Fundamentals of magnetic field

Transcription

1 Fundmentls of mgnetic field The forces between sttic electric chrges re trnsmitted vi electric field (ccording to the Coulomb's lw). Forces between moving chrges (e.g. in current crrying conductors) lso pper, those re trnsmitted vi mgnetic field. Chrges moving with constnt speed (direct current) cuse constnt mgnetic field, while chrges moving with vrible speed (ccelerting or slowing) cuse vrible mgnetic field. When conductor is moving in mgnetic field or when the mgnetic field chnges round wire, physicl force cts to the chrges contined in the conductor nd seprtes them by polrity, which results in n electric field, the induced voltge. The mgnetic field Description nd understnding the mgnetic field is simpler when dc current ssumed. Consider in vcuum (or ir) two stright long prllel wires of smll cross-section compred to their length. f the chrges in the wires move with constnt speed (constnt currents re flowing) the forces between wires re constnt. Using nottion F 1 =F =F the vlue of these forces expressed s F k 1 = l (N). 1 F 1 F l Forces between current crrying stright conductors VAs J f 1 = =1 A nd l==1 m, then F = 10 7 N = = m m, 7 Vs 4π10 µ consequently k = , this cn be written s k = = Am π π. 7 Vs Here µ 0 = 4π10 the mgnetic permebility of vcuum. Am The formul of force bove my be used to define the current of 1 A s well. Using the vlue of permebility µ 0 the formul of force expressed s: F = µ 0 1 l (N). π The direction of the forces bove re ttrctive in the cse of unidirectionl currents nd they re repulsive in the cse of opposite direction of currents.

2 VVEMA13 Alternting current systems 018 nterpret force F in the figure s follows: the moving chrges of current 1 produce specil stte of the spce the mgnetic field round the wire nd this field cts on the chrges moving inside the second wire crrying current. H 1 B 1 1 Mgnetic field round long stright conductor crrying current 1 One of the spce vrible vectors which describe the mgnetic field is the mgnetic field intensity H. n homogeneous substnce the mgnetic field intensity (field strength) H 1 produced by current 1 defined s 1 H1 =, π The current in the wire produces mgnetic field tht circles round the wire nd frther from the wire gets weker. The mgnetic field strength decreses s 1/, where is the distnce from the wire nd pointed perpendiculr to n imginry line from the wire to the investigted loction. At distnce from wire crrying current 1 the vectoril formul of field intensity H 1 1 H1 = l 0 0, π where 0 unit vector from the wire to the point investigted, l 0 unit vector long the wire in the direction of current flow. l H 1 l 0 0 H 1 llustrtion of field intensity vector Henceforth simplifying nottion the vector symbol 1 denotes current (which is sclr volume). The direction of this vector coincides the direction of current flow nd the quntity of the vector is proportionl to the current mgnitude, ctully 1 = 1l 0. n n inhomogeneous nd ferromgnetic substnce the clcultion of the mgnetic field intensity H is more complicted, the Ampère's excittion lw hs to be used. The mgnetic field intensity H is vector, its direction in ech point of the spce follows the north (N) side of compss needle. The direction of field intensity round single wire meets the right-thred screw turns. The S unit of mgnetic field intensity

3 Fundmentls of mgnetic field [ H ] = A m. Mgnetic field intensity in ech point of the spce is usully illustrted by directed lines. These lines form closed pths, they do not rise nd do not end. S N N S H B Direction of the mgnetic field, the definition Consider wire of length l crrying current in field of mgnetic field intensity H. The force exerted F = µ 0 l H, where the direction of is the sme s the movement of positive chrges inside the conductor. n the cse shown on the figure F = µ l H. 0 1 H 1 B 1 H 1 B 1 1 F 1 F Force to current-crrying wire in the field of nother conductor Exmple A conductor crries 1 A, the mgnetic field intensity t 1 m distnce from the conductor is H = A m. The mgnitude of the force on n 1 m long piece of wire crrying 1 A, being in mgnetic field of H = 1 A m is F = N π 10 7 m. 3

4 VVEMA13 Alternting current systems 018 The second spce vrible vector describing the mgnetic field is the mgnetic flux density B. The mgnitude of flux density depends on the substnce in the spce, its S unit in honour of Tesl's 1 scientific ctivity Vs [ B ] = T= tesl = m. n cse of mgnetic field strength H the flux density B = µµ 0 r H, here µ r the reltive permebility, substnce specific, non-dimensionl fctor. The reltive permebility is often not constnt, its vlue my depend on the mgnetic field intensity nd lso on the initil mgnetic conditions. Exmple The mgnetic flux density t mgnetic field intensity H = 1 A m in free spce B=4π10-7 T. The direction of flux density vector B nd tht of field intensity H usully coincide: ligned with the direction of the compss needle to the north pole t ech point of the spce. The field lines re directed from the south pole to north inside mgnet (e.g. inside compss needle) nd from the north to south outside it. Consequently, mgnetic flux-lines leve the mgnet t the north pole nd go on towrds the south. The compss needle in ordinry use directed to the geogrphicl north pole of Erth. nside prticulr mterils the ferromgnetic mterils the mgnetic flux density is significntly incresed in contrst to free spce. Simple nd illustrtive explntion of this phenomenon is the contribution of the moleculr mgnets (or circulr currents) of such mterils to the flux density of externl mgnetic field. The reltive permebility µ r expresses the rtio of the flux density in comprison with tht in free spce. For ferromgnetic mterils t the sme field intensity H 1 µ r The vlue of µ r is usully determined by mesurements or by sophisticted clcultions. The mgnetic flux density lso illustrted by directed lines. The force F cting to piece of current-crrying wire with length l nd current in rbitrry mteril substnce round cn be expressed s: F = l B. n the cse ccording to the figure F = l B. 1 Exmple The force cting to wire crrying 1 A current plced in mgnetic field of 1 T flux density is F = 1 N m. The mgnetic flux of n re A defined s the surfce integrl of the flux density to tht re Φ = BdA. A n homogeneous field when the flux density vector is in right ngle to the surfce A: Φ=BA. The flux is sclr vlue, the S unit in honour of Weber's scientific ctivity [Φ]=Wb =weber=vs. 1 Tesl, Nikol ( ) engineer, investigtor, Serbin origin Weber, Wilhelm Edurd ( ) Germn physicist 4

5 Fundmentls of mgnetic field The mgnetic field is often visulised s lines of mgnetic flux tht form closed pths. The lines re close together where the mgnetic field is strong nd frther prt where the field is weker. By convention the flux lines leve the north-seeking end (N) of mgnet (e.g. compss needle or horse-shoe mgnet) nd enter its south-seeking end (S). Exmple Through n re of 1 m in perpendiculr homogeneous field of 1 T flux density the flux is 1 Wb. The Ampère's 3 lw of excittion This is the most importnt rule for clcultion of mgnetic circuit. The line integrl of the mgnetic field intensity vector H round rel or imginry closed pth is equl to the ggregted current flowing through tht surfce. This ggregted (net) current termed the excittion Θ of the re A. Hdl = JdA = Θ, here J the current density through the surfce A. A f the pth of integrtion in the left side goes long seprte prts with homogeneous field intensity thn H my be extrcted from the dot product nd the integrl, thus the integrl my be converted to sum. Wheres if the chrge flow concentrted in one-dimensionl thin wires thn the integrl in the right side of eqution turns into sum: H l =. n substnce of constnt permebility µ r the lw of excittion my lso be written s: B 1 Hd l = dl = Bdl =, or Bdl µ µ = µ, here µ=µ 0 µ r. i i i j j The Ampère's lw of excittion, relting stedy electric current to mgnetic field, ws pplying the line integrl of flux density in free spce (in n ironless surrounding) nd hs the originl form: B dl = µ 0, l where is the net conduction current through ny surfce A bounded by pth l. Mxwell's 4 ddition extended the pplicbility of Ampère's lw to time-dependent conditions involving the displcement current. According to Ampère's lw the enclosed net current includes ll currents tht penetrte ny surfce for which l is boundry. n the cse of cpcitor chrging the chrge ccumultes t the cpcitor plte. The integrl of the mgnetic field round the loop l must be the sme either surfce A 1 or A is choosen s shown in the figure. C C l l A 1 A 3 Ampère, Andrè-Mrie ( ) French physicist, mthemticin, chemist 4 Mxwell, Jmes Clerk ( ) Scottish physicist 5

6 VVEMA13 Alternting current systems 018 nterprettion of lw of excittion with conduction current nd displcement current As the cpcitor is chrging the electric field between the electrodes lso chnges with time. d The displcement current d determined from the chnge in chrge: () qt d =, the chrge is the surfce integrl of the displcement vector D, ccording to Guss's lw: Q = D da. The displcement vector with the electric field intensity D = ε 0 E, if free spce considered between the electrodes of the cpcitor. Thus the formul for displcement current: d d = ε 0 E da. A The extended form of lw of excittion: d B dl = µ 0 + ε 0 E da. l A The Ampère-Mxwell lw of excittion is mentioned s 4th Mxwell's eqution. t expresses the reltion between the mgnetic field nd the chnge of electric field. When mgnetic circuit or the necessry excittion is clculted the mgnetic field produced by the conduction current considered. Clculting with mgnetic field intensity preferred to flux density becuse the B(H) reltion determined from mgnetistion curves. A Exmple Consider current-crrying wire with current 1 A, in distnce from tht wire the field intensity H =. π f the surrounding substnce is non-ferromgnetic nd the closed pth investigted is concentric circle with rdius nd the direction of integrl grees to the direction of field intensity H thn Hd d l = l = π =. π π l l 3 l 4 r 1 r H l 1 Demonstrtion of the lw of excittion Similr result is obtined when (in non-ferromgnetic substnce) the pth is closed vi different rcs ccording to the next figure: 6

7 Fundmentls of mgnetic field long l 1 the field intensity H1 =, π r1 long l 3 nd l 4 the field intensity H is perpendiculr to the pth of integrl, therefore the sclr product Hdl = 0, long l the field intensity H =. π r The result of integrl 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 When the existing or desired field intensity H is known the excittion Θ which cn produce this field my be clculted. Visulistion of mgnetic field (flux-lines) Current loop B Flux-lines of current loop (turn) Solenoidl nd toroidl coil Since the length of solenoidl coil l is much greter thn its dimeter d, i.e. l» d, the field inside the coil my be considered uniform nd outside the coil it my be neglected. The sme ide used for toroidl coil, if verge dimeter D» d. l d d Mgnetic field of solenoidl nd toroidl coils At these coils the single turns re in series, nd the sme current flows in ll turns. Applying the Ampère's lw of excittion Θ=Hl=N, where N the number of turns (number of wires, number of currents). D 7

8 VVEMA13 Alternting current systems 018 At given direction of current flow the direction of the mgnetic field produced in coil depends on the direction of the twist of turns. B B Mgnetic field of right-thred nd left-thred coils Force to current-crrying wire in externl mgnetic field n homogeneous mgnetic field ccording to the formul for force: F = l B. F B B F llustrtion of force to current-crrying wire in homogeneous mgnetic field "Useful" mgnetic field nd lekge The lekge mgnetic field is formed by field lines leving the desired pth. f consider two linked coils (like the coils in trnsformers or the sttor-rotor coils in rotting electricl mchines) only portion of produced mgnetic field is linking to both coils, the rest of the field is leking, linking only to the coil tht produced it. The lst prt termed s flux lekge or mgnetic field lekge. The mesure of lekge is defined with coefficient of lekge σ: φ σ = l (0 σ 1), φ t where φ t the totl flux, while φ l the flux lekge. n certin cses the mgnetic lekge is importnt, e.g. the lekge rectnce cn limit the short circuit current. The lw of flux refrction (the boundry conditions) f mgnetic field psses boundry lyer of substnces with different permebility the field intensity H nd the flux density B chnge different wy. Let us investigte the conditions t the boundry between two mterils of permebility µ 1 nd µ. Consider the flux lines crossing the boundry s shown in figure with ngle of incidence α 1 nd ngle of refrction of α. The flux mgnitude through n elementl re inside the boundry lyer da pproched from both side of the boundry must be identicl, ssuming tht no mgnetic flux emerges from 8

9 Fundmentls of mgnetic field elementl surfce s it pproches to zero da 0. Since the flux-lines re closed, the overll fluxes in both substnces must be identicl: Φ = BdA =B 1n da=b 1 cosα 1 da=b cosα da= B n da, da tht is the B n norml component of flux density vector B remins unchnged, it crosses boundry continuously. Refrction of flux density vector On the other hnd, ccording to the excittion lw the line integrl of the mgnetic field intensity round closed pth crossing the boundry lyer of wih dl is equl to zero if no excittion (no current flowing) inside the boundry. Around the closed pth the integrl Refrction of field intensity vector Hdl = 0 since no current linked: Hdl = H 1t dl- H t dl= H 1 sinα 1 dl- H sinα dl=0, hence H 1t =H t, tht is the H t tngentil component of field intensity vector H remins unchnged in both side. 9

10 VVEMA13 Alternting current systems 018 The tngentil component of flux density vector B nd the norml component of field intensity vector H re chnging through the boundry lyer. From the sttements bove H 1 sinα 1 = H sinα, or substituting flux density for field intensity: B1 B sinα1 = sinα sinα1 sinα tgα1 µ r1 µµ 0 r1 µµ 0 r = =. µµ tg B B 0 r1cosα1 µµ 0 rcosα α µ r 1cosα1 = cosα The flux-lines t iron-ir boundry lyer Suppose tht µ r1»µ r (e.g. t the iron-ir boundry, where µ riron =10 6, µ rir =1), thn tgα 1»tgα, α 1»α, i.e. α 1 ~ 90 while α ~ 0. Consequently the mgnetic flux emerges into ir norml to the surfce of iron with pproximtely infinitely permebility, the flux-lines leve the iron t right ngles. The Frdy's 5 induction lw This lw is one of the most importnt sttement of electricl engineering, discovery of the phenomenon described in it mde (nd mke) possible the genertion nd wide public use of electricl energy. Consider wire loop ( single turn of coil). f in ny cse the flux through the surfce enclosed by the loop is chnging, it produces electricl field nd voltge ppers (inducing) in the loop. The mgnitude of induced voltge u i (t) is proportionl to the time derivtive of flux φ(t): d ( t) ui () t = φ. The flux chnge occurs either becuse the mgnetic field itself is chnging with time (trnsformer induction) or becuse the wire loop is moving reltive to mgnetic field (motionl induction). The Frdy's induction lw describes both phenomen. ) f wire loop is fixed nd the flux is vrying with time becuse the excittion current or the mgnetic circuit is chnging the phenomen clled trnsformer induction. b) n the cse of motionl induction conductor (or wire loop) during its displcement crosses the mgnetic flux lines i.e. the motion hs component perpendiculr to the fluxlines. The min point of induction is tht the chnge in mgnetic field cuses electricl field. Using the term of induced voltge phenomenon in mgnetic field my be replced with phenomenon in electricl circuit. n rel equipment the two types of induction (trnsformer nd motionl) often pper simultneously (e.g. in rotting electricl mchines). 5 Frdy, Michel ( ) English physicist 10

11 Fundmentls of mgnetic field mportnt notice: f the spce contins both sttic electric nd chnging mgnetic fields, the electric field become non-conservtive, becuse the line integrl is no more pth-independent, since exist such closed pths which enclose chnging mgnetic field nd the integrl long such pths is not zero. n this cse the electric potentil s sclr descriptor is unusble. n closed loop the induced voltge produces current ccording to the resistnce of the loop. The voltge-drop on the resistnce is blncing the induced voltge if no other voltge source in the loop. The Kirchhoff's voltge lw for dc circuits: R + U = 0, hve to be extended: j R + U + U = 0, j j k ik here U i is the induced voltge, U b is the non induced voltge (e.g. from kind of glvnic or bttery type source). For sinusoidlly chnging lternting current the voltge lw bove is vlid for phsors nd the impednces re tken into ccount insted of resistors. Trnsformer induction The reference direction of the flux chnging d φ nd tht of the chrge-seprting electricl field intensity E re ccording to the figure, Ui = Edl. E n bn j dφ > 0 j j k ik - + U i Reference directions for trnsformer induction According to the Frdy's lw the induced voltge depends not on the mgnitude of mgnetic flux φ but on the mgnitude nd the direction of the derivtive of mgnetic flux d φ. 11

12 VVEMA13 Alternting current systems 018 φ φ U i - + U i + - φ t dφ > 0 φ t dφ < 0 Polrity of the induced voltge t different directions of derivtive of flux (φ > 0) U i - + U i + - φ φ φ t dφ > 0 φ t dφ < 0 Polrity of the induced voltge t different directions of derivtive of flux (φ < 0) The flux linkge Usully the chnging flux is enclosed by not single wire loop, but rther by coil of N series turns (nd ll turns re connected in series) so the induced voltges of the single turns re 1

13 Fundmentls of mgnetic field ggregted long the coil. f ech turn of the wire linked with the sme flux mgnitude then the resultnt induced voltge ( ) u() t N d φ t i =. The sum of the fluxes linked with the single turns gives the resultnt flux linkge of the coil ψ=nφ which cn be used for clcultion of the resultnt induced voltge: d ( t) ui () t = ψ. The physicl unit of the flux linkge Ψ is the sme s tht of the flux Φ: [Ψ]=Wb=Vs. Lenz's 6 lw As follows form the principle of energy conservtion the currents nd forces produced by induction hve such n effect, which is decresing the process generting them. d The induced voltge ui = φ hs such polrity tht produces current i through n externl resistnce which opposes the originl chnge of flux linkge, decresing the inducing effect. f coil is moving inside mgnetic field then such induced voltge is generted, tht the current produces force which cts ginst the movement (s brking force). n other words the mgnetic field produced in the process of induction cts for conservtion of the initil stte, it tends to mintin the existing flux. This principle ppers in the phenomenon of self induction. φ dφ > 0 i R U i - + i Mgnetic effect of the current produced by induced voltge Motionl induced voltge Motionl induced voltge is generted in wire when it moves in sttic mgnetic field. This voltge my be explined with force of interction between the sttic mgnetic field nd the chrges trvelling inside the wire. (Remember, for current-crrying conductor the produced force expressed s: F = l B. This force cts to the chrges inside the conductor. The direction of this force is perpendiculr to the current flow.) 6 Lenz, Heinrich Friedrich Emil ( ) physicist, Germn origin 13

14 VVEMA13 Alternting current systems 018 Consider the chrges trvelling together with the wire. Let us define this chrge movement s current which is not rel current, nywy it helps to clculte force cosed by the rel movement of chrges. The force F ccording to : h Q F = h B = dhq B = Qv B, becuse = t t dh h nd v = t dh. l +Q dh B +Q F + v E U i - F + R - h A possible illustrtion of motionl induction n the Figure the conductor is moving in homogeneous mgnetic field B nd the speed v is perpendiculr to the mgnetic field. Hence the direction of displcement dh is perpendiculr both to mgnetic field nd to the conductor. nside the conductor mgnetic force F cts to the chrges. This force seprtes the chrges thus produces n electric field E long the conductor. The direction of the electric field is the sme s the force F (cting to the positive chrges) E = = F Q v B. This electric field results in the ppernce of induced voltge. The voltge u i between the ends of the conductor with length l in homogeneous mgnetic field expressed s ui = E l = v Bl = l B v, if the reference of voltge directed from chrges (+) to chrges (-). This voltge is n induced voltge, produced by the chrgeseprting electric field E, often mentioned s electromotive force (emf). For this electric field dφ Edl=. The induced voltge cuses (rel) current in closed circuit. The interction of the (rel) current nd the mgnetic field B produces physicl force F ginst the movement, ccording to the Lenz's lw. (Another explntion: the density of flux lines incresing in the direction of movement.) This men tht in cse of closed circuit the continuous movement of wire requires continuous force, energy investment. There re two forces discussed: - force F tht cts to the chrges trvelling with the conductor, the consequence of which is the chrge-seprting effect, n electric field E nd the induced voltge u i, - due to the current cused by this induced voltge force F cts to the conductor ginst it's movement. The direction of the two forces F nd F re different. 14

15 Fundmentls of mgnetic field The integrl form of Frdy's lw usully written s: d E dl = B da. l A This eqution involves two different phenomen: producing of both trnsformer nd motionl induced voltge. n cse of trnsformer induction in the right side of the form the flux density is chnging with time, thus cn be used n lternte form: E dl = t B da. E represents the electric l A field in sttionry reference frme of ech segment dl of the pth of integrtion. This induced electric field is very different from the fields produced by electric chrge. The induced electric field lines form complete loops long the pth l, the boundry of surfce through which the mgnetic flux is chnging over time. The induced electric field is nonconservtive, no electric potentil cn be defined. n cse of motionl induction the mgnetic flux is chnging t stedy flux density e.g. by chnging surfce A. As fr s the flux φ = B da, the induced voltge proportionl to the A d rte of chnge with time of the mgnetic flux through surfce A: ui = φ. E in cse of motionl induction represents the electric field t ech segment dl of the pth of integrtion in the reference frme in which tht segment is sttionry (i.e. in moving reference frme). The negtive sign in Frdy's lw shows tht the induced voltge opposes the chnge in flux (Lenz's lw). The mgnetic field of long stright wire of finite dimensions The mgnetic field produced by low-frequency lternting current or direct current flowing through long conductor with finite size of cross section exists not only outside the conductor but is lso significnt within the conductor. The internl flux links only frction of the current, this linkge must therefore be treted seprtely from externl. Consider such conductor with rdius r c in free spce, crrying current. Externl field The externl mgnetic field my be clculted pproximtely by the sme wy s in the cse of conductor with infinitely smll size (one dimensionl conductor). Assum tht the current distribution within the conductor is uniform, thus the current density J nd the current : J = = Ac r, nd = JdAc = Jr c cπ π, where r c the rdius of conductor, A c the cross-section re of conductor. 15

16 VVEMA13 Alternting current systems 018 l da R r c d Clcultion of mgnetic flux n the rnge of distnce from the centre of conductor >r c the field intensity H e of the externl mgnetic field He ( ) =, while r c. π µ 0 The externl flux density in free spce is Be( ) = µ 0He( ) =. π The nnulus externl flux dφ e enclosed by n nnulus re da=ld determined s µ 0 dφ ( ) B ( ) da d e = e = l. π Since the single wire is considered s one turn the flux linkge is equl to the flux dψ e =dφ e, nd the totl externl flux is R R µ 0l d µ 0l R ψ e = dφ = π = ln, π r r r c c c here R is sufficient finl distnce where the field considered prcticlly zero (theoreticlly R but nturl log of infinity gives infinity). The externl inductnce derived from the externl flux: ψ e φ e µ 0l R Le = = = ln. π r c 16

17 Fundmentls of mgnetic field H H i H e R r c d The mgnetic field intensity vs. distnce Exmple µ l n free spce µ=µ 0, when R=5r c then L e = 16, = 3, 10 π 0 7 l (H), or L e = 0.3 µh/m. nternl field Consider conductor tht consists of n infinite number of prllel thin wires nd the current distribution is uniform (the current density is constnt throughout the cross-section re). Clculte the current flowing inside rdius <r c s frction of totl current : J = π = for r c. rc The internl mgnetising force H i is due only to the current within rdius Hi( ) = = =, r c. π πrc πrc At the surfce of the conductor the formuls for externl nd internl fields give the sme results. The internl flux density Bi( ) = µ Hi( ) = µ π r, c where µ the totl permebility of the conductor mteril, if it is non-ferromgnetic then usully considered s µ=µ 0. The internl flux for nnulus re da: µ dφ i( ) = BidA= d π r l. c f the whole cross section re of the conductor is considered s one turn, then the flux dφ i within rdius r c links only turn N frction of 1, proportionl to the cross-section re 17

18 VVEMA13 Alternting current systems 018 N = r, r c. v Hence the elementl internl flux linkge for r c 3 µ µ dψ i( ) = Ndφi( ) = ld = l d. 4 rc π rc π rc The totl internl flux linkge: rc µ l 3 µ l ψ i = d= π r. 4 8π The inductnce derived from the internl flux: L c 0 i ψ i µ l = = 8π Exmple µ 0 l 7 if µ=µ 0, then L i = = 05. l 10 (H), or L i = 0.05 µh/m. 8π As mentioned bove the current in the segment of conductor inside n rbitrry rdius r c is only the frction of the totl current : =, wheres the sme segment of the conductor is linked with the totl internl flux φ i. Hence the internl inductnce of segment of con- rc ductor derived from the internl flux: l i L i r c Chnge of internl inductnce l i ( ) ψ i = = c L r i for r c. rc The inductnce of the conducting elements nerer to the centre is greter then ner to the surfce of wire. 18

19 Fundmentls of mgnetic field For lternting current the inductive rectnce of the conductive elements ner to the centre is greter then ner to the surfce, nd hence more current flows in the elements ner to the surfce. This phenomenon clled skin effect or current displcement. Skin effect According to complex, sophisticted clcultions in circulr conductor the current density decresing from the surfce to the centre of conductor exponentilly. A depth of penetrtion δ used to define the distnce t which the current density decreses to 1/e of its vlue t the outer surfce. Approximtion: in cses the rdius of the conductor r c > (3-5)δ, the current density usully considered uniform from the surfce to the depth of penetrtion nd zero within the rdius <r c -δ. The influence of skin effect is obviously more pronounced t lrger inductive rectnce which vries with the ngulr frequency ω of the current, nd with the inductnce. The inductnce vries with the permebility of the conductor µ, consequently the lrger the permebility, the smller the depth of penetrtion. On the other hnd, the greter the resistivity of the conductor ρ, the smller will be the effect of the vrition in inductive rectnce (within the impednce) cusing non-uniform current distribution. The reltionship between the depth of penetrtion δ nd ω, µ nd ρ expressed in formul ρ ρ 1 ρ ρ δ = = = = 5033,. ωµ πµ 0 fµ r πµ fµ fµ 0 r r The eqution for the depth of penetrtion bove my be pplied to flt conductors (r c ) e.g. power buses. The sme considertions pply to the penetrtions of lternting mgnetic flux in conductors becuse no mgnetic field cn be produced without current. The proximity effect Similr phenomenon occurs when two conductors crrying lternting currents re in ech other's mgnetic fields, there will lso be redistribution of currents. The currents re crowding to those prts of the conductors, which link the lest mount of flux nd therefore hve less inductnce. The illustrtion of proximity effect with skin effect, the current flow identicl Exmple The vlues for copper: ρ Cu = Ωm, µ rcu =1, the depth of penetrtion t frequency f=50 Hz δ Cu =9.49 mm (δ Cu =8.66 mm t f=60 Hz, δ Cu =0,949 mm t f=5 khz). The vlues for luminium: ρ Al = Ωm, µ ral =1, the depth of penetrtion t frequency f=50 Hz δ Al =1 mm (10.95 mm t f=60 Hz). The vlues for iron: ρ Fe = Ωm, µ rfe =5000, 19

20 VVEMA13 Alternting current systems 018 the depth of penetrtion t frequency f=50 Hz δ Fe =0.35 mm (0.3 mm t f=60 Hz). δ (mm) Al Cu Fe f (Hz) The depth of penetrtion vs. frequency From the clcultions of penetrtion depth it is obvious tht t frequency f=50 Hz uneconomic the use of copper wire with dimeter more then bout 0 mm or luminium wire with dimeter more then bout 5 mm. For iron the depth of penetrtion highly depends on permebility thus lso on sturtion, in sturted iron (µ r 1) the depth of penetrtion more thn long the liner zone of mgnetistion curve. For wire with cross-section of rectngle shpe the depth of penetrtion depends on the rtio of the sides. n rectngle wire (e.g. power bus) the current density long the longer side distributed s in the figure when the geometric rtio δ nd b» δ. J x b Distribution of current density in rectngle wire n overhed power lines to increse the trnsmitted power t the sme voltge nd current density is preferble to use bundled conductors. Bundles consist of severl conductors connected prllel. The distnce from ech other is kept by spcers. The conductors often form cylindricl configurtion round circle of cm dimeter. Widely used the two- threend four-conductor bundles, 765 kv lines ws built by Americn Electric Power using six conductors per phse in bundle. 0

21 Fundmentls of mgnetic field The electric nd mgnetic field of bundle conductors is similr to tht of single conductor with dimeter of the bundle. Two- three- nd four-conductor bundles Further dvntges of bundle conductors: - hve lower rectnce, compred to single conductors, - reduce the voltge grdient in the vicinity of the line reducing the coron loss, - lines hve higher cpcitnce in comprison with single lines, thus, they hve higher chrging currents, which helps in improving power fctor. Skin effect in ferromgnetic surrounding The phenomen in conductor surrounded by ferromgnetic mteril re bsiclly the sme s the skin nd proximity effects in ny system of conductors with lternting current. ) b) Representtive shpes of slots in squirrel-cge rotor ) deep-br b) double cge The figure shows in cross section deep, nrrow br of squirrel-cge rotor nd double cge of induction motor. Next figure shows the generl chrcter of the slot-lekge field produced by the current in the br within such slot, in prticulr when the depth of the slot exceeds its wih. Slot-lekge field in deep-br nd double cge rotor f the rotor iron ssumed of infinite permebility, ll the lekge-flux lines would close in pths below the slot, s shown. f imgine the br to consist of lyers, which re electriclly connected in prllel, the lekge inductnce of the bottom lyers is greter thn tht of the 1

22 VVEMA13 Alternting current systems 018 top lyers becuse the bottom lyer is linked by more lekge flux. Consequently, the volume of lternting current in the low-rectnce upper lyers will be greter thn tht in the highrectnce lower lyers. The current will be forced towrd the top of the slot, in ddition the current in the upper lyers will led the current in the lower ones. The non uniform current distribution results in n increse in the effective resistnce nd smller decrese in the effective lekge inductnce of the br. J J h f r =f 1 h f r 0 Distribution of current density in the brs of double cge rotor Since the distortion in current distribution depends on n inductive effect, the effective resistnce is function of the frequency (i.e. of the slip) nd lso function of the depth of the br nd of the permebility nd resistivity of the br mteril. The effect cn be explined s follows: the prticulr prts or lyers of the wire re linked by different mgnetic flux, therefore the inductnce l = ψ i ( π l) nd the impednce z = r + f r lso differ depending on the distnce from the top of the slot. Applying double cge or deep slot increses the frequency-dependence of the rotor resistnce. n double cge rotor the upper brs (close to the rotor surfce) re mde from mteril of less cross-section nd more resistivity (e.g. brss) while lower brs re mde from mteril of more cross-section nd less resistivity (e.g. copper). At blocked rotor or strt-up (the slip S=1) the frequency of the rotor current is equl to tht of the sttor current (e.g. f r =50-60 Hz), due to the skin effect the rotor current flows minly in the upper cge of more resistnce (nd less rectnce), while round the rted (nominl) speed (e.g. S n , f rn =1.5-3 Hz) the distribution of rotor current determined by the rtio of the resistnces of the cges. f r =f 1 J f r 0 h Distribution of current density in the wire of deep slot rotor

23 Fundmentls of mgnetic field By mens of construction my be chieved tht the effective resistnce of rotor coil is more t strt-up nd less t nominl speed. The phenomen nd its influence is similr in rotor with deep slot, but vry of the current density J is continuous in function of the distnce from the top of the slot. The influence of skin effect to the slot-lekge field (pproximtion) Exploiting the frequency dependence of the effective rotor resistnce some improvement my be chieved: the strting torque incresing, the speed fll t nominl performnce decresing. w w 1 R r (t S n ) R r (t S=0) T T bg MT i1 s T l T s1 Sttic speed-torque chrcteristic of induction mchine The strting torque T s t the smller resistnce wouldn't be enough to strt with lod torque T l. On the other hnd, t the greter resistnce the speed fll t lod torque T l would be greter in stedy stte. The two chrcteristics re simplifictions, the trnsition between the chrcteristics is continuous during ccelerting of the rotor, the vry of lekge inductnce is neglected. n the figure: w 1 synchronous speed, T bg brekdown torque (breking mode), T bm brekdown torque (motor mode), T s1, T s1 strting torque, T l lod torque, S n nominl slip. 3

24 VVEMA13 Alternting current systems 018 w w 1 1 b c T T n Representtive torque-speed curves of induction mchines with skin effect ) proper curve for mchine strting without lod, b) curve for mchine strting with rted lod, during strt-up pproximtes the double rted torque, c) the strting torque exceeds the rted torque. nfluence of eddy currents to the distribution of the mgnetic field Due to the chnge in the mgnetic flux n induced emf ppers nd eddy currents flow in the mgnetic core. According to Lenz's lw the eddy currents oppose the originl chnge in the flux, the inducing process produces such field, which displces the flux to the surfce of the mgnetic conductor. The depth of penetrtion of the mgnetic field my be clculted s tht of current density. dψ eddy nfluence of eddy currents to the distribution of mgnetic field n tht ferromgnetic prts of mchines which re determining the pth of mgnetic field the eddy currents re not desired becuse they displce the flux nd increse the iron losses. The eddy currents cn be decresed using lmintion of iron or using ferrite core. Composed by: Kádár stván April

25 Fundmentls of mgnetic field Questions for self-test 1. Explin the force on current-crrying conductor in mgnetic field.. Review the vribles describing the mgnetic field. 3. Define the mgnetic field intensity, flux nd flux density. 4. Explin the mgnetic permebility. 5. Explin the Ampere's lw of excittion. 6. Explin the Frdy's induction lw Frdy's induction lw. 7. llustrte the flux lekge. 8. llustrte pproximtely the mgnetic field of current-crrying conductor nd conductor loop. 9. llustrte pproximtely the mgnetic field of solenoidl nd toroidl coils. 10. How pproximted the clcultion of mgnetic field of solenoidl nd toroidl coils? 11. Define the flux linkge. 1. Explin the motionl induction. 13. Explin the trnsformer induction. 14. Explin the Lenz's lw for motionl nd trnsformer induction. 15. llustrte the mgnetic field of current-crrying conductor of finite dimension. 16. llustrte the current distribution in conductor of finite dimension. 17. Explin the inductnce of conductor of finite dimension. 18. Explin the skin effect. 19. Explin the depth of penetrtion in current-crrying conductor. 0. How pproximted the current distribution with respect to skin effect? 1. Explin the proximity effect.. Why bundle conductors re used? 3. llustrte the skin effect in ferromgnetic surrounding. 4. Explin the eddy current nd it's influence on the distribution of mgnetic field. 5. How cn be reduced the eddy current? 5