cos θ π / cos θ π / Lsf cos θ π /, θ π / L Th g stod s: W L Lbb b Lcc c L ff f Lbc b c Lc c Lb b Lf f Lbf b f Lcf c f W omttd th tms wth cos to smlf th sso. If w cosd th followg vls, cosodg to stt wth: b Iˆ c Iˆ f wthot fld ct. gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC d 7

285 FINITE ELEMENTS FO ELECTICL ENGINEEING PPLICTIONS W obt th followg sso: W I L L cosθ Lσ s I 5 Th lst tm s slt to th dsso ffct d wll b ot cosdd fo th m dctc clclto. If w cosd two slctd ostos fo th oto,.. cos θ Dct fld otto cos θ qdt fld otto Som ft lgbc mltos, w obt: W I L L L cosθ Lmd [ L L ] W I Lmd Lσ s I W I L L L cosθ Lmq [ L L ] W I Lmq Lσ s I C. Idctc dtmto b ms of fl coctto oth tchq fo th clclto of dctc s b th s of fl coctto b col. If w cosd mgtc fld dstbto log th g, d ts fst hmoc, w c clclt th fl coctto d th m dctc: ds dl S L N N I I I If w cosd slt ol mch, w s fl otd ov th dct d qdt s sctvl, fo th dtmto of dct d qdt dctcs.. Pctcl lctos Th followg ghs show th mls of dtmto of m dctc. Two of thm comd wth mtl slts.. schoos mch:.5 W; 5 Hz; / 8 ; 6. /.7 ; cosϕ.85; m - ; F clss; J.5 gm;. Coo. Gomtc d lctcl dt: 6/8 slots; codctos/slot; D 8 mm; g.75 mm; L mm. W cosdd tht. d ξ c st σs σs I I gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 5 d 7

286 FINITE ELEMENTS FO ELECTICL ENGINEEING PPLICTIONS Fg. FEM modl fo schoos mch. Ol ¼ of th mch hs b modld. Tbl II. M dctc fo schoos mch. Mthod L m H ltcl clclto. FE. Emtl slts.55. Schoos mch: 6 ; ; 5.8 ; 5 Hz; 5 m - ; Y coo. Gomtc d lctl dt: slt ol wth fom g d th ol g mm; D mm; L mm; ψ.55; 6 slots; dobl l l wdg,; 5 codctos slot d l. W cosdd tht. d ξ c st Fg. FEM modl fo schoos mch. Dct fld otto. Fg 5. Qdt fld otto Tbl III. M dctc fo schoos mch. Mthod L d mh L q mh ltcl clclto FE.7. Emtl slts dcd sl tst C. Schoos mch wth mt mgts. 5. Nm; 5 m - ; I N.56 ; F clss Gomtcl dt: D 8 mm; L 68.9 mm; 6 slots; 6 ol; 5 codctos slot; sgl l l wdg; ζ.96; mt mgt hght h mm; g.5 mm; ψ.65;. ; sfc mt mgt. c st I ths cs to mos f w chg th PM chctstc fom o-mgtc mtl wth th sm mgtc mblt of th PM. Ths mch s cosdd s fom g mch d to th vl of col mblt of th PM to.. gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 6 d 7

287 FINITE ELEMENTS FO ELECTICL ENGINEEING PPLICTIONS Fg Çs 6 d 7. FEM modl fo schoos mch. Dct fld otto. Qdt fld otto Fo ths mch w dtmd th dctc b th mthod of fl coctto. W obt th mgtc fld dstbto d hmoc comots showd th fgs 8 d 9. Fg 8. Mgtc fld dstbto log th g. Fg 9. Hmoc dstbto of mgtc fld. Th followg tbl I shows th clcltd vls. Tbl I. M dctc fo schoos mch wth PM. Mthod L mh ltcl clclto 6.55 FE g 5.5 Fl mthod FE 6.. Coclsos. W ld som mthods to dtm th m dctcs fo lttg ct mchs dctol vomt. W cosdd cocto fctos tht ddts o th ol-sh cofgto. FEM s mo cs th ltcl clclto d s ot ddt o mcl o gomtcl fctos. Som of ths mtl slts dscodt wth thotcl slts d to stmto of som gomtcl mss d mgtc chctzto. blogh [] J. Cols Mtí. Cálclo Idstl d Máqs Eléctcs. Tomo. Ed. Mcombo. 98 []. Chlms,. Wllmso. C Mchs. Elctomgtcs d Dsg. Ed. sch Stds Pss Ltd. 99. [] J.F. Gs, E. St, M. Wg. Clclto of schoos mchs of smll mt mgt lttg ct motos: comso of ltcl och d FEM wth msmts. IEEE Tsctos of Mgtcs, vol, º 5, Stmb 988. [] M.. Hsszdh, : oms. ltcl clclto of mgtzg dctcs to mt mgt motos. ICEM, Ccow, Stmb. [5]. glló. Dsño d Máqs Eléctcs. Tm 8. Dtmcó d ámtos. EUETI... gllo. ELECTICL ENGINEEING DEPTMENT. EUETI-UPC 7 d 7

288 Std Stt Pfomc Evlto of Pmt Mgt Schoos Moto sd o FE Ld Ptovs, So Mmb IEEE d Gog Cvtovs, Mmb IEEE ho: , Ss. Cl d Mthods Uvst Fclt of Elctcl Egg os II b.b., P.O.o 57, So Mcdo f: , -ml: [email protected] ; [email protected] bstct. Th dls wth th dtmto d vlto of std stt fomc chctstcs of schoos moto wth sfc motd mt mgts. t th bgg, mcl clclto of th mgtc fld dstbto of mt mgt schoos moto PMSM, d cosdto s cd ot. Fo ths os, th Ft Elmt Mthod FEM s ld. sg ott dt fom th fld comtto, ll lvt chctstcs of th moto dtmd. Th slts of th mcl clcltos std b dgms. Wh ossbl, clcltd std stt chctstcs comd wth mtll obtd os; th show v good gmt. vlto of th std stt bhvo of mt mgt schoos moto, bsd o th Ft Elmt lss FE s std.. Obct of Std Th obct of vstgto s oc moto t EM 9M-6, wth td dt: 8, Nm, m. Th moto s sld fom C soc t 5 Hz, b ct s wvs. S mt mgt ols md of SmCo5 sfc motd o th oto. Th sd vw of th moto d ts gomtcl coss scto std Fg. d Fg., sctvl. Wods Pmt Mgt Schoos moto, FEM, FE, Mgtc fl dst, Cog, Elctomgtc toq.. Itodcto Th -hs mt mgt C moto, ctg s covtol schoos t moto, hs fod wd tst th lst two dcds [], []. Th ct dvlomt of hgh g mgts hs hcd th lcto wd g of s. Th blt- of mt mgts th oto co of schoos motos s ctto, d tcl th s of smm-coblt o odmm-boo-o mgts hs chllgd ovtos th mt mgt schoos moto PMSM dsg d lss. I th, mts d std stt fomc chctstcs of PMSM dtmd d lsd. Th m ts s lws to clclt std stt chctstcs, s ct s ossbl. It hs b fod s th comlctd ss. It s obvos tht th stss shold b t o th ct dtmto of th mts, s th "lg" mott ol th ccc wth whch ll th chctstcs of th PM schoos moto d cosdto wll b dvd. Fg.. Sd vw of PMSM t EM 9M-6 Fg.. Coss-scto of th moto EM 9M-6

289 . FEM Clclto of PMSM Th Ft Elmt Mthod FEM hs b sd tsvl th mcl clclto of th mgtc fld lctcl mchs, gl. Th ott slts, d ossblt to s thm fo clclto of both lctomgtc d lctomchcl chctstcs, cllt bss fo cg ot Ft Elmt lss FE. M schs ll ov th wold, cldg th thos of ths, hv do lot of wo ths. M s ths toc hv b blshd []-[]. Dfft softw cgs st s. Th std slts th comtd b sg s fdl softw cg FEMM []. I th fst st, sll cosdd s -ocssg stg, ddg o th s choc th msh of ft lmts wth ot dst s gtd fll tomtcll. I th FE of th PMSM t s cosstd of 7,9 ods d, lmts. Fo th oss of FEM clcltos of th mgtc fld th moto d cosdto, th msh s sd ov th whol coss-scto of th moto, s c b s Fg.. I th -ocsso, md fmm. ll qstd t dt cldd: th ct gomtcl coss scto of stto d oto mgtc co; ct dst th ctd stto wdgs; ll bod codtos of th go whch s gog to b lsd; ll mtl chctstcs of th moto mt mgts, co w, -H mgtsg cv. Th mcl FEM modl of th PMSM, bg comltd, s d fo ctcl s. Wh lg th softw cg FEMM fo lss of th mt mgt schoos moto, th mgtc oblm s cosdd to b th tm ddt hmoc oblm. Hc, th clcltos of th mgtc fld fomd t td fqc f 5Hz. Th fld soltos obtd b g th FEMM solv, clld f.. s th whol coss scto of th moto s sd, ol th fst od Dchlt's bod codtos ld; o th ot stto l d th oto l t s st to b. mt cts th stto wdgs vd fom I to th td vl I 8. oto s fl movg ottg th -g, cotosl chgg osto, d th d-s of th oto s cotosl tg dfft gls θ gst th ftl s of th stto, fml ld wth o of th wdg s. ft th ocssg st s ctd, th vls of mgtc vcto ottl v od of th moto dom obtd. Lt, o c s thm fo m oss. Th t fmmvw. th FEMM cg s offg s fdl clcltos d ghcl sttos of th most mott lctomgtc d lctomchcl qtts.. Mgtc Fld Dstbto Th bst w to dstd th hom vstgtd moto s "to gt sd d to s" th mgtc fld dstbto. Ghcl stto d vslzto of th FEM slts gv th mgtc fl dstbto th coss-scto of mt mgt schoos moto. t of th most tstg slts of th clcltos gv cotto. Th mgtc fld dstbto PMSM s std Fg., t followg gms: o-lod codto,.. zo mt ct, mg mgtc fld obtd b th mt mgts ol; b td-lod wth td stto wdg ct I 8 d td lod gl δ 9 [dg.l.],.. θ [dg.mch.]; c lodg codto t ll-ot mmm vl toq, mg lod gl δ m 9 [dg.l.],.. θ [dg.mch.]. Fg.. Ft lmts msh of th PMSM o-lod t I d θ dg.

290 o-lod t I d θ dg. b td lod t I 8 d θ dg. b td lod t I 8 d θ dg. δ 9 dg.l. b ll-ot lod t I 8 d θ dg. Fg.. Mgtc fl lots th mddl coss-scto of PMSM d tcl otg codtos Th FEMM softw cg bls comhsv stto of th stl dstbto of mgtc fl dst log bt slctd l, s wll. Th dstbto log th md-g l s std Fg. 5, b d c, th sm w s cdt, t th sm otg gms d lodg codtos of th PMSM. Th followg dgms sd to o ol tch. Ths dgms c b sd fo cg ot ofod lss of th -g mgtc fld ots gdg both tst d sh. Wh th mt mgt schoos moto s lodd, th flc of mt cto mgtc fld s cll show, th fgs. Th flc of stto co tth, o th -g fld dstbto s lso cll dctd th fgs. c ll-ot lod t I 8 d θ dg. δ m 9 dg.l. Fg. 5. Md-g mgtc fl dst stl dstbto. -g Fl Lg Th mcl clclto of fls s bsd o th fld tho, ld o bodd d closd sstms. If th clcltos fomd of ctd ols, t s: g otds d ds Σ C Σ

291 Fo N ctd ts, th -g fl lg s: Ψg N g ds S C. Comtto of Idctcs It s v mott mtt to clclt s cct s ossbl th vls of th mts of th PMSM. Of th most mott sgfcc th dct- d th qdt- s dctcs, s th dtmg cosodg schoos ctcs []; t s wll ow tht th th most sgfct mts wh dlg wth std stt d/o dmc fomc lss of PMSM. Th mcl clclto of dctcs s bsd o FEM slts. It s fomd stl fo d- d q- s. I ths cs, t s fod to b sffct to clclt mgtc fld ol log o ol tch. Nm's bod codtos of th scod od, mosd o th sd ls of th ct []. Th fld shold ot b ctd; t ms tht mt mgts hv to b lcd wth ft lmts ltd ol wth cosodt mblt µ.5, bt ot cg th mgtc mc.95 T. Ol th mt wdg s gzd ot w [5], [6], s ld bllow: d-s: Wh clcltg th dct s dctc L d, cts th mt wdg dstbtd to t qdt s, odcg fld wth t dct s. Th mgtc fld dstbto s std Fg. 6. Fg. 7. Mgtc fl dstbto fo L q clclto I gl, th dctc s clcltd s to of th fl lg to th mt ct, ldg to: L ψ I Th vos qto s ld fo th comtto of th d-s d q-s dctc. Th cosodg fl lg ψ, fo ch stdd cs s clcltd b lg sctv slts FEM Eq. ; th mgtc fld clcltos cd ot w s vosl hs b ld, ccodc wth Fg. 6 d Fg. 7. s t ws ssmd th fcttos dct/qdt wdg to hv th sm mb of ts s th l stto hs wdg, t s qstd to todc oth fcto to fd th dct- d qdt- s dctc, sctvl []. Fo -hs C mch, th mt ct th dct/qdt s wold hv to b / tms s gt s th hs ct to odc th sm mgto-motv foc log th sctv s s th th hs wdg. Hc, L L d, q / Th clcltos ld to slts: L d 5.86 mh d L q 5.8 mh 5. Moto Pmts Dtmto Fg. 6. Mgtc fl dstbto fo L d clclto q-s: Th qdt s dctc L q, s clcltd sml w s L d. I ths cs, th mt fld s movd fowd sc fo 9 l. d odcs t qdt s. It ms tht mt cts of th stto wdgs t th dct s. I ths cs, th mgtc fl dstbto fo o ol tch, d wth th sm bod codtos wh clcltg d- s mgtc fld, s std Fg. 7. Th two-s modl of th schoos mchs s wll stblshd clsscl och fo ltc-ghc vstgto of th bhvo []. M schs wdl s ths mthod fo fst dcto of th tl dt fo fth mo dtld d dd lss [7]. Th bsc d s to dvlo d to s st of qtos, dscbg th moto fomc d,q fc fm d tms of th lodg gl δ. Th ol qst s to hv vlbl th moto mts. Th ccc, wth whch th fomc chctstcs of th PMSM wll b dtmd, s th dct ddc of th ccc wth whch th moto mts clcltd.

292 Som of th moto mts cold b sl msd; som vlbl fom th odc's dt; bt v oft th vls ow, d t s qstd c d sll to l th bst w stg d wll ow mcl, mtl o ltcl clclto mthods. Dfft ochs ossbl. Sttg wth th mcl ocd, th d,q mts of th PMSM d cosdto dtmd. sg th FEM slts comtd fo L d d L q gv th vos hdg wth Eq. 5, o c dtm th vls fo d,q ctc of PMSM, t 5 Hz s: X d.87 [Ω] d X q.8 [Ω] Th ld ow fct tht, tht schoos motos wth sfc motd mt mgts, th s lmost o dffc btw ctc log d- d q-s hs b lso ovd ths cs. mt wdg sstc d lg dctc L s hs dtmd fom mtl tstg vstgto of th mt mgt schoos moto t EM 9M-6 []. Th msd vls gv bllow:. [Ω] L s. [mh] X s.69 [Ω] Hvg vlbl th mts of th PMSM, th hso dgm t td otg codtos s costctd [7], d s fod th td lodg gl to b: δ 9. [dg.l.] Ths vl of th lodg gl of th cosdd moto t td otg codtos, wll b dtmd b sg mcl clclto of th std-stt chctstcs v Ft Elmt Mthod. 5. Std Stt Chctstcs I th gg ctc, th tto of schs, odcs d ss s lws focssd to stmto, lss d vlto of th lctc mch bhvo. Fo tht os, t s qstd to hv vlbl fomc chctstcs, s cct s ossbl. Th mt cts I d oto ostos θ log o ol tch, bt slctd. Th otto s sosd to b cot clocws. Th fc s s slctd to b th -hs s of th stto wdgs; th tl oto osto d θ dg. mch. s dfd wh stto -s d oto N-ol s d-s ccodc. Th PMSM s lsd t dfft otg codtos. Nmcl clcltos of th most lvt lctomgtc d lctomchcl qtts, bsd o th FEM ost-ocssg slts, std th followg sbsctos.. Mgtc Fl Dst Th fl dst s clcltd fom th bsc lto sd th dfto d todcto of th mgtc vcto ottl, th comttos of th mgtc fld wth Ft Elmt Mthod. Th qto dfg th l btw d s: 6 lg th mcl ocd fo ts solto th -g dom, mgtc fl dst g of ols s comtd. I Fg. 8, chctstcs of th fl dst, fo th tcl mt cts I zo, hlf of th td d td d dfft oto ostos θ log o ol tch 6 dg. mch. of th moto std. Mddl -g fl dst g [T] Fg. 8. Mgtc fl dst chctstcs g fθ,i. Mgtc Fld Cog I [] I 9 [] I 8 [] oto osto Θ [dg. mch.] I l mgtc fld oblms, th mgtc g W d th cog W' ql. t, th most css, th oblm s o-l, so th cog s comtd b sg: W' J d 7 I fct, ths qtt hs o hscl lto, bt t s v sfl fo clclto of th lcto-mgtomchcl qtts wh g coct s ld. Fo th qs sttc modl of th PMSM, lctomgtc cog W' s clcltd mcll fom th followg sso: I W' θ, I ψ I, θd I θ cost Th mgtc cog s clcltd ddc of th osto of movg ts th dom th oto t bt slctd mt ct. Th clcltd chctstcs std Fg. 9. 8

293 g mgtc fld cog W ' [ J ] oto osto Θ [dg. mch.] Fg. 9. Mgtc cog chctstcs W' fθ,i C. Elctomgtc Toq Th owldg of th sttc toq chctstcs s v mott ss fo cg ot lss d vlto of bhvo of lctc motos. Fo clclto, vos ochs st. I tho, th toq s comtd fom th fld solto mb of vos ws. Th ochs fo clclto ctcl s: Fl- Ct Mthod, th Mwll Stss Mthod d tl Wo Mthod. I ths, th g coct fo mcl clclto of toq th PMSM s ld. Th lctomgtc toq T m s ffctd b th vto of th mgtc fld cog th -g dom, t vtl dslcmt of th oto, whl th mt ct s focd to b costt. Th qto fo clclto s dvd th fom: T m,i θ,i W' θ I cos. 9 θ Th slts of clcltos, fomd fo td ct I 8 d I / 9, std Fg.. Elctomgtc toq Tm [Nm] T δ I [] I 9 [] I 8 [] I 9 [] I 8 [] oto osto Θ [dg. l.] Fg.. Elctomgtc toq chctstcs T m fθ,i 7. Pfomc Evlto of PMSM Th oosd mthodolog b mlmtg dfft mthods fo clclto of std-stt chctstcs d dfft otg codtos, bls to c ot dd fomc lss of th mt mgt schoos moto, d vlto of ts bhvo t vos lods. Th Ft Elmt lss FE, bsd o th comttos fomd b sg FEM, bls to vlt th mgtc fld ots th whol vstgtd dom of th PMSM. Th Fgs 5. wh th stl dstbto of th mgtc fl dst s std showg th ffct of th mt cto fld o th m PM ctto fld, th most tl d vdt w. Th sm homo s lso cogsd Fg. 8., std b th ots wh th mgtc fl dst chctstcs g fθ,i ssg thogh zo vls. Th tcl FEM clclto s fomd, d th dctcs,.. th ctcs log th d,q s dtmd. sg thm, th hso dgm of th PMSM d td otg codtos s costctd; th td lodg gl s fod to b δ 9. [dg.l.]. t th sm tm, th FEM slts fo th lctomgttc toq clcltos, d th cosodg chctstcs T m fθ,i std Fg., llow to dtm mcll th td lodg gl, too. Fom th chctstc clcltd t th td mt ct I 8 [], fo th td vl of toq T m [Nm] o c s fod lmost th sm vl fo δ 9 [dg.l.]. Th fomc chctstcs of th cosdd PMSM vfd two ws, ddg o th vlbl dt. Som of th comtd slts comd wth th dt obtd dctl fom th odc, d th oths, wth th mtll obtd os. Th mt wdgs mts clcltd two ws: th sstc hs s clcltd ltcll; th lg dctc L s hs s dtmd b sg th-dmsol mgtc fld clcltos th whol vstgtd dom of th PMSM []. Ths mts lso msd. Showg v good gmt, th ov th ld mthodologs s cct d lbl. s vfcto of ths wo, h bllow s std ol bf comso of mt ct I, t td lod toq [Nm], dtmd b dfft mthods: Clcltd: Msd: I clc 8[] I ms 7.6[] Th bov std lss s stfg th ld mthodolog fo clclto fomc chctstcs of th PMSM t EM 9M-6, s cct d coct. Cosqtl, t c b commdd fo sml clcltos of t of schoos motos.

294 Coclso Ft Elmt lss s th bst w fo fomc vlto of th lctcl mchs gl. Pstd och, wh ld o sfc motd mt mgt schoos moto s ovg th sttmt. Obvosl, th hom otsd of th mgtc co.. d gos ot showg mott flc, so th std stt chctstcs, clcltd b sg Ft Elmt Mthod, th D dom, wth th stsfcto ccc. Ths fct tht th td lodg gl, dtmd wth two dfft ochs s wth th lmost sm vl, s ovg two mott cotbtos std th : fst, th moto mts clcltd b FEM och qt cct; scod, th s of th hso dgm fo dtmg th td lodg gl s ovd to b coct. owg tht th hso dgm s dw wth FEM clcltd vls of th ctc X d d X d, th dct coclso s tht th vls c b tctd s cct. O th oth hd, th sttc lctomgtc toq chctstcs lso dtmd b sg FEM, bt qt dfft ocd. oth ocds gvg th sm slts obvosl coct. Msd vls d th tstg slts th bst w to cofm both ltcll d mcll clcltd mts d chctstcs. Th mtl gmt std th, s ovg th oosd och d mthodolog s cct. Th thos fosg th ft ts s tst fomc d dmc lss of th cosdd PMSM. Ths wo d tcl th std slts, showg cllt gmt, c b sd s good bss d lvt gd. fcs [] S. Low d W. H. L, "Chctstcs d Pfomc lss of Pmt Mgt moto wth Mltstcd Imbctd oto", IEEE Ts. o Eg Covso, vol. EC-, No., Octob 987, [] L. Ptovs: " Cotbto to lss of Pmt Mgt Ectto Fld Iflc o Chctstcs of Elctocll Otd Schoos Moto, v D Mgtc Fld Clclto",., PhD Thss, So, Mcdo, 99. [] N. Esosto,. Msolo,. Tll: Nol lss of Elctomgtc Flds Pmt Mgt Mchs v Itgl fomlto", Jol Stds ld lctomgtcs d Mchcs, ol.,. -, IOS Pss, Ohmsh, mstdm, Nthlds, 996. [] L. Ssc, Z. Hzd: "Elctomgtc lss of dl-fl Schoos Mch Ectd b Sfc Motd Pmt Mgts", Ittol Smosm o Elctomgtc Fld Comttos-ISEF',. -, Ccow, Pold,. [5] L. Ptovs, M. Cdv, G. Cvtovs d. Sc, Dfft scts of Mgtc Fld Comtto Elctcl Mchs, oo of bstcts of th th Ittol Smosm o Nmcl Fld Clclto Elctcl Egg - IGTE,.. 7; o CD. -6, Gz, st,. [6] L. Ptovs, M. Cdv,. Sc: "FEM lss of smmtcl Mgtc Fld Elctcl Mchs", Pocdgs of th Scod COMEN Ittol Cofc, msct blshd o CD.-, Lg, lgm,. [7] L. Ptovs, M. Cdv, G. Cvtovs,. Stolov: Modllg th Cofgto of Elctcl Mchs fo D Mgtc Fld Clclto, Jol o ld Elctomgtcs, JMSEM '98, Ho Pss ld Scc Ss,.. 5-5, Sof, lg,. [8] G. Cvtovs, L. Ptovs, M.Cdv,. Stolov: D Mgtc Fld Clclto Pmt Mgt Mchs, Jol o ld Elctomgtcs, JMSEM '98, Ho Pss ld Scc Ss,. 9-, Sof, lg,. [9] G. Cvtovs, L. Ptovs, M. Cdv: PM Dsc Moto Pmts Evlto Usg FEM Dt, Pocdgs o th 6th Ittol Smosm o Elctc d Mgtc Flds - EMF, Smosm ots,.. 9-9, ch, Gm,. []. Sc, L. Ptovs, M. Cdv, "No-l Tm Hmoc lss of Shdd-Pol Mcomoto" Ittol Smosm o Elctomgtc Fld Comttos-ISEF', vol. /,. 7-, Mbo, Slov,. [] Q. Zho, Z., Z. L,. Tg: "lss of Fl Lg Coffct of Pmt Mgt Schoos Moto wth U-shd Mgts oto", Pocdgs of th 6 th Ittol cofc o Elctcl Mchs d Sstms ICEMS', ol. /, , g, Ch,. [] D. M, "Ft Elmt Mthod Mgtcs - FEMM", Us's Ml,.., Fost-Mll, M, US,. [] C. s, O. Wsch d S.D. Sdhoff, "lss of Elctcl Mch", boo, IEEE Pss, d Edto, Nw Yo, US, 99. [] S. J. Slo, Ft Elmt lss of Elctcl Mchs, boo, lw cdmc Pblshs, Nowll, M, US, 995. [5] Z. olodzovs d L. Ptovs, Idtfcto of Schoos Gto Pmts Ft Elmt lss, oo of bstcts of th th Ittol Smosm o Nmcl Fld Clclto Elctcl Egg - IGTE,.. 96; o CD. -6, Sggbg Gz, st,. [6]. Shm,. Id d M. Thsh, Ft-Elmt Clclto of Lg Idctcs of Sttd Slt-Pol Schoos Mch wth Dm Ccts, IEEE Ts. O Eg Covso, vol. 7, No., Dcmb, [7] G. Cvtovs, L. Ptovs d S. G, "Pfomc lss of Pmt Mgt Dsc Moto fo Dct Elctc hcl Dv", Jol ELECTOMOTION, ol., No., Jl-Stmb,

295 Iflc d l Gomtí dl oto l Cot dl Estto d MSIPs J.. Güms loso,.m. Iolgot Iodo Dtmto d Igí Eléctc E.U.I.T.I., Uvsdd dl Pís sco Plz d l Csll,, 8 lbo Esñ Tlf.: 966, 9667, f: 96 -ml: [email protected],[email protected] sm. E st tbo, s lz l flc dl tmño d los ms l comotmto d moto sícoo d ms mts MSIP d b vlocdd, co los ms motdos l sfc dl oto. S com dos motos co dstts gomtís dl oto modfccó dl tmño d los ms gl stto s obsv l flc ls cctístcs cls d l máq. Hmos comobdo q moto d 5 W, l mt l volm d los ms 66 %, l vlo d l cot dl stto s dc 9 %. El stdo dl comotmto dl moto s lz tlzdo l método d lmtos ftos MEF. El método costt lccó dstl l dsño modldo d st to d máqs d s útl dtm fómls css l cálclo d l cot dl stto,, fcto d otc dmto d l máq, cdo s modfc l tmño d los ms. Plbs llv. Ims mts, método d lmtos ftos, modldo, moto sícoo, smlcó.. Itodccó Los motos d ms mts d c d dvds dos gds gos: los motos bshlss dc NLDC q t fom d od d l fz lctomotz tzodl los motos sícoos d ms mts MSIP q t fom d od d l fz lctomotz sodl. El tés d los MSIPs stá ccdo mlo bco d lccos. Ls zos cls dl cmto d lccos d los MSIPs so m lg, l fcc ls bs édds dl oto sgdo lg, l dccó dl co d los ms NdF. E témos gls, los MSIPs so fbls los motos sícoos, tods qlls lccos q q vlocdd costt co q sv fcomto slcoso. E gí léctc, omlmt ls máqs léctcs so stds o ccto qvlt. El coocmto d los dfts ámtos dl ccto qvlt d l máq, os mt cooc s comotmto, bo cts codcos d fcomto, o lccó d ls ls d los cctos léctcos. El MEF, costt hmt m mott los gos léctcos, l álss dl comotmto d máqs léctcs, l mt cooc co mg d o cotdo, los dfts ámtos d l máq, sí como s comotmto stcos tms d fcomto. E l álss o lmtos ftos d motos sícoos d ms mts, l m so s sml l comotmto d l máq oso s cot o los dvdos dl stto, lo q os mt cooc l vlo d l fz lctomotz stm l vlo d l cot o los dvdos dl stto l otc útl d l máq. E l vlo d l fz lctomotz, fl d fom otbl l gomtí dl oto l tmño d los ms. L otmzcó d l gomtí dl oto s m mott, ddo q odmos cosg l msmo co dsttos vlos d l cot o los dvdos dl stto, dsmdo coscc l co dl cotoldo dl moto. E los últmos ños, s h blcdo dsttos tbos los q s std l flc d l fom tmño d los ms gomtí dl oto l: dsño d cctos mgétcos [], b stdo d dos vbcos d motos [], c comotmto d motos léctcos [] [6]. E st tbo s com dos motos co dstts gomtís d oto gl stto comobmos s flc ls cctístcs cls d l máq. D cdo co lo to, l obtvo dl tbo s obsv cómo fl l tmño d los ms modfccó d l gomtí dl oto ls

296 cctístcs dl moto. L smlcó l stdo dl comotmto d l máq, s lz tlzdo l MEF.. El método osto, d s m útl dtm fómls css, d lccó dstl, q mt clcl l cot dl stto,, fcto d otc dmto d l máq, cdo s modfc l tmño d los ms.. Método L fz lctomotz oso o fs, los dvdos dl stto, d dtms o mdo d l só: Dod: E π Nf w φ q E q fz lctomotz dcd o l flo d ctcó dl oto s cosd l ccó d dcdo; N úmo d ss s o fs dl dvdo dl stto; f fcc; w fcto d dvdo dl stto; φ flo mámo. Cosddo l lcó stt t l flo l vcto otcl mgétco. φ dl P modlos los dos dmsos, l ccó d os l fom: E q π NfP Dod P s l ofddd dl modlo logtd dl qt d chs. Po ot t, como s coocdo, l lctomgétco d moto sícoo tfásco d olos slts d dtms o l ccó: Dod: ω X d X q I d I q T [ E I X X I I ] q q d q d q ω úmo d s d olos; vlocdd gl; ctc dct; ctc cdt; tsdd dct; tsdd cdt. l m témo d l ccó, s dom ldo co los ms l sgdo d lctc. E los motos sícoos d ms mts, co los ms motdos l sfc dl oto, ls ctcs dct cdt d dmts gls. X X X 5 d E sts codcos l d lctc s lo l ccó d scbs: q s T Eq I q 6 ω D cdo co lo to, l otc lctomgétc d l máq d dtms o mdo d l só: π P T E 6 Dod s l vlocdd dl moto m. Flmt l otc útl s clcl, stdo l otc lctomgétc, ls édds l ho ls édds mcács. Ls édds mcács s clcl fcó d l vlocdd dl oto ls édds l ho t d l dsdd d flo mgétco los lmtos dl modlo d lmtos ftos [7]. D l ccó 7 obsvmos q l otc d moto sícoo d ms mts, co los ms motdos l sfc, l otc útl dd d l cot l fz lctomotz oso. E coscc cosg dtmdo vlo d otc/ d l máq, d s tst dsñ máqs q tg lto vlo d l fz lctomotz oso, dsmdo d st fom l cot o los dvdos dl stto coscc dcdo, vsblmt l otc, l costo dl dsostvo cotoldo dl moto. Los fctos q fl l vlo d l fz lctomotz oso, so l úmo d ss, l fcc l flo. L fcc v dtmd o l vlocdd d l máq. s vz, l vlo dl flo fl l gomtí dl stto, l logtd d tho, l gomtí dl oto, l fom, tmño dccó d mtcó d los ms. E st tbo stdmos como fl ls cctístcs d l máq, l vlo dl flo modfcdo l volm d los ms mcdo costt l úmo d ss. scto l fom d los ms s cosbl q tg l c to, clmt l to d fom ccl do sml l dl oto, o sto cc otblmt s co, o lo q tlzmos ms d sccó ctgl. q I q 7

297 . Moto Ls cctístcs cls dl moto obto d stdo so: Moto tfásco Tsó d td dl cotoldo: Fcc d l tsó dl cotoldo: 5 Hz Fcc d l tsó dl moto:.5 Hz locdd oml fd.5 Hz: 5 m Númo d ms mts: ; 6 Númo d s dl stto: 7 Potc útl: 5 W To d ms: NdF. L smlcó stdo dl comotmto d los motos oto oto, s lz mdt l método d lmtos ftos tlzdo modlos dos dmsos.. Modlo El modlo d lmtos ftos stá costtdo o sccó ct dl moto sco d q lo od. L fg, mst l mlldo tlzdo. L fg mst cto d l gomtí dl moto. oto Fg.. Moto oto Ls cctístcs d, otc, dmto fcto d otc dl moto dcdo, so comds co ls dl moto q s obt l sstt l oto o l mostdo l fg mcdo costt l logtd d tho. El volm d los ms s cmt 66 %, mcdo costt l lt logtd d los ms; l stdo d mtcó, l to d ms. b oto Fg.. Mll d lmtos ftos cto dl modlo Ls odds d los mtls s df como sg: Fg.. Moto oto P l tho, comzo s dl oto to l cob bobdo dl stto o mdo d l mbldd mgétc.

298 P l úclo mgétco dl stto oto o mdo d l cv H, l fcc d tbo, dl mtl co l q stá costds l chs mgétcs. Los ms o mdo d s mbldd mgétc cmo coctvo ss dos comots H CX H CY. Ls ctcos s df o mdo d ls dsdds d cot los dvdos dl stto. Podmos obsv q l vcto otcl mgétco, s cmt d fom mott l moto co l oto scto l moto co l oto. El vlo d l fz lctomotz o fs, clcld o mdo d l ccó, s l m moto l sgdo moto cmto dl %. L otc lctomgétc sd d cd o d los motos, vlos d < I q <, clcld d cdo co l só 7, s mst l fg 5. L codcó d cotoo s vcto otcl mgétco lo l f dl modlo. El to d álss lzdo s mgtostátco. 5. álss E m lg s dtm l fz lctomotz o fs, dcd los bobdos dl stto o los ms E q, lo cl s lz l smlcó dl comotmto d l máq oso. U vz lzdo l álss s t los ms d vcto otcl mgétco mostdos l fg. Fg. 5. Potc lctomgétc Dl gáfco to, s ddc q otc útl d 5 W vz stmds ls édds mcács l ho, l tsdd sd s d 5,6 l m moto d 7,6 l sgdo moto dccó l vlo d l cot dl 9 %. L fg 6, mst l lctomgétco clcldo d cdo co l só 6, vlos < I q <. oto Fg. 6. P lctomgétco E sgdo lg, s h smldo l comotmto dl moto oto vlos I q d 5,, 5. U vz lzdos los álss los vlos d cot dcdos, s dtm l fz lctomotz sltt cg, o mdo d l ccó: E π Nf P 8 Dod s l vlo mámo dl vcto otcl mgétco sltt dbdo los ms cot o l stto. b oto Fg.. cto otcl mgétco oso L fg 7 mst l dstbcó dl vcto otcl mgétco, mbos motos, 5.

299 L ctc síco, s dtm o mdo d l só: X X s X cb Dod: X X cb ctc d ccó d dcdo más l ctc d dssó dl qt d chs; ctc d dssó d cbzs d bob. L ctc d dssó d cbzs d bob, d clcls o mdo d l só [8]: oto Dod: X cb πf Z λ L f fcc; úmo d s d olos; f úmo d s o fs olo; Z úmo d codctos o ; λ cb mc o dd d logtd; L logtd dl dcdo. f cb Flmt l fcto d otc s dtm o l ccó: Eq I q cos ϕ U b oto Fg. 7. cto otcl mgétco cg I q 5 L fg 9 mst l vcó dl fcto d otc, fcó d l cot, cd o d los motos stddos. El áglo d tso dl oto scto dl cmo sltt, s dtm o mdo d l ccó: φq δ φ cos 9 φ Dod φ s l flo sltt dbdo los ms cots. L fg 8 mst l vcó dl áglo d tso dl oto scto dl cmo mgétco sltt, fcó d l cot, los dos motos lzdos. Fg. 9. Fcto d otc Podmos obsv q l otc dsd, l fcto d otc s,8 m bo l m moto oto d,9 l sgdo moto oto. L tsó d lmtcó o fs, s d 8 l m moto d 7 l sgdo. Fg. 8. glo d tso dl oto. D cdo co lo dcdo tomt odmos fm q l msmo vlo d otc 5 W, l tsdd sd o los bobdos dl stto l moto co l oto s 9,7 % mo q l moto co l oto, mts q l fcto d otc co l oto s,5 % so q co l oto.

300 6. Coclsos E st tbo s h smldo l comotmto d moto sícoo d ms mts, obsvdo como s modfc ls cctístcs dl moto l modfc l tmño d los ms. Hmos vfcdo q l ototo cl otc d 5 W, l fcto d otc s m bo l tsó d lmtcó s fo los. l cmt l volm d los ms 66 % scto d s tmño l ototo cl, l cot dl stto s dc l msm otc 9 %. El método sto st tbo d s m útl dtm fómls fbls l cálclo d l cot dl stto, fcto d otc dmto d l máq, cdo s modfc l tmño d los ms, l logtd d tho o l úmo d ss. fcs [] T. Nt N. Thssh, Nw dsg mthod of mt mgts b sg ft lmt mthod, IEEE Ts.Mgtcs, vol. 6, o. 6,. 9-97, ovmb 98. [] G.H. Jg D.. L, Th ffct of mgt gomt o lctc moto vbto, IEEE Ts. Mgtcs, vol. 7, o. 6,. 5-5, ovmb 99. [] D. Pvl,.. Gg, J.. J. Wss, ft lmt tchq fo clcltg th mgt szs d dctcs of mt mgt mchs, IEEE Ts. Eg Covso, vol., o.,. 6-, mzo 988. [] Sog-Po Hog, H-Sm Cho, H-So L H- Cho, Effct of th mgtzto dcto mt mgt o moto chctstcs, IEEE Ts. Mgtcs, vol. 5, o.,. -, mo 999. [5] Mm. Dbos, H Pold J. F, Mgt shg fo mml mgt volm mchs, IEEE Ts. Mgtcs, vol. 8, o. 5, , stmb. [6] Yog L, Jb Zo, Yogg L, Otm dsg of mgt sh mt-mgt schoos motos, IEEE Ts. Mgtcs, vol. 9, o. 6,. 5-56, ovmb. [7] Jc F. Gs Mtchll Wg, Pmt Mgt Moto Tcholog, Ed. Mcl D, Nw Yo, [8] J. Cols Mtí, Cálclo dstl d máqs léctcs. Tomo I, Mcombo o Edtos. clo 98.

301 Dtmcó d l ctc d Dssó d Cbzs d ob MSIPs J.. Güms loso,.m. Iolgot Iodo Dtmto d Igí Eléctc E.U.I.T.I., Uvsdd dl Pís sco Plz d l Csll,, 8 lbo Esñ Tlf.: 966, 9667, f: 96 -ml: [email protected],[email protected] sm. E st tbo, s st método d omcó áctc l dtmcó d l ctc d dssó d cbzs d bob, lcdo moto sícoo d ms mts MSIP. El álss dl comotmto dl moto s lz tlzdo l método d lmtos ftos MEF mdt modlos ts dmsos D. El flo d cbzs d bob d s clcldo mdt modlos dos dmsos D, o s cso solv g úmo d modlos. E l tbo s dscb método clcl l ctc d dssó d cbzs d bob comdo l gí clcld modlo D comlto cl ls cbzs d bob, co l gí clcld modlo D s cbzs d bob, co l gí clcld modlo D. El método osto d tlzs dtm ccos d so dstl lcds l cálclo d l ctc d dssó d cbzs d bob. Los sltdos obtdos dc q los vlos d ls ctcs d dssó d ls cbzs d bob so lgo mos q los obtdos o cálclo lítco. Plbs llv. Ims mts, método d lmtos ftos, modldo, moto sícoo, ctc d cbzs d bob, smlcó.. Itodccó El g dsollo d ls lcos d lto mgtsmo h hcho osbl q los motos d ms mts MIP s ho dí lttv m tst. E los últmos ños s h dsolldo vs fmls d ms mts, co odds mgétcs físcs m dstcbls. Los tos d ms mts más tlzdos l ctldd l cso d máqs léctcs, so los ms d ts s d Nodmo Ho oo NdF. Los MIP d cot lt so smts los motos sícoos covcols los q l dvdo d cmo gtoo s sstt o ms mts. S l fom d l od d l tsó dcd s sodl s dom motos sícoos d ms mts MSIP, cdo l fom d l od s tzodl so coocdos como motos cc s scoblls o bshlss dc [], st tmologí s dbd l smz co ls cctístcs d los motos d cot cot. El tés d los MSIPs stá ccdo mlo bco d lccos. Ls zos cls d st cmto so m lg l fcc sgdo lg l dccó dl co d los ms NdF. E témos gls, los MSIPs so fbls los motos sícoos, tods qlls lccos q q vlocdd costt, q sv fcomto slcoso. El MEF s ocdmto d álss mtmátco q om los vlos d ls mgtds físcs, q d dscbs co ccos dfcls válds dtmd gó. Iclmt s lcó sob todo l álss mcáco, o dsd hc s décds s v tlzdo co éto l á d gí léctc, tclmt l stdo d máqs léctcs ottvs. El so dl MEF como hmt d cálclo smlcó dt l ocso d dsño d moto léctco, mt o t, cooc s comotmto, o ot, modllo o mdo d ccto qvlt. E gí léctc, omlmt ls máqs léctcs so stds o ccto qvlt. El coocmto d los dfts ámtos dl ccto qvlt d l máq, os mt cooc s comotmto, bo cts codcos d fcomto, o lccó d ls ls d los cctos léctcos. U ámto mott dl ccto qvlt d MSIP s l ctc síco. Como s coocdo, dch ctc s l sm d l ctc d ccó d dcdo, l ctc d dssó dl úclo l ctc d dssó d cbzs d bob

302 L ctc d ccó d dcdo l ctc d dssó dl úclo d clcls smldo l comotmto dl moto mdt modlos D vcío cg Cosddo q ls zos d cbzs d bob, l flo mgétco s clz clmt o l, md ccs dl moto q ls cbzs d bob stá fomds o t ct ot ccl cocétc co l, l ctc d cbzs d bob s odí clcl o mdo d modlos D. Pmo s lz l t ct d ls cbzs co modlo lo D ostomt los tmos ccls mdt modlos D smétcos. Est método d álss cst sml g úmo d modlos D smétcos oq cd tmo ccl l úmo d codctos cots s dft. El dsollo d los qts fomátcos d lmtos ftos l mo otc d los ctls oddos sols, mt solv sstms co mo úmo d ccos más ádmt q hc os ocos ños. Co l técc d modldo sóldo D, s d solv gomtís comls. bob, fcó d l gomtí d ls cbzs d bob.. Dsccó dl moto Ls cctístcs cls dl moto obto d stdo so: Moto tfásco Tsó d lmtcó dl cotoldo: Fcc d l tsó dl cotoldo: 5 Hz Fcc d l tsó dl moto:.5 Hz locdd oml fd.5 Hz: 5 m Númo d ms mts: ; 6 Númo d s dl stto: 7 Potc útl: 5 W To d ms; NdF. L fg mst cto d l gomtí dl moto. El obtvo fdmtl d st tbo s dtm ls ctcs d dssó d cbzs d bob, d moto sícoo d ms mts. L smlcó stdo dl comotmto dl moto s lz mdt l MEF tlzdo modlos sóldos D. L moí d los stdos lzdos tlzdo l MEF l smlcó álss dl comotmto d ls máqs léctcs gl, d los MSIPs tcl, so dos dmsos, sdo ltvmt ocos los lzdos ts dmsos. Et ls blccos d stos últmos ños odmos ct l tbo d Dmo [], q clcl ls dctcs d cbzs d bob d moto d ms mts tlzdo l método dl lmto ldo, mdt smlcos D D l tbo d Egstöm [], q m l fcto d l dssó máqs d ms mts co scl tcó l odccó dl, l cso d máqs s s, comdo los sltdos obtdos smlcos mdt modlos D D. El método osto, s bs com l gí obtd modlo comlto D cl ls cbzs d bob, co l gí clcld, o t, modlo s cbzs d bob D, o ot, modlo D. Los sltdos d l ctc d cbzs d bob clcldos t d ls gís tomt ctds, s com co l vlo dtmdo t d l gí modlo D, dod sólo s sml l comotmto d ls cbzs d bob, tmbé co l vlo obtdo o cálclo lítco. El método stdo st tbo, d s m útl dtm fómls css, d lccó dstl, l cálclo d l ctc d cbzs d. Modlos Fg.. Moto S tlz dos tos d modlos d lmtos ftos, D D.. Modlos D Dd ls smtís stts, los modlos stá costtdos o º d l mtd dl moto sco d oddo ls cbzs d bob. S tlz ts modlos: comlto, s cbzs d bob sólo cbzs d bob. E ls fgs,, s mst sctvmt los ts modlos ctdos.

303 El mlldo s h lzdo co lmtos ttdos d odos, stdo scl tcó ls zos cítcs tho zos óms. E l fg 5 s mst, modo d mlo, l mlldo tlzdo l modlo comlto. Fg.. Modlo comlto Fg. 5. Mll d lmtos ftos D Ls odds d los mtls s df como sg: Fg.. Modlo s cbzs d bob P l tho, comzo s dl oto to l cob bobdo dl stto o mdo d l mbldd mgétc. P l úclo mgétco dl stto oto o mdo d l cv H, l fcc d tbo, dl mtl co l q stá costds ls chs mgétcs. Los ms o mdo d s mbldd mgétc cmo coctvo ss dos comots H CX H CY. Ls ctcos s df o mdo d l cot cd o d los dvdos dl stto. L codcó d cotoo s flo tgcl l to dl moto.. Modlo D El modlo d lmtos ftos stá costtdo o sccó ct tsvsl dl moto sco d q lo od. Ls ctcos s df o mdo d ls dsdds d cot los dvdos dl stto. Fg.. Modlo sólo cbzs d bob Ls odds d los mtls s df gl q l modlo D.

304 L codcó d cotoo s vcto otcl mgétco lo l f dl modlo. L fg 6, mst l mlldo tlzdo. Fg. 7. Modlo comlto D Fg. 6. Mll d lmtos ftos D cto dl modlo. Mtodologí álss Los sos sgdos l stdo q s st st tbo, so los sgts: Sml l comotmto dl moto mdt los cto modlos dsctos tomt comlto D, s cbzs d bob D, s cbzs D sólo cbzs d bob D. Cálclo d l gí mgétc d cd o d los modlos. Dtm l gí d dssó d ls cbzs d bob o mdo d ls gís obtds los dsttos modlos. Clcl l ctc d dssó d cbzs d bob t d ls gís d dssó d ls cbzs obtds l so to. Dtm l ctc d dssó d cbzs d bob o método lítco. 5 Com los sltdos.. Método d lmtos ftos E todos los modlos, l álss dl comotmto dl moto s h lzdo l stcó cosodt l fcomto d l cg cosddo cot o l stto tsvsl. L fgs 7 8 mst sctvmt l dstbcó d l dsdd d flo mgétco obtd l smlcó dl modlo comlto D, l modlo s cbzs d bob D. Fg. 8. Modlo D L gí d cbzs d bob, tdo ct q los modlos D lzdos st /6 dl moto, s clcl co ls sgts ccos: Dod: W W cb cb 6W W cd D 6W W P cd D W cb 6W sd W cb gí d cbzs d bob clcld o l m ocdmto; W cb gí d cbzs d bob clcld o l sgdo ocdmto; W cb gí d cbzs d bob clcld o l tc ocdmto; W cd gí clcld l modlo comlto D; W D gí clcld l modlo s cbzs D; W D gí clcld l modlo D;

305 P Pofddd dl modlo logtd dl qt d chs. W sd gí clcld l modlo D d sólo cbzs d bob. L ctc d dssó o fs d ls cbzs d bob, s dtm o mdo d l sgt só: Dod: f I π f Wcb X cb I fcc; Cot o fs.. Método lítco Co l oósto d com los sltdos obtdos o l método osto, s h clcldo l ctc d dssó d ls cbzs d bob tlzdo cálclo lítco []. P llo, mmt s dtm l mc o dd d logtd d ls cbzs d bob t d ccó: Dod: λ cb 7 Lcb τm π f,6, 5 L L λ cb mc o dd d logtd; f úmo d s o olo fs L cb logtd md d cbz d bob; L logtd dl dcdo; γ so d bob s; τm so mdo d ; L ctc d cbzs d bob, d clcls o mdo d l só: Dod: Z X cb 5. sltdos πf Z λ L 6 úmo d s d olos; úmo d codctos o ; U vz lzdos los álss los dsttos modlos, s t los sltdos mostdos l tbl I TL I.- Egí mgétc d los modlos Modlo Egí J Comlto D W cd 85,798 S cbzs D W D 85,8 S cbzs D W D 5, Sólo cbzs d bob D W sd,776 L tbl II mst l gí d dssó d cbzs d bob clcld o cd o d los ocdmtos vmt dsctos,,. f cb TL II.- Egí d cbzs d bob Pocdmto Egí J Modlo comlto s cbzs d bob W cb,876 Modlo comlto D W cb 5,7 Modlo sólo cbzs W cb,566 Flmt, l tbl III mst l vlo d l ctc d dssó d cbzs, clcld o mdo d l gí l clcld o l método lítco 6. TL III.- ctc d dssó d cbzs d bob Pocdmto X cb Ω Modlo comlto s cbzs d bob,569 Modlo comlto D,87 Modlo sólo cbzs,66 Cálclo lítco 6,9 D los sltdos obtdos odmos dc q: El m ocdmto d cálclo, ofc vlo so % l clcldo o l método lítco. Ddo q o s do cotst los sltdos mtlmt o odmos fm cl d stos vlos s l mo omcó. Co l sgdo ocdmto s obt vlos m ltos, o lo q o c odo. L cs d s como coscc d com dos modlos co foms tmños d lmtos m dstts. El vlo obtdo co los sltdos d l smlcó dl modlo q solmt cog ls cbzs d bob, s m qño, q o t ct q t dl flo d dssó d ls cbzs s c tvés dl úclo. D lo q s ddc q st tc ocdmto, s d q sí l más scllo d lc t tmo d solcó mo, tmoco sí váldo l comt motts os. 6. Coclsos E st tbo s h smldo l comotmto d MSIP mdt modlo d lmtos ftos D. S h osto dsttos ocdmtos l cálclo d l ctc d dssó d cbzs d bob. El ocdmto más dcdo s l mo, obtdo l ctc d cbzs d bob t dl modlo comlto l modlo s cbzs d bob D. El cálclo d l ctc d dssó d cbzs d bob t dl modlo d sólo cbzs d bob, o s dcdo ddo q codc os motts. El método d s útl dtm fomls css q mt clcl l ctc d dssó d cbzs d bob fcó d l fom dl bobdo.

306 fcs [] J. Gs d M. Wg, Pmt mgt moto tcholog. Ed. Mcl D, Ic. Nw Yo,. []. Dmo, L. Now d W. Ptws, Clclto of d-t lg dctcs of lctcl mchs sg th dg lmt mthod. Th Ittol Jol fo Comtto d Mthmtcs Elctcl d Elctoc Egg, vol, o, -9,. [] J. Egstöm, Effcts of l Lg o toq fomc PM mchs, Pocdgs of th Ittol Cofc o Elctcl Mchs, ICEM, gg - lgm, [] J. Cols Mtí, Cálclo dstl d máqs léctcs, d. Mcombo o Edtos, clo 98. [5] Chg-Cho Hwg d Y. H. Cho, Effcts of lg fl o mgtc flds o to mt mgt schoos motos. IEEE Ts. Mgtcs, vol 7,. -,.

307 àls d Elmts Fts lct Moto Síco d Imts Pmts D. Sltv UNIESITT POLITÈCNIC DE CTLUNY DEPTMENT D ELECTÒNIC E-Ml: [email protected] bstct qst tbll std l dtmcó d ls cctístqs gm mt d moto síco mb mts mts sfcls, mtçt l mètod dls lmts fts FEM. Pmmt, s st l dstbcó dl cm mgètc dl moto síco d mts mts PMSM std, tbllt bt codcos omls. t dl càlcl d l g mgètc mmgtzmd s clcl ls dctàcs ls os dct qdt. Flmt, s st gàfcmt l ll lctomgètc l ll d coggg dl moto sct l gl d càg. Q s com ls sltts clclts sct ls obtgts mtlmt, s dmost q qst mètod d càlcl s fbl.. Itodccó El moto tfàsc d lt d mts mts, fcot com moto síco covcol, h tgt tès ct ls ds ds dècds. El ct dsvolmt d mts d lt g, h mlt sgfctvmt ls ss cms d lccó. L scó d mts mts d SmCo o NdF l oto dls motos sícos c l ctcó, h sost ovcos l dss àlss dls PMSM. E qst tbll, s dtm ltz ls àmts ls cctístqs d moto síco d mts mts sfcls. í docs, l tsc cl és dtm mb l màm csó ls cctístqs gm mt dl PMSM, ò s h d clcl ls ss àmts d l m més ct ossbl, mtçt l mètod dls lmts fts. El ogm tltzt ltz l àls d lmts fts FE és l FEMM. dl D. Dvd M. qst és softw d ll dstbcó s ot tob l àg wb: htt://fmm.fost-mll.t.. Moto síco d mts mts El PMSM stdt és moto comcl ts UMC d Cotol Tchqs, ls svs cctístqs s st ls Tls I II. El moto s lmt t d ts cots ssoïdls l oto s tob ss ols d mts mts d NdF mtts sfclmt. l Fg. s ot v vst lçd d qst moto stl lt bcd mtl.

308 Tl I: Escfccos costctvs dl PMSM Ièc,5 g m Logtd dl oto 9,5 mm s l stto 8 Codctos Mtl mgètc NdF mç mgètc, T Cocvtt mgètc 95 /m Tl II: Escfccos lèctqs dl PMSM Ts d coó Estll Númo d ols 6 Potc oml,8 W Costt d volttg,98 ms/m Costt d ll t,6 N m/ms Cot oml 7,65 ms loctt oml m Pll oml, N m Fg.. st lçd dl PMSM Cotol Tchqs UMC

309 Fg.. Sccó tsvsl dl PMSM Cotol Tchqs UMC. Càlcl dl PMSM mb l mètod d lmts fts El mètod d lmts fts s h tltzt tsvmt l càlcl mèc dl cm mgètc d màqs lèctqs gl. L ossbltt d clcl ls cctístqs lctomgètqs lctomcàqs t dls sltts obtgts, stfc l tltzcó d l àls d lmts fts. Molts vstgdos d dl mó tbll qst cm h blct molts s mb qst toc. Est molts qts d softw ltz l FE, lgs só gls lts st més otts màqs lèctqs. Els sltts stts qst tbll h stt obtgts tltzt l ogm FEMM. E l m s, cosdt com t d océs, s db l sccó tsvsl dl PMSM s df l mll d lmts fts mb dstt od. Nomlmt, l md dls tgls d l mll s d scoll tomàtcmt l ogm, c q lgs csos s tss slcco mlmt md més tt obt s sltts més cts, com l cs d l tfo d l màq. E l FE ltzt obt l dstbcó dl cm mgètc, l mll st fomd 8 ods 688 lmts, dstbïts tot l sccó tsvsl dl PMSM, com s ot v l Fg.. E l ocssdo, omt fmm dto, s tod tots ls dds qds ltz l àls: l sccó tsvsl ct dl cl mgètc d l stto dl oto, ls mtls dl PMSM mts mts, co,, c ls svs cctístqs cob d mgttzcó, mbltt mgètc, cocvtt mgètc, codctvtt lèctc, ls codcos d coto d l gó ltz l cot q ccl ls bobts d l màq. Com s h dbt l sccó tsvsl comlt dl moto, omés s lc l codcó d coto d m od d Dchlt l ccl més t d l stto. U vgd cbd l t d océs, l modl dl PMSM s tob t s ltzt. L solcó dl oblm s cosg ctt l solv dl FEMM, í

310 s obt ls vlos dl otcl vcto mgètc cd od. Dsés s od f sv qsts sltts dfts oòsts. Mtçt l ostocssdo dl FEMM, omt fmm vw, s ossbl ltz ls càlcls ls stcos gàfqs d ls vbls lctomgètqs lctomcàqs més motts.. Dstbcó dl cm mgètc U d ls mllos ms d td l fcomt d qlsvol moto s v l dstbcó dl s cm mgètc. L stcó gàfc dls sltts obtgts mtçt FEM, s ooco l dstbcó dl fl mgètc l sccó tsvsl dl moto síco d mts mts stdt. U t dls sltts més tssts s st cotcó. L dstbcó dl cm mgètc l PMSM s st l Fg., ls dos sgüts ègms: tbllt bt, és d, cot d md co, tt l cm mgètc obtgt és l q ové úcmt dls mts; b tbllt codcos omls, és d, ll oml T N, N m mb cot oml ls bobts d l stto I N 7,65. E bt b E codcos omls Fg.. stcó dl fl mgètc l sccó tsvsl dl PMSM El ostocssdo dl FEMM tmbé s mt db l dstbcó scl d l dstt d fl mgètc l llg d lí scolld. l Fg.. s st l sv dstbcó l llg d mcàcs 6 lèctcs d tfo.

311 E bt b E codcos omls Fg.. stcó dl fl mgètc l sccó tsvsl dl PMSM Mtçt qsts dgms s od ltz ofdtt ls otts dl cm mgètc l tfo d l màq, tt comt l sv tstt fom. l Fg. b s ot c l fct d l ccó d md, q l moto síco d mts mts tbll mb càg. ls fgs tmbé s ot c l fct d ls dts d l stto sob l dstbcó dl cm mgètc.. Càlcl d ls dctàcs És molt mott clcl l més cdmt ossbl ls vlos dls àmts dl PMSM. Els més sgfcts só ls dctàcs l dct L d l qdt L q, q d lls d ls ctàcs sícos X d X q. qsts só ls àmts més motts ltz l fomc ègm mt /o dàmc dl PMSM. El càlcl mèc d ls dctàcs s bs ls sltts obtgts mb FEM és f st ls os dct qdt. El cm o h d st ctt, ò sos q ls mts mts s h d sbstt lmts fts ssocts l sv coso mbltt ltv µ.8, ò ss cocvtt mgètc H c. Només ls bobts dl stto h d st ctts d m od, com s lc ls sgüts àgfs. E d: Q s clcl l dctàc l dct L d, ls cots sttòcs h d st dstbïts t l s màm l qdt, odt l màm d cm d md l dct. L dstbcó dl cm mgètc s st l Fg. 5.

312 Fg. 5. Dstbcó dl cm mgètc l càlcl d L d E q: L dctàc l qdt L q, s clcl d m sml L d. E qst cs, l cm d md s dslç 9 lèctcs, odt l s màm l qdt. ò sos q l màm dls cots sttòcs t l s màm l dct. L dstbcó dl cm mgètc qst cs s st l Fg. 6. Fg. 6. Dstbcó dl cm mgètc l càlcl d L q Ls dctàcs s clcl t d l g mgètc l tfo l cot sttòc, mtçt l sgüt foml: W L I Com q s ssm q ls bobts fctcs dls os dct qdt t l mt úmo d volts q l bobt l fs, màq tfàsc d lt,

313 l cot sttòc ls os dct qdt h d s / vgds més g q l cot sttòc fs, od l mt foç lctomot. L cosot g mgètc W l tfo, s obté cd cs t dls sltts d l àls d lmts fts dl PMSM, d cod mb l Fg. 5 l Fg. 6. Els sltts fls só: L L W,77, mh I 7,65 d W,9 5,5 mh I 7,65 q. Dtmcó dls àmts dl PMSM El modl dos os d ls màqs sícos és l`omcó més clàssc ltz vstg l s comotmt. Molts vstgdos tltz qst mètod com m omcó, dsés ltz àls més dtllt ofdtt d l màq. L d bàsc és dsvol tltz sè d qcos q dscg l comotmt dl moto ls os d q. L úc qmt és dsos dls ss àmts. L csó d ls cctístqs d fcomt dl PMSM obtgds tltzt qst mètod, dè d m dct d l cttd l càlcl dls àmts dl moto. lgs dls àmts dl moto ls dó l fbct l fll d cctístqs, ò molt sovt ls ss vlos só dscogts. P tt, omlmt s h d bsc l mllo ocó od clcl-los mtçt mètods mècs, lítcs o mtls. E qst tbll ls dctàcs cls dl PMSM, s h clclt lct ocdmts mècs. t dls sltts obtgts ls qcos, s od clcl ls ctàcs X d X q dl PMSM tbllt 5 Hz. Xd π f Ld π 5,,57 Ω Xq π f Lq π 5 5,5,68 Ω 5 El ft d q moto síco d mts mts sfcls, o h h g dfc t ls ctàcs ls os dct qdt, s dmost qst cs. Ls ctàcs cls tmbé s h obtgt mtçt ssg. Els sltts só ls sgüts:

314 Xd π f Ld π 5,5,66 Ω 6 Xq π f Lq π 5 5,9,85 Ω 7 Comt ls sltts clclts 5 ls mtls 6 7, odm obsv q l dscàc és mím. 5. Cctístqs dl PMSM gm mt E l scto d l g d màqs lèctqs, l obct d vstgdos costctos, st sm ctt stm, ltz vl l comotmt d l màq. P tl oòst, s csst dsos d s cctístqs d fcomt, l més cds ossbl. E qst tt, l PMSM s ltz dfts codcos d fcomt s st ls cctístqs d ll lctomgètc ll d coggg.. Pll lctomgètc El comt d l cctístc d ll stàtc és molt mott od ltz l àls l vlcó dl comotmt dls moto lèctcs. P clcl-l s od f sv dfts omcos. E to, l ll s clcl t d l solcó dl cm mgètc d dfts foms. Ts omcos s tltz hbtlmt: l mètod d fl-cot, l mètod dl tso d Mwll l mètod d tbll vtl. E qst tbll, s h tltzt l mètod dl tso d foç clcl l ll lctomgètc l voltt d l tfo, mtçt l ost-ocssdo dl FEMM. P obt l cctístc d ll comlt, s h t dslçt l fso scl d cot sttòc d 8 lèctcs, cmts d lèctcs, mtt l dct dl oto lt mb l d l fs d l stto. qst càlcl h stt ltzt l cot oml d l màq tt l cctístc q s most l Fg. 7, és l oml. Fg. 7. Cctístc oml d ll lctomgètc dl PMSM

315 t d l gàfc d l Fg. 7 sbt l ll oml T N, N m, s ot obt l gl d càg oml δ N 57 lc dl moto.. Pll d coggg Es ot obt l ll d coggg l lt l cot d md ft g l oto, tltzt l mt mètod d càlcl d l tt to. E qst cs, s h slccot l d l fs d l stto com d fèc, l oscó cl és q l d l stto l d dl oto s tob fs, tt l ll d coggg és co. P obt bo cctístc, s h t gt l oto cmts d, mcàcs, d mcàcs, és d, d lt. El ll màm d coggg obtgt és, N m omdmt l sv fom és fomtlmt ssoïdl, com s ot obsv l Fg. 8. Coclsos Fg. 8. Pll d coggg dl PMSM L àls d lmts fts és d ls mllos solcos vl l comotmt d màqs lèctqs gl. L omcó std, lcd moto síco d mts mts sfcls, dmost qst fmcó. E qst cs, ls lmts costcts fo dl cl mgètc, o só d g flc. P tt, ls cctístqs gm mt clclds t dl mètod dls lmts fts, dos dmsos, t sfct csó. Els sltts clclts mtls só l mllo m d cofm ls àmts ls cctístqs obtgds. L cocodç t lls, std qst tbll, ob q l mtodolog tltzd és coct fbl.

316 Mgtc fld dcto scg sstm fo mgtc fld soc sd dstl lctos H.lt,.Fst, M.Gcí Isttt d Tcolog Elèctc. vd. J d l Cv, Pc Tcològc Pt 698 Pt lc. S Pho: F: E-ml: [email protected] [email protected], [email protected] bstct. Ths sts th dscto of th dsg d solto gv s mgtc sc fo 5Hz dstl lcto, combg dfft mtls to obt th otmm dcto of th fld. Imott ds std o th mgtc fld bhvo, th sos of th dfft mtls sbctd to mgtc flds, th ffcts d vtos th sh d tst of th fld, todcd b th sc, d lso, th dffcs ths flcs odcd b th sc s fcto of th mtl ots, dmsos o ostos. g th mgtc fld wth ct go of th sc wthot dstbg th fld th oth gos s ot s ts. Tht s wh smlto d l msmts hv to b combd. Wth th dgtl modl, lg mb of smltos cd ot modfg th sc st b st to obt th otml fld dcto. Th fl msmts hv vldtd th movmts fomd b th sc. wods Mgtc shldg, low fqc lctomgtc flds, hm os, sc mts, dstl lcto. Fo occtol os: 5µT Fo gl blc os: µt Th lcto todcd ths dls wth th mgtc scg fomd dstl ocss tg to dc th mgtc fld mttd b o of th comots blow th stblshd lmts.. Poblm dscto Plcd dstl vomt, s t of th odctv ocss, dstl col s sd to dmgtz th wocs bg fbctd whch cold mt mgtc flds o ts sd s slt of th odctv ocss mlto d tsfomto. Ths flds stg th wocs ot sfl t ll, sltg v dmgg fo th coct fctog of th cs oc th stlld d hv to stt wog. Thfo, th hv to b lmtd.. Itodcto Dg th lst s, lctomgtc flds hv b mt ot of coflct d to th gowg wss of th hlth ss b th gl blc. lthogh lots of bomdcl stds hv b cd ot o ths dom, o of thm hs b bl to stblsh cl lto cofmg lctomgtc flds s cs of d of llss. Howv, t s cl th hs to b lmts od to cotol ossbl oss. Eo thots hv gltd ths scts fg lmts fo th dtd mssos odcd b dfft d of lctomgtc socs []. Ths lmts v s fcto of th fqc d, fo th cs of th dstl lctos 5Hz th : Fg - Col wth woc d mgtc fl ls.

317 Dmgtzto s th dcto of mgtc fld fom ts mmm mgtzto tst to lmost zo, chvd b td olt vsl t gv fqc. Fo dmgtzto, th mltd of ld lttg fld mst b cotosl dcd s show fg. Th tl dmgtzto fld stgth mst b t lst ql to th mgtzto fld stgth stg th sml. Th dcto of fld stgth wth th woc c b chvd lctcll b dcg th mgtc fld ogssvl whl t cclts thogh th col, o mchcll b slowl wthdwg th woc fom th fld of costtl gzd dmgtzto col. Th fg blow c gv d of th ocd. Fg - M of msmts od th col. Th vls gstd vls of mgtc fld dst µt d osto mts bg th, fc ot th ct of th col: Fg - Dmgtzto ocss. Th dmsos of th col whch hs b shldd : Lgth mm. Itl dmt 5mm. Etl dmt 5mm. Th col s coctd to th low voltg lctc two wth sl stblshg vl fo th fld stgth o ts ct od /m. Th lvtd fld stgth mls th stc of hgh fld lvls ot ol th ct of th col bt ll th sodgs. Ths lvls to b chcd d dcd thos zos cosdd css. Ths wll b do thogh th stllto of mgtc sc whch hs s m qmt, t fom shldg th mgtc fl th otsd, ot to lt th fld stgth th ct of th col sc t cold dmg th fctog of th col, thfo vtg th gol t hs th dstl ocss, whch s, th movl of mt mgtc fld sd th wocs. Tbl - ls of mgtc fld dst wthot sc. Ths tl msmts md t ossbl to chctz th fld ctd b th col. Oc ths fld ws ow, dgtl modl ws stblshd. Th mgtc fld ctd b ths modl hs th sct dctd fg d ws dvlod to cosod ctl wth tht ctd b th l col. Ths ws obtd b th comso of mgtc fld vls th ots cotolld st d th fld smltd b th modl thos ots.. Pocd. Itl msmts. Ths msmts llow dtmg th tl stto d oml codtos, st. Fo th gstto of th mgtc fld vls totl mb of ots w cotolld. Th stto of th mgtc fld ths ots cold gv d of th sh d sct of th fld. Th schm of msmts s dsld fg. ll th msmts w sbmttd t th col s hght, sc th w cldd l tht th sm l, d th o wh th vls of mgtc fld mmm fd to th ct of th col. Fg - D vw of th dgtl modl wth th mgtc fld fl ls gtd b th col. Th vls of mgtc fld sfl ot ol to dvlo th modl bt lso to hv d of th tl stto. Wth ths owldg t s s to oct th t of sc whch s gog to b css od to f th mgtc fld th sodgs d th lmts.

318 . Dsg of th sc. I vw of th mgtc bhvo of th dfft mtls []-[] o d lmm w slctd s th otml mtls to mlo th costcto of th sc. oth mott dcso to t ws th thcss of th lts to stll [5], fom [],[6], mm w slctd fo both lmm d o, tg wth ths sz to otmz th fld dcto d th stct wght. t fom tht, th sh fo th sc s wll s th osto hd to b chos, oth stds [],[7] hld to dcd t shold b lcd s clos to th col s ossbl, tg to clos t sd th sc. Th w sc lmttos d to th dstl vomt wh th dmgtzg col ws lc, thogh th wdth of th sc cold ot b lg. Th fst oto ws to todc o lts, o o ch sd of th col, to lz th bsoto lvl of mgtc fld th w bl to fom. Th lcto of o loctd o th sds of th col wh th fld s lll to th sfc of th lt s bcs of ts good bhvo tht osto []. Th sct of th fld wth ths fst sc ws: It ws dcdd, std of cmtg th ltl volm of th sc, d d to oblms of sc, to stll oth ts of th sc th dcto. So, lmm ws sd to clos th sc od th col lcg wgs of ths mtl, wth gl of 6º, o ch of th to o lts. Th sltg stct ws s follows: Fg 7 - Scto of th fl dsg of th sc sodg th col. os gls fo th wgs w smltd. Eqll, fg thm to th to o lt ws td too. Fom ll th ossblts, th vos dstbto ws fod to b th bst. Ths solto dotd s dftv cts dstoto of th mgtc fld s s vslzd blow. Fg 5 Fl ls dstbtos wth th fst sc. slts fom ths dstbto of th mgtc fld w ot sccssfl, v csg th wdth of th o fom th tl.5m to l m th dcto ws ot ogh. Th scod oto ws th th todcto of scod o lt o ch ltl. Dfft smltos w fomd, vg th wdth of th lts s wll s th dstc btw thm. Th otml dstc ws cocldd to b 5cm. Wth ths w dsosto, gt dcto ws obtd o th dcto bt t ws stll oo th dcto. Th sct of th fld dstbto c b obsvd fg 6. Fg 8 - Mgtc fl ls smltd wth th col shldd. Oc smltd d th dsgd vls obtd d th lmts d wth ct sct mg, th sc ws costctd d stlld. C. Fl msmts. ft th dsg, costcto d stllto of th sc, w msmts w fomd od to chc th ffcc of th shldg d vf th clcltos d smltos cd ot dg th dsg. Th slts smmzd tbl. Oc g th mgtc fld dst s ssd µt d th osto mts hvg s th, fc th ct of th col. Fg 6 - Fl ls dstbtos wth scod sc.

319 Th mgtc fld dcto chvd b ms of th stllto of th sc gos bod th 88% th most ctcl ots fd to th tl stto. Th combto of two ds of mtls s wll s th good slcto of th ltv osto hs b fdmtl od to obt sch lg dcto. fcs Tbl - ls of mgtc fld dst wth sc. comg tbls d, t s cll obsvd tht th vls of mgtc fld hv gstd bg dcto ll th ot, ct ot mb 9. Ths s d to th fct tht th sc dcs th fld bsobg mgtc fl ls d cofg ths g to th mtl, bt lso dflcts th bsobd fl ls. Ths homo ms t ossbl to cs th mgtc fld som gos of th sc d to th coctto of mgtc ls of hgh fld tst. Ths s th cs of ot 9 s hs b s dg th dsg. It s loctd clos of th of th col w mgtc fld ls coctd. t fom tht, th gol of th dcto of mgtc fl dst d th lmts hs b ccomlshd fo ll th ots. No of thm sts bov µt whch ws th qmt. Th ctg of dcto vs fom th 88% obtd ot to th % of ot. Th smll th fld ws t th bgg, th smll th dcto obtd. w, fo th st of th ots th dctos ll mott bt dfft, d to th dfomto gstd b th sc.. Coclsos Th whol dsg of mgtc sc hs b fomd thoghot msmts d smltos. [] Dctv //CE dl lmto oo dl coso d 9 d bl d sob ls dsoscos míms d sgdd d sld ltvs l oscó d los tbdos los sgos dvdos d los gts físcos cmos lctomgétcos [] Estdo dsño d sstms d tllmto ctos d tsfomcó ft los cmos lctomgétcos. Itl docmt blshd b UNION FENOS collboto wth th Isttto d Tcologí Eléctc. J 5. [] L.Hsslg, J.Lom, Gomtcl scts of Mgtc Shldg t Etml Low Fqcs, IEEE tscto o lctomgtc comtblt, ol.7, N., gst 995. [] O.ottsco, M.Chm, D. Chbglo, F. Follo, L. occho, d M. Zcc, ol of mgtc mtls ow fqc shldg:mcl lss d mts,. Poc.Ist.Elct.Eg., vol.8,. -, M.. [5] U.do, O.ottsco, M.Zcc, Mtl Effcc Mgtc Shldg t low d tmdt fqc, IEEE tscto o mgtcs. ol.9. N.5, Stmb. [6] H.lt,.Fst,.Qo, "Otml Shldg Thcss of Low Fqc Mgtc Flds" XII Ittol Smosm o Elctomgtc Flds, mchtocs, lctc d lctoc gg. o S 5. [7] E.Sls, Codctv d fomgtc scg of 5Hz mgtc flds fom th hs sstm bsbs Jol of Mgtsm d Mgtc Mtls

320 álss dl cmo mgétco lí léctc d T. Moo, J.. Güms I. Pco Dtmto d Igí Eléctc E.U.I.T.I., Uvsdd dl Pís sco Plz d l Csll,, 8 lbo Esñ Tlf.: 966, f: 96 -ml: [email protected] sm. El cmto los últmos ños d l occó o los osbls fctos q d t los cmos mgétcos, odcdos o ls lís léctcs d lt tsó los ss vvos, h ogdo mlttd d stdos foms sclc st osbl ft odcto d fmdds. Cmos q d slt d tldd, ftos dsños d lís d tsot d gí léctc, ml hmt d cálclo como s l método d lmtos ftos, l cálclo d l dstbcó mgtd dl cmo mgétco gdo o ls lís léctcs. D st fom, s d cooc co tlcó s l lí odc cmo mgétco so l stmdo o l comdcó dl Coso Eoo 999/59/CE, q stblc l límt d tolbldd l cmo mgétco µt, cso cso tom ls mdds coctos oots. E st tbo tlzdo l método d lmtos ftos, s clcl l dsdd d flo mgétco gdo o lí d d dobl ccto s com los sltdos co los mddos mtlmt l lí. Iglmt s sml l comotmto d dfts cofgcos gométcs d lís léctcs co l f d dtm cls odc mo cotmcó mgétc. Plbs llv. Cmo mgétco, cotmcó mgétc, método d lmtos ftos, lís léctcs,. Itodccó lgos mtos d lbotoo h dtctdo q, bo dtmds codcos, lgos modlos mls, los cmos mgétcos d t fctos bológcos. S mbgo, ls vstgcos " vto", ls lzds mls sos h dmostdo q dchos cmos s ocvos l sld. Tmoco h mtdo stblc l mcsmo mdt l cl, los cmos mgétcos odí ct sob los ss vvos, lo q s más mott, t d qé doss odí hbls d sgo ls sos. El Msto d Sdd Cosmo d Esñ Jlo d stblc: "No d fms q l oscó cmos lctomgétcos dto d los límts stblcdos l comdcó dl Coso Eoo 999/59/CE odzc fctos dvsos l sld hm. Po tto, l comté cocl q l cmlmto d l ctd comdcó s sfct gtz l otccó d l oblcó" L comdcó d l Uó Eo, d d Jlo d 999, f l límt d tolbldd µt, l cl h tdo ct l co d ccó. Est comdcó s tfcd o mlttd d ogsmos, sttcos, foms ctífcos, médcos, tc. Et llos dstcmos: - L soccó Itcol l Potccó dológc IP. - El Isttto Ncol d Nomtv d Estdos Udos NSI. - L Comsó Itcol l Potccó cot l dcó o Iozt ICNIP. - El Coso Ncol d Potccó dológc dl o Udo NP. - El Coso Ncol d Potccó dológc Mdds d Estdos Udos NCP. - El Comté Eoo d Nomlzcó Elctotécc CENELEC. E los últmos ños, s h blcdo dsttos tbos los q s std l cmo mgétco odcdo o lís d tsot d gí léctc [] [5] l fom d dclo [6] [7]. El m cl obtvo dl tbo, s sml l comotmto mgétco d lí d lt tsó, tlzdo l método d lmtos ftos mdt modlos dos dmsos, com los sltdos obtdos co ls mdds mtls lzds l lí. El sgdo obtvo s sml lz l dstbcó d l dsdd d flo mgétco fcó d l cofgcó gométc d los codctos d l lí, l scc d fs l dsfs d cots t cctos, co l f d dtm q cofgcos odc mo cotmcó mgétc.

321 . Lí d lt tsó El método stdo st tbo, s h lcdo l dtmcó mdd d l dstbcó d l dsdd d flo mgétco d lí léctc d lt tsó d ª ctgoí, d dobl ccto co codcto o fs d lmo co L-8. L cofgcó d l lí s to bl, sdo l lt dl codcto más bo l cto dl vo d m. E l fg, d vs l cofgcó d l lí. 6 Com los vlos obtdos ls mdcos d cmo co los mddos l smlcó dl comotmto d l lí tlzdo l método d lmtos ftos. 7 Sml l comotmto d dstts cofgcos d lís co l f d dtm q cofgcos odc mo cotmcó mgétc.. Modlo El modlo d lmtos ftos tlzdo s lo D st costtdo o sccó ct d los 6 codctos l sco d q los od. El sco d tomdo l zqd, b dch d los codctos s d 5 m db s sfctmt mlo dmt q l cotoo dl modlo, l dsdd d flo mgétco s dscbl. E l fg, s mst l modlo tlzdo.. Mtodologí Fg.. Lí léctc d lt tsó. Los sos sgdos l comccó vld l método osto so los sgts: Md l dsdd dl flo mgétco ct std dcl l lí d lt tsó, l cto dl vo mto dl lo dl to. Ls mdds s lz mbos ldos dl cto d l lí co mímo d tos d mdd hst dstc d 5 m dl d l lí. Cooc l hstóco d cots otcs l tmo d ls mdds. Utlzdo l método d lmtos ftos, sml l comotmto d l lí l cot, l momto d lz ls mdds, fcltd o l comñí smstdo l hstóco d otcs. Md l vlo d l dsdd d flo mgétco l modlo d lmtos ftos los msmos tos dcdos l to. 5 fc q l mgtd dstbcó dl cmo mgétco obtdo d l smlcó stá cods co ls ls físcs fdmtls l cálclo dl cmo mgétco. Fg.. Modlo Los lmtos tlzdos so los d odos. E l fg, s mst l mlldo tlzdo l zo d los codctos. Ls ctcos s df o mdo d l dsdd d cot cd o d los codctos. L dsdd d cot s clcld t dl hstóco d otcs, l odo l q s lzo ls mdds d cmo. Ls odds s df o mdo d l mbldd mgétc dl, codctos léctcos l t. L codcó d cotoo s vcto otcl mgétco lo l f dl modlo. El to d álss lzdo s mgtostátco.

322 Fg.. Dsdd d flo mgétco zo óm los codctos Fg 5. cto otcl mgétco b zoom d codcto Fg.. Mlldo L smlcó dl comotmto d l lí, s h lzdo stts d tmo d ;,5; 5; 7,5; ;,5, 5; 7,5 ms cd 5º dt odo d l fom d od cosodt l cot o l fs. E l fg, s mst l fom d od d l dsdd d flo mgétco l cto d l lí cto dl vo, mto d lt d l sfc dl to, obtd d l smlcó dl comotmto d l lí tlzdo l método d lmtos ftos 5 o ccto. Podmos obsv q, l cot d so, l fom d od d l dsdd d flo mgétco st dsfsd,5º scto d l fom d od d l cot o l fs. E ls fgs 5 6 s mst sctvmt, l dstbcó dl vcto otcl mgétco dsdd d flo mgétco, l zo d los codctos, l stt l q l cot s mám l fs. Fg. 6. Dsdd d flo mgétco 5. Comcó d sltdos U vz lzd l mdd d cmo l smlcó dl comotmto d l lí ls cots clclds 5 o ccto, s t los vlos d dsdd d flo mgétco mostdos l fg 7. Podmos obsv como los sltdos obtdos o smlcó s sm los mddos mtlmt. Úcmt los tos stdos bo l léctc lí, s obt o mdcó vlo

323 d l dsdd d flo mgétco lgmt so l obtdo o smlcó. 6. Smlcó dl cmo mgétco co dstts cofgcos /o sccs d fs. E vst d los sltdos obtdos tomt, st tdo smlmos l comotmto mgétco d: dstts cofgcos d lís, b dstts scc o dsfss d ls cots t cctos. E mbos csos l stdo s lz l vlo mámo d l cot o l lí o ccto.. Dstts cofgcos E l fg 8, s mst ls dstts cofgcos d lís stdds. Fg. 7. Dsdd d flo mgétco L dfc stt t los vlos mddos ls áctcs d cmo co los obtdos o smlcó, d s dbdo clmt q l cot l, l odo d mdcó, f lgmt fo l clcld o l hstogm d otcs fcltdo o l comñí smstdo tmbé l csó dl stmto d mdd. E l cv d mdds d l ctcs d cmo Fg 7, s obsv q los vlos d dsdd d flo mgétco l ldo dcho so lgo sos l msm dstc l ldo zqdo, lo q dmst q l cot l o l ccto, l momto d lz ls mdds, so l cot o l ccto. Fg. 8. Dstts cofgcos U vz lzd l smlcó dl comotmto d ls dstts cofgcos, s t l sltdo mostdo l fg 9. El cmo mgétco cd o d los s s h mddo co gímto co go d mdd d, µt µt mt, co csó d ± % cho d bd d Hz. Ls mdds d dsdd d flo mgétco, s lzo tos sdos dos mtos t s stdos ct loclzd dcl l lí d lt tsó, l cto dl vo mto dl lo dl to. E cd o d los tos s lz ts mdds,, z, clcldo l vlo sltt d l dsdd d flo mgétco s to, o mdo d l ccó: z lzdo vmt l smlcó dl comotmto mgétco d l lí, l vlo mámo d cot, o ccto, s t vlo mámo d l dsdd d flo mgétco d µt mdo l cto dl vo mto d lt sob l lo dl to. El vlo obtdo s m fo los µt stblcdos o l comdcó d l Uó Eo 999/59/CE. Fg. 9 Dsdd d flo mgétco dstts cofgcos d los codctos d l lí S d obsv q l vlo d l dsdd d flo mgétco dsm omdmt l mtd, tlzdo ls cofgcos táglo hozotl, mts q, como sdo, l cofgcó dobl c vtcl d sltdos m smls l cofgcó bl. E l fg, s mst como ví l dstbcó d l dsdd d flo mgétco fcó d l cofgcó gométc d los codctos d l lí, l stt l q l cot s mám l fs.

324 dl cmo s más csd 7 %. E cmbo s úcmt ls cots d ccto v dsfsds º scto d ls dl oto cofgcó, o s obt dccó dl cmo mgétco cbl. l tcl Hozotl Táglo Fg.. Dstbcó d l dsdd d flo mgétco. Modfccó d l scc d fss o dsfs d cots t cctos. Flmt s h stddo como fct l dstbcó d l dsdd d flo mgétco, los sgts fctos: Cmbo dl od d scc d fss dl ccto cofgcó vés fg. Cmbo dl od d scc d fss dlto d º d ls cots dl ccto co scto ls cots dl ccto cofgcó - vés fg b dlto d º d ls cots dl ccto scto dl ccto cofgcó - vés fg c. Fg.. Dsdd d flo mgétco dstts sccs dl od d scsó d fss /o dsfss d cots L fg, mst l dstbcó d l dsdd d flo mgétco cd d ls cofgcos tomt dcds l stt q l cot s mám l fs. Ogl Cofgcó Cofgcó Cofgcó Fg.. Dstbcó d l dsdd d flo mgétco Fg.. Dstts sccs /o dsfss d cots t cctos U vz lzdo l álss s t los sltdos mostdos l fg. S d c como l cmb l od d scc d fss o d los cctos cofgcó, s dc l cmo mgétco d fom mott 66 %. S dmás d sto, ls cots t mbos cctos stá dsfsds cofgcó, l dccó D los sltdos tos, odmos dc q co dcdo lto dl od d scc d fss dl dsfs d cots t los dos cctos s d dc los vls d dsdd d flo mgétco d fom mott. E st tbo s h stddo dstts cofgcos d lís s h dcdo cl s l más cosbl scto d l mím cotmcó mgétc. No obstt l dsño d lí, fl otos fctos como l ccdd t codctos o t stos t, l tmño d ls tos, tc.

325 El método stdo st tbo s d g d los gos léctcos q s ddc l cálclo d lís léctcs, s mt cl ss octos dmás d los dtos hbtls, l cv d dsdd d flo mgétco sd. Ot lccó mott q d t l método s l cálclo d l ccdd t codctos t d l lí ost. 7. Coclsos Sgú los sltdos d ls mdds lzds st, s costt q l cmo mgétco odcdo o st lí d d dobl ccto, stá o dbo dl límt d tolbldd stblcdo o l comdcó d l Uó Eo, 999/59/CE. L smlcó tvés d método d lmtos ftos, d sltdos ctbls slt hmt m útl l lzcó d stdos d cmo mgétco. El método stdo st tbo mt lz comcos t los ms d dsdd d flo mgétco, gdos o dsttos tos d cofgcos gométcs d los codctos d l lí, co o s cmbo d l sccs d fs /o dsfs d ls cots t cctos. S h costtdo q l cofgcó tglo hozotl ofc mo sltdo q l cofgcó bl vtcl. Co coct dstbcó d ls fss s cosg dc d fom mott l vlo dl cmo. No obstt l dsño d lí h d cosds otos fctos tls como ls ccdds t codctos, ls édds o fcto coo, l costo d l lí, tc. El método sto d tlzs dtm fomls d lccó dstl, clcl l vlo dl cmo mgétco odcdo dstts cofgcos d lís fcó d l cot l scc d fss. fcs [] W.T. d L.E. Zffll, lss of mgtc flds odcd f fom lctc ow ls, IEEE Ts. Pow Dlv, vol. 7, o.,. 8-9, 99. [].G. Ols, Elctomgtc flds fom ow ls, IEEE Ittol Smosm o Elctomgtc Comtblt,. 8-, 99. [].. Mshv d.d. ssll, Msmt of Mgtc flds th dct omt of ow l codctos, IEEE Ts. Pow Dlv, vol., o.,. -6, 995 [] C. Gdo,.F. Oto d J. Cdás, Low-fqc mgtc flds fom lctcl lcs d ow ls, IEEE Ts. Pow Dlv, vol. 8, o.,. -9,. [5] P. Sm Mvd,. Tgo, D.L. Golt d C. Cdl, mtl of sdtl mgtc flds th vct of tsmsso ls, vol., o.,. 8-, 998 [6].. Mm d W. Jschws, Mtgto of mgtc fld ow ls, IEEE Ts. Pow Dlv, vol., o., , 996 [7].S. shs d. lod, Mgtc fld dcto mthods: ffcc d cost, vol., o., , 998 [8] IEEE Stdd ocds fo msmt of ow fqc lctc d mgtc flds fom C Pow l, IEEE Std [9] os, Mgtc flds fom lctc ow ls tho d comso to msmts, IEEE Ts. Pow Dlv, vol., o.,. 7-6, 988. []P.P. Slvst,.L. F, Elmtos Ftos Igí Eléctc. []comdcó dl coso Eoo sob los límts d oscó dl úblco cmos lctomgétcos Hz GHz. 999/59/CE []Cmos léctcos mgétcos d 5hz. álss dl stdo ctl d coocmtos EE UNES,.