Fundamentals of magnetic field

Transcription

1 Fundmentls of mgnetc feld The forces between sttc electrc chrges re trnsmtted v electrc feld (Coulomb's lw). Forces between movng chrges (current crryng lnes) lso pper, those re trnsmtted v mgnetc feld. Chrges movng wth constnt speed (drect current) cuse constnt mgnetc feld, whle chrges movng wth vrble speed (ccelertng or slowng) cuse vrble mgnetc feld. n the cse of movng wre n mgnetc feld or when the mgnetc feld chnges round wre, physcl force cts to the chrges of wre seprtng them by polrty, whch results n electrc feld, n nduced voltge. The mgnetc feld Consder two strght long prllel wres n vcuum (or r) of smll cross-secton compred to ther length. f the chrges n the wres move wth constnt speed (constnt currents re flowng) the forces between the wres re constnt. Usng notton F 1 =F =F the vlue of these forces re expressed s F = k l 1 (N). 1 F 1 F l Forces between current crryng strght conductors f 1 = =1 A nd l==1 m, then F = VAs 10 7 N = m, 7 7 Vs 4π10 µ 0 consequently k = 10 = = Am π π, here µ π 7 Vs 0 = 4 10 the mgnetc permeblty of vcuum. Ths formul my be used to defne the current of 1 A. Am Usng the vlue of permeblty µ 0 for clculton of force: F = µ 0 1 l (N). π The drecton of the forces re ttrctve n the cse of undrectonl currents nd they re repulsve n the cse of opposte drecton of currents. Descrpton nd understndng the mgnetc feld s smpler when dc current ssumed. nterpret force F n the fgure s follows: the movng chrges of current 1 crete specl stte of the spce (the mgnetc feld) round the wre whch feld cts on the chrges movng nsde the second wre crryng current.

2 VVEM111 Alterntng current systems 014 H 1 B 1 1 Mgnetc feld round long strght conductor crryng current 1 The frst spce vrble vector whch descrbes the mgnetc feld s the mgnetc feld ntensty H. n homogeneous substnce the mgnetc feld ntensty (feld strength) H 1 from current 1 defned s: 1 H1 =, the force F to the wre crryng current expressed s: π F =H 1 µ 0 l. n homogeneous nd ferromgnetc substnce the clculton of the mgnetc feld ntensty H s more complcted, the Ampère's exctton lw hs to be used. The mgnetc feld ntensty H s vector, ts drecton n ech pont of the spce s the sme s the north (N) sde of compss needle. Around sngle wre the drecton of feld ntensty s the sme s rght-thred screw turns. The S unt of mgnetc feld ntensty [ H ] = A m. Mgnetc feld ntensty n ech pont of the spce s llustrted by drected lnes. These lnes form closed pths, they do not rse nd do not end. H 1 B 1 H 1 B 1 1 F 1 F Force to current crryng wre n the feld of nother conductor Consder wre of length l crryng current n feld of mgnetc feld ntensty H. The force exerted F = µ 0 l H, where the drecton of s the sme s the movement of postve chrges nsde the conductor. n the cse shown on the fgure F = µ l H. 0 1

3 Fundmentls of mgnetc feld Exmple A conductor crres 1 A, the mgnetc feld ntensty from 1 m of the conductor s H = A m. The mgntude of the force on wre beng n mgnetc feld of H = 1 A, crryng 1 A s m F = π 10 7 N m. The other spce vrble vector descrbng the mgnetc feld s the mgnetc flux densty B. The mgntude of flux densty depends on the substnce n the spce, ts S unt n honour of Tesl's 1 scentfc ctvty Vs [ B ] = T= tesl = m. At mgnetc feld strength H B = µµ 0 r H, here µ r the reltve permeblty, substnce specfc, non-dmensonl fctor. The reltve permeblty s often not constnt, ts vlue my depend on the mgnetc feld ntensty nd lso on the ntl mgnetc conons. Exmple The mgnetc flux densty t mgnetc feld ntensty H = 1 A m n free spce B=4π10-7 T. The drecton of flux densty vector B s usully concdes wth tht of feld ntensty H : the drecton of the compss needle to the north pole t ech pont of the spce. The feld lnes re drected from the south pole to north nsde mgnet (e.g. nsde compss needle) nd from the north to south outsde the mgnet. Consequently, mgnetc flux-lnes leve the mgnet t the north pole nd go on towrds the south. The compss needle n ordnry use drected to the geogrphcl north pole of erth. S N N S H B Drecton of the mgnetc feld, the defnton nsde prtculr mterls the ferromgnetc mterls the mgnetc flux densty s sgnfcntly ncresed n contrst to free spce. Smple nd llustrtve explnton of ths phenomenon the contrbuton of the moleculr mgnets (or crculr currents) of such mterls to 1 Tesl, Nkol ( ) engneer, nvestgtor, Serbn orgn 3

4 VVEM111 Alterntng current systems 014 the flux densty of externl mgnetc feld. The reltve permeblty µ r expresses the rto of the flux densty n comprson wth tht n free spce; 1 µ r The vlue of µ r s usully determnng by mesurements or complcted clcultons. The mgnetc flux densty lso llustrted by drected lnes. The force F ctng to pece of current-crryng wre wth length l nd current n rbtrry mterl substnce cn be expressed s: F = l B. n the cse ccordng to the fgure F = l B. Exmple 1 The force to current-crryng wre wth 1 A nsde mgnetc feld of 1 T s F = 1 N m. The mgnetc flux of n re s sclr vlue, defned s the surfce ntegrl of the flux densty to the re of nterest: Φ = BdA, n homogeneous feld Φ=BA, the S unt n honour of Weber's scentfc ctvty A [Φ]=Wb =weber=vs. The mgnetc feld s often vsulsed s lnes of mgnetc flux tht form closed pths. The lnes re close together where the mgnetc feld s strong nd frther prt where the feld s weker. By conventon the flux lnes leve the north-seekng end (N) of mgnet (e.g. compss needle) nd enter ts south-seekng end (S). Exmple The flux through n re of 1 m n perpendculr homogeneous feld of 1 T s 1 Wb. The Ampère's 3 lw of exctton The most mportnt rule for clculton of mgnetc crcut The lne ntegrl of the mgnetc feld ntensty vector H round closed pth s equl to the ggregted current through re A enclosed by tht pth, ths ggregted current termed the exctton Θ of the re A. Hdl = JdA =Θ. A f the pth studed goes long seprte prts of homogeneous feld ntensty thn the ntegrl of the left sde of equton s smplfed s sum. Wheres f the chrge flow concentrted n wres thn the ntegrl of the rght sde of equton s smplfed s sum: H l =. n the cse of constnt permeblty µ r the lw of exctton my lso be wrtten s: B 1 Hd l = dl = Bdl =, or Bdl µ µ = µ, here µ=µ 0 µ r. Exmple Consder current-crryng wre wth current 1 A, n dstnce from t the feld ntensty: H =. π j j Weber, Wlhelm Edurd ( ) Germn physcst 3 Ampère, Andrè-Mre ( ) French physcst, mthemtcn, chemst 4

5 Fundmentls of mgnetc feld f the surroundng substnce s non-ferromgnetc nd the closed pth nvestgted s concentrc crcle wth rdus nd the drecton of ntegrl grees to the drecton of feld ntensty H thn Hd d l = l = π =. π π Smlr result s obtned when (n non-ferromgnetc substnce) the pth s closng v dfferent rcs ccordng to the next fgure: l l 3 l 4 r 1 r H l 1 Demonstrton of the lw of exctton long l 1 the feld ntensty H1 =, π r1 long l 3 nd l 4 the feld ntensty H s perpendculr to the drecton of pth of ntegrl, tht s why the sclr product Hdl = 0, long l the feld ntensty H =, π r 3 Hd 1 r1 r1 4 3 l = π = π 4 l 1 Hdl 1 Hd r r 4 1 =. l = π = π 4 l Knowng the exstng or desred feld ntensty the exctton producng t my be clculted. The ntegrl long pth wth H = const. Hdl = Hdl. f the pth consdered conssts of peces wth H = const. then Vsulston of mgnetc feld (flux-lnes) Current loop Hdl = H l =Θ. B Flux-lnes of current loop (turn) 5

6 VVEM111 Alterntng current systems 014 Solenodl nd torodl col Snce the length of solenodl col l s much greter thn ts dmeter d,.e. l» d. the feld nsde the col my be consdered unform nd outsde t my be neglected. The sme de used for torodl col, f verge dmeter D» d. At these cols the sngle turns re n seres, they hve the sme current. Applyng the Ampère's lw of exctton Θ=Hl=N, where N the number of turns (number of wres, number of currents). l d d Mgnetc feld of solenodl nd torodl cols D At gven drecton of current flow the drecton of the mgnetc feld produced n col depends on the drecton of the twst of turns. B B Mgnetc feld of rght-thred nd left-thred cols Force to current-crryng wre n mgnetc feld n homogeneous mgnetc feld ccordng to the formul for force: F = l B. F B B F llustrton of force to current-crryng wre n homogeneous mgnetc feld 6

7 Fundmentls of mgnetc feld "Useful" mgnetc feld nd lekge f consder lnked cols (lke the cols of trnsformers or the sttor-rotor cols of rottng electrcl mchnes) only porton of produced by one col mgnetc feld s lnkng to the other col, the rest of the feld s lekng. The lst prt termed s flux lekge or mgnetc feld lekge. The mesure of lekge s defned wth coeffcent of lekge σ: σ = l (0 σ 1), t where t the totl flux, whle l the flux lekge. n certn cses the mgnetc lekge s mportnt, e.g. the lekge rectnce cn lmt the short crcut current. The lw of flux refrcton (the boundry conons) The feld ntensty H nd the flux densty B pss boundry lyer of substnces wth dfferent permeblty dfferent wy. Refrcton of flux densty vector Let us exmne the conons t the boundry between two mterls of permeblty µ 1 nd µ. Consder the flux lnes crossng the boundry s shown n fgure wth ngle of ncdence α 1 nd ngle of refrcton of α. The flux mgntude through n elementl re da pproched from ech sde of the boundry must be dentcl, ssumng tht no mgnetc flux emerges from surfce s da 0. Snce the flux-lnes re closed, the overll fluxes n the two substnces re dentcl: Φ = BdA =B 1n da=b 1 cosα 1 da=b cosα da= B n da, da tht s the norml component of flux densty vector B remns unchnged, t crosses boundry contnuously. Accordng to the exctton lw the lne ntegrl of the mgnetc feld ntensty round closed pth of wh dl s equl to zero f no exctton (no current flowng) on ether sde. Around the closed pth Hdl = 0 snce no current lnked: Hdl = H 1t dl- H t dl= H 1 snα 1 dl- H snα dl=0, hence H 1t =H t, 7

8 VVEM111 Alterntng current systems 014 tht s the tngentl component of feld ntensty vector H remns unchnged, t crosses boundry contnuously. Refrcton of feld ntensty vector The tngentl component of flux densty vector B nd the norml component of feld ntensty vector H re chngng through the boundry lyer. From the sttements bove H 1 snα 1 = H snα, or substtutng flux densty for feld ntensty: B1 B snα1 = snα snα1 snα tgα1 µ r1 µµ 0 r1 µµ 0 r = =. B B µµ 0 r1 α1 µµ 0 r α tgα µ r 1cosα1 cos cos cos = α The flux-lnes t ron-r border-lyer Suppose tht µ r1»µ r (e.g. t the ron-r boundry, where µ rron =10 6, µ rr =1), thn tgα 1»tgα, α 1»α,.e. α 1 ~ 90 whle α ~ 0. Consequently the mgnetc flux emerges nto r norml to the surfce of ron wth pproxmtely nfntely permeblty, the flux-lnes leve the ron t rght ngles. The Frdy's 4 nducton lw Ths lw s one of the most mportnt sttement of electrcl engneerng, dscovery of the phenomenon descrbed n t mde (nd mke) possble the generton nd publc use of electrcl energy. 4 Frdy, Mchel ( ) Englsh physcst 8

9 Fundmentls of mgnetc feld Consder wre loop ( sngle turn of col), f n ny cse the flux enclosed by tht loop s chngng, t produces electrcl feld, voltge ppers (nducng) n the loop. The mgntude of nduced voltge u (t) s proportonl to the flux chnge (t) n the tme unt: d () t u () t =. The flux chnge occurs ether becuse the mgnetc feld s chngng wth tme (trnsformer nductnce) or becuse the wre loop s movng reltve to mgnetc feld (motonl nductnce). The Frdy's nducton lw descrbes both phenomen. ) f wre loop s fxed nd the flux s vryng wth tme becuse the exctton current or the mgnetc crcut s chngng the phenomen clled trnsformer nductnce. b) n the cse of motonl nductnce conductor (or wre loop) durng ts dsplcement crosses the mgnetc flux lnes.e. the moton hs component perpendculr to the fluxlnes. The mn pont of nducton s tht the chnge of mgnetc feld cuses electrcl feld. Usng the term of nduced voltge phenomenon n mgnetc feld my be replced wth phenomenon n electrcl crcut. n rel equpment the two types of nductnce (trnsformer nd motonl) often pper smultneously (eg. n rottng electrcl mchnes). mportnt notce: f the spce contns both sttc electrc nd chngng mgnetc felds, the electrc feld become non-conservtve, becuse the lne ntegrl s no more pth-ndependent, snce exst such closed pths whch enclose chngng mgnetc feld nd the ntegrl long such pths s not zero. n ths cse the electrc potentl s sclr descrptor s unusble. n closed loop the nduced voltge produces current ccordng to the resstnce of loop. The voltge on the resstnce s blncng the nduced voltge f no other voltge source n the loop. The Krchhoff's voltge lw for dc crcuts: R + U = 0, hve to be extended j j j j R + U + U = 0, j j k here U s nduced voltge, U b s non nduced voltge (e.g. one of glvnc source). (For snusodlly chngng lterntng current the voltge lw s vld for phsors nd mpednces re tken nto ccount nsted of resstors.) k Trnsformer nducton The reference drecton of the flux chngng nd tht of the chrge-seprtng electrcl feld ntensty re ccordng to the fgure, U = Edl. E n bn k d > 0 k - + U Reference drectons for trnsformer nducton 9

10 VVEM111 Alterntng current systems 014 The nduced voltge depends not of the mgntude of mgnetc flux but the mgntude nd the drecton of the dervtve of mgnetc flux. U - + U + - t d > 0 t d < 0 Polrty of the nduced voltge t dfferent drectons of flux nd dervtve of flux ( > 0) U - + U + - t d > 0 t d < 0 Polrty of the nduced voltge t dfferent drectons of flux nd dervtve of flux ( < 0) 10

11 Fundmentls of mgnetc feld The flux lnkge Usully the chngng flux s encrcled by col of N seres turns (nd the turns exctng n the sme drecton) so the nduced voltges of the sngle turns re ggregted for the col. f ech turn of wre lnked wth flux of the sme mgntude then the resultnt nduced voltge () u() t N d t =. The sum of the fluxes lnked wth the sngle turns gves the resultnt flux lnkge ψ=n whch cn be used for clculton of the resultnt nduced voltge: d () t u () t = ψ. The physcl unt of the flux lnkge Ψ s the sme s tht of the flux Φ: [Ψ]=Wb=Vs. Lenz's 5 lw As follows form the conservton of energy prncple the currents nd forces produced by nducton hve such n effect, whch s decresng the process genertng them. d The nduced voltge U = hs such polrty tht produces current through n externl resstnce whch opposng the orgnl chnge of flux lnkge, decresng the nducng effect. The force ctng to current-crryng wre movng n mgnetc feld brkes the movement. n other words the mgnetc feld produced n the process of nducton cts for conservton of the ntl stte. Ths prncple ppers n the phenomenon of self nducton. d > 0 R U - + Mgnetc effect of the current produced by nduced voltge Motonl nduced voltge Motonl nduced voltge s genertng n movng wre becuse force of ntercton ppers between the sttc mgnetc feld nd the chrges trvellng wth the wre. (For currentcrryng conductors the produced force expressed s: F = l B. Ths force ct to the chrges whch forwrd t to the conductor.) Defne current whch s not rel one, but helps to clculte force snce the movement of chrges. The force F ccordng to : 5 Lenz (Lenc), Henrch Fredrch Eml ( ) physcst, Germn orgn 11

12 VVEM111 Alterntng current systems 014 h F = h B = t Q B = Qv B Q h, becuse = nd v =. t t nsde conductor, movng n homogeneous mgnetc feld B wth speed v perpendculr both to t's own drecton nd to B, mgnetc force cts to the chrges n the conductor. Ths force seprtes the chrges nsde the conductor thus produces n electrc feld E. The drecton of the electrc feld s the sme s the force ctng to the postve chrges. l +Q dh B +Q F E + v u (t) F (t) + R - - h A possble llustrton of motonl nductnce F E = = v B. Due to ths electrc feld the chrges ccumultng by polrty t the ends of Q conductor, whch results the ppernce of nduced voltge. The voltge u between the ends of conductor wth length l n homogeneous mgnetc feld expressed s u = El = v Bl = l B v, f the reference of voltge s from chrges (+) to chrges (-). Ths voltge s n nduced voltge, produced by the chrge-seprtng electrc feld E, often mentoned s electromotve force (emf). d Edl =. The nduced voltge cuses (rel) current n closed crcut. The ntercton of the current nd the mgnetc feld B produces physcl force F gnst the movement, ccordng to the Lenz's lw. (Another explnton: the densty of flux lnes ncresng n the drecton of movement.) Ths men tht n cse of closed crcut the movement of wre requres contnuous force, energy. There re two forces dscussed: - force cts to the chrges trvellng wth the conductor, the consequence of whch s the chrge-seprtng effect of n electrc feld E nd the nduced voltge U, - due to the current cused by ths nduced voltge force cts to the conductor. The drecton of the two forces re dfferent. Flux lnkge of long strght wre of fnte dmenson The mgnetc feld produced by low-frequency lterntng current or drect current flowng through long crculr conductor wll be not only externl to the conductor but lso exsts wthn the conductor. The nternl flux wll lnk only frcton of the current, ths lnkge must therefore be treted seprtely from externl. Consder such conductor n free spce of rdus r c, crryng current. 1

13 Fundmentls of mgnetc feld Externl flux The externl mgnetc feld my be clculted pproxmtely by the sme wy s tht of conductor wth nfntely smll sze (one dmensonl conductor). Assumng unform current dstrbuton wthn the conductor, the current densty J or the current : J = = Ac r, or = JdAc = Jrc cπ π, where r c the rdus of conductor, A c the cross-secton of conductor. n rnge of rdus >r c the feld ntensty H e of the externl mgnetc feld He () =, whle r c. π µ 0 The externl flux densty n free spce s Be() = µ 0He() =. π The nnulus externl flux d e enclosed by n re da=ld determned by nnulus d through length l of conductor (nto the plne of the pper) s µ 0 d () B () da d e = e = l. π Snce the sngle wre s consdered s one turn the flux lnkge s equl to the flux dψ e =d e, the totl externl flux s R R µ 0l d µ 0l R ψ e = d = π = ln, π r r r c c c where R s suffcent dstnce to gve zero feld (theoretclly R ). H H H e R r c d The mgnetc feld ntensty vs. dstnce The externl nductvty derved from the externl flux: ψ e e µ 0l R Le = = = ln. π r c 13

14 VVEM111 Alterntng current systems 014 Exmple µ 0 l 7 n r µ=µ 0, then L e = = 05. l 10 (H), or L b = 0.05 µh/m. 8π nternl flux The conductor s consdered to be mde up of n nfnte number of prllel conductng elements nd the current dstrbuton s unform (the current densty s constnt throughout the cross-secton). The current flowng nsde rdus <r c clculted s frcton of current : J = π = for r c. rc The nternl mgnetsng force H s due only to the current wthn the rdus H() = = =, r c. π π rc π rc (At the surfce of the conductor the formuls for externl nd nternl felds gve the sme results.) The nternl flux densty B() = µ H() = µ π r, c where µ the totl permeblty of the conductor mterl, for non-ferromgnetc mterls usully consdered s µ=µ 0. The nternl flux for nnulus re da: µ d () = BdA= d π r l. c f the whole cross secton of the conductor s consdered s one turn, then the flux d lnks only turn N frcton of 1, proportonl to the cross-secton for rdus r c N = r. v Hence the elementl nternl flux lnkge for r c 3 µ µ dψ () = Nd() = ld = l d. 4 rc π rc π rc The totl nternl flux lnkge: rc µ l 3 µ l ψ = d= π r. 4 c 8π 0 ψ µ l The nductvty derved from the nternl flux: L = = 8π Exmple µ 0 l 7 f µ=µ 0, then L = = 05. l 10 (H), or L = 0.05 µh/m. 8π 14

15 Fundmentls of mgnetc feld As mentoned bove the current n the segment of conductor nsde n rbtrry rdus r c s only the frcton of the totl current : = r c, wheres the sme segment of the conductor s lnked wth the totl nternl flux. Hence the nternl nductvty of segment of conductor derved from the nternl flux: c l () L r ψ = = for r c. r c l L r c Chnge of nternl nductvty The nductnce of the conductng elements nerer the centre s greter then tht of those ner the outsde. When the current s lterntng the nductve rectnce of the elements ner the centre s greter then tht of those ner the outsde, nd hence more current flows n the elements ner outsde. Skn effect Accordng to complex, complcted clcultons n crculr conductor the current densty decresng from the surfce to the centre. A depth of penetrton δ used to defne dstnce t whch the current densty decreses to 1/e of ts vlue t the outer surfce. n the cse f the rdus of the conductor r c > (3-5)δ, the current densty cn be consdered unform up to the depth of penetrton nd zero wthn the rdus r c -δ. The nfluence of skn effect s obvously more pronounced t lrger nductve rectnce whch vres wth the ngulr frequency of the current ω, nd wth the nductnce. The nductnce vres wth the permeblty of the mterl of the conductor µ, consequently the lrger the permeblty, the smller the depth of penetrton. On the other hnd, the greter the resstvty of the conductor ρ, the smller wll be the effect of the vrton n nductve rectnce n cusng non-unform current dstrbuton. The reltonshp between the depth of penetrton δ nd ω, µ nd ρ expressed n formul 15

16 VVEM111 Alterntng current systems 014 ρ ρ 1 ρ ρ δ = = = = 5033,. ωµ πµ 0 fµ r πµ fµ fµ 0 r r The equton for the depth of penetrton bove my be ppled to flt sheets (r c ). The sme consdertons pply to the penetrtons of lterntng mgnetc flux nto conductors. The proxmty effect Smlr phenomenon occurs when two conductors crryng lterntng currents re n ech other's mgnetc felds, there wll lso be redstrbuton of currents. The currents re crowdng to those prts of the conductors, whch lnk the lest mount of flux nd therefore hve less nductvty. The llustrton of proxmty effect wth skn effect, the current flow dentcl Exmple The vlues for copper: ρ Cu = Ωm, µ rcu =1, the depth of penetrton t frequency f=50 Hz δ Cu =9.49 mm (8.66 mm t f=60 Hz). The vlues for lumnum: ρ Al = Ωm, µ ral =1, the depth of penetrton t frequency f=50 Hz δ Al =1 mm (10.95 mm t f=60 Hz). The vlues for ron: ρ Fe = Ωm, µ rfe =5000, the depth of penetrton t frequency f=50 Hz δ Fe =0.35 mm (0.3 mm t f=60 Hz). δ (m) Al Cu f (Hz) The depth of penetrton vs. frequency 16

17 Fundmentls of mgnetc feld Skn effect n ferromgnetc surroundng The phenomen n conductor surrounded by ferromgnetc mterl re bsclly the sme s the skn nd proxmty effects n ny system of conductors wth lterntng current n them. The fgure shows n cross secton of deep, nrrow br of squrrel-cge rotor nd further shows the generl chrcter of the slot-lekge feld produced by the current n the br wthn ths slot, n prtculr when the depth of the slot exceeds ts wh. f the rotor ron hs nfnte permeblty, ll the lekge-flux lnes would close n pths below the slot, s shown. f mgne the br to consst of lyers, whch re electrclly n prllel, the lekge nductnce of the bottom lyers s greter thn tht of the top lyers becuse the bottom lyer s lnked by more lekge flux. Consequently, the lterntng current n the lowrectnce upper lyers wll be greter thn tht n the hgh-rectnce lower lyers. The current wll be forced towrd the top of the slot, n don the current n the upper lyers wll led the current n the lower ones. The non unform current dstrbuton results n n ncrese n the effectve resstnce nd smller decrese n the effectve lekge nductnce of the br. ) b) Mgnetc feld of current-crryng wre n deep slot Representtve shpes of slots n squrrelcge rotor ) double cge b) deep-br Snce the dstorton n current dstrbuton depends on n nductve effect, the effectve resstnce s functon of the frequency (.e. of the slp) nd lso functon of the depth of the br nd of the permeblty nd resstvty of the br mterl. J J h f r =f 1 h f r =0 Dstrbuton of current densty n the brs of double cge rotor 17

18 VVEM111 Alterntng current systems 014 The effect cn be explned s follows: the prtculr prts or lyers of the wre re lnked by dfferent mgnetc flux, therefore the nductvty l = ψ ( π l) nd the mpednce z = r + f r lso dffer dependng on the dstnce from the top of the slot. To ncrese the frequency-dependence of the rotor resstnce double cge or deep slot re ppled. n double cge rotor the upper brs (close to the rotor surfce) re mde from mterl of less cross-secton nd more resstvty (e.g. brss) whle lower brs re mde from mterl of more cross-secton nd less resstvty (e.g. copper). At stndstll or strtup (S=1) the frequency of the rotor current s equl to tht of the sttor current (e.g. f r =50-60 Hz), due to the skn effect the rotor current flows mnly n the upper cge of more resstnce (nd less rectnce), whle round the rted (nomnl) speed (e.g. S n , f rn =1.5-3 Hz) the dstrbuton of rotor current determned by the rto of the resstnces of the cges. By mens of constructon my be cheved tht the effectve resstnce of rotor col s more t strtup nd less t nomnl speed. f r =f 1 J f r =0 h Dstrbuton of current densty n the wre of deep slot rotor The phenomen nd ts nfluence s smlr n rotor wth deep slot, but vry of the current densty J s contnuous n functon of the dstnce from the top of the slot. Explotng the frequency dependence of the effectve rotor resstnce some mprovement my be cheved: the strtng torque ncresng, the speed fll t nomnl performnce decresng. w w 1 R r (S n ) R r (S=0) T T bg MT 1 s T l T s1 Sttc speed-torque chrcterstc of nducton mchne 18

19 Fundmentls of mgnetc feld The strtng torque T s t the smller resstnce wouldn't be enough to strt wth lod torque T l. On the other hnd, t the greter resstnce the speed fll t lod torque T l would be greter n stedy stte. The two chrcterstcs re smplfctons, the trnston between the chrcterstcs s contnuous durng ccelertng, the vry of lekge nductnce s neglected. n the fgure: w 1 synchronous speed, T bg brekdown torque (brekng mode), T bm brekdown torque (motor mode), T s1, T s1 strtng torque, T l lod torque, S n nomnl slp. nfluence of eddy current to the dstrbuton of the mgnetc feld Due to the chnge of the mgnetc flux nduced emf ppers nd eddy currents flow n the conductor. Accordng to Lenz's lw the eddy currents oppose the orgnl chnge of the flux, the nducng process, producng such feld, whch dsplces the flux to the surfce of the conductor. The depth of penetrton of the mgnetc feld my be clculted s tht of current densty. dψ eddy nfluence of eddy currents to the dstrbuton of mgnetc feld n ferromgnetc prts conductng, controllng mgnetc feld the eddy currents re not desred becuse of the dsplcng the flux nd ncresng the losses. The eddy currents cn be decresed usng lmnton of ron or usng or ferrte core. Composed by: Kádár stván Mrch

20 VVEM111 Alterntng current systems 014 Questons for self-test 1. Expln the force on current-crryng conductor n mgnetc feld.. Revew the vrbles descrbng the mgnetc feld. 3. Defne the mgnetc feld ntensty, flux nd flux densty. 4. Expln the mgnetc permeblty. 5. Expln the Ampere's lw of exctton. 6. Expln the Frdy's nducton lw Frdy's nducton lw. 7. llustrte the flux lekge. 8. llustrte pproxmtely the mgnetc feld of current-crryng conductor nd conductor loop. 9. llustrte pproxmtely the mgnetc feld of solenodl nd torodl cols. 10. How pproxmted the clculton of mgnetc feld of solenodl nd torodl cols? 11. Defne the flux lnkge. 1. Expln the motonl nducton. 13. Expln the trnsformer nducton. 14. Expln the Lenz's lw for motonl nd trnsformer nducton. 15. llustrte the mgnetc feld of current-crryng conductor of fnte dmenson. 16. llustrte the current dstrbuton n conductor of fnte dmenson. 17. Expln the nductvty of conductor of fnte dmenson. 18. Expln the skn effect. 19. Expln the depth of penetrton n current-crryng conductor. 0. How pproxmted the current dstrbuton wth respect to skn effect? 1. Expln the proxmty effect.. llustrte the skn effect n ferromgnetc surroundng. 3. Expln the eddy current nd t's nfluence on the dstrbuton of mgnetc feld. 4. How cn be reduced the eddy current? 0