2.3 Magnetostatic field

Transcription

1 37.3 Mageosaic field I a domai Ω wih boudar Γ, coaiig permae mages, i.e. aggregaes of mageic dipoles or, from ow o, sead elecric curre disribued wih desi J (m - ), a mageosaic field is se up; i is defied b field iesi (m - ) as well as flu desi (Wbm - T). I geeral, he lik bewee ad, i.e. he cosiuive law of he medium, is complicaed. Neglecig hseresis, he law is sigle-valued ad ca be epressed, for a isoropic medium i he absece of permae mageiaio, b μ (.3.) where μ is called permeabili ( m - ) ad, i he mos geeral case, is a fucio of ; he iverse of μ is called relucivi. The medium is supposed o be a res wih respec o he observer of he field..3. Mawell s equaios for mageosaics The equaios goverig he mageic field are i Ω ad alog Γ if Γ is a flu lie (flu lies parallel o Γ), or 0 (.3.) J (.3.3) 0 (.3.4) μj S (.3.5) if curre of surface desi J S ( m - ) is prese, or 0 (.3.6) if flu lies are perpedicular o Γ. For a isoropic ad liear medium, i erms of, he equaios become i Ω wih 0 ; μj (.3.7)

2 38 or or 0 (.3.8) μj S (.3.9) 0 alog Γ (.3.0) The equaios wrie above uambiguousl defie he mageosaic field which, because of (.3.), is soleoidal. If boh J S ad J are give, he i mus be J s dγ J dω (.3.) Γ Ω i.e. he oal curre sums up o ero: herefore, desiies J S ad J cao be idepede. I a o-homogeeous domai a he ierface bewee wo media of permeabili μ ad μ, from (.3.) i holds ( ) 0 (.3.) (Fig..) so ha he ormal compoe of is alwas coiuous. If here is a curre of desi J S ( m - ), he from (.3.3) ( ) s J (.3.3) If J s 0, he ageial compoe of is coiuous. Equaios (.3.) ad (.3.3) are called rasmissio codiios. I he case of a o-homogeeous medium, he followig remark ca be pu forward. fer (.3.) ad (.3.), cosiderig vecor idei (.4), oe has μ μ μ 0 (.3.4) I he case of a o-homogeeous medium, field is soleoidal if μ ad are orhogoal vecors; his meas ha lies separaig laers of differe μ are parallel o field lies of. Coversel, afer (.3.) ad (.3.3), cosiderig vecor idei (.6), i urs ou o be μ μ μ J (.3.5)

3 39 I appears ha, i a curre-free medium (i.e. J 0), field is irroaioal if μ ad are parallel vecors; his meas ha lies separaig laers of differe μ are orhogoal o field lies of. If μ 0 (homogeeous curre-free medium), he is alwas irroaioal. ad J 0 Fiall, a eesio of cosiuive law (.3.) is cosidered. I he presece of a permae mageiaio 0 i he mageic maerial (permae mage) he cosiuive law is μ 0 I his case he field equaios are (.3.6) 0 (.3.7) μj (.3.8) 0 I paricular, he field iside a permae mage is described b (.3.8) wih J 0; i follows ha he mage ca be modelled b a equivale disribuio of curre give b J μ 0. eq.3. Mageosaic poeials i) From (.3.), sice, for a vecor, ( ) 0 holds (see.8), i is possible o defie a vecor fucio (Wb m - ) called vecor poeial b meas of ad (.3.9) 0 (gauge codiio) (.3.0) This wa (.3.) is fulfilled, while (.3.3) becomes ( ) J (.3.) μ For a homogeeous domai, afer (.) ad (.3.0) i urs ou o be μj (.3.)

4 40 This is he (Poisso s) vecor equaio goverig. I a ssem of recagular coordiaes i correspods o he followig scalar equaios J μ μj μj (.3.3) I geeral, he gradie of a harmoic fucio ma be added o, havig all he equaios saisfied. Of course, suiable boudar codiios o Γ mus be added i order o defie he field i a uique wa. I paricular, afer (.3.) ad (.3.8), he poeial iside a permae mage is give b 0. ii) I a wo-dimesioal domai, vecors J ad so have ol oe oero compoe; hece, vecor poeial ca be reaed as a scalar quai. The boudar codiios (.3.8) ad (.3.0), i erms of ( ), alog he boudar Γ wih ormal versor ( ), ad ageial versor ( ) ( ),,, become, i erms of, sice ad : 0 (.3.4) i.e. cos alog Γ ad ( ) i i ( ) 0 i i (.3.5)

5 4 i.e. 0 alog Γ, respecivel. iii) If J 0 i Ω ad Ω is simpl coeced, he, alog wih, he field ca be described b a scalar fucio ϕ (poeial, ) defied as ϕ (.3.6) I fac, (.3.3) is auomaicall saisfied, while from (.3.) we obai μ ϕ 0 i Ω (.3.7) The laer is he Laplace s equaio goverig mageic scalar poeial ϕ wih suiable boudar codiios. The codiio of simpl coeced domai ca be obaied b suiable cus, if ecessar. If his codiio is o fulfilled, everheless ϕ ca be sill defied, apar from muliples of a cosa. iv) Whe i (.3.) permeabili μ depeds o, oe has μ( ) ad for he soluio of (.3.) oe should resor o a ieraive procedure. ccordig o he Newo-Raphso mehod, he residual r() of he goverig equaio (.3.) is developed i Talor s series, rucaig he developme a he firs order dr r d k ( k ) r( ) k ( k k ) o( k ) (.3.8) If a predicio of he soluio k- a he (k-)-h ieraio is available, he subseque predicio k a he k-h ieraio is give b (.3.8) imposig r( k ) 0. I resuls k r( k dr k ) (.3.9) d k The, μ is updaed b meas of he ew esimaio of ad so ad he problem is solved agai. The procedure sops whe he error bewee wo successive soluios is less ha he prescribed hreshold. I is ecessar o dr kow a iiial predicio 0 ad he value of he derivaive a each d ieraio; uder hese assumpios, i ca be prove ha he procedure coverges quadraicall.

6 4.3.3 Mageosaic eerg Give a mageosaic field characeried b iesi ad flu desi i a liear medium, he specific eerg (Jm -3 ) of he field is defied as ; if he medium is isoropic, he eerg W (J) sored i a ubouded regio Ω is give b dω Ω W (.3.30) If he cosiuive relaioship of he mageic maerial is o-liear, he specific eerg is d' ad he oal eerg is 0 W d' dω Ω 0 I some cases i is coveie o iroduce he specific co-eerg ad he oal co-eerg is W' d' dω Ω 0 (.3.3) d' 0 (.3.3) I he case of liear medium WW resuls. I he liear case, akig io accou he followig idei (see.3) ( ) ( ) ( ) J ( ) (.3. 33) ad (.3.3), he oal eerg sored i a regio Ω of boudar Γ is W dω J dω dγ (.3.34) Ω Ω ( ) Γ

7 43 The equaio above reduces o 0 alog Γ. W J dω if eiher 0 or Ω.3.4 Field of a lie curre i a hree-dimesioal domai: differeial approach curre I (), coceraed a r 0 ad direced alog he ais i a ssem of clidrical coordiaes (r, ϕ, ), is cosidered (Fig..9). I r P The smmer implies ( 0,,0) Fig..9 - Lie curre. ad he field equaio is from (.3.3): r Iδ(r), r > 0 (.3.35) r r r r where vaishes as r approaches ifii. The geeral soluio (see Secio..7) is: r () r I ρ ρ ρδ ( )d k (.3.36) r 0 The Dirac's δ i a clidrical geomer ca be approimaed b:

8 44 δ lim α 0 δ α, α > 0 (.3.37) wih δ α, r α ad δ α 0 elsewhere. Cosequel, he field πα ca be approimaed as For r α oe has lim α 0 α r I r α I ρδ ρ αd kα k α r r πα 0 (.3.38) Ir k α (.3.39) πα r Sice δ α is a regular fucio ear he origi, also will be regular ear ero; herefore k α 0. For r α oe has α α α I I ρ ρ ρδαdρ kα d r r 0 πα 0 I α I r πα πr, r > 0 (.3.40) The io-savar s law follows: I πr () lim () r, r > 0 r α α 0 (.3.4) leraivel, he Sokes s heorem ca be applied o (.3.3), givig dl I, if l is a closed lie likig he coducor oce. Thaks o he l field geomer, l ca be ake as a circular lie cered a r 0; hece, (.3.4) resuls. From (.3.4) ad (.3.9) he vecor poeial resuls: I l ri, r > 0 (.3.4) π

9 Eerg ad forces i he mageosaic field i) Priciple of virual work Give a srucure i he field regio, o which force F is o be calculaed, a virual liear displaceme ds i he direcio of F, supposig ha he mageic ssem is supplied b a cosa curre I creaig a likage flu Φ, he sum of mechaical work Fds ad variaio of mageic eerg dw is equal o he ipu eerg IdΦ so ha F ds Fds dw IdΦ d(iφ W) d F ( IΦ W) (.3.43) ds I he case of a agular displaceme d ϑ, he orque T wih respec o he roaio ais is d T (IΦ W) (.3.44) dϑ The quai IΦ-W, deoed b W, is he complemear eerg or coeerg of he ssem. O he oher had, if he mageic ssem is isolaed, mechaical work Fds ad variaio of mageic eerg dw ake place so ha ece he force ca be evaluaed as while he orque is If he ssem is liear, W ad W coicide. F ds dw 0 (.3.45) dw F (.3.46) ds dw T (.3.47) dϑ

10 46 ii) Lore s mehod I is based o he defiiio of flu desi; he force F eered o curre disribued wih desi J i he regio Ω is F J dω (.3.48) Ω where is he eeral field, i.e. he flu desi i he absece of curre. Direcios of force, flu desi ad curre desi are muuall orhogoal. iii) Mehod of Mawell s sress esor Defied a closed surface Γ eclosig he srucure, he force F is evaluaed as where is he ouward ormal versor. F T dω T dγ (.3.49) Ω Γ The Mawell s mageic sress esors T, assumig a ssem of recagular coordiaes, i a hree-dimesioal domai ca be represeed i mari form as ( ) T ( ) ( ) (.3.50) I order he esor be uiquel defied, surface Γ should o be coicide wih he ierface bewee maerials havig differe permeabili. Remark There is a lik bewee Lore s ad Mawell s approach o force calculaio. I fac, usig (.3.),(.3.3) ad (.3.48), he force desi f (Nm -3 ) akes he epressio ( ) f J (.3.5)

11 47 I paricular, he -direced compoe is f (.3.5) fer addig ad subracig he erm oe has f ( ) (.3.53) I follows ( ) ( ) f ( ) (.3.54) ( ) ( ) f (.3.55) Due o (.3.) he las erm of (.3.55) is ero; he, if vecor,, v ( ),, (.3.56) is defied, f ca be viewed as is divergece, apar from a cosa k which ca be se o ero, amel v f (.3.57) similar resul holds for force desi compoes f ad f ; oe has (,, v ) (.3.58)

12 48 such ha f v (.3.59) ad v 3,, ( ) (.3.60) such ha f v 3 (.3.6) respecivel. Therefore, accordig o (.3.49), he force F (N) ca be wrie as he iegral of he divergece of esor T represeed b mari (.3.50), i which he row eries are he compoes of vecors v k, k, Force o a elecromage Le us cosider a elecromage wih a movable par (Fig..0). IRON NI IRON Γ Fig..0 Model of he elecromage. The iro core is supposed o have ifiie permeabili. The air gaps i he direcio are supposed o be much smaller ha he air gap i he direcio. The circulaio of he mageic field, alog a lie likig he eciaio curre NI ad crossig he air gap i he ormal direcio, reduces o NI (.3.6) Therefore a he air gap

13 49 NI (.3.63) while i he iro par 0. Followig (.3.3), he co-eerg sored i he air gap is give b ( NI) μ0 S W' μ0 S (.3.64) where S is he cross-secio of he ceral limb ad μ 0 is he air permeabili. If NI is cosa, accordig o (.3.43), he force acig o he movable par is F W' μ0s NI (.3.65) The force is egaive, i.e. opposie o he direcio of icreasig ; herefore, i is aracive, regardless of he sig of I. I order o appl he mehod of Mawell s sress esor, a iegraio surface Γ eclosig he movable par is cosidered havig as is ouward ormal versor. Takig io accou he field disribuio, oe acuall has: Therefore i resuls: 0 T (.3.66) 0 F T dγ 0, S Γ (.3.67) F NI μ 0 S μ0s (.3.68) The force is aracive, because variables ad are orieed i opposie direcios.

14 Tes problems Throughou he book, he problem of he compuaio of he mageic field i some es cases is cosidered. The firs case is ha of he air-gap of a sigle-side sloed elecrical machie. The model of oe half of he field regio is show i Fig... The slo, of widh a ad heigh b, accommodaes a coceraed or disribued curre, whereas he surrouded iro is assumed of ifiie permeabili. 0 iro core air 0 slo μ 0, J iro core 0 0 Fig.. Sigle-side sloed elecrical machie: oe half of he field regio. I erms of vecor poeial, he followig boudar codiios for he field regio are se up: fied values of alog a flu lie ( 0) ad vaishig ormal derivaive a he smmer lie 0. Oher es cases are preseed i Fig..a ad.b.

15 slo 0 μ 0, J iro core 0 slo μ 0, J 0 iro core 0 0 a) b) Fig.. Sigle slo: oe half of he field regio. The correspod o a sigle slo produced i a iro core wih oe side facig he air-gap (mageicall ope slo: case a) ad o a slo full embedded i a iro core (mageicall closed slo: case b). The releva boudar codiios ca be approimael se up as show. This es problem proposed migh represe a bechmark because i deals wih a simple, clear ad meaigful eample; i elecrical egieerig, i fac, a broad class of devices icludes a mageic pole formed b a currecarrig slo.