Chapter 1

Introduction to Electrical machines-chapter 1 magnetics

1. Chapter One
Magnetics
Introduction
Magnetic circuits
Magnetic Materials and Their Properties
Magnetically Induced Emf and Force
Ac Operation of Magnetic Circuits
Hysteresis and eddy current losses
By :Yimam A.(MSc)

2. Introduction
Anelectricalmachineisadevicewhichconvertselectricalpower(voltages
andcurrents)intomechanicalpower(torqueandrotationalspeed),and/or
viceversa.
Amotordescribesamachinewhichconvertselectricalpowertomechanical
power;agenerator(oralternator)convertsmechanicalpowertoelectrical
power.
Almostallpracticalmotorsandgeneratorsconvertenergyfromoneform
toanotherthroughtheactionofamagneticfield.
Transformersareusuallystudiedtogetherwithgeneratorsandmotors
becausetheyoperateonthesameprinciple,thedifferenceisjustinthe
actionofamagneticfieldtoaccomplishthechangeinvoltagelevel.
2

3. Principle of Electromagnet
Theprinciplesofmagnetismplayanimportantroleintheoperationofan
electricmachine.
Thebasicideabehindanelectromagnetisamagneticfieldaroundthe
conductorcanbeproducedwhencurrentflowsthroughaconductor.In
otherword,themagneticfieldonlyexistswhenelectriccurrentisflowing
Byusingthissimpleprinciple,youcancreateallsortsofthings,including
motors,solenoids,read/writeheadsforharddisksandtapedrives,
speakers,andsoon.
3

4. Magnetic Field
magnetic field encircle their current
source.
field is perpendicular to the wire and
that the field's direction depends on
which direction the current is
flowing in the wire.
A circular magnetic field develops
around the wire follows right-hand
rules.
4

5. Properties of Magnetic Lines of Force
Magneticlinesofforceare
directedfromnorthtosouth
outsideamagnet.
Magneticlinesofforceare
continuous.
Magneticlinesofforceinthesame
directiontendtorepeleachother.
Magneticlinesofforcetendtobe
asshortaspossible.
Magneticlinesofforceoccupy
three-dimensionalspaceextending
(theoretically)toinfinity.
Magneticlinesofforceenteror
leaveamagneticsurfaceatright
angles.
Magneticlinesofforcecannot
crosseachother.
5

6. Cont.…
magneticfieldsarethefundamentalmechanismbywhichenergyis
convertedfromoneformtoanotherinmotors,generators,and
transformers.Fourbasicprinciplesdescribehowmagneticfieldsareusedin
thesedevices:
1.A current-carrying wire produces a magnetic field in the area around it.
2.A time-changing magnetic field induces a voltage in a coil of wire if it
passes through that coil. (This is the basis of transformer action.)
3.A current-carrying wire in the presence of a magnetic field has a force
induced on it. (This is the basis of motor action.)
4. A moving wire in the presence of a magnetic field has a voltage induced in
it.(This is the basis of generator action.)
6

7. Example of Electromagnet
Anelectromagnetcanbemadeby
windingtheconductorintoacoiland
applyingaDCvoltage.
Thelinesofflux,formedbycurrent
flowthroughtheconductor,combine
toproducealargerandstronger
magneticfield.
Thecenterofthecoilisknownas
thecore.Inthissimpleelectromagnet
thecoreisair.
7

8. Cont.…
Ironisabetterconductorofflux
thanair.Theaircoreofan
electromagnetcanbereplacedbya
pieceofsoftiron.
Whenapieceofironisplacedin
thecenterofthecoilmorelinesof
fluxcanflowandthemagnetic
fieldisstrengthened.
8

9. Cont.…
Because the magnetic field around a
wire is circular and perpendicular to
the wire, an easy way to amplify the
wire's magnetic field is to coil
the wire.
Thestrengthofthemagneticfieldin
theDCelectromagnetcanbe
increasedbyincreasingthe
numberofturnsinthecoil.
Thegreaterthenumberofturnsthe
strongerthemagneticfieldwillbe. 9

10. Basics of Magnetic Circuits
1. Magnetic flux(ϕ):
The magnetic lines of force produced by a magnet is called magnetic flux.
It is denoted byϕand its unit is Weber.
1 weber = 10
8
lines of force
2. Flux density(B)
The total number of lines of force per square metreof the cross-
sectional area of the magnetic core is called flux density.
Its SI unit is Tesla (weber per metresquare).
B=ϕ/AWb/m2or Tesla
Whereϕ-total flux in webersA -area of the core in square metres
B-flux density in weber/metresquare.
10

11. Cont.…
3. Magneto-Motive Force
Theamountoffluxdensitysetupinthecoreisdependentuponfive
factors-thecurrent,numberofturns,materialofthemagneticcore,
lengthofcoreandthecross-sectionalareaofthecore.
Morecurrentandthemoreturnsofwireweuse,thegreaterwillbethe
magnetizingeffect.
Thisabilityofacoiltoproducemagneticfluxiscalledthemagneto
motiveforce.
mmf = NI ampere -turns
Where mmfis the magneto motive force in ampere turns
N is the number of turns.
11

12. Cont.…
4. Magnetic field Intensity(H)
The magnetic field intensity is the mmf per unit length along the path of
the flux.
Is also known as magnetic flux intensity and is represented by the letter
H.Its unit is ampere turns per meter.
H= mmf/ Length
H = NI/l AT/m
Where His magnetic field intensity
N is the number of turns
l is average path length of the magnetic flux 12

13. Cont.…
5. Magnetic Flux Linkage(??????):
The product of magnetic coupling to a conductor, or the flux thru a single
turn times the number of turns in coils.
�=�∅
Which also relates to define inductance as
�=????????????
Where �=
�
�??????
�and �=
�
�??????
????????????, L is inductance
13

14. Cont.…
6. Reluctance [S] or
It is the opposition of a magnetic circuit to setting up of a magnetic flux in
it.
����=∅=��;??????=���=��;�=��
∅
�
=
��
�??????
=
??????
????????????
??????
??????
;ℎ����∅=
??????
????????????
??????�
??????
F
∅=
�
??????
??????????????????????????????
=
�
�
; �=
�
∅
�ℎ����=
??????
????????????????????????�
Where, S –reluctance of the magnetic circuit
l -length of the magnetic path in meters
μo-permeability of free space µr-relative permeability
14

15. Cont.…
7. Permeability [μ]
A property of a magnetic material which indicates the ability of
magnetic circuit to carry electromagnetic flux.
Ratio of flux density to the magnetizing force, μ = B / H
Unit: henry / meter
Permeability of free space or air or non magnetic material
�
�=4??????×10
−7
��
Relative permeability [�
??????]:
�
??????=
�
�
�
15

16. Cont.…
8.Residual Magnetism
Itisthemagnetismwhichremainsinamaterialwhentheeffective
magnetizingforcehasbeenreducedtozero.
9. Magnetic Saturation
The limit beyond which the strength of a magnet cannot be increased is
called magnetic saturation.
16

17. Cont.…
10.EndRule
Accordingtothisrulethecurrentdirectionwhenlookedfromoneendof
thecoilisinclockwisedirectionthenthatendisSouthPole.Ifthecurrent
directionisinanticlockwisedirectionthenthatendisNorthPole.
11.Lenz’sLaw
Whenanemfisinducedinacircuitelectromagneticallythecurrentsetup
alwaysopposesthemotionorchangeincurrentwhichproducesit.
17

18. Cont.…
12. Electro magnetic induction
Electromagnetic induction means the electricity induced by the magnetic
field.
Faraday's Laws of Electro Magnetic Induction
There are two laws of Faraday's laws of electromagnetic induction.
They are,
1) First Law
2) Second Law
18

19. Cont.…
FirstLaw
Wheneveraconductorcutsthemagneticfluxlinesanemfisinducedin
theconductor.
SecondLaw
Themagnitudeoftheinducedemfisequaltotherateofchangeofflux-
linkages
�=−??????
�∅
��
WhereVisinducedvoltage Nisnumberofturnsincoil
�∅ischangeoffluxincoil��istimeinterval
19

20. Magnetic Materials
FerroMagneticMaterials:thesematerialsarestronglyattractedbya
magnet.example:iron,steel,nickel,cobalt,somemetallicalloys.The
relativepermeabilityofthesematerialsisveryhigh.
ParaMagneticMaterials:thesematerialsareattractedbyamagnetbut
notverystrongly.example:aluminum,tin,platinum,magnesium,
manganeseetc.Therelativepermeabilityofthesematerialsisslightlymore
thanone.
DiaMagneticMaterials:thesematerialsarenotatallattractedbyany
magnet.Therelativepermeabilityofthesematerialsislessthanone.
example:zinc,mercury,lead,sulfur,copper,silveretc.
20

21. Magnetic Circuit
The complete closed path followed by any group of magnetic lines of flux
is referred to as magnetic circuit.
Equivalent electrical circuit
21

22. Analogy with Electric circuits
Similarities
Electric circuit
oEmf (volt)
oCurrent(ampere)
oResistance(ohm)
oCurrent density(A/�
2
)
oConductivity
Difference
Current actually flows
Circuit may be open or closed
Magnetic circuit
om.m.f(AT)
oFlux(weber)
oReluctance(A/Wb)
oflux density(T or Wb/�
2
)
oPermeability
flux is created, but does not flow
Circuit is always closed
22

23. Cont.…
23
Electric circuit magnetic circuit

24. Cont.…
Theequivalentreluctanceofa
numberofreluctancesinseriesis
justthesumoftheindividual
reluctances:
Similarly,reluctancesinparallel
combineaccordingtotheequation
Important formulas
24

25. Leakage Flux and Fringing
LeakageFlux:themagneticflux
whichdoesnotfollowthe
particularlyintendedpathina
magneticcircuit.
Whenacurrentispassedthrougha
solenoid,magneticfluxisproduced
byit.
25

26. Cont.…
Mostofthefluxissetupinthecoreofthesolenoidandpassesthrough
theparticularpaththatisthroughtheairgapandisutilisedinthemagnetic
circuit.ThisfluxisknownasUsefulflux∅
??????
Practicallyitisnotpossiblethatallthefluxinthecircuitfollowsa
particularlyintendedpathandsetsupinthemagneticcoreandthussome
ofthefluxalsosetsuparoundthecoilorsurroundsthecoreofthecoil,
andisnotutilisedforanyworkinthemagneticcircuit.Thistypeofflux
whichisnotusedforanyworkiscalledLeakageFluxandisdenotedby
∅
??????.
ThetotalfluxΦproducedbythesolenoidinthemagneticcircuitisthe
sumoftheleakagefluxandtheusefulflux. 26

27. Cont.…
Leakage coefficient
The ratio of the total flux produced
to the useful flux set up in the air
gap of the magnetic circuit is called
leakage coefficient or leakage
factor. It is denoted by (λ).
λ=
���??????�����(����??????��ℎ�??????����??????�ℎ)
����������(����??????��ℎ�????????????��??????�)
Fringing
Theusefulfluxwhensetsupinthe
airgap,ittendstobulgeoutwardat
(bandb’)asshowninfigure,
becauseofthisbulgingthe
effectiveareaoftheairgap
increasesandthefluxdensityof
theairgapdecreases.Thiseffectis
knownasFringingandthelonger
theairgapthegreateristhe
fringing.
27

28. Series magnetic circuits
Magnetic circuit composed of
various materials of different
permeabilities.
Whencompositemagneticcircuit
partsareconnectedoneafterthe
otherthecircuitiscalledseries
magneticcircuit.
Consideracircularringmadeupof
differentmaterialsoflengths
�
1,�
2??????���
3andwithcrosssectional
areas??????
1,??????
2??????��??????
3withabsolute
permeabilities�
1,�
2and�
3.
28

29. Cont.…
Equivalent electric circuit
29

30. Series magnetic circuit with air gap
Consider a ring having mean
length of iron part as �
??????
Where �
??????=reluctance of iron path
�
??????=reluctance of air gap
�
??????=
??????
??????
????????????
??????
�
??????=
??????
??????
????????????????????????
�
�=
�
??????
�??????
??????
+
�
??????
�
�??????
??????
∅=
�.�.�
������??????���
=
??????�
�
�
Total �.�.�=??????�AT
Total reluctance �
�=�
??????+�
??????
30

31. Parallel magnetic circuit
A magnetic circuit which has more
than one path for the flux is
known as a parallel magnetic
circuit.
At point A the total flux ∅divides
into two parts ∅
1??????��∅
2.
∅=∅
1+∅
2
The fluxes ∅
1??????��∅
2have their
paths completed through ABCD
and AFEDrespectively
Magnetic core Equivalent electrical circuit
31

32. Cont.…
Total �.�.�=??????�AT
����=
�.�.�
������??????���
�.�.�=∅×�
For path ABCDA
??????�=∅
1�
1+∅�
�
For path AFEDA
??????�=∅
2�
2+∅�
�
Where
S
1=
l
1
μa1
,S
2=
l
2
μa2
and S
c=
l
c
μac
For parallel circuit
Total m.m.f= m.m.frequired by central limb
+ m.m.frequired by any one of outer limbs.
??????�=(??????�)
��+(??????�)
������(??????�)
����
??????�=∅�
�+[∅
1�
1��∅
2�
2]
32

33. Parallel magnetic circuits with air gap
Consider a parallel circuit with air
gap in the central limb
Theanalysisofthiscircuitisexactly
similartotheparallelcircuit.
Theonlychangeistheanalysisof
centrallimb.Thecentrallimbis
seriescombinationofironpathand
airgap.
Thecentrallimbismadeupof
PathGD=ironpath=�
�
PathGA=airgap=�
??????
33

34. Cont.…
∅=∅
1+∅
2
The reluctance of central limb is
�
�=�
??????+�
??????=
�
�
μ??????
�
+
�
??????
μ
�??????
�
m.m.fof central limb is
(�.�.�)
��=(�.�.�)
��+(�.�.�)
��
The total m.m.fcan be expressed as
(??????�)
??????�??????????????????=(??????�)
��+(??????�)
��+??????�
������(??????�)
����
Examples:?
Equivalent electrical circuit
34

35. Magnetic Behavior of Ferromagnetic Materials
Toillustratethebehaviorofmagneticpermeabilityinaferromagnetic
material,applyadirectcurrenttothecore,startingwith0Aandslowly
workinguptothemaximumpermissiblecurrent.
Atfirst,asmallincreaseinthemagnetomotiveforceproducesahuge
increaseintheresultingflux.Afteracertainpoint,though,further
increasesinthemagnetomotiveforceproducerelativelysmallerincreasesin
theflux.Finally,anincreaseinthemagnetomotiveforceproducesalmost
nochangeatall.
Thegraphbetweenthefluxdensity(B)andthemagneticfieldintensity(H)
forthemagneticmaterialiscalleditsmagnetizationcurveorB-Hcurve.
Itisalsocalledasaturationcurve.
35

36. Cont.…
36
Experimental set up to obtain B-H curve
Knee
unsaturation
saturation

37. Cont.…
MagneticSaturationisThelimitbeyondwhichmagneticfluxdensityina
magneticareadoesnotincreasesharplyfurtherwithincreaseofmmf.
Residualmagnetismistheamountofmagnetizationleftbehindafter
removingtheexternalmagneticfieldfromthecircuit.Inanotherwordthe
valueofthefluxdensityretainedbythemagneticmaterialiscalledResidual
Magnetismandthepowerofretainingthismagnetismiscalledretentivity
ofthematerial.or
Residualfluxdensityisthecertainvalueofmagneticfluxperunitarea
thatremainsinthemagneticmaterialwithoutpresenceofmagnetizing
force(i.e.H=0).
37

38. Cont.…
38
The magnetization curve expressed
in terms of flux density and
magnetic field intensity.
Magnetization curve of
different magnetic materials

39. Cont.…
Theregionofthisfigureinwhichthecurveflattensoutiscalledthe
saturationregion,andthecoreissaidtobesaturated.
Incontrast,theregionwherethefluxchangesveryrapidlyiscalledthe
unsaturatedregionofthecurve,andthecoreissaidtobeunsaturated.
Thetransitionregionbetweentheunsaturatedregionandthesaturated
regionissometimescalledthekneeofthecurve.
Thevalueofrelativepermeabilitymainlydependsonthevalueofflux
density.Butforthenon-magneticmaterialslikeplastic,rubber,etc.andfor
themagneticcircuithavinganairgap,itsvalueisconstant,denotedby(µ
0).
Itsvalueis4πx10
-7
H/mandcommonlyknownasabsolutepermeabilityor
permeabilityoffreespace.
39

40. Magnetic Hysteresis
1:WhensupplycurrentI=0,sonoexistenceoffluxdensity(B)and
magnetizingforce(H).Thecorrespondingpointis‘O’inthegraphbelow.
2:Whencurrentisincreasedfromzerovaluetoacertainvalue,magnetizing
force(H)andfluxdensity(B)botharesetupandincreasedfollowingthe
patho–a.
3:Foracertainvalueofcurrent,fluxdensity(B)becomesmaximum(B
max).
Thepointindicatesthemagneticsaturationormaximumfluxdensityof
thiscorematerial.Allelementofcorematerialgetalignedperfectly.Hence
H
maxismarkedonHaxis.SonochangeofvalueofBwithfurther
incrementofHoccursbeyondpoint‘a’.
40

41. Cont.…
41
Hysteresis Loop
Experimental set up to obtain Hysteresis Loop

42. Cont.…
4:Whenthevalueofcurrentisdecreasedfromitsvalueofmagneticflux
saturation,HisdecreasedalongwithdecrementofBnotfollowingthe
previouspathratherfollowingthecurvea–b.
5:Thepoint‘b’indicatesH=0forI=0withacertainvalueofB.This
laggingofBbehindHiscalledhysteresis.Thepoint‘b’explainsthatafter
removingofmagnetizingforce(H),magnetismpropertywithlittlevalue
remainsinthismagneticmaterialanditisknownasresidualmagnetism
(B
r).Hereo–bisthevalueofresidualfluxdensityduetoretentivityofthe
material.
42

43. Cont.…
6:IfthedirectionofthecurrentIisreversed,thedirectionofHalsogets
reversed.TheincrementofHinreversedirectionfollowingpathb–c
decreasesthevalueofresidualmagnetism(B
r)thatgetszeroatpoint‘c’
withcertainnegativevalueofH.ThisnegativevalueofHiscalledcoercive
force(H
c).
7:Hisincreasedmoreinnegativedirectionfurther;Bgetsreverses
followingpathc–d.Atpoint‘d’,againmagneticsaturationtakesplacebut
inoppositedirectionwithrespecttopreviouscase.Atpoint‘d’,BandH
getmaximumvaluesinreversedirection,i.e.(-B
mand-H
m).
43

44. Cont.…
8:IfwedecreasethevalueofHinthisdirection,againBdecreases
followingthepathde.Atpoint‘e’,HgetszerovaluedbutBiswithfinite
value.Thepoint‘e’standsforresidualmagnetism(-B
r)ofthemagnetic
corematerialinoppositedirectionwithrespecttopreviouscase.
9:IfthedirectionofHagainreversedbyreversingthecurrentI,then
residualmagnetismorresidualfluxdensity(-B
r)againdecreasesandgets
zeroatpoint‘f’followingthepathe–f.AgainfurtherincrementofH,the
valueofBincreasesfromzerotoitsmaximumvalueorsaturationlevelat
pointafollowingpathf–a.
The path a –b –c –d –e –f –a forms hysteresis loop.
[NB: The shape and the size of the hysteresis loop depend on the nature of
the material chosen]
44

45. Cont.…
Hysteresis:Thephenomenonoffluxdensity(B)laggingbehindthemagnetizingforce
(H)inamagneticmaterialisknownasMagneticHysteresis.
Coerciveforceisdefinedasthenegativevalueofmagnetizingforce(-H)thatreduces
residualfluxdensityofamaterialtozero.
Retentivity:Itisdefinedasthedegreetowhichamagneticmaterialgainsitsmagnetism
aftermagnetizingforce(H)isreducedtozero.
Thehysteresislossinanironcoreistheenergyrequiredtoaccomplishthe
reorientationofdomainsduringeachcycleofthealternatingcurrentappliedtothe
core.
Theareaenclosedinthehysteresisloopformedbyapplyinganalternatingcurrentto
thecoreisdirectlyproportionaltotheenergylostinagivenaccycle.
45

46. Hysteresis Loss
The work done by the magnetizing force against the internal friction of the
molecules of the magnet, produces heat. This energy which is wasted in the
form of heat due to hysteresis is calledhysteresis loss.
Where, P
h–hysteresis loss in watts
Ƞ–hysteresis or Steinmetz’s constant in J/m
3
,
B
max–maximum value of the flux density in the magnetic material in
wb/m
2
�–number of cycles of magnetization made per second
�-volume of the magnetic material (part in which magnetic
reversal occur) in m
3
46

47. Cont.…
Soft magnetic material
Thesoftmagneticmaterialhasanarrowmagnetic
hysteresisloopwhichhasasmallamountof
dissipatedenergy.Theyaremadeupofmaterial
likeiron,siliconsteel,etc.
Itisusedinthedevicesthatrequirealternating
magneticfield.
Ithaslowcoercivity
Lowmagnetization
Lowretentivity
47

48. Cont.…
Hard magnetic material
TheHardmagneticmaterialhasa
widerhysteresisloopandresultsina
largeamountofenergydissipation
andthedemagnetizationprocessis
moredifficulttoachieve.
It has high retentivity
High coercivity
High saturation
48

49. Importance of Hysteresis Loop
Smallerhysteresisloopareasymbolizeslesshysteresisloss.
Hysteresisloopprovidesthevalueofretentivityandcoercivityofa
material.Thusthewaytochooseperfectmaterialtomakepermanent
magnet,coreofmachinesbecomeseasier.
FromB-Hgraph,residualmagnetismcanbedeterminedandthuschoosing
ofmaterialforelectromagnetsiseasy.
Magneticmaterialhavingawiderhysteresisloopisusedinthedeviceslikemagnetic
tape,harddisk,creditcards,audiorecordingsasitsmemoryisn’teasilyerased.
Magneticmaterialshavinganarrowhysteresisloopareusedaselectromagnets,solenoid,
transformersandrelayswhichrequireminimumenergydissipation.
49

50. Eddy Current Loss
Whenanalternatingmagneticfieldisappliedtoamagneticmaterialanemf
isinducedinthematerialitself.
Sincethemagneticmaterialisaconductingmaterial,theseEMFscirculates
currentswithinthebodyofthematerial.Thesecirculatingcurrentsare
calledEddyCurrents.Theywilloccurwhentheconductorexperiencesa
changingmagneticfield.
Asthesecurrentsarenotresponsiblefordoinganyusefulwork,andit
producesaloss(�
2
�loss)inthemagneticmaterialknownasanEddy
CurrentLoss.Similartohysteresisloss,eddycurrentlossalsoincreasesthe
temperatureofthemagneticmaterial.
50

51. Cont.…
Thehysteresisandtheeddy
currentlossesinamagnetic
materialarealsoknownbythe
nameironlossesorcorelossesor
magneticlosses.
Whenthechangingfluxlinkswith
thecoreitself,itinducesemfin
thecorewhichinturnssetsupthe
circulatingcurrentcalledEddy
Currentandthesecurrentinreturn
producesalosscallededdycurrent
lossor(�
2
�)loss.
whereIisthevalueofthecurrentand
Ristheresistanceoftheeddycurrentpath.
51

52. Cont.…
If the core is made up of solid iron of larger cross-sectional area, the
magnitude of I will be very large and hence losses will be high. To reduce
the eddy current loss mainlythere are two methods.
By reducing the magnitude of the eddy current.
Themagnitudeofthecurrentcanbereducedbysplittingthesolidcoreintothin
sheetscalledlaminations,intheplaneparalleltothemagneticfield.Eachlamination
isinsulatedfromeachotherbyathinlayerofcoatingofvarnishoroxidefilm.By
laminatingthecore,theareaofeachsectionisreducedandhencetheinducedemf
alsoreduces.Astheareathroughwhichthecurrentispassedissmaller,the
resistanceofeddycurrentpathincreases.
Theeddycurrentlossisalsoreducedbyusingamagneticmaterialhaving
thehighervalueofresistivitylikesiliconsteel.
52

53. Cont.…
Itisdifficulttodeterminetheeddycurrentlossfromtheresistanceand
currentvalues,butbytheexperiments,theeddycurrentpowerlossina
magneticmaterialisgivenbytheequation
Where,P
e=eddycurrentlossinwatts
K
e=coefficientofeddycurrent.
B
m=maximumvalueoffluxdensityinΤWbm
2
�=thicknessoflaminationinmeters
�=frequencyofreversalofmagneticfieldinHz
�=volumeofmagneticmaterialin�
3
. 53

54. Faraday's law-induced voltage from a time-changing magnetic field
Fromthevariouswaysinwhichanexistingmagneticfieldcanaffectits
surroundings,thefirstmajoreffectisFaraday'slaw.
Itstatesthatifafluxpassesthroughaturnofacoilofwire,avoltagewill
beinducedintheturnofwirethatisdirectlyproportionaltotherateof
changeinthefluxwithrespecttotime.
Where�
??????��isthevoltageinducedintheturnofthecoiland
∅isthefluxpassingthroughtheturn.
IfacoilhasNturnsandifthesamefluxpassesthroughallofthem,thenthe
voltageinducedacrossthewholecoilisgivenby
54

55. Cont.…
Where �
??????��= voltage induced in the coil
N = number of turns of wire in coil
∅=flux passing through coil
TheminussignintheequationsisanexpressionofLenz'slaw.
Lenz'slawstatesthatthedirectionofthevoltagebuildupinthecoilissuch
thatifthecoilendswereshortcircuited,itwouldproducecurrentthat
wouldcauseafluxopposingtheoriginalfluxchange.Sincetheinduced
voltageopposesthechangethatcausesit,aminussignisincluded.
55

56. Cont.…
The above equation assumes that
exactly the same flux is present in
each turn of the coil. Unfortunately,
the flux leaking out of the core into
the surrounding air prevents this
from being true.
The magnitude of the voltage in the
??????
??????ℎ
turn of the coil is always given
by
If there are N turns in the coil of
wire, the total voltage on the coil is
56
Where
�=Flux linkage

57. Production of induced force on a wire
Asecondmajoreffectofamagneticfield
onitssurroundingsisthatitinducesaforce
onacurrent-carryingwirewithinthefield.
Theforceinducedontheconductoris
givenby
Where i= magnitude of current in wire
??????=lengthofwire,withdirectionofI
definedtobeinthedirection
ofcurrentflow
B = magnetic flux density vector
57
Fleming's Right-hand Rule

58. Cont.…
Thedirectionoftheforceisgiven
bytheright-handrule:Iftheindex
fingeroftherighthandpointsin
thedirectionofthevectorIand
themiddlefingerpointsinthe
directionofthefluxdensityvector
B,thenthethumbpointsinthe
directionoftheresultantforceon
thewire.
Themagnitudeoftheforceis
givenby
where ??????is the angle between the
wire and the flux density vector.
58current-carrying wire in the
presence of a magnetic field

59. Induced voltage on a conductor moving in a magnetic field
If a wire with the proper orientation moves through a magnetic field, a
voltage is induced in it. The voltage induced in the wire is given by
Where �= velocity of the wire
B = magnetic flux density vector
�= length of conductor in the magnetic field
59

60. Cont.…
Vector�pointsalongthedirection
ofthewiretowardtheendmaking
thesmallestanglewithrespectto
thevector��.
The voltage in the wire will be built
up so that the positive end is in the
direction of the vector ��.
60

61. 61