Properties of ferromagnetic materials, magnetic circuits principle of calculation

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1 Proprts of frromagntc matrals magntc crcuts prncpl of calculaton Frromagntc matrals Svral matrals rprsnt dffrnt macroscopc proprts thy gv dffrnt rspons to xtrnal magntc fld Th rason for dffrnc s crtan dffrnt mcroscopc proprts lk lctron shll confguraton movmnt and spn of lctrons or rathr th nflunc of xtrnal magntc fld on ths proprts In physcs da- para- and frromagntc matrals ar dstngushd whl n th practc of lctrcal ngnrng all non-frromagntc substancs consdrd as fr spac wth rlatv prmablty r Magntud of th rlatv prmablty of frromagntc matrals (ron nckl cobalt and thr alloys) may b vry hgh vn n ordr Frromagntc matral may b alloyd from non-frromagntc substancs In frromagntc matrals th rlatonshp btwn B and H (magntsaton curv) s nonlnar multvalud hstory dpndnt and oftn tm dpndnt ts charactrstcs can not b dscrbd analytcally hnc t has to b dtrmnd xprmntally for ach typ of matral Magntsaton curv Whn an ntally non-magntsd pc of frromagntc matral s magntsd by ncrasng th xtrnal fld slowly thn th flux dnsty wll also ncras Th curv B(H) showng ths vary of flux dnsty B vs chang of magntc fld ntnsty H calld prmary orgnal dc or normal magntsaton curv B B max B r b c d -H c a H H max Typcal shap of magntsaton curv In th prmr magntsaton curv 4 sctons dstngushd: a - startng zon b - lnar zon c - kn of th curv (ntrmdat zon) d - saturatd zon

2 VIVEM Altrnatng currnt systms 04 As fld ntnsty achvd saturatd zon and H s dcrasd from ts maxmum valu H max to zro slowly th flux dnsty B whl also dcrass t wll not rturn along th orgnal curv but on a nw (somwhat hghr) on and a B r rmannt magntsaton (rsdual rmannt magntc flux dnsty or rmannc rtntvty) rmans although H bcom zro Th chang of B dlayd wth rspct to chang of H (hystrss dlay) To rduc rmannt magntc flux dnsty to zro nds a fld ntnsty n th oppost drcton calld corctv forc -H c As th fgur shows th valu of prmablty th valus of rlaton B/H s not sngl-valud ts chang s non-lnar t dpnds of th prvous valu of fld ntnsty H th rat of chang In th saturatd zon r ~ If th magntsng forc s altrnatng (AC) th flux dnsty changs along a loop known as hystrss loop Th largst hystrss loop blongs to th pak valus B max and H max dtrmnd by th flux dnsty of saturaton Th hystrss loops of lss pak valus locatd nsd th largst loop If th magntsng forc changs slowly th loop s calld statc hystrss loop B B max H max H Hystrss curvs of dffrnt maxmum valu of H max Dynamc hystrss curvs In cas of altrnatng magntc fld gnratd by altrnatng currnt of ntwork frquncy (or hghr) th ponts of th B(H) plan form a full hystrss loop durng ach prod In addton du to th altrnatng flux an nducd voltag appars whch producs ddy currnts nsd th frromagntc matral Accordng to Lnz's law th magntc fld of ths ddy currnts maks th chang of th flux dnsty mor dlayd thrfor ncrasng th frquncy th dynamc hystrss loops bcom spacous compard to statc ons

3 Proprts of frromagntc matrals magntc crcuts prncpl of calculaton statc dnamc B H Statc and dynamc hystrss curvs Rlatv prmablty In ach pont of magntsaton curv th absolut prmablty B H and th rlatv pr- B mablty r may b dtrmnd Bcaus of th non-lnarty of th curv n ngnrng practc svral smplfcatons 0H appld B P α d α t α s 0 H Explanaton of th full th dffrntal and th ntal prmablty - full (ordnary) prmablty: th slop of th ln from orgn to th ponts of prmary magntsaton curv (g to pont P) B r α t H tg 0 - ntal prmablty: th rlatv prmablty at low xctaton lvl th slop of th startng zon of prmary magntsaton curv rs tgα s - dffrntal prmablty: th slop of prmary magntsaton curv at a gvn pont (g at pont P) db rdff α d dh tg 0 3

4 VIVEM Altrnatng currnt systms 04 B ncrmntal B rvrsbl 0 H 0 H 0 H Illustraton of ncrmntal and rvrsbl prmablts - ncrmntal prmablty: spcfc for th narrow loop du to small cyclc changs n a pont of th magntsaton curv whn a small altrnatng fld s suprmposd on a dc or slowly changng fld B rnc H 0 - rvrsbl prmablty: th sam as ncrmntal on but th cyclc chang so small that th lmntary loop forms a sngl ln In lctrcal powr ngnrng practc mostly th full (ordnary) prmablty s usd Calculaton of magntc crcuts Th magntc crcut s a closd sgmnt of th magntc fld n whch th flux consdrd constant flux lns do not lav such a sgmnt Actually all closd flux ln form a magntc crcut In magntc crcuts usually frromagntc componnts dtrmn th path of th flux lns Th calculaton s smpl n cass th path of flux (th arrangmnt) s known Magntc crcuts xampls Th rqurd xctaton for a dsrd fld of a complx magntc crcut composd from dffrnt sgmnts may b asly calculatd whras to answr th nvrs problm calculaton th rsult of known xctaton s dffcult bcaus of th non-lnarty of magntsaton curv Th flux lakag may b consdrd as a rsult of calculaton or stmaton but may b nglctd as wll 4

5 Proprts of frromagntc matrals magntc crcuts prncpl of calculaton Along a magntc crcut usually thr ar sgmnts of dffrnt proprts (gomtry prmablty) and may occur ramfcatons Th Ampèr's law of xctaton s vald for slow changs only: for drct currnt and tm nstants of altrnatng currnt In cas of fast changs th ffcts of ddy currnts hav to b takn nto account too Srs magntc crcuts Th srs magntc crcuts usually consst of sgmnts of dffrnt cross-sctons and matrals Calculaton th xctaton ncssary to gnrat spcfd magntc flux Suppos th Φ flux s gvn spcfd along th magntc crcut nvstgatd and th lakag nglctd Φ l 0 accordng to th fgur B H A B H l / l l / B δ H δ 0 A δ A δ Outln of a srs magntc crcut In ths cas th flux dnsty n th ar gap s B δ Φ and n th furthr frromagntc sg- A mnts B Φ B Φ and so on A A Bδ Th fld ntnsty n th ar gap s smply calculatd: Hδ whl n th frromagntc 0 sgmnts th fld ntnsts H H tc or th rlatv prmablts r r tc ar usually dtrmnd from th magntsaton curv B Φ B Φ H and H 0 r 0 ra 0 r 0 ra Applyng th law of xctaton th rsultant xctaton for th whol crcut usng trms 0 r : Bδ Φ l Θ Θ Hl H l H l K l Φ δ 0 A A snc Φ provdd to b constant Th rsultant xctaton can b dfnd as sum of partal xctatons for th sngl sgmnts of crcut In such cass whn th most of rsultant xctaton blongs to th ar gap th frromagntc (ron) parts of magntc crcut may oftn b nglctd ( ron» 0 thus H δ»h ron ) δ 5

6 VIVEM Altrnatng currnt systms 04 Exampl Gvn B δ B ron T δ mm l ron m from th magntsaton curv rron 0 6 Bδ Bδ 6 6 A Th fld ntnsty nsd th ar gap: Hδ Bδ m Bvas Bδ A nsd th ron sgmnts: Hron 08 Bδ 08 0 rron 0 rron m Th rsultant xctaton s th sum of th rqurd xctaton for ron and ar gap: ΘΘ ron Θ δ Th xctaton for ron Θ ron H ron l ron 08 A whl that for ar gap Θ δ H δ l δ 800 A Θ 8008 Th currnt ndd whn th numbr of turns n th col N: I ( A ) N N Th xctaton nds for ron sgmnt s 0% of th rsultant xctaton Usng ron of lss prmablty th ncssary xctaton for ron ncrass and s not nglgbl For xampl f rron 0 3 H ron 800 A m Θ ronh ron l ron 800 A that s 50% of th full xctaton (Θ600 A) Th nvrs problm whn th flux or flux dnsty hav to b answrd for known currnt I s dffcult to solv bcaus th dstrbuton of full xctaton dpnds on th rlaton among prmablts of ach sgmnts howvr to dtrmn th prmablts th fld ntnsty ndd bcaus of th non-lnarty of magntsaton curv A practcal soluton s th calculaton of xctatons or currnts whch gnrat dffrnt fluxs and thn th drawn Φ(Θ) or Φ(I) rlatonshp may gv th answr of problm Paralll magntc crcuts Snc th flux lns ar closd th total flux at nput and output ar dntcal: ΦΦ Φ Φ H A l Φ Φ Φ H A l Outln of a paralll magntc crcut Accordng to xctaton law for th l -l closd loop: H l - H l 0 hrby H l H l Θ p that s th xctatons of paralll sgmnts Θ p ar qual Substtutng fld ntnsts 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 Th total flux Φ wth xctaton Θ p : A A A Φ Φ Φ Θ p Θ p l l l 6

7 Proprts of frromagntc matrals magntc crcuts prncpl of calculaton Th magntc Ohm s law Whn modfyng th formula of xctaton law Θ Hdl bcaus of formal analogy th quatons for complx magntc crcuts oftn mntond as magntc Ohm s law Wth substtutons H B and B Φ nto th ln ntgral of th magntc fld ntnsty th A formula for th xctaton of srs magntc crcut Φ l Θ l Φ A A s smlar to th formula for th rsultant voltag of srs conductor sgmnts wth fnt rsstancs: I l U l I I R γ A γ A whr γ s th spcfc lctrc conductvty th rcprocal of th spcfc rsstvty ρ ρ Th total xctaton of srs magntc crcut may b wrttn as Um Φ Rm whr U m Θ th total magntc voltag (xctaton) Rm l th magntc rsstanc (magntc rluctanc) of th sgmnt Th total magntc rsstanc of th srs sgmnts: R R hrwth U m ΦR m A m A Physcal unts of varabls abov: [U m ] A [Φ] VsWb [ R ] A m Vs Wb Th mor th prmablty th lss th magntc rsstanc and th xctaton (magntc voltag) of a sgmnt n a magntc crcut Th xctaton of a sgmnt n a magntc crcut may b trmd as magntc voltag of that sgmnt for th th sgmnt: U m Φ l A Th xctaton law may b formulatd as follows: th ln ntgral of th magntc voltag around a closd path s qual to th xctaton Θ of th ara nclosd by that path Θ U m Th formula drvd for th total flux of paralll crcut A Φ Θ p l may b wrttn usng th abov rlatons as Φ U mp Λ A whr Λ th magntc conductvty (magntc prmanc) of th -th sgmnt th rcprocal of magntc rsstanc Physcal unt of magntc conductvty s l Rm Vs [ Λ m ] Wb A A m 7

8 VIVEM Altrnatng currnt systms 04 Th total magntc conductvty of th paralll sgmnts: Λ Λ whrby ΦU m Λ m ΘΛ m From th analogy abov may b buld up substtutng lctrc crcuts of magntc crcuts Such substtuton hav to b usd carfully bcaus th analogy s only formal th physcal phnomna ar ssntally dffrnt a) Th lctrc currnt I s a ral flow of th chargs (charg-carrr partcls) whl th magntc flux Φ dscrbs th stat of th spac wthout any movmnt of partcls b) Kpng on th lctrc currnt (dc currnt as wll) producs losss whl kpng on th magntc flux dos not rqur nrgy (only th buldng up or th changng of magntc fld) c) Th ln ntgral of th lctrc voltag around a closd path s qual to zro f th path dos not ncrcl changng flux d Φ 0 whl th ln ntgral of th magntc voltag dt around a closd path qual to zro f th path dos not ncrcl currnt ΣI0 d) Th lctrc conductvty γ s usually constant (at constant tmpratur) and currntndpndnt whl th prmablty of frromagntc matrals vary wth flux (flux dnsty) sgnfcantly ) Th rlaton btwn th conductvts of lctrc conductors and nsulators s about 0 0 thus currnt lakag s usually nglgbl Ths rlaton for magntc conductors and nsulators s about thus flux lakag and ts nflunc s oftn has to b takn nto account f) Th mthod of suprposton unusabl n crcuts contanng frromagntc componnts n most cass th xctatons may b summarsd only summarsng s not allowd for th magntc flux dnsts of sngl xctng componnts Slf nducton slf nductanc Whn a currnt I flows n a col t bulds up a magntc flux Φ nsd th col changng currnt causs changng th magntc flux Accordng to Faraday's nducton law th changng d () t magntc flux producs an nducd voltag u () t ψ In ths procss th col nducs a dt voltag n tslf and thrfor ths voltag s calld slf nducd voltag As follows from Lnz's law th ffct of nducd voltag opposs th changng what has producd that voltag (back mf) In gnral takng nto account that th flux lnkag s a functon of currnt ψ ψ( t ()) th nducd voltag: dψ ( () t ) dψ () t d() t u () t dt d() t dt Th rlaton btwn th flux lnkag and th currnt s rprsntd by th slf nductanc d () t L ψ Th SI unt of nductanc n honour of Hnry's scntfc actvty d t () Vs [ L ] H hnry Ω s A () Th formula for nducd voltag bcom as u() t L d t By mans of th changng n dt magntc fld may convrtd to changng n lctrc crcut Hnry Josph ( ) Amrcan physcst m m 8

9 Proprts of frromagntc matrals magntc crcuts prncpl of calculaton Ψ const Ψ Ψ const Ψ L L I I Currnt-dpndnc of th nductanc th magntsaton curv s lnar non-lnar In non-frromagntc substanc th rlaton ψ() s lnar thus () t () d L ψ d t Ψ I const n frromagntc substanc L const Snc th fld outsd th col s nglgbl by th law of xctaton for th homognous fld nsd solnod: Φ NΦ Ψ Ψ A NI Hl l l l and L N N 0 Λ A N A N A I l H φ N l A Approxmat calculaton of nductanc n a solnodal col Th nductanc of solnodal col dpnds on th numbr of turns th gomtrc dmnsons and th matral In frromagntc substanc th nductanc s currnt-dpndnt Explanaton th squar of N: on th on hand th magntc fld s gnratd by currnt I n N turns of col on th othr hand th magntc fld nducs voltag also n N turns of col To fabrcat a crcut componnt (g wound rsstor) wth low nductanc (pur-rsstv nductanc-fr componnt) th doubl-wound (or bflar) tchnology may b usd Thr ar two cols a rght-thrad and a lft-thrad accordngly th gnratd fluxs ar n oppost drcton thy dstroy ach othr and th rsult s a vry low (dally zro) flux thus dψ s small (th voltag of slf-nducton u s small) thrfor th nductanc L s also small Schm of nductanc-fr (bflar) col 9

10 VIVEM Altrnatng currnt systms 04 Mutual nducton mutual nductanc Consdr two cols placd nar ach to othr A porton of th magntc flux gnratd by th currnt of th frst col affcts th scond col or part of t Changng th currnt (t) of th frst col (prmr) th chang of th flux φ (t) lnkd wth th turns of th scond col (scondary) nducs voltag n th scond col: dψ () t d () t d() t ψ u() t dt d() t dt whr ψ (t)n φ (t) Th frst ndx shows th col what s affctd by th magntc fld of th col wth th scond ndx: φ - th flux lnkd wth th scond col producd by th currnt n th frst col φ N N u l A H Illustraton of coupld cols Th drvatv d ψ s trmd mutual nductanc usually markd as M or L th SI unt s d th sam as that of slf nductanc: [M]H (hnry) Th mutual nductanc dpnds on th numbrs of turns th dmnsons and th matral In frromagntc substanc th mutual nductanc s currnt-dpndnt If th prmablty s constant (g n ron-lss col) n stady stat th mutual nductanc s constant M Ψ Applyng th law of xctaton to th I closd path of th magntc fld dtrmnd by flux φ rsults (smplfd): φ θ N Hl l φr m A φ NΛ ψ N φ M NN Λ In th last formula ncluds th product of th numbrs of turns N N bcaus N turns ar magntsng and th voltag nducd n N turns Th lnk xsts n th oppost drcton too whn xctng th scond col th voltag nducd n th frst col In sotropc substanc M M snc Λ Λ Th mutual nductanc oftn has to b dtrmnd by masurmnt data Th voltag nducd n th scondary col u (t): dψ u M d dt dt whn th currnt (t) s snusodal wth tm thn U rms X I rms (usng lnar approxmaton) and th mutual nductanc: M U rms fπ I rms 0

11 Proprts of frromagntc matrals magntc crcuts prncpl of calculaton Magntc fld of coupld cols dcomposton Th cols ar coupld (n most smpl cas cols) whn thy ar placd nto th magntc fld ach othrs and th mutual nflunc of ths flds s not nglgbl Dpndng on th applcaton th am may b th strong lnkag (g for nrgy convrson) or th wak lnkag (g to rduc th lctromagntc nos) φ φ l φ φ m φ l φ φ Dcomposton of magntc fld Dpndng on th arrangmnt and gomtrc dmnsons th only xstng complt magntc fld s lnkd wth th ndvdual cols not qually To mak th llustraton and xplanaton mor smpl th flux of th magntc fld usually dcomposd nto four parts: - th part φ of flux φ gnratd by th currnt of th col s lnkd wth th col th rst s th flux lakag φ l of th col (whch lnkd only wth th col ) φ φ φ l - th part φ of flux φ gnratd by th currnt of th col s lnkd wth th col th rst s th flux lakag φ l of th col (whch lnkd only wth th col ) φ φ φ l Th complt flux bcoms: φφ φ φ φ l φ φ l Ths componnts usually dvdd accordng to ts orgn or functon If th componnts ar slctd by orgn (coupld-crcut modl) th rsultant flux of ach col s th sum of th whol own flux and th addtonal part from th othr col: th rsultant flux lnkd wth col s φ φ φ φ φ l φ th rsultant flux lnkd wth col s φ φ φ φ φ l φ If th componnts ar slctd by functon (magntc fld thory) th rsultant flux of ach col s th sum of th mutual part of th complt flux φ m and th own flux lakag φ l : th rsultant flux lnkd wth col s φ φ m φ l φ φ φ l th rsultant flux lnkd wth col s φ φ m φ l φ φ φ l Th mutual flux φ m has two componnts: φ m φ and φ m φ thus φ m φ m φ m φ φ Th complt flux of th systm by both ntrprtatons s crtanly qual For dscrpton of lctrcal machns (g transformrs nducton motors) th magntc fld s usually approachd n accordanc wth fld thory and th flux componnts ar consdrd as thos cratd n sutabl nductancs: flux lakags ar cratd n lakag nductancs magntsng (or mutual) flux s cratd n magntsng (or mutual) nductanc 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

12 VIVEM Altrnatng currnt systms 04 φ l φ l Ll L l m φ φ m φ L m Equvalnt crcut for dcomposd magntc fld Couplng and lakag coffcnts Gnrally th qualty of nductv couplng that xsts btwn th two cols s xprssd as a fractonal numbr btwn 0 and whr 0 ndcats zro or no nductv couplng and ndcatng full or maxmum nductv couplng Th couplng coffcnt k xprsss that th porton of flux gnratd n col by currnt whch lnkd wth col : k φ φ Smlarly couplng coffcnt k xprsss th porton of flux gnratd n col by currnt lnkd wth col : k φ φ Th unty couplng coffcnt mans that all th flux lns of on col cuts all th turns of th othr col Th lakag coffcnt σ xprsss that th porton of flux gnratd n col by currnt dos not lnkd wth col thus th complmnt of couplng coffcnt k Smlarly dfnd σ for th col φ l φ φ φ φ l φ φ σ k and σ k φ φ φ φ φ Coupld cols n srs Du to srs conncton a common currnt flows through both cols Say th rsstancs ar nglgbl compard to nductancs If th fluxs of th cols tnd towards th sam drcton (adng or cumulatv couplng) thn th rsultant voltag u qual to th sum of slf nducd voltags and mutual nducd voltags: u L d M d L d M d ( L L M M ) d L d dt dt dt dt dt dt L - th quvalnt nductanc M L M u u Cumulatv coupld srs cols In cas th numbrs of turns n th cols N and N ar qual and M M M thn L L L M whl wthout couplng M M 0 and L L L

13 Proprts of frromagntc matrals magntc crcuts prncpl of calculaton Wth prfct couplng σ0 k φ φ ψ φ ψ Nφ L N ψ φ M ψ Nφ M N N M N L L N N L L N N N N L Provdd M M thn L N L N or L L N N N N substtutng nto th formula of th quvalnt nductanc N L N L N N ψ φ If N N thn L 4 L whl ψ N φ hrby L N For clos coupld dntcal cols L L M M L thus L 4L whl wthout couplng M M 0 and L L If th fluxs of th cols tnd towards th oppost drcton (opposng or dffrntal couplng) thn th mutual nducd voltags hav to b subtractd from th sum of slf nducd voltags: u ( L L M M ) d L d dt dt If M M M thn L L L -M M L M u u Opposng coupld srs cols Wth prfct couplng th quvalnt nductanc L 0 (qual to that of bflar col) whl wthout couplng whn th mutual nductancs ar nglgbl th quvalnt nductanc s also L L L Th mutual nductanc may b dtrmnd xprmntally by masurmnt data: th quvalnt nductancs masurd whn coupld n th sam drcton mnus that n oppost drcton: L L M - (L L -M) 4M Coupld cols n paralll Du to paralll conncton th voltag of both cols s common whras th currnts ar ordnarly dffrnt Th voltag quaton f th couplng s adng: 3

14 VIVEM Altrnatng currnt systms 04 u L d M d L d M d dt dt dt dt In cas M M M thn d ( L M) d ( L M) consquntly d d L M and d dt dt dt dt L M dt wth rstrcton L M and L M d L dt L M M u L u M M L L Cumulatv coupld paralll cols Substtutng th currnt drvatvs nto th voltag quaton d u L M L M d LL M d L M L M d LL M dt L M dt L M dt L M dt L M Intgratng th currnt drvatvs d and d th rsultant currnt bcom: dt dt L M LL M udt L M LL M udt L L M udt udt LL M L Thus th quvalnt nductanc L : LL M L L L M LL In cas th mutual nductanc nglgbl M 0 thn L L L L L L If L L L thn L u L u M M L L Opposng coupld paralll cols 4

15 Proprts of frromagntc matrals magntc crcuts prncpl of calculaton If th couplng s opposng thn th nducd voltag: u L d M d L d M d dt dt dt dt In cas M M M thn d ( L M) d ( L M) consquntly d d L M and d d L M dt dt dt dt L M dt dt L M Substtutng th currnt drvatvs nto th voltag quaton d u L M L M d LL M d L M L M d LL M dt L M dt L M dt L M dt L M Intgratng th currnt drvatvs d and d th rsultant currnt bcom: dt dt L M LL M udt L M LL M udt L L M udt udt LL M L Thus th quvalnt nductanc L : LL M L L L M For clos coupld dntcal cols L L M M L th quvalnt nductanc usng th l'hosptal's rul L L In cas th mutual nductanc nglgbl M 0 thn th rsult crtanly not dpnds on th drcton of th producd fluxs: LL L L L L L L f L L L thn L Composd by: István Kádár March 04 5

16 VIVEM Altrnatng currnt systms 04 Qustons for slf-tst Rvw th man proprts of frromagntc matrals Illustrat th rprsntatv sctons of normal magntsaton curv 3 Illustrat and ntrprt th proprts of hystrss curv 4 Intrprt th statc and dynamc hystrss curvs 5 Rprsnt som prmablty dfntons 6 Dfn th absolut prmablty 7 Dfn th dffrntal prmablty 8 Dfn th ntal prmablty 9 Dfn th ncrmntal and rvrsbl prmablts 0 Dfn th magntc crcut Rvw th concpt for calculaton of srs magntc crcut Rvw th concpt for calculaton of paralll magntc crcut 3 Rvw th prncpl of magntc Ohm s law and th lmtatons of analogy 4 Rvw th phnomnon of slf nducton 5 Explan th slf nductanc 6 What approxmaton usd n calculaton th nductanc of a solnodal col wthout frromagntc cor 7 Illustrat th currnt-dpndnc of nductanc 8 Illustrat th usd componnts of coupld cols 9 Explan th ways usd to group th flux componnts of coupld cols 0 Rvw th phnomnon of mutual nducton Explan th mutual nductanc How dtrmnd th quvalnt nductanc of cumulatv coupld srs cols 3 How dtrmnd th quvalnt nductanc of opposng coupld srs cols 4 How dtrmnd th quvalnt nductanc of cumulatv coupld paralll cols 5 How dtrmnd th quvalnt nductanc of opposng coupld paralll cols 6