Properties of ferromagnetic materials, magnetic circuits principle of calculation

Transcription

1 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton Ferromagnetc materals Several materals represent dfferent macroscopc magnetc propertes they gve dfferent response to external magnetc feld The reason for dfference are certan dfferent mcroscopc propertes lke electron shell confguraton movement and spn of electrons or rather the nfluence of external magnetc feld on these propertes In physcs da- para- and ferromagnetc materals are dstngushed whle from pont of vew of electrcal engneerng practce usually all non-ferromagnetc substances consdered as free space wth relatve permeablty r agntude of the relatve permeablty of ferromagnetc materals (ron nckel cobalt and ther alloys) may be very hgh even n the order of Ferromagnetc materal may be alloyed also from non-ferromagnetc substances (eg Ag-n-Al) In ferromagnetc materals the relatonshp between B and H (the magnetsaton curve) s a functon wth multple value t s non-lnear hstory dependent and often tme dependent ts shape can not be descrbed analytcally hence t has to be determned for each type of materal expermentally or wth sophstcated calculatons eg wth fnte element method agnetsaton curve When an ntally non-magnetsed pece of ferromagnetc materal s magnetsed by slowly ncreasng the external feld then the flux densty wll also ncrease The curve B(H) showng ths vary of flux densty B vs change of magnetc feld ntensty H called prmary orgnal dc or normal magnetsaton curve (dashed lne n the fgure) B B max B r b c d -H c a H H max Typcal shape of magnetsaton curve In the prmary magnetsaton curve 4 sectons are dstngushed: a - startng zone b - lnear zone c - knee of the curve (ntermedate zone) d - saturated zone

2 VIVEA3 Alternatng current systems 08 As feld ntensty acheved the zone of saturaton and H decreases from ts maxmum value H max to zero slowly the flux densty B whle also decreases the workng pont wll not return along the orgnal fallng curve but on a new (somewhat hgher) one and a B r remanent magnetsaton (resdual remanent magnetc flux densty or remanence retentvty) remans although H becomes zero The change of B delayed wth respect to change of H (hysteress delay) To reduce remanent magnetc flux densty to zero needs a feld ntensty n the opposte drecton -H c called coerctve force As the fgure shows the value of permeablty the values of relaton B/H s not sngle-valued ts change s non-lnear t depends on the prevous value of feld ntensty H and the rate of change For changes above the flux densty of saturaton B max the relatve permeablty usually consdered as r ~ B B max H max H Hysteress curves of dfferent maxmum value of H max If the feld ntensty s alternatng (AC) the flux densty changes n the B(H) plane along a loop known as hysteress loop The largest hysteress loop belongs to the peak values B max and H max determned by the flux densty and feld ntensty of saturaton The hysteress loops of less peak values located nsde the largest loop If the magnetc feld ntensty changes slowly the loop s called statc hysteress loop Dynamc hysteress curves In case of alternatng magnetc feld generated by alternatng current of network frequency (or hgher) the ponts of the B(H) plane form a full hysteress loop durng each perod In adon due to the alternatng flux an nduced voltage appears whch produces eddy currents nsde the ferromagnetc (conductor) materal Accordng to Lenz's law the magnetc feld of these eddy currents makes the change of the flux densty more delayed therefore ncreasng the frequency the dynamc hysteress loops become spacous compared to statc ones

3 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton statc dnamc B H Statc and dynamc hysteress curves Relatve permeablty In each pont of magnetsaton curve the absolute permeablty B H and the relatve per- B meablty r may be determned Because of the non-lnearty of the curve n engneerng practce several smplfcatons 0H appled B P α d α f α s 0 H Explanaton of the full the dfferental and the ntal permeablty - full (ordnary) permeablty: the slope of the lne from orgn to the ponts of prmary magnetsaton curve (eg to pont P) B rf α f H tg Ths s the most wdely used snglevalued approxmaton however for H0 t gves meanngless value 0 - ntal permeablty: the relatve permeablty at low exctaton level the slope of the startng zone of prmary magnetsaton curve rs tgα s - dfferental permeablty: the slope of tangent at a gven pont of prmary magnetsaton curve (eg at pont P) db rdff α d dh tg 0 3

4 VIVEA3 Alternatng current systems 08 B ncremental B reversble 0 H 0 H 0 H Illustraton of ncremental and reversble permeabltes - ncremental permeablty: specfc for the narrow loop due to small cyclc changes around a pont of the magnetsaton curve when a small alternatng feld s supermposed on a dc or B slowly changng feld rnc 0 H - reversble permeablty: the same as ncremental one but the cyclc change s small enough and the elementary loop forms a sngle lne In electrcal power engneerng practce mostly the full (ordnary) permeablty s used B-H saturaton curves of three magnetc materals Saturaton curves of magnetc and nonmagnetc materals - all curves become asymptotc to the B-H curve of vacuum where H s hgh Calculaton of magnetc crcuts The magnetc crcut s a closed segment of the magnetc feld n whch the flux consdered constant flux lnes do not leave such a segment Actually all closed flux lne form a magnetc crcut In magnetc crcuts usually ferromagnetc components determne the path of the flux lnes The calculaton s smple n cases the path of flux (the arrangement) s determned 4

5 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton agnetc crcuts examples The requred exctaton (sometmes mentoned as mmf) for a desred feld of a complex magnetc crcut what s composed from dfferent segments may be easly calculated Whereas to answer the nverse problem that s calculaton the result of known exctaton s dffcult because of the non-lnearty of magnetsaton curve because the dstrbuton of total exctaton depends on change n B/H relaton The flux leakage may be consdered as a result of calculaton or estmaton but may be neglected as well Along a magnetc crcut usually there are segments of dfferent propertes (geometry permeablty) and may occur ramfcatons The Ampère's law of exctaton s vald for slow changes only: for drect current and tme nstants of alternatng current In case of fast changes the effects of eddy currents have to be taken nto account too Seres magnetc crcuts The seres magnetc crcuts usually consst of segments of dfferent cross-sectons lengths and materals Calculaton the exctaton necessary to generate specfed magnetc flux Suppose the Φ flux s gven specfed along the magnetc crcut nvestgated and the leakage neglected Φ l 0 accordng to the fgure B H A B H l / l l / B δ H δ 0 A δ A δ Outlne of a seres magnetc crcut In ths case the flux densty n the ar gap s B δ Φ and n the further ferromagnetc seg- A δ 5

6 VIVEA3 Alternatng current systems 08 ments B Φ B Φ and so on A A Bδ The feld ntensty n the ar gap s smply calculated: Hδ whle n the ferromagnetc 0 segments the feld ntenstes H H etc or the relatve permeabltes r r etc are usually determned from the magnetsaton curve B Φ B Φ H and H 0 r 0 ra 0 r 0 ra Applyng the law of exctaton the total exctaton Θ for the whole crcut usng terms 0 r : Bδ Φ l Θ Θ Hl + H l + H l + K l Φ δ 0 A A snce Φ provded to be constant The resultant exctaton can be defned as sum of partal exctatons for the sngle segments of crcut In such cases when the most of resultant exctaton belongs to the ar gap the ferromagnetc (ron) parts of magnetc crcut may often be neglected (f ron» 0 thus H δ»h ron ) Example Gven B δ B ron T δ mm l ron m from the magnetsaton curve rron 0 6 Bδ Bδ 6 6 A The feld ntensty nsde the ar gap: Hδ Bδ m Bron Bδ A nsde the ron segments: Hron 08 Bδ 08 0 rron 0 rron m The resultant exctaton s the sum of the requred exctaton for ron and ar gap: ΘΘ ron +Θ δ The exctaton for ron Θ ron H ron l ron 08 A whle that for ar gap Θ δ H δ l δ 800 A Θ 8008 The current needed when the number of turns n the col N: I ( A ) N N The exctaton needs for ron segment n ths case s only 0% of the total exctaton Usng ron of less permeablty the necessary exctaton for ron ncreases and s not neglgble For example f rron 0 3 H ron 800 A m Θ ronh ron l ron 800 A that s 50% of the full exctaton (Θ600 A) The nverse problem when the flux or flux densty have to be answered for known current I s dffcult to solve because the dstrbuton of full exctaton depends on the relaton among permeabltes of each segments however to determne the permeabltes the feld ntensty needed because of the non-lnearty of magnetsaton curve A practcal soluton s the calculaton of exctatons or currents whch generate dfferent fluxes and then a plot of Φ(Θ) or Φ(I) relatonshp may gve the answer of problem Parallel magnetc crcuts Snce the flux lnes are closed the total flux at nput and output sdes are equal when the leakage s neglected: ΦΦ +Φ Accordng to exctaton law for the l -l closed loop: H l - H l 0 hereby H l H l Θ p 6

7 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton that s the exctatons of all parallel segments Θ p are equal Substtutng feld ntenstes H and H : Φ Φ A A Θ p l l from whch follow Φ Θ p and Φ Θ p A A l l Φ H A l Φ Φ Φ H A l Outlne of a parallel magnetc crcut The total flux Φ wth exctaton Θ p : A A A Φ Φ + Φ Θ p + Θ p l l l The magnetc Ohm s law When modfyng the formula of exctaton law Θ Hdl because of formal analogy the equatons for complex magnetc crcuts often mentoned as magnetc Ohm s law Wth substtutons H B and B Φ nto the lne ntegral of the magnetc feld ntensty the A formula for the exctaton of seres magnetc crcut Φ l Θ l Φ A A s smlar to the formula for the resultant voltage n an electrcal crcut whch conssts of seres conductor segments wth fnte resstances: I l U l I I R γ A γ A where γ s the specfc electrc conductvty the recprocal of the specfc resstvty ρ ρ I R U R U R U γ l A γ l A U γ l A Illustraton of Ohm's law - seres connecton The total exctaton of seres magnetc crcut may be wrtten as 7

8 VIVEA3 Alternatng current systems 08 Um Φ Rm where U m Θ the total magnetc voltage (exctaton) Rm l the magnetc resstance (magnetc reluctance) of th segment The total magnetc resstance of the seres segments: R R herewth U m ΦR m A m A A Physcal unts of varables above: [U m ] A [Φ] VsWb [ Rm ] Vs Wb The more the permeablty the less the magnetc resstance and the exctaton (magnetc voltage) of a segment n a magnetc crcut The exctaton of a segment n a magnetc crcut may be termed as magnetc voltage of that segment for the th segment: l U m Φ Φ Rm A The exctaton law may be formulated as follows: the lne ntegral of the magnetc voltage around a closed path s equal to the exctaton Θ of the area enclosed by that path Θ U m The formula derved for the total flux of parallel crcut A Φ Θ p l s smlar to the formula for the resultant current n an electrcal crcut whch conssts of parallel conductor segments wth fnte resstances: I m U I R γ l A I R γ l A I R γ l A Illustraton of Ohm's law - parallel connecton A I U G U γ l The total flux of parallel magnetc crcut may be wrtten usng the above relatons as: Φ U mp Λ A where Λ the magnetc conductvty (magnetc permeance) of the th segment the recprocal of magnetc resstance Physcal unt of magnetc conductvty s l Rm Vs [ Λ m ] Wb A A 8

9 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton The total magnetc conductvty of the parallel segments: Λ Λ whereby ΦU m Λ m ΘΛ m From the analogy above may be buld up electrc crcuts substtutng the magnetc crcuts Such substtuton have to be used carefully because the analogy s only formal the physcal phenomena are essentally dfferent n the two crcuts a) The electrc current I s a real flow of the charges (charge-carrer partcles) whle the magnetc flux Φ descrbes the state of the space wthout any movement of partcles b) A steady electrc current produces losses on the resstors of crcut whle a steady magnetc flux does not requre energy (only the buldng up or the changng of magnetc feld) c) The lne ntegral of the electrc voltage around a closed path s equal to zero f the path does not enclose changng flux d Φ 0 whle the lne ntegral of the magnetc voltage around a closed path equal to zero f the path does not encrcle current ΣI0 d) The electrc conductvty of electrc conductors γ s usually constant (at constant temperature) and current-ndependent whle the permeablty of ferromagnetc materals (magnetc conductors) vary wth flux (flux densty) sgnfcantly e) The relaton between the conductvty of electrc conductors and that of nsulators s about 0 0 thus current leakage s usually neglgble Ths relaton for magnetc conductors and nsulators s about thus flux leakage and ts nfluence s often has to be taken nto account f) The superposton method s unusable for crcuts contanng ferromagnetc components n most cases the exctatons may be summarsed only summarsng of the magnetc flux denstes of sngle exctng components s not allowed but n lnear electrc crcuts ths method allowed Self nducton self nductance When a current I flows n a col t bulds up a magnetc feld descrbed wth flux Φ nsde the col The current change causes magnetc flux change Accordng to Faraday's nducton law d ( t) the magnetc flux change produces nduced voltage u () t ψ In ths process the change of magnetc flux enclosed by the col nduces a voltage n the col tself and therefore ths voltage s called self nduced voltage As follows from Lenz's law the effect of nduced voltage opposes the changng what has produced that voltage (back emf) In general takng nto account that the flux lnkage s a functon of current ψ ψ( t ()) the nduced voltage: dψ ( ( t) ) dψ ( t) d( t) u () t d() t The relaton of change n the flux lnkage to the change of current s represented by the (self) d () t nductance L ψ The nductance s symbolsed by L t's SI unt n honour of Henry's d() t scentfc actvty s Vs L H henry A [ ] Ω s m m Henry Joseph ( ) Amercan physcst 9

10 VIVEA3 Alternatng current systems 08 The formula for nduced voltage may become as u () t L d t Usng the nductance as parameter the changng n magnetc feld can be converted to changng n electrc crcut: dψ d ( ) Ψ const Ψ Ψ const Ψ L L I I Current-dependence of the nductance the magnetsaton curve s lnear non-lnear In non-ferromagnetc substance the relaton ψ() s lnear thus () t () d L ψ d t Ψ I const n ferromagnetc substance L const Snce the feld outsde a solenodal col s neglgble accordng to the law of exctaton for the unform feld nsde solenod: NI B N H Φ A Φ l l l l N A Ψ Ψ A l and L N N 0 Λ N A I l H φ N l A Approxmaton for calculaton the nductance of a solenodal col The nductance of solenodal col depends on the number of turns the geometrc dmensons and the substance nsde Wth ferromagnetc substance the nductance s current-dependent Explanaton N squared: on the one hand the magnetc feld s generated by current I n N turns of col on the other hand the magnetc feld nduces voltage also n N turns of col To fabrcate a crcut component (eg wound resstor) wth low nductance (pure-resstve nductance-free component) often used the double-wound (or bflar) technology Actually there are two cols wth the same current but n opposte drecton Accordngly the generated fluxes are n opposte drecton they destroy each other and the result s a very low (deally zero) flux thus dψ s small (the voltage of self-nducton u s small) therefore the nductance L s also small 0

11 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton Scheme of nductance-free (bflar) col agnetc feld of coupled cols decomposton The cols (n most smple case cols) are sad coupled when they are placed nto the magnetc feld each others and the mutual nfluence of these felds s not neglgble Dependng on the specfc applcaton the am may be the strong lnkage (eg for energy converson) or the weak lnkage (eg to reduce the electromagnetc nose) φ φ l φ φ m φ l φ φ Decomposton of magnetc feld of two coupled cols The arrangement the surroundngs and the geometrc dmensons determne that the only exstng complete magnetc feld how s lnked wth the ndvdual cols To make the llustraton and explanaton more smple the flux of the magnetc feld usually decomposed nto four parts Indexng: the frst dgt of ndex denotes that col the feld component lnked wth whle the second dgt denotes the source col the current of whch produces the feld component - the part φ of flux φ generated by the current flowng n col s lnked wth the col the rest s the flux leakage φ l of the col (whch lnked only wth the col ) φ φ +φ l - the part φ of flux φ generated by the current flowng n col s lnked wth the col the rest s the flux leakage φ l of the col (whch lnked only wth the col ) φ φ +φ l The complete flux becomes: φφ +φ φ +φ l +φ +φ l These components usually dvded accordng to ther orgn or functon If the components are selected by orgn (coupled-crcut model) the resultant flux of each col s the sum of the whole own flux and the adonal part from the other col: the resultant flux lnked wth col s φ φ +φ (φ +φ l +φ ) the resultant flux lnked wth col s φ φ +φ (φ +φ l +φ ) If the components are selected by functon (magnetc feld theory) the resultant flux of each col s the sum of the mutual part of the complete flux φ m and the own flux leakage φ l : the resultant flux lnked wth col s φ φ m +φ l (φ +φ +φ l ) the resultant flux lnked wth col s φ φ m +φ l (φ +φ +φ l )

12 VIVEA3 Alternatng current systems 08 The mutual flux φ m has two components: φ m φ and φ m φ thus φ m φ m +φ m φ +φ The complete flux of the system by both nterpretatons s certanly equal For descrpton of electrcal machnes (eg transformers nducton motors) the complete magnetc feld s usually approached n accordance wth feld theory and the flux components are consdered as they created n sutable nductances: flux leakages ψ l are created n leakage nductances L l magnetsng (or mutual) ψ m flux s created n a magnetsng (or mutual) nductance L m ψ l L l ψ l L l ψ m L m ψ m L m ψ m ψ m +ψ m m L m ( + )L m ψ l ψ l L l m L l ψ ψ m ψ L m Equvalent crcut for decomposed magnetc feld (feld theory approach) Couplng and leakage coeffcents Generally the qualty of nductve couplng that exsts between two cols s expressed as a fractonal number between 0 and where 0 ndcates zero or no nductve couplng and ndcates full or maxmum nductve couplng The couplng coeffcent k expresses what porton of flux φ generated n col by current lnked wth col : k φ φ Smlarly couplng coeffcent k expresses what porton of flux φ generated n col by current lnked wth col : k φ φ The unty couplng coeffcent means that all the flux lnes of one col cuts all the turns of the other col The leakage coeffcent σ expresses what porton of flux generated n col by current does not lnked wth col thus t s the complement of couplng coeffcent k Smlarly defned σ for the col φ l φ φ φ φ l φ φ σ k and σ k φ φ φ φ φ utual nducton mutual nductance Consder two cols placed near each to other A porton of the magnetc flux generated by the current of the frst col affects the second col or part of t Changng the current (t) of the frst col (prmer) the change of the flux φ (t) lnked wth the turns of the second col (secondary) nduces voltage n the second col: dψ ( t) d ( t) d( t ψ ) u() t d() t where ψ (t)n φ (t) The frst ndex shows the col what s affected by the magnetc feld of the col wth the second ndex: φ the flux lnked wth the second col produced by the current n the frst col

13 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton φ l l N N u A l A Illustraton of coupled cols The dervatve d ψ s termed mutual nductance usually symbolsed by or L t's SI d unt s the same as that of self nductance: []H (henry) The mutual nductance depends on the numbers of turns the dmensons and the materal In a ferromagnetc substance the mutual nductance s current-dependent If the permeablty s constant (eg n ron-less col) n steady state the mutual nductance s constant Ψ Applyng the law of exctaton to I the closed path of the magnetc feld determned by flux φ gves wth approxmaton l l +l : φ φ l l Θ N l + l φ + φ R m A A A A Θ Θ φ Θ Λ N Λ R l l m + A A ψ ψ N φ N N Λ consequently NN Λ The last formula ncludes the product of the numbers of turns N N because N turns are magnetsng and the voltage s nduced n N turns The flux lnk exsts n the opposte drecton too Whle exctng the second col the nduced voltage appears n the frst col va φ The mutual nductance often has to be determned by measurement data The voltage nduced n the secondary col u (t): dψ u d when the current (t) s a snusodal functon of tme then U rms πf I rms (n lnear crcuts) and the mutual nductance s: that s the nduced voltage U as- U rms fπ I rms sumed to be also a snusodal functon In sotropc substance snce Λ Λ In engneerng practce ths s the usual case therefore notaton wll used hereafter Coupled cols n seres Due to seres connecton a common current flows through both cols Say the resstances are neglgble compared to reactances If the fluxes of the cols tend towards the same drecton (adng or cumulatve couplng) then the resultant voltage u equal to the sum of self nduced voltages and mutual nduced voltages: 3

14 VIVEA3 Alternatng current systems 08 u L d d L d d ( L L ) d L d e L e the equvalent nductance L e L d d u L d d u Cumulatve coupled seres cols In case the numbers of turns n the cols N and N are equal then L e L +L + whle wthout couplng 0 and L e L +L Wth perfect couplng σ0 k φ φ ψ φ ψ Nφ L N ψ φ N ψ φ N L N N N N L N N Le L + L N + + N L e L d L d u d d u Opposng coupled seres cols If the fluxes of the cols tend towards the opposte drecton (opposng or dfferental couplng) then the mutual nduced voltages have to be subtracted from the sum of self nduced voltages: 4

15 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton u ( L L ) d L d + e Wth perfect couplng the equvalent nductance L e 0 (equal to that of bflar col) whle wthout couplng when the mutual nductances are neglgble the equvalent nductance s also L e L +L The mutual nductance may be determned expermentally by measurement data: the equvalent nductances measured when coupled n the same drecton mnus that n opposte drecton: L +L + - (L +L -) 4 Coupled cols n parallel Due to parallel connecton the voltage of both cols s common whereas the currents are ordnarly dfferent The voltage equaton f the couplng s adng: u L d d L d d + + d ( L ) d ( L ) consequently d wth restrcton L and L d L L and d d L L u L u L e L Cumulatve coupled parallel cols Substtutng the current dervatves nto the voltage equaton d u L L d LL d L L d LL + + L L L L Integratng the current dervatves d and d the resultant current become: L LL u L LL u L + L + + u u LL L e Thus the equvalent nductance L e : LL Le L + L For close coupled dentcal cols L L L the l'hosptal's rule must be used because the denomnator gets zero Thus the equvalent nductance L e L LL In case the mutual nductance neglgble 0 then Le L L L + L 5

16 VIVEA3 Alternatng current systems 08 L If L L L then Le If the couplng s opposng then the nduced voltage: u L d d L d d u L u L e L Opposng coupled parallel cols d ( L ) d + ( L + ) consequently d d L + and d d L + L + L + Substtutng the current dervatves nto the voltage equaton d u L L d LL d L L + + d LL + L + L + L + L + Integratng the current dervatves d and d the resultant current become: L + LL u L + LL u L + L u u LL L e Thus the equvalent nductance L e : LL Le L + L + For close coupled dentcal cols L L L the equvalent nductance L e 0 as far as cols destroy the feld components each others (lke n bflar cols) In case the mutual nductance neglgble 0 the result certanly not depends on the drecton of the produced fluxes: LL Le L L L + L L f L L L then Le Composed by: István Kádár Aprl 08 6

17 Propertes of ferromagnetc materals magnetc crcuts prncple of calculaton Questons for self-test Revew the man propertes of ferromagnetc materals Illustrate the representatve sectons of normal magnetsaton curve 3 Illustrate and nterpret the propertes of hysteress curve 4 Interpret the statc and dynamc hysteress curves 5 Represent some permeablty defntons 6 Defne the absolute permeablty 7 Defne the dfferental permeablty 8 Defne the ntal permeablty 9 Defne the ncremental and reversble permeabltes 0 Defne the magnetc crcut Revew the concept for calculaton of seres magnetc crcut Revew the concept for calculaton of parallel magnetc crcut 3 Revew the prncple of magnetc Ohm s law and the lmtatons of used analogy 4 Revew the phenomenon of self nducton 5 Explan the self nductance 6 What approxmaton used n calculaton the nductance of a solenodal col wthout ferromagnetc core 7 Illustrate the current-dependence of nductance 8 How to make nductance-free col? 9 Illustrate the used magnetc feld components of coupled cols 0 Explan the ways used to group the flux components of coupled cols Revew the phenomenon of mutual nducton Explan the mutual nductance 3 How determned the equvalent nductance of cumulatve coupled seres cols 4 How determned the equvalent nductance of opposng coupled seres cols 5 How determned the equvalent nductance of cumulatve coupled parallel cols 6 How determned the equvalent nductance of opposng coupled parallel cols 7