Electromagnetic Theory - J. R. Lucas

Transcription

1 Eectmagnetic They - J. R. Lucas One f the basic theems in eectmagnetism is the Ampee s Law which eates, the magnetic fied pduced by an eectic cuent, t the cuent passing thugh a cnduct. Ampee s Law Ampee s Law states that the ine intega f the magnetic fied cuent magnetic fied taken aund a csed path is equa t the tta cuent encsed by the path. cuent i.e. d I F a unifm fied, is a cnstant and we have. Σ I if is cnstant ve sectins, with diffeent sectins having diffeent, then Σ. Σ I Magnetic Fied The magnetic fied at a pint is defined as being equa t the fce acting n a unit magnetic pe paced at that pint. [Unit f magnetic fied is ampee pe mete (A/m)] Magnetmtive fce (mmf) Magnetmtive fce is the fux pducing abiity f an eectic cuent in a magnetic cicuit. [It is smething simia t eectmtive fce in an eectic cicuit]. [Unit f magnetmtive fce is ampee (A)] - Nte: Athugh sme bks use the tem ampee-tuns, it is sticty nt cect as tuns is nt a dimensin] mmf I Σ I Cnside a ci having N tuns as shwn. It wi ink the fux path with each tun, s that Fux path tta cuent inking with the fux wud be Σ I N.I Thus fm Ampee s Law, the mmf pduced by a ci f N tuns wud be N I, and N I. Fied pduced by a ng staight cnduct If a cicua path f adius is cnsideed aund the cnduct caying a cuent I, then the fied ang this path wud be cnstant by symmety. by Ampee s Law, 1.I.π I at a adia distance fm the cnduct. π I Eectmagnetic They Pfess J R Lucas 1 Nvembe 001

2 Fied pduced inside a tid Cnside a tid (simia t a ing) wund unifmy with N tuns. If the mean adius f the magnetic path f the tid is a, then the magnetic path ength wud be π.a, and the tta mmf a I pduced wud be N I. Thus fm Ampee s Law tid N tuns magnetic fied NI π a inside the tid. [vaiatin f the magnetic fied inside the css sectin f the tid is usuay nt necessay t be cnsideed and is assumed unifm]. Magnetic fux density The magnetic fied gives ise t a magnetic fux φ, which has a magnetic fux density f a given aea A. The eatinship between and is given by the pemeabiity f the medium µ.. µ, whee µ µ µ, µ is the eative pemeabiity and µ is the pemeabiity f fee space µ 4π 10-7 /m [pemeabiity f ai is geneay taken t be equa t that f fee space in pactice] [Unit f pemeabiity is heny pe mete (/m)]. [Unit f magnetic fux density is the tesa (T) ] φ.a [Unit f magnetic fux is the webe (Wb)] Reuctance f a magnetic path A magnetic mateia pesents a Reuctance S t the fw f magnetic fux when an mmf is appied t the magnetic cicuit. [This is simia t the esistance shwn by an eectic cicuit when an emf is appied] Thus mmf Reuctance fux I S. φ F a unifm fied, I N I., and φ. A µ. A. S. µ. A s that the magnetic euctance S, whee ength and A css-sectin µ A [Unit f magnetic euctance is heny -1 ( -1 ) ] 1 Magnetic Pemeance Λ is the invese f the magnetic euctance. Thus Λ S [Unit f magnetic pemeance is heny ()] µ A Eectmagnetic They Pfess J R Lucas Nvembe 001

3 Sef Inductance Whie the euctance is a ppety f the magnetic cicuit, the cespnding quantity in the eectica cicuit is the inductance. Induced emf e N d φ L di, N φ L i, L Nφ i The sef inductance L f a winding is the fux inkage pduced in the same winding due t unit cuent fwing thugh it. F a ci f N tuns, if the fux in the magnetic cicuit is φ, the fux inkage with the ci wud be N.φ. as since N I S φ, L N N µ A S Thus the inductance f a ci f N tuns can be detemined fm the dimensins f the magnetic cicuit. Mutua Inductance When tw cis ae pesent in the vicinity f each the s magnetic cicuit, mutua cuping can take pace. One ci pduces a fux which inks with the secnd ci, and when a cuent in the fist ci vaies, an induced emf ccus in the secnd ci. Induced emf in ci due t cuent in ci 1: e N d φ 1 M di 1, N 1 φ 1 M 1 i 1, N M φ 1 1 i1 The Mutua inductance M 1, f ci due t a cuent in ci 1, is the fux inkage in the ci due t unit cuent fwing in ci 1. as since N 1 I 1 S φ 1, and a factin k 1 f the pimay fux wud ink with the secnday, φ 1 k 1. φ 1 k 1N1N k 1N1Nµ A M1, S k 1 is knwn as the cefficient f cuping between the cis. k 1 k 1 s that M 1 M 1. F gd cuping, k 1 is vey neay equa t unity. Magnetisatin Chaacteistic In the anaysis s fa, the eatinship between and has been assumed t be inea. That is the pemeabiity has been assumed t be cnstant. This is nt the case with magnetic mateias, due especiay t satuatin. The cuve dawn shws a typica shape f the φ satuated egin magnetisatin chaacteistic f a magnetic mateia. They ae chaacteised by an anke pint f vey w knee pint eves f fux density, a inea egin and a satuatin egin. The knee pint appeas between the inea egin inea egin and the satuated egin. In the inea egin, the pemeabiity is much geate than that f fee space, wheeas in the satuated egin it is cse t that f fee anke pint space. The chaacteistic may be dawn with eithe I φ ptted n the y-axis and I n the x-axis. Eectmagnetic They Pfess J R Lucas 3 Nvembe 001

4 The tems, anke pint and knee pint ae used t define the cuve as the diagam ks quite a bit ike a bent eg with the anke and the knee ccuing in thse psitins. Magnetic mateias ae usuay peated in thei inea egin, and f best utiisatin f the mateia, they ae usuay peated just bew the knee pint. [Maximum peating fux densities in stee ae in the egin f 1.6 T]. Since pemeabiity is the ati f t, it is n nge cnstant f a nn-inea magnetisatin chaacteistic. In these egins an incementa pemeabiity is defined which cespnds t the ate f change f with. Incementa pemeabiity d d Absute pemeabiity µ F magnetic mateias, the eative pemeabiity in the inea egin can vay fm abut,000 t abut 100,000 dependant n the mateia. - p If the magnetic fied is inceased, at a cetain stage the magnetic fux shws negigibe incease. This situatin is satuatin. If the magnetic fied is appied t an initiay demagnetised magnetic mateia and then subsequenty emved, the mateia etains sme f the magnetisatin. That is, the magnetic fux density pduced by the magnetic fied des nt cmpetey vanish. The amunt f emaining fux density is knwn as the Remnance m. This emnance can be emved by an appicatin f a magnetic fied in the ppsite diectin. The amunt f demagnetising magnetic fied equied is knwn as the cecivity ( cecive fce) c. m E m A c E d m This f cuse must mean that the demagnetisatin is nt ccuing ang the igina magnetisatin cuve. An incease in the magnetic fied in the negative diectin wud esut in satuatin in the evese diectin. C D m If the pcess is cntinued, a p wud be fmed. This p is knwn as the - p the hysteesis p. This p nmay ccus when the magnetic fied is atenating and is assciated with a ss in enegy, knwn as the hysteesis ss. This may be cacuated as fws. pwe suppied t the magnetic fied P e.i d but fm the aws f eectmagnetic inductin, e N φ Eectmagnetic They Pfess J R Lucas 4 Nvembe 001

5 s that P e.i N d φ.i d. A. N. i d A d i.e. P vume d enegy suppied pe unit vume d [Nte:.d is as the aea f the eementa aea indicated n the - p] Thus enegy suppied/unit vume/cyce d aea f - p This enegy ss is knwn as the hysteesis ss. Thus the enegy suppied/unit vume/secnd pwe wud be the abve mutipied by cyces/secnd fequency. Thus pwe ss due t hysteesis P h aea f - p fequency i.e. P h f It is as seen that the aea f the p inceases as the maximum fux density m eached inceases. The incease in m as causes a cespnding incease in m, but nt pptina t the incease in m. Thus the hystesis ss wud as be pptina t a pwe f m. Steinmetz Law Steinmetz Law states that the enegy ss pe cyce due t hysteesis is pptina t n m. Geneay, a 1.6. a i.e. enegy ss/unit vume/cyce aea f hysteesis p η. m whee the cnstant η is knwn as the Steinmetz cnstant and the index a is knwn as the Steinmetz index. Thus f an atenating suppy with fequency f, pwe ss/unit vume η. 1.6 m.f The Steinmetz cnstant f sme cmmn magnetic mateias ae ad cast stee 7000, cast stee 750 t 3000, cast in 760 t 4000,vey sft in 500, siicn sheet stee (with 0.% Si) 530, siicn sheet stee (with 4.8% Si) 191. Enegy sted in a magnetic fied Enegy sted in an induct v i L di i L i di 1 Li Enegy sted in a unit vume in magnetic fied N d φ i N i dφ vume A. N i φ d d A 1 1 d µ µ J/m 3 Eectmagnetic They Pfess J R Lucas 5 Nvembe 001

6 Fce exeted in an magnetic fied Cnside mving the eectmagnet s that the spacing changes by dx change in enegy sted ½.Η (change in vume) ½.Η A. dx As, change in enegy sted wk dne F. dx, F. dx ½.Η. A. dx F ½.Η. A i.e. Fce exeted n unit aea in an eectic fied F/A ½.Η Ν/m Eddy Cuent Lss The tem eddy cuent is appied t an eectic cuent which cicuates within a mass f cnduct mateia, when the mateia is situated in a vaying magnetic fied. These A eddy cuents esut in a ss f pwe. Eddy cuent sses tgethe with the hysteesis dx sses cause heating f magnetic mateias. T educe these sses, nt ny ae mateias with fux d w Steinmetz cefficients chsen but the magnetic ce is made in the fm f aminatins t educe the eddy cuent ss. D C Cnside the magnetic fux incident n the face t ACD f the css-sectin f a aminatin f thickness t. [The thickness f the aminatin has been gssy exageated t shw detais f m m cs ωt eddy cuent paths]. If the fux is atenating with fux density m cs ωt, then eddy cuents wi estabish t themseves und the peiphey, as shwn shaded n the diagam. m The eddy cuents pduced wi as stive t change the fux, s that the fux at the peiphey wud be cmpaativey me. weve, as vey thin pates ae cnsideed (factin f a mm), this effect can be negected. Cnside an eddy cuent path at a distance x fm the cente-ine f the css-sectin and penetating the fu ength f the pates. If the fequency f the suppy is f and a eementa path f thickness dx is cnsideed, then the aveage ate f change f fux density ve haf a cyce wud be given by diffeence in the psitive and negative peaks divided by the time f haf a cyce. d i.e. aveage ve haf cyce m ( m ) 4m 1 T T 4 m f aveage ate f change f fux 4 m f. aea F a given path, at distance x, the aea wud be the aea encsed by that path. aveage ate f change f fux 4 m f. x. d 8 m. f. x. d i.e. aveage induced emf in the eddy cuent path 8 m. f. x. d and, ms induced emf in the eddy cuent path 8 m. f. x. d. k e whee k is the fm fact [f a sinusida wavefm, k 1.111] Eectmagnetic They Pfess J R Lucas 6 Nvembe 001

7 Since the thickness f the aminatin is negigibe, the ength f each eddy cuent path can be cnsideed as cnstant and equa t d, and the aea thugh which cuent fws as dx. esistance f eddy cuent path ρ.d. dx pwe ss in eementa stip due t eddy cuents 3 m f x d k. dx ρ, whee ρ is the esistivity f the mateia t e (8m f xd k) ρ.d. dx tta pwe ss in pate f thickness t 3 3m f x d k 4. dx m f t d k 0 ρ 3 ρ vume f the pate d..t, s that 4 Eddy cuent ss pe unit vume m f t k 3 ρ It is thus seen that the eddy cuent ss pe unit vume is pptina t the squae f the thickness f the individua pates. Thus f a given vume, if the thickness f the aminatins is made vey sma, the eddy cuent sses can be minimised. Cmpaisn f ppeties f eddy cuent and hysteesis sses 1. Eddy cuent ss is pptina t the squae f the fequency, whie the hysteesis ss is pptina diecty t the fequency.. Eddy cuent ss is pptina t the squae f the peak fux density, whie the hysteesis ss is usuay pptina t the 1.6 th pwe f the peak fux density. 3. Eddy cuent ss is pptina t the squae f the thickness f the aminatins whie the hysteesis ss des nt depend n thickness f aminatins. 4. Eddy cuent ss is dependant n the esistivity f the mateia, whie the hysteesis ss is dependant n the Steinmetz cnstant f the mateia. Anaysis f Eectmagnetic Cicuits Eectmagnetic cicuits can be anaysed in a manne simia t the anaysis f esistive cicuits. Cnside the fwing tw winding tansfme wund n a thee imb ce. y I 1 N 1 I A m N A A y Css sectin aeas f the ce, and the effective engths f magnetic paths ae indicated. Eectmagnetic They Pfess J R Lucas 7 Nvembe 001

8 It is assumed that the css-sectin des nt change at the cnes. m.m.f.s I 1 and I ae pduced in the tw windings and equa t N 1 I 1 and N I. euctances f each ute imb S, µ µ A y euctance f each pat f tp and bttm ykes, µ µ A euctances f midde imb S m, [ength f midde imb same as ute] µ µ Am If the fuxes fwing in the paths ae φ (ute imbs, ykes) and φ m (cente imb), then an equivaent cicuit simia t the eectica equivaent cicuit may be dawn as fws. y S m I 1 N 1 I 1 S I N I S φ S I 1 S m φ m φ S I The fuxes can be cacuated using aws simia t Ohm s aw and Kichff s aw as fws. φ m φ + φ simia t Kichff s cuent aw I 1 + I S m φ m + ( + S ) φ simia t Kichff s vtage aw and Ohm s aw Ony ne p was cnsideed as bth ute imbs ae identica and must theefe have the same fux. If the ute imbs wee diffeent, then thee wud have been ne additina fux tem and ne additina equatin. The ny unknwns ae φ m and φ which can be cacuated. In the case f thee phase tansfmes, the winding cuents wud have diffeent phase anges, s that the cespnding mmfs t wud have diffeent phase anges. The anaysis f this wud be simia t the anaysis f thee phase pbems, but n equivaent being thee f inductances and capacitances in the cespnding equatins. The abve anayses ae vaid ny in the inea egin f the magnetisatin chaacteistic whee the pemeabiity can be assumed t be cnstant. weve, when satuatin ccus, the anaysis is me cmpicated. Anaysis in the pesence f a nn-inea magnetisatin chaacteistic Ony a simpe cicuit having a nn-inea magnetic chaacteistic and a seies ai gap wi be cnsideed t iustate the methd f anaysis. It is assumed that thee is n finging f fux aund the ai gap s that the fux density wi be the same in bth the ai gap as we as the magnetic ce. m a The chaacteistic f the magnetic ce is as knwn. The ai gap has a inea chaacteistic with pemeabiity µ. Eectmagnetic They Pfess J R Lucas 8 Nvembe 001

9 Let the css sectin the ce (and ai gap) be A, the ength f the magnetic path be m in the magnetic mateia and a in the ai gap. Let the numbe f tuns in the ci be N and the cuent in the winding be I. Then fm Ampee s aw m m a m + b N I m m + a a f the ai gap, a a µ µ m m m N I m m + µ a, m - a m + b Since this equatin has been witten in tems f the paametes f the magnetic mateia, intesectin f this staight ine with the magnetisatin chaacteistic wud give the peating psitin. Eectmagnetic They Pfess J R Lucas 9 Nvembe 001