Table Of ContentMAGNETICC IRCUITS AND TRANSFORMERS
PRINCIPLES OF
ELECTRICAL ENGINEERING
SERIES
MagnetiCci rcuits
Transforlners
and
A FIRST COURSE
FOR POWER AND COMMUNICATION
ENGINEERS
By Memberso ft heS tafofft he
DepartEmleencttE rnofig cianle ering
MassachusIentsttsi touftT ee chnolyo g
1111111
THE M.LT.P RESS
MASS ..\ CIIUSETTSI �STITUTE OF TECII1\"OLOGY
CA:-'1BRIDGE,i\ IASSACIICSETTS
COPYRI1G9H43T ,
BY
THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY
AU RighRte88 erved
Thibso ok01 a'n yp artt heremoufs tn ot
ber eproduciend an y form, t'ilthohuet
mUm permission ofl he publisher.
FOURTEENTH PRISTISG,J UI.Y,1 962
F1ITF.ENTli PRISTISG,j .�SUARY. 1965
PRINTIID IN THB UNITB'> STATIIS or "'UIIRleA
Foreword
Thes tatTo ft heD epartmeonftE lectriEcnagli neeraitnt ghe :\Iassa
chusetItnss titouftT ee chnolohgays f ors omey earbse ene ngageidn a n
extensipnro:g raomf r evisiansag unistu bstantiiatlsel nyt iprree senta
tiono ft heb asitce chnologpircianlc ipolfee sl ectreincgailn eerTiInnsg .
yolumies o neo fa sericeosy eritnIgn rse visperde sentation.
The decisitoonu ndertaskoec omprehensai pvlea nr athetrh ant o
addh erea ndp atcht hercea mef romt heb elitehfa tt heD epartment's
largset affw,i thi tvsa rieidn tereisntt se achianngd r elatreeds earccohu,l d
effeac tn ew synthesoifes d ucatiomnaatle riianlt hefi eldo fe lectrical
engineerainndge vole\ a· seto ft extboo,viktsh a breadtohf v iewn ot
easialpyp roachbeyd a na uthowro rkinign dhidually.
Sucha comprehenssievrei ietsw ,a sf elsth,o ulbde f refer omt hed upli
cationrse,p etitiaonnds u,n bahncesso oftepnr esenitn u nintegrated
serioefst extbooIktss .h oulpdo sseas usn itayn da breadtahr isifnrgo m
theo rganizaotfia o snu bjeacsta wholeI.ts houladp peatlo t hes tudent
ofo rdinaprrye paratainodna lspor midae deptahn dr igocrh allengtion g
thee xceptiosntauld eanntd a cceptatbolt eh ea dvancsecdh olIatrs .h ould
comprias bea sicco ursaed equaftoera lslt udenotfse lectreincgailn eering
regardloefst sh eiurl timastpee cialRteys.t ricttoem da teriwahli chi so f
fundamentiamlp ortantcoe a llb rancheosf e lectriecnI,ai,>ln eeritnhge,
coursseh oullde anda turailnltyao n yoneb ranch.
Thisb ooka ndt her eorganipzreodb 'Taomf t eachionugto fw hiciht h as
growna rct husp roducotfsa majorr esearpcrho jetcoti mproveed uca
tionmaelt hodTsh.e f actth atth ibso oka ndi tcso mpanioanrsct her esult
ofc ollaboraetnidveea voimr· oh·isnugb stantitahleel nyt irset affo ft he
Institute's DepoafEr ltemcetnrti Ecnabl> ineearnidnt gh att hicso llabo
ratiohna sb eenc arrioeudt , ,,itshu stained enthiusss tiraiskimnt ge sti
monyt oe achc ontributdoerv'ost itoont hibsr oaodb jectoifvi em proved
instructTioo tnh.o soef u sw hoh aveo bserntxhle p rI,o',Teosfst hew ork
fromt heo utsidteh,id se monstratoifuo nni tayn de sprdiect o ropns t he
parto fa largset affi sp erhaptsh em osti mpressaisvpee cotf t hee ntire
eiTort.
KARLT . COMPTON
v
Preface
Thibso oki sth es econvdo lumien t heP rnicipeslo fE lectrEingcianle er
ings erideess cribientd h eF oreworadn,d e xtenidnst aon othefire ltdh e
circutihte orbeyg un int hefi rsvto lumeC.om putationo fm agnetic-circuit
performparnicnec,i apslescsoia tedw itthh eco nceptso fi nterlienlkceetdr ic
andm agneticci rcuits,a ndt heiarp plicant tioot hea nalyosfit sr ansformers
aret hee ssentsiuablj emcatt ter.
Liket heo ther volumeisnt hisse rietsh,ib so oki sin tended fora first
basicco urseE.m phasi,sh encea,g aiinsp laceodnf undamentalp rinciples
important tos tudenotfse l ectircale ngineerreignagr desls ofw hatt heir
specifiaell dusl tmiately mayb e;b otpho \\a"nedrc ommunicatpiroobnle ms
arec onsiderReidg.oo rf t houghatn da nalysriast hetrha n extensiveness
ofs copiesl ikewitshee in tendde featoufrt eh ibso ok.
Thet reatmeansts umetsh atth e reahdaeskr n owledogfee lectric-circuit
theory asg iveinn t hefi rsvto lumoef t hes erioersi no thetre xtboooknas
similar level\lI.a themast thircough diffetrielaen q uatioinsfs r eye ulseid
noth,o wevears,a substitute fort hee xplanatoifpo hny sicpahle nomena
butr athears a means ofd ecsribing quantitatively theicro nsequences.
Paraltloet lh em athematiacnaallys i,se xplanatinos areg iveinn t erms of
physicianlt erpretations.
Thet exitsd ividiendt tow op artPsa.rt I ," .:\IagnetCiicr cuitsst,a"r ts
witha discussoifot nh ec urrent theoroyf f erromagnetdiesrmi,vt ehse
magnetic-cciornccueipta tn,d c ontinu\e\s" idtehv elopmeonftt h ef unda
mentaplri ncipels forc omputatoifot nh eb ehavioofrm agnetciircc uits.
;\r ha!1tised rev otetdod iscussion ofi ron-croeraec tobryms eanosf m odel
theor-ya method ofa nalsyis ofg reapto werw henevenro nlinpehaer
nomenaa rei nvolved. A briesfu mmaryof t hermal-ciprrocbuleimts is
also includPe,du.t I I," Transformebresg,i"nw si tha shortd icsusison
pointingo utt hatth et heorfireosm " 'hitchhe hcaracteristciso fa lell ectric
apparatcuasnb ec omputeda red eveloptehdr ougthh ec ombination oft he
fundamenctoanlec pts oft hee lectrdiice,l ectmraignce,t ci, thermaaln,d
mechaniccailr cuciotnst ainientd h ema,n dt hacto stasn do theprra ctical
matterasr eo fp aramouinmtp ortnace int heiri nfluenocnet hed evelop
mento fe lectirca pparatuPsa.rt I It henc ontinuweist ha studoyf t he
appliatcionso fth eseg enerparli nciptloet sr ansformTehrest .h eoroyf
transfomrersi sd eveloped bothf romt her esolutoifot nh eimra gnetic
fieldinst oJ eakaagnedr esultmauntuta lc omponenatnsdf romt hec lassical
VH
PREFACE
theoroyfc ouplceidr cuithtesi ,n terrelabteitowneset nh et wom ethoodfs
analsyisbe nige mphasiezd.
Although thet exits b yn o meanse xhaustiinva en yonefi eldn,o ri si t
intendteodb ea desigbno oka,n attempth asb eenm adet od iscuas s
varieotfyr epresentaprtaictviceal p roblemins b otht heh eavy-current
powera ndt hel giht-current control, measuremeanntd,c ommunicaiton
applicatoifom nasg netimcat erialsa ndt ransformeArss i.n t heo ther
volumeosf t hese rise, morem ateritahla ni su sualcloyve red ina first
coursei spr esentde, top rovidfeo ra dditiosntuadly b yp articulaaprtol ry
advancsuetdde nt,sa ndt oi ncreaset heu sefulnoefts hset exats a reference
book.
CAMBRIDGEM, ASSACHUSETTS
Janua1r9y1,3
Contents
PAGE
FOREWOR.D V
PREFAC.E • vii
TABLEO FS nmOLS xix
PARTI . MAGNETIC CIRCUITS
CHAPTER I
PROPERTIOEFSF ERRmlAGNETl\IICA TERI.A LS. 3
Art.1 .I mportance ofF erromagnMeatitce rial; 3
2.T heoroyf M agnetis.m . . 4
2a.E lectraonndsK ucle.i 5
2b.T heA tom . 5
2c.T heD omain . . . . 7
2d.T he Crystal 8
2e.P olycrystSaulblstiancnee s 10
3.G eneraPlr opertoifMe sa gnetci Materials. 15
4.C haratceritsicsa ndM agnetizationC urveosf� Iagnetci l\laterials1 8
5.M agneticT esting . . . . . . . . . . . 2.9 . . .
6.E ngineerUitnigl izaotfFi eorrno magnetiMca terials 34
Problems. .. . . . . . . . . . . . 4.0 . . . .
CHAPTER II
TIll\El AGKETIc-CIRCCOlJXICTE PT. . . . . . . .• . . • 41
Art.1 .T heG enrealP roblem. . . . . . . . . . 41. . .
2.A nalogies betweeEnl ectraincdM agnetiCci rcuits 44
CHAPTER III
PRIXCIPLOEFSM AGNETICC IRCUICTO �Il'UTATIO::\'S 54
Art.1 .U nitfso rP ractiMcaaglne tci Calculati. o.n s . 54
2.C alculatfioorMn asg neticP athWsh olliyn I ron 55
3.Il lustratEixvaem ploef C alculatfioorMn asg netSitcru ctureo fU ni-
formM ateriaanld C ross-SectArieoan. a l. . . . .. .. 5 9. . .
4.I llustrEaxtaimvpele so fC alculatifoonMrs a gneticS tructoufrU en i-
formM ateriaanldN onunifoCrrmo ss-SectAiroean al 61
5.O ccurreonfcA ei rG apsi nF erormagnetic Circuits 65
6.F ringianngd L eakagFelu xes. . . . . . . . . 6.7 .
7.M ethdoso fC alculaftoirSo hno rAti rG aps. . . . . 68
8.I llustraEtxiavmep loef C alculatifoonars S implFee rromagnCeitri-c
cuiwti tah ShorAti rG ap . . . . . . •. . . . .. ..7 4
zx
CONTENTS
PAGE
9. Illustrxaatmiplveeso fEC alculatfioornS se ries-PaIrraollnaen ld
ShorAti r-GaPpa ths . . ..... . .. . .. . ..... . 78
10.C alculatfioorLn osn gA irG apsa ndf orLe akage FluxP aths. . . 8 1
11.I llustrEaxtaimvpel oef C alculaotfiL oena kagFelu xf otrh el\ Iagnetic
Circuoifta nO sciolglraph . . . . . . . . . 8.2 . . . . . . . .
12.C alculaotfit ohneM agnetizaCtuirovnoe f a D irect-CuDryrnaemnot 84
Problems. .. . . •. . . . . . . . . .. .. 9 4. . . . . . .
CHAPTER IV
PERMANENTM AGNETA}.S"D ENERGYIN THEl\ IAGNETFlICE LD 98
Art.1 .M agnetCiicr cuIintvso vlingP ermanel\nIta gnet.s 99
2.D esigfno rM inimum Amounto fM agneticM aterial 103
3.F ringianngdL eakage.. .. . . . 105
4.S tabilizoaftP ieornm aneMnatg net.s 111
5.E nergy int heM agnetFiicel d. 114
6.M agneticP ull 117
Proble.ms. . . .. 121
CHAPTER V
LOSSEISN MAGNETIC CORESC ONTAIXIKTGD IEV-ARYING FLUXES 124
Art.1 .H ystreesis osLs . . . .. ... . 125
2.E ddy-CurrLeonstsa ndS kiEnfI ect. . .. ... . 132
3.T otaClo reL oss.. . .. . . .. . .. . .. . . 138
4.R eduction ofT otalC oreL ossD;e sigCno nsiderations 142
5.l\ leasuremento fT otaClo reL oss. . .. . . . . . 144
6.E xperimnetalL ossC urveEsx,p erimentalC oefficients 148
7. epSarationo fH ysterseisa ndE ddy-CurrLeonsts es 151
Problems. . . . . . . . . . . . .... .. 154
CHAPTER VI
ALTERNATING-CURRENT EXCITATIONCHARA CTERISTICS OF IRON-COREREA C-
TORS AND TRANSFORMERS . • . • • • . 156
Art. 1.G eneralP roperties ofI ron-CoRree actors 156
2.I ronC oresi nT ransformers. 157
3.A ssumption.s . . 158
3a. Capacitance 158
3b.R esistan.c e 158
4.R elatiboentsw eeInnd uced VoltagFel,u xa,n dC urrent 159
5.A lternatAipnpgl iVeodl tag.e . . . 160
6.S inusoiEdxacilit ngC urrent. . . . . . . . . 1.6 4. .
7.V oltaIgned ucebdy a SinusoiFdlaulx. • . . . . . . 166
8.W aveformo ft heE xcitiCnugr refnotr Si ne-Wave Flux 169
9.G enerParlo pertoife sF ouSreireire s 173
9a.T heC onsatnt Component. 174
9b.E vena ndO ddF unctions . 175
9c.E ven andO ddH annonic.s 176
9d.P ower . . . . . 177
ge.E ffectiVvael ues.. . . . 180
CONTENTS X2
PAGE
10.F ouriAenra lysoifst heE xcitiCnugr renfto rS ine-WaFvleu x 181
lOaE.fI ect aonfA irG ap . . . .. .. . 184
11.V ectoRre presentaotfit ohneE xcitiCnugr ren.t . . . . 186
12., \lternating-CEuxrcrietnatt Cihoanr actteirciss. . . . . 189
12a.I llustraEtxiavmep loef I ron-CoRreea ctoDre sign 193
13.E quivaleCnitr cui.t s. . . . . . . . . . 19.5
14.C haracteriosfIt riocSnsu bjectteoSd u perposDeidr ecatn dA lternat-
ing::\ IagnctiFzoirncge s. .. . . . .. . .. .. .... .19 7
15.I ron-CoRreea ctowrist hS uperposDeidr ecatn dA lternaitngE xcita-
tions. . .. . . . . . . . .. . . . . . . . . . . . . . .2 02
1530I.l lustraEtxiavmep loef C alculatioofAn p parent Inductanc2e0 3
Problems. .. . . . . . . . . . . . 20.7 . . . . . . . . . . .
CHAPTER VII
::\10DETLH EORYA XDD ESIGNO F IROX-CORREE ACTORS 212
.\rt1.. G eneraRle latio. n.s . . .. . .. . . 212
2.E fIectosf C hangeisn t heW inding 216
3. EfIectosf C hangesi nL inearD imension.s 218
4.E nergCyo nisderat.i o.n s. . . .. . . 221
5.E fIectosf a n AirG ap . . . . . . .. . 228
Sa.R eductioofnN" onlineEafTre ctbsy Meanso fa nA irG ap 228
5b. EfTectosf a nA irG apo nQ ualFiatcyto r 230
6.M odelT heorAyp plietdoE ntirKee twork.s . . . . . . . 232
6a.R elatioanmso ngt heV ariable.s . . .. ... . .2 33
6b.R elatiobnest weeCno rrespondLiinnge aPra ramete.r s 235
6c.R elation betthweDe eerni veadn dP rototeyR peactors 236
Problems. . . . . ... .. .. . . . . . .. . . . 238
CHAPTER Vln
TIIER�IPARLO PERTIAEXSD HEAT FLOW 242
Art.1 .E fTectosf H eato nM aterials 242
2.C onductioofHn e at 244
3.R adiatioofHn e at.. . . . 2-16
4.C onvectioofHn e at . . . . 248
5. CoolinofgE lectriAcpapla ratus 250
Sa.T ransforme.r . s . 251
5b.R otatiMnagc hines 252
6.T ransielnIte ating 253
PART II.T RANSFORMERS
CHAPTER IX
GENERAL DESIGN AXDC OSTC ONSIDERATIONS 259
Art.1 .E lectriMcaaclh inery 259
2.P owerS ystem.s . . . . . . 261
3.E lectrocnDi evices. . . . . 262
4.E lectriocan ndO theCri rcu.i ts 263
XII
CHAPTER X
PAGE
TRAXSFOR�IERSG;E XERALP RIXCIl'LES 265
Art.1 .E lementaTrrya nsforTmheero ry 265
2.T heI deaTlr ansformer 268
3. Useso fT ransforme.r s . 272
4.E volutioofTn r ansformers 278
5.T ransforPmreorb lems 279
5a.L osses. . 280
5b.C ooling 2g0
5c.;, IagnetLieca kage 280
5lIE.x ctii nCgu rrctn 281
5e.T hcE lectrFiice ld 282
5f.� ranufactuPrrionbgl ems 282
5g.O thcCro nsiderations 282
Problems. .. . . 283
CIL\PTERX I
PHYSICAFLE ATCREOSF TIL\XSFOIL.\ IERS 286
.r\t .1 .C ores. . .. .. . . 21\6
2.W inding.s .. . . . . 296
3.C oolianngd I nsulation 297
3a.c \iCro oli.n g. . 2lJlJ
3b.A ir-blCaosotl ing 29lJ
3c.T ransforOmie.lr 2lJlJ
3d.O il-immerseSdel,l- Cooled 2lJ9
3e.O il-ImmersedF,o rced-Air-Cooled 300
3f.O il-ImmerWseda,t er-Cool.e.d. 300
3g.F orced-COooililn g . . . . .. 300
3h.P yranoIln,e rteeonr,C hlore.x tol 300
4.T ank.s . . .. . 300
S.B ushing.s .. . . 302
6.P rotectDievvei ces 30-l
7.S hieldin.g ... . 310
CHAPTER XII
MAGNETILCE AKAGE IN TRANSFORMERS. .. .. . . . . . . 313
Art. 1.M agnetLieca kagaetN o Load. . . ... ... . 313
2.V oltaEgqeu atioannsdF luxD istribuutnidoeLonra d 317
2a.C omponeFnltu xes.. . . . 31g
2b.R esultFalnutDx i stribut'io"n . . . . . 320
3.Le akageI nductance.s . . .. . .. .. ... . 32-t
4.E xcitianngdL oadC omponenotfts h eP rimarCyu rrte n 325
S.An EquivalCeinrtc u.it. .... . . ... . . .. 326
6.E ffectso ft heI ronC orea ndS impyilifngA pproximations 328
Problem.s . . ..... .. . . . . . . ... .. 329
CHAPTER XIII
LEAKAGIEX DUCTAXCE.S . . . 332
Art.1 .V ectoDri agraamnsd a n EquivaleCnitr cuit 332
