J. Electrical Systems xx-xx (xxxx): x-xx. Regular paper
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1 .ZZOUI K. RIRI.E.H Bboud J. Elccl ym xx-xx (xxxx): x-xx Rgul pp F Volum gc odlg fo Jl-ho cl Hy odl Opmg by Gc lgohm Jl ho modl pg h mgc duco dpd vbl popod od o b dcly ud f-volum clculo ppld o h mgc vco pol fomulo. Th pm of h Jl ho hy modl dfd by ug l codd gc lgohm. Th pm dfco pfomd by mmg h m qud o b xpml d muld mgc fld cuv. Th mhod vfd by pplyg o x-ymmcl fomgc ym. Th pocdu vldd by compg xpml d muld ul. Kyod: gc hy, Jl-ho modl, F volum mhod, Gc lgohm, m dfco. Ioduco Numcl lcomgc h hoy d pcc of olvg lcomgc fld poblm o dgl compu. I flc h gl d cc d gg o fomul h l of u compu lgohm d o mul phycl poc o dgl compu. Whl hoy d xpm m h o dol pll of cc d gg, umcl modlg d mulo p hd pll h uppo, complm, d omm plc hm. Fo h o, h objcv of h ok hfo o udy fo modlg, mgc hy, o g o compu cod fld. Dff hy modl vlbl fo copoo o h f volum fmok. lhough h ch modl pp o b h hy modl of choc, h J modl cv bcu of mplcy d of mplmo. Th bu of h J hy modl cludg lvl of ccucy fo vl pccl ml, of mplmo o h FV (F Volum hod), d compuol ffcc mk vbl choc fo mplmo o -D f volum modl [], []. Th udy ll choo h modl b ud fom h dpo ccucy, pocg pd d of mplmo. Th okg hypoh cd o h c of c gm d h quo h olv x-ymmcl o dmo -D, h mgodymc o-l. Thu, h f lm mhod h povd lf ffcv ool olvg dffl quo, llo oh o k o ccou complx gom d o- ly pobl, oly mplmo g fly complcd. o choo ou udy fo h mhod of f volum, hch l dffcul o chv d mpl dg.. FINITE VOLUE FORULTION INCLUDING GNETIC HYTEREI. Bc Fld x-ymmcl Equo Th dvo of h f volum quo bg h xll fld quo. B () Copodg uho :.ZZOUI Dpm of Elccl Egg, Uvy of Djlf, Djlf 7, lg. ou_ddk@yhoo.f Copygh JE 9 o-l : joul.goup.og/j
2 Wh H J () B : gc flux dy, H : gc fld y, J : Tol cu dy. J h h ol cu dy hch h um of h coducv cu dy J d h dplcm cu dy J. d Th xll fld quo xdd o llo m of hy by cludg h couv quo fo mgc ml. Th gl quo fo fomgc ml c b xpd μ H (3) B h h ol mgo fuco, d μ h pmbly f pc []. Th flux dy B c b xpd h cculo of pol v vco, B, h h mgc vco pol ully fyg xll quo B c h dvgc of h cul o. Th mo obvou y o dcly u μ (o μ by pplyg h vco fom of (3) mp l. Th fld quo h gv by ) d h fld quo ( ) J ( ) ( ) (4), J ( ) d ( ), pcvly, h mgc vco, h m h ( ) pol, h cu dy, d h mgo vco m ( ) p. Wh codg J d glgbl h muh dco (x-ymmcl Fomulo), (4) ug h o-dmol (-D) mgc vco pol uko cyldcl cood c
3 σ J ϕ x (5). Jl-ho odl of Hy Th ogl J modl pd [3] gv h mgo vu h xl mgc fld H. Th modl bd o h mgc ml po hou hy lo. Th h hyc bhvo hch (H) cuv c b dcbd h modfd Lgv quo H ( H ) coh (6) H h HHα h ffcv fld xpcd by h dom: H h xl ppld fld d α h m fld pm pg -dom couplg. Th co cg fuco of h mpu. Th mgo dcompod o vbl compo v d vbl compo. v (7) Th lohp b h o compo d h hyc mgo obd fom phycl codo of h mgo poc d gv by Wh v d dh c ) (8) ( (9) k h, α, c, k d h uo mgo pm hch dmd fom mud hy chcc [4], dcol pm d k h vlu fo dh/d > d - fo dh/d <. Ug h mhod, h mgo commoly obd fom h mgc fld H. Wh h popod v Jl ho modl, ll b clculd fom h mgc duco B, gg dffl quo m of d/. To ob uch lohp, ll ubug (8) (7) d dffg h ulg m h pc o h ffcv flux dy BµH. c( ) ( c) c () d d ) d c ( c () Ug h ffcv flux dy xpo gv O c h m B B μ α μ 3
4 d d μ( α d () ) Dvg (6) h pc o H d coh μ d, h dvg (9) ul d μ k H H (3) (4) I hch fo /d > d - fo /d <. Th m (4) obd pplyg (8) (7) c c Flly, () ug () d (3), d olg d/ gv h m quo of h popod v Jl-ho modl d d ( c) d c (5) (6) d d μ( c)( α) μc( α) 3. Gc lgohm 3. Ioduco Gc lgohm dvlopd fo h pupo of opmo. Thy llo h ch fo globl xm. Th lgohm bd o ul lco mchm (D) d h gc voluo. gc lgohm chgg populo of g ug h mchm. I u co fuco bd o pfomc co o clcul (f). Tho mo "og" ll b bl o poduc d hv mo offpg h oh. Ech chomoom co of of lm clld chcc o g. Th gol o fd h opmum combo of h lm hch gv "f" mxmum. ch o (go of populo), populo cd fom h pvou populo. Th my y o pocd. Ech u cully lv dm b o dp h poblm. F, ll dvdul vlud, clculd o h b of h ubly (f) d hy kd fom b o o. o dvdul h op of h l, h mo lkly o cu, h ph of poduco do vl g of muo d cobdg. Th f o domly chg o o mo g fom o chomoom (chcc of dvdul), hl h cog do by xchgg vl g b o p. 4
5 Oglly, h codg of dvdul cbg by pm o opm o fom g. Th g h pu ogh o fom h chomoom. Th, hov, ppoch clld l codg, h h fuco of chg d pg o pply dcly o h vco of pm hou gog hough h by. W hv dfd codg l, mo flxbl d pc. Th vod om poblm du o by codg. Th cul codg lo povd dc v of g houghou h voluo of h populo. Th modfd gc opo ud h pp ll h mpovm ool pd [7]. 3. m Idfco ocdu Th chmc po of h pm dfco pocdu ho [8], [9], h f p h chco of h dvdul h ll fom h populo. Th dvdul θ compod by h fv pm of h J modl ( l codg, o cy o cod h vbl by po) [6], []. W cod h c h h populo gv by pop,,,,, k c α (7), p, p, p, p, p k c α h ch l p dvdul ( po h opmo pc), h go, d p h populo. Th l vlu gd o h populo dom vlu h llobl g, ho Tbl I. Ech dvdul of h populo vlud ug h f b clculd d xpml ul. Th mm h f fuco gv by [] ff ( θ ) ( ) (, θ ) (8) ( ) Wh d (, θ ) p h mud d md mgo, pcvly. Th opml pm vco obd olvg lo o mxmum llod umb of go. G θ m( ff ( θ )) Fgu ho h vo of h fuco of dpo (f) ccodg o h umb go, d bl gv h fl ul of h gc lgohm. TBLE : l m m Dg Vbl Rg Opmd Vlu k c α d 5
6 .5 x F vlu Go Fg.. Evoluo of h ol o. 3.3 Th oluo ocdu Th f volum mhod dco mhod hch ll ud fo h umcl mulo of vou yp (llpc, pbolc o hypbolc, fo c) of covo l; h b xvly ud vl gg fld, uch flud mchc, h d m f o polum gg []. om of h mpo fu of h f volum mhod ml o ho of h f lm mhod, my b ud o by gom, ug ucud o uucud mh, d ld o obu chm. ddol fu h locl covvly of h umcl flux, hch h umcl flux, covd fom o dco cll o ghbo. Th l fu mk h f volum mhod qu cv h modlg poblm fo hch h flux of mpoc, uch flud mchc, m-coduco dvc mulo, h d m f Th f volum mhod loclly covv bcu bd o "blc" ppoch: locl blc o ch dco cll hch of clld "cool volum"; by h dvgc fomul, gl fomulo of h flux ov h boudy of h cool volum h obd. Th flux o h boudy dcd h pc o h dc uko. Th oluo of h pod poblm coucd ug f volum mhod. Wh h pupo l u mulply mgodymc quo (5) by fuco of pojco β d g obd quo ov dom Ω, g β ddd μ μ σ β J x ddd β dd d (9) 6
7 Wh Β fuco of lcd pojco /. O c (9) x d dd J d dd μ μ σ () f go, h xpo c b follo ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] J N W E σ σ () Th quo () dcd oc follo [( ) ( ) ( ) ( ) [ ] d p p p N N W W E E ] σ () Th dc, W, N, E d f o h vlu of h od d dc p,,, d f o h vlu of h fc of volum of cool (Fgu ). Th coffc W, N, E, d d gv by ( ) ( ) ( ) ( ). ) (,, J d x N W E N W E σ μ μ μ μ (3) Oc h h vou fomulo f volum gg h modl of hy blhd, mhod of oluo of ol poblm mu b cho. Th gy copod h fo coculy olvg h ol J ml quo (6), log h h plly dpd fld quo (5), h fxd-po mhod. Th fxd-po mhod h pfd mhod fo mplmg h complx mhmcl ml modl h f volum mhod (FV) [] c bl mhod d lvly y o mplm. 7
8 N N E E jd W W d Fg.. Cool volum x-ymmcl cyldcl cood.. Ilo : (m), (pc), gv ε. Numcl cod Volum cool.. Ilo : Tbl : Iv p lgohm.. σ J ( ) B H H B μ ω. H μ ( ω). H.5. lxo cœffc.6. f ( H ) fom Jl-ho hy modl.7. Clcul h dco of.8. Clcul h pco τ.9. If ε τ covgc : go o., El go o. 3. Rul I h c of h bl, h fxd-po o chm fo mulouly olvg h ol fld quo (5), d h J modl (6) p ho. 8
9 4. Rul 4. ud Cuv Th dmo of h mgc quly of ml pmly o h u of h ym of mum ud. Th voluo of h dd h fld of h chco of ml gfc fco fo h kg o ccou of h phycl u of mgc ml d h codo of h u. Th poducbly of mum d h fcly of hdlg lo fco hch mk pobl o choo h yp of mgc ccu o mplm. ccodgly ou choc ld o h lo of fmok T (gl h T) 5*5 mm. Th dvc pld fo chco of h h o od g mu mk pobl o k mum by mpl oduco of h mpl, o lco 3% o od d lv, h pfc poog d hou doo of h pol fc of h mgc ccu of clog of flux. Th chco of ml udd do by dmg h follog qu xpd m of chco of h fm d mug oupu volg mud by ocllocop: V, V H, d V H (look fgu 3). Th xco pk fld, hch ubmd h mpl obd by polo fom o mud h ml of o col gl H d H locd dc dff fom h mpl h d dh ( ) d dh( ) d dh ( ) d d d d d d d dh d (4) V μ ( ) H( ) dh d (5) V μ ( ) H ( ) (6) Th mgc duco obd by m go of h V () volg cody col mug B by T V Bxp() d V 4 N f V( ) d (7) T h d, d : Dc o mpl h col Fld, (m). N : Numb of u of h col mug B., : Numb of u of h col Fld No, No. : co of h mpl (m )., : ufc of h col H, H (m ). V H, V H : po oupu of h col H, H (V). 9
10 Fg. 3. gl h T (T) 5*5 mm. I mpo o mk h h u of h v J modl fo h pm dfco h ddol dvg compd h h ogl modl: h pu of h v modl h mgc duco vfom. c h mgc duco obd fom go, ully fld, h f ocllo h ho of h mgc fld vfom. Th o p h fld vfom bg ddol dffcul o h pm dfco pocdu. Th obd of pm vld fo modl, ogl d v, llog good gm b mud d clculd d [6]. 4. Compo h mulo Th co of cyld fomgc h lgh of 4 cm d cm dm chcd by cycl of hy (.653 6, k93.3, c47.7-3, 596.7, α.), h cyld uoudd by col of h m lgh vd by m of dy J 5 /m. Th dv hch cou h duco hv dm D cm d 5 cm lgh. Th gp E cm. Th gomy of h ym udd, p o ymm, h f xlly (o) d h cod ccodg o h pl (o). W c h cod mgc poblm cyldcl cood ym, qu of dom. Fo umcl modlg h hocl lm (h f, ) bough bck o f dc hch c vy ccodg o h dd pco. I h udy, h lm fxd dc L 5 cm of h udd dvc. Th boudy codo ocd h h mgc quo h codo of Num d h codo of h Dchl of pg o fgu gp Lgh, [m].3 l Iduco Dm, [m] Fg. 4. Fld of udy of h x-ymmcl poblm h h boudy codo.
11 Fo h vldo of h pm obd, o upmpod o h fgu 5, h cycl xpml d h cycl of mulo obd g fom h dfd pm. Th uppoo ho h dg of ccucy of h cycl dfd by gc lgohm. No h h dffc b h muld d h cycl mud pp, hch flcd h vlu of h LE o. I my b od h h cycl ll blcd. Fgu 6 ho, fo opo fqucy of 5 H, h xpml d muld fld cuv of h ml h ubmd o.75 T pk vlu uodl duco. 5 gc Iduco, B (T) ud muld gc Fld, H [/m ] muld ud gc Fld, H (/m) Tm, [] Fg. 5. ud d mulo B (H) Fg. 6. Fld mud d. loop ul 5 H. modld 5 H ddol ul gv fgu 7, h mud d clculd mgc duco fo 5 H volg fqucy ly uppod h focu of h dvc lod..5 Flux dy, B (T) ud muld fo.5 m d. m muld fo.5 m d. m Tm, [] Fg. 7. Flux dy mud d modld 5 H.
12 Fgu (8) d (9) llu h vo of dl d xl mgc vco pol. W o h h vlu of vco pol mxml h duco d h gdully dc o o h boud of h fld. odul d pol mgéqu, [/m].8 x cm 3 cm 4 cm 6 cm odul d pol mgéqu, [/m].5 x cm cm 5 cm 5 cm Coodoé dl, [m] Coodoé xl, [m] Fg. 8. Rdl vo of h mgc. Fg. 9. xl vo of h mgc pol vco pol vco. 5. Cocluo I h pp, ogl mhodology o mplm mgc hy modl f-volum clculo h b pd. Th mhodology, bd o v Jl ho cl modl, h dfco of h pm Jl-ho modl dffcul poc h o l, bu h u of h chqu of opmo by gc lgohm mk pobl o f h dffculy. Rul obd by pplyg ou modl o h xymmc mgodymc dvc bd o v Jl ho modl d dffl lucvy h good pfomc h gd o umcl covgc d gv vy fcoy ul o T fm.. I h fuu, h u of dymc modl of mgc hy o udy h bhvo of ml h ym hgh fquc d h pplco o oh dvc ll b udk. Rfc [] H. L. Tom, R. G. Colcl,.. Kf, To Dmol F Elm gc odlg fo cl Hy Effc, IEEE Tco o gc, vol. 37, pp , ch. [] N. dok, N. J. Bl, J... Bo, d. Ljo-c, Iv Jl ho odl o Tk Io ccou Hy Tm-ppg F-Elm Clculo, IEEE Tco o gc., vol. 38, pp 797-8, ch.. [3] D. C. Jl d D. L. ho, Thoy of fomgc hy, Joul of gm d gc l, vol. 6, pp. 48 6, 986. [4] D. C. Jl, J. B. Tholk, d. K. Dv, Numcl Dmo of Hy m fo h odlg of gc opy Ug h Thoy of Fomgc Hy, IEEE Tco o gc, vol. 8, pp. 7 35, J. 99. [5]. Bbou,. Clé d F. ou, Compo of ch d Jl ho modl o k o ccou hy phomo fo f lm ly, Joul of gm d gc l, vol. 6, pp. 39 6, 3. [6] J. V. L,. L.vl, d oh, Rl Codd Gc lgohm fo Jl-ho odl m Idfco, IEEE Tco o gc, vol. 34, pp , ch.4. [7] J.. Vcoclo, J.. Rm, R. H. C. Tkhh, R. R. ldh, Impovm Gc lgohm, IEEE Tco o gc, vol. 37, pp , pmb.. [8] [8]. R. Wlo, N. Ro,. D. Bo, Opmg h Jl-ho odl of hy by Gc lgohm, IEEE Tco o gc, vol. 37, pp , ch..
13 [9]. Glldo, D.. Loh, om pc of Nchg Gc lgohm ppld o Elcomgc Dvc Opmo, IEEE Tco o gc, vol. 36, pp76-79, July.. [] K. Chk, J. cyglok, Idfco of Hy odl m h gc lgohm, Joul of hmc d Compu mulo, vol. 7, pp. 6-, 6. [] L.. L. lmd, G.. Dp,..N. Lm d H. Nff, odlg mgocv duc ug gc lgohm, Joul of gm d gc l, vol. 66-3, pp. 6-64,. []. V. k, Numcl H Tf d Flud Flo, compuol mhod mchc d hml cc. 3
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