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8/14/2019 INDUCTANCE Thestudyofinductancepresentsaverychallengingbutrewardingsegmentofelectricity.Itis Cha
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INDUCTANCE
Thestudyofinductancepresentsaverychallengingbutrewardingsegmentofelectricity.Itis
challenginginthesensethat,atfirst,itwillseemthatnewconceptsarebeingintroduced.You
willrealizeasthischapterprogressesthatthese"newconcepts"aremerelyextensionsand
enlargementsof
fundamental
principles
that
you
learned
previously
in
the
study
of
magnetism
andelectronphysics.Thestudyofinductanceisrewardinginthesensethatathorough
understandingofitwillenableyoutoacquireaworkingknowledgeofelectricalcircuitsmore
rapidly.
CHARACTERISTICSOFINDUCTANCE
Inductanceisthecharacteristicofanelectricalcircuitthatopposesthestarting,stopping,ora
changeinvalueofcurrent.Theabovestatementisofsuchimportancetothestudyof
inductancethatitbearsrepeating.Inductanceisthecharacteristicofanelectricalconductor
thatOPPOSES
CHANGE
in
CURRENT.
The
symbol
for
inductance
is
Land
the
basic
unit
of
inductanceistheHENRY(H).OneHenryisequaltotheinductancerequiredtoinduceonevolt
inaninductorbyachangeofcurrentofoneamperepersecond.
Youdonothavetolookfartofindaphysicalanalogyofinductance.Anyonewhohaseverhad
topushaheavyload(wheelbarrow,car,etc.)isawarethatittakesmoreworktostarttheload
movingthanitdoestokeepitmoving.Oncetheloadismoving,itiseasiertokeeptheload
movingthantostopitagain.ThisisbecausetheloadpossessesthepropertyofINERTIA.Inertia
isthecharacteristicofmasswhichopposesaCHANGEinvelocity.Inductancehasthesame
effecton
current
in
an
electrical
circuit
as
inertia
has
on
the
movement
of
amechanical
object.
Itrequiresmoreenergytostartorstopcurrentthanitdoestokeepitflowing.
Q.1whatisthebasicunitofinductanceandtheabbreviationforthisunit?
ELECTROMOTIVEFORCE(EMF)
Youhavelearnedthatanelectromotiveforceisdevelopedwheneverthereisrelativemotion
betweenamagneticfieldandaconductor.
Electromotiveforceisadifferenceofpotentialorvoltagewhichexistsbetweentwopointsinan
electricalcircuit.
In
generators
and
inductors
the
emf
is
developed
by
the
action
between
the
magneticfieldandtheelectronsinaconductor.Thisisshowninfigure21.
Figure21. Generationofanemfinanelectricalconductor.
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Whenamagnetic
field
moves
through
astationary
metallic
conductor,
electrons
are
dislodged
fromtheirorbits.Theelectronsmoveinadirectiondeterminedbythemovementofthe
magneticlinesofflux.Thisisshownbelow:
Theelectronsmovefromoneareaoftheconductorintoanotherarea.Theareathatthe
electronsmovedfromhasfewernegativecharges(electrons)andbecomespositivelycharged.
Theareatheelectronsmoveintobecomesnegativelycharged.Thisisshownbelow:
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Thedifferencebetweenthechargesintheconductorisequaltoadifferenceofpotential(or
voltage).Thisvoltagecausedbythemovingmagneticfieldiscalledelectromotiveforce(emf).
Insimpleterms,theactionofamovingmagneticfieldonaconductorcanbecomparedtothe
actionofabroom.Considerthemovingmagneticfieldtobeamovingbroom.Asthemagnetic
broommovesalong(through)theconductor,itgathersupandpusheselectronsbeforeit,as
shownbelow:
Thearea
from
which
electrons
are
moved
becomes
positively
charged,
while
the
area
into
whichelectronsaremovedbecomesnegativelycharged.Thepotentialdifferencebetween
thesetwoareasistheelectromotiveforceoremf.
Q.2Anemfisgeneratedinaconductorwhentheconductoriscutbywhattypeoffield?
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SELFINDUCTANCE
Evenaperfectlystraightlengthofconductorhassomeinductance.Asyouknow,currentina
conductorproduces
amagnetic
field
surrounding
the
conductor.
When
the
current
changes,
themagneticfieldchanges.Thiscausesrelativemotionbetweenthemagneticfieldandthe
conductor,andanelectromotiveforce(emf)isinducedintheconductor.Thisemfiscalleda
SELFINDUCEDEMFbecauseitisinducedintheconductorcarryingthecurrent.Theemf
producedbythismovingmagneticfieldisalsoreferredtoasCOUNTERELECTROMOTIVEFORCE
(cemf).Thepolarityofthecounterelectromotiveforceisintheoppositedirectiontothe
appliedvoltageoftheconductor.Theoveralleffectwillbetoopposeachangeincurrent
magnitude.ThiseffectissummarizedbyLenz'slawwhichstatesthat:THEINDUCEDEMFIN
ANYCIRCUITISALWAYSINADIRECTIONTOOPPOSETHEEFFECTTHATPRODUCEDIT.
Ifthe
shape
of
the
conductor
is
changed
to
form
aloop,
then
the
electromagnetic
field
around
eachportionoftheconductorcutsacrosssomeotherportionofthesameconductor.Thisis
showninitssimplestforminfigure22.Alengthofconductorisloopedsothattwoportionsof
theconductorlienexttoeachother.Theseportionsarelabeledconductor1andconductor2.
Whentheswitchisclosed,current(electronflow)intheconductorproducesamagneticfield
aroundALLportionsoftheconductor.Forsimplicity,themagneticfield(expandinglinesofflux)
isshowninasingleplanethatisperpendiculartobothconductors.Althoughtheexpanding
fieldoffluxoriginatesatthesametimeinbothconductors,itisconsideredasoriginatingin
conductor1anditseffectonconductor2willbeexplained.Withincreasingcurrent,theflux
fieldexpands
outward
from
conductor
1,
cutting
across
aportion
of
conductor
2.
This
results
in
aninducedemfinconductor2asshownbythedashedarrow.Notethattheinducedemfisin
theoppositedirectionto(inOPPOSITIONto)thebatterycurrentandvoltage,asstatedinLenz's
law.
Figure22. Selfinductance.
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ThedirectionofthisinducedvoltagemaybedeterminedbyapplyingtheLEFTHANDRULEFOR
GENERATORS.Thisruleisappliedtoaportionofconductor2thatis"lifted"andenlargedfor
thispurposeinfigure22(A).Thisrulestatesthatifyoupointthethumbofyourlefthandinthe
directionofrelativemotionoftheconductorandyourindexfingerinthedirectionofthe
magnetic
field,
your
middle
finger,
extended
as
shown,
will
now
indicate
the
direction
of
the
inducedcurrentwhichwillgeneratetheinducedvoltage(cemf)asshown.
Infigure22(B),thesamesectionofconductor2isshownaftertheswitchhasbeenopened.
Thefluxfieldiscollapsing.Applyingthelefthandruleinthiscaseshowsthatthereversalofflux
MOVEMENThascausedareversalinthedirectionoftheinducedvoltage.Theinducedvoltage
isnowinthesamedirectionasthebatteryvoltage.Themostimportantthingforyoutonoteis
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thattheselfinducedvoltageopposesBOTHchangesincurrent.Thatis,whentheswitchis
closed,thisvoltagedelaystheinitialbuildupofcurrentbyopposingthebatteryvoltage.When
theswitchisopened,itkeepsthecurrentflowinginthesamedirectionbyaidingthebattery
voltage.
Thus,from
the
above
explanation,
you
can
see
that
when
acurrent
is
building
up
it
produces
an
expandingmagneticfield.Thisfieldinducesanemfinthedirectionoppositetotheactualflow
ofcurrent.Thisinducedemfopposesthegrowthofthecurrentandthegrowthofthemagnetic
field.Iftheincreasingcurrenthadnotsetupamagneticfield,therewouldhavebeenno
oppositiontoitsgrowth.Thewholereaction,oropposition,iscausedbythecreationor
collapseofthemagneticfield,thelinesofwhichastheyexpandorcontractcutacrossthe
conductoranddevelopthecounteremf.
Sinceallcircuitshaveconductorsinthem,youcanassumethatallcircuitshaveinductance.
However,
inductance
has
its
greatest
effect
only
when
there
is
a
change
in
current.
Inductance
doesNOTopposecurrent,onlyaCHANGEincurrent.Wherecurrentisconstantlychangingasin
anaccircuit,inductancehasmoreeffect.
Q.3Defineinductance.
Q.4Whatismeantbyinducedemf?Bycounteremf?
Q.5StateLenz'slaw.
Q.6Whateffectdoesinductancehave(a)onsteadydirectcurrentand(b)ondirectcurrent
whileitischanginginamplitude?
Toincrease
the
property
of
inductance,
the
conductor
can
be
formed
into
aloop
or
coil.
A
coil
is
alsocalledaninductor.Figure23showsaconductorformedintoacoil.Currentthroughone
loopproducesamagneticfieldthatencirclestheloopinthedirectionasshowninfigure23(A).
Ascurrentincreases,themagneticfieldexpandsandcutsalltheloopsasshowninfigure2
3(B).Thecurrentineachloopaffectsallotherloops.Thefieldcuttingtheotherloophasthe
effectofincreasingtheoppositiontoacurrentchange.
Figure23. Inductance.
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Inductorsareclassifiedaccordingtocoretype.Thecoreisthecenteroftheinductorjustasthe
coreofanappleisthecenterofanapple.Theinductorismadebyformingacoilofwirearound
a
core.
The
core
material
is
normally
one
of
two
basic
types:
soft
iron
or
air.
An
iron
core
inductoranditsschematicsymbol(whichisrepresentedwithlinesacrossthetopofitto
indicatethepresenceofanironcore)areshown,infigure24(A).Theaircoreinductormaybe
nothingmorethanacoilofwire,butitisusuallyacoilformedaroundahollowformofsome
nonmagneticmaterialsuchascardboard.Thismaterialservesnopurposeotherthantohold
theshapeofthecoil.Anaircoreinductoranditsschematicsymbolareshowninfigure24(B).
Figure24. Inductortypesandschematicsymbols.
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FactorsAffectingCoilInductanceThereareseveralphysicalfactorswhichaffecttheinductance
ofacoil.Theyincludethenumberofturnsinthecoil,thediameterofthecoil,thecoillength,
thetypeofmaterialusedinthecore,andthenumberoflayersofwindinginthecoils.
Inductancedependsentirelyuponthephysicalconstructionofthecircuit,andcanonlybe
measuredwithspeciallaboratoryinstruments.Ofthefactorsmentioned,considerfirsthowthe
numberofturnsaffectstheinductanceofacoil.Figure25showstwocoils.Coil(A)hastwo
turnsandcoil(B)hasfourturns.Incoil(A),thefluxfieldsetupbyoneloopcutsoneotherloop.
Incoil(B),thefluxfieldsetupbyoneloopcutsthreeotherloops.Doublingthenumberofturns
inthecoilwillproduceafieldtwiceasstrong,ifthesamecurrentisused.Afieldtwiceas
strong,cuttingtwicethenumberofturns,willinducefourtimesthevoltage.Therefore,itcan
besaidthattheinductancevariesasthesquareofthenumberofturns.
Figure25. Inductancefactor(turns).
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Thesecondfactoristhecoildiameter.Infigure26youcanseethatthecoilinviewBhastwice
thediameterofcoilviewA.Physically,itrequiresmorewiretoconstructacoiloflarge
diameterthanoneofsmalldiameterwithanequalnumberofturns.Therefore,morelinesof
forceexisttoinduceacounteremfinthecoilwiththelargerdiameter.Actually,theinductance
ofacoilincreasesdirectlyasthecrosssectionalareaofthecoreincreases.Recalltheformula
fortheareaofacircle:A=pr2.Doublingtheradiusofacoilincreasestheinductancebyafactor
offour.
Figure26. Inductancefactor(diameter).
Thethirdfactorthataffectstheinductanceofacoilisthelengthofthecoil.Figure27shows
twoexamplesofcoilspacings.Coil(A)hasthreeturns,ratherwidelyspaced,makingarelatively
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longcoil.Acoilofthistypehasfewfluxlinkages,duetothegreaterdistancebetweeneach
turn.Therefore,coil(A)hasarelativelylowinductance.Coil(B)hascloselyspacedturns,
makingarelativelyshortcoil.Thisclosespacingincreasesthefluxlinkage,increasingthe
inductanceofthecoil.Doublingthelengthofacoilwhilekeepingthesamenumberofturns
halves
the
value
of
inductance.
Figure2 7. Inductancefactor(coillength).CLOSELYWOUND
Thefourthphysicalfactoristhetypeofcorematerialusedwiththecoil.Figure28showstwo
coils:Coil(A)withanaircore,andcoil(B)withasoftironcore.Themagneticcoreofcoil(B)isa
betterpathformagneticlinesofforcethanisthenonmagneticcoreofcoil(A).Thesoftiron
magneticcore'shighpermeabilityhaslessreluctancetothemagneticflux,resultinginmore
magneticlinesofforce.Thisincreaseinthemagneticlinesofforceincreasesthenumberof
linesofforcecuttingeachloopofthecoil,thusincreasingtheinductanceofthecoil.Itshould
nowbe
apparent
that
the
inductance
of
acoil
increases
directly
as
the
permeability
of
the
core
materialincreases.
Figure28. Inductancefactor(corematerial).SOFTIRONCORE
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Anotherwayofincreasingtheinductanceistowindthecoilinlayers.Figure29showsthree
coreswithdifferentamountsoflayering.Thecoilinfigure29(A)isapoorinductorcompared
totheothersinthefigurebecauseitsturnsarewidelyspacedandthereisnolayering.Theflux
movement,indicatedbythedashedarrows,doesnotlinkeffectivelybecausethereisonlyone
layerofturns.Amoreinductivecoilisshowninfigure29(B).Theturnsarecloselyspacedand
thewirehasbeenwoundintwolayers.Thetwolayerslinkeachotherwithagreaternumberof
fluxloopsduringallfluxmovements.Notethatnearlyalltheturns,suchasX,arenexttofour
otherturns(shaded).Thiscausesthefluxlinkagetobeincreased.
Figure29. Coilsofvariousinductances.
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Acoilcanbemadestillmoreinductivebywindingitinthreelayers,asshowninfigure29(C).
Theincreasednumberoflayers(crosssectionalarea)improvesfluxlinkageevenmore.Note
thatsome
turns,
such
as
Y,
lie
directly
next
to
six
other
turns
(shaded).
In
actual
practice,
layeringcancontinueonthroughmanymorelayers.Theimportantfacttoremember,however,
isthattheinductanceofthecoilincreaseswitheachlayeradded.
Asyouhaveseen,severalfactorscanaffecttheinductanceofacoil,andallofthesefactorsare
variable.Manydifferentlyconstructedcoilscanhavethesameinductance.Theimportant
informationtoremember,however,isthatinductanceisdependentuponthedegreeoflinkage
betweenthewireconductor(s)andtheelectromagneticfield.Inastraightlengthofconductor,
thereisverylittlefluxlinkagebetweenonepartoftheconductorandanother.Therefore,its
inductanceisextremelysmall.Itwasshownthatconductorsbecomemuchmoreinductive
whentheyarewoundintocoils.Thisistruebecausethereismaximumfluxlinkagebetween
theconductorturns,whichliesidebysideinthecoil.
Q.7Listfivefactorsthataffecttheinductanceofacoil.
Bendingastraightpieceofwireintoalooporcoilhaswhateffectontheinductanceofthewire?
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Doublingthenumberofturnsinacoilhaswhateffectontheinductanceofthecoil? Decreasingthediameterofacoilhaswhateffectontheinductanceofthecoil? Insertingasoftironcoreintoacoilhaswhateffectontheinductanceofthecoil? Increasingthenumberoflayersofwindingsinacoilhaswhateffectontheinductance
ofthecoil?
UNITOFINDUCTANCE
Asstatedbefore,thebasicunitofinductance(L)istheHENRY(H),namedafterJosephHenry,
thecodiscovererwithFaradayoftheprincipleofelectromagneticinduction.Aninductorhas
aninductanceof1henryifanemfof1voltisinducedintheinductorwhenthecurrentthrough
theinductorischangingattherateof1amperepersecond.Therelationshipbetweenthe
inducedvoltage,theinductance,andtherateofchangeofcurrentwithrespecttotimeis
statedmathematicallyas:
whereEindistheinducedemfinvolts;Listheinductanceinhenrys;andIisthechange
incurrentinamperesoccurringintseconds.Thesymbol(Greekletterdelta),
means"achangein....".Thehenryisalargeunitofinductanceandisusedwithrelativelylarge
inductors.Withsmallinductors,themillihenryisused.(Amillihenryisequalto1X103
henry,
andone
henry
is
equal
to
1,000
millihenrys.)
For
still
smaller
inductors
the
unit
of
inductance
is
themicrohenry(H).(AH=1X106H,andonehenryisequalto1,000,000
microhenrys.)
GROWTHANDDECAYOFCURRENTINANLRSERIESCIRCUIT
Whenabatteryisconnectedacrossa"pure"inductance,thecurrentbuildsuptoitsfinalvalue
ataratedeterminedbythebatteryvoltageandtheinternalresistanceofthebattery.The
currentbuildupisgradualbecauseofthecounteremfgeneratedbytheselfinductanceofthe
coil.Whenthecurrentstartstoflow,themagneticlinesofforcemoveoutwardfromthecoil.
Theselines
cut
the
turns
of
wire
on
the
inductor
and
build
up
acounter
emf
that
opposes
the
emfofthebattery.Thisoppositioncausesadelayinthetimeittakesthecurrenttobuildupto
asteadyvalue.Whenthebatteryisdisconnected,thelinesofforcecollapse.Againtheselines
cuttheturnsoftheinductorandbuildupanemfthattendstoprolongtheflowofcurrent.
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Avoltagedividercontainingresistanceandinductancemaybeconnectedinacircuitbymeans
ofaspecialswitch,asshowninfigure210(A).SuchaseriesarrangementiscalledanLRseries
circuit.
Figure210. GrowthanddecayofcurrentinanLRseriescircuit.
WhenswitchS1isclosed(asshown),avoltageEsappearsacrossthevoltagedivider.Atthis
instantthecurrentwillattempttoincreasetoitsmaximumvalue.However,thisinstantaneous
currentchangecausescoilLtoproduceabackEMF,whichisoppositeinpolarityandalmost
equaltotheEMFofthesource.ThisbackEMFopposestherapidcurrentchange.Figure210(B)
showsthatattheinstantswitchS1isclosed,thereisnomeasurablegrowthcurrent(ig),a
minimumvoltagedropisacrossresistorR,andmaximumvoltageexistsacrossinductorL.
Ascurrentstartstoflow,avoltage(eR)appearsacrossR,andthevoltageacrosstheinductoris
reducedbythesameamount.Thefactthatthevoltageacrosstheinductor(L)isreduced
meansthatthegrowthcurrent(ig)isincreasedandconsequentlyeRisincreased.Figure2
10(B)showsthatthevoltageacrosstheinductor(eL)finallybecomeszerowhenthegrowth
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current(ig)stopsincreasing,whilethevoltageacrosstheresistor(eR)buildsuptoavalueequal
tothesourcevoltage(ES).
Electricalinductanceislikemechanicalinertia,andthegrowthofcurrentinaninductivecircuit
canbelikenedtotheaccelerationofaboatonthesurfaceofthewater.Theboatdoesnot
moveat
the
instant
aconstant
force
is
applied
to
it.
At
this
instant
all
the
applied
force
is
used
toovercometheinertiaoftheboat.Oncetheinertiaisovercometheboatwillstarttomove.
Afterawhile,thespeedoftheboatreachesitsmaximumvalueandtheappliedforceisusedup
inovercomingthefrictionofthewateragainstthehull.
Whenthebatteryswitch(S1)intheLRcircuitoffigure210(A)isclosed,therateofthecurrent
increaseismaximumintheinductivecircuit.Atthisinstantallthebatteryvoltageisusedin
overcomingtheemfofselfinductionwhichisamaximumbecausetherateofchangeof
currentismaximum.Thusthebatteryvoltageisequaltothedropacrosstheinductorandthe
voltage
across
the
resistor
is
zero.
As
time
goes
on
more
of
the
battery
voltage
appears
across
theresistorandlessacrosstheinductor.Therateofchangeofcurrentislessandtheinduced
emfisless.Asthesteadystateconditionofthecurrentisapproached,thedropacrossthe
inductorapproacheszeroandallofthebatteryvoltageis"dropped"acrosstheresistanceof
thecircuit.
Thusthevoltagesacrosstheinductorandtheresistorchangeinmagnitudeduringtheperiodof
growthofcurrentthesamewaytheforceappliedtotheboatdividesitselfbetweentheeffects
ofinertiaandfriction.Inbothexamples,theforceisdevelopedfirstacrosstheinertia/inductive
effectandfinallyacrossthefriction/resistiveeffect.
Figure2 10(C)showsthatwhenswitch
S2isclosed(sourcevoltageESremovedfromthecircuit),thefluxthathasbeenestablished
aroundtheinductor(L)collapsesthroughthewindings.ThisinducesavoltageeLintheinductor
thathasapolarityoppositetoESandisessentiallyequaltoESinmagnitude.Theinduced
voltagecausesdecaycurrent(id)toflowinresistorRinthesamedirectioninwhichcurrentwas
flowingoriginally(whenS1wasclosed).Avoltage(eR)thatisinitiallyequaltosourcevoltage
(ES)isdevelopedacrossR.Thevoltageacrosstheresistor(eR)rapidlyfallstozeroasthevoltage
acrosstheinductor(eL)fallstozeroduetothecollapsingflux.
Justastheexampleoftheboatwasusedtoexplainthegrowthofcurrentinacircuit,itcanalso
beusedtoexplainthedecayofcurrentinacircuit.Whentheforceappliedtotheboatis
removed,theboatstillcontinuestomovethroughthewaterforawhile,eventuallycomingtoa
stop.Thisisbecauseenergywasbeingstoredintheinertiaofthemovingboat.Afteraperiod
oftimethefrictionofthewaterovercomestheinertiaoftheboat,andtheboatstopsmoving.
Justasinertiaoftheboatstoredenergy,themagneticfieldofaninductorstoresenergy.
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Becauseofthis,evenwhenthepowersourceisremoved,thestoredenergyofthemagnetic
fieldoftheinductortendstokeepcurrentflowinginthecircuituntilthemagneticfield
collapse.
Q.8WhenvoltageisfirstappliedtoaseriesLRcircuit,howmuchoppositiondoesthe
inductancehave
to
the
flow
of
current
compared
to
that
of
the
circuit
resistance
Inaseriescircuitcontainingaresistor(R1)andaninductor(L1),whatvoltageexistsacrossR1
whenthecounteremfisatitsmaximumvalue?
WhathappenstothevoltageacrosstheresistanceinanLRcircuitduringcurrentbuildupinthe
circuit,andduringcurrentdecayinthecircuit?
L/RTimeConstant
TheL/RTIMECONSTANTisavaluabletoolforuseindeterminingthetimerequiredforcurrent
inaninductortoreachaspecificvalue.Asshowninfigure211,oneL/Rtimeconstantisthe
timerequiredforthecurrentinaninductortoincreaseto63percent(actually63.2percent)of
themaximumcurrent.Eachtimeconstantisequaltothetimerequiredforthecurrentto
increaseby63.2percentofthedifferenceinvaluebetweenthecurrentflowingintheinductor
andthemaximumcurrent.MaximumcurrentflowsintheinductorafterfiveL/Rtimeconstants
arecompleted.Thefollowingexampleshouldclearupanyconfusionabouttimeconstants.
AssumethatmaximumcurrentinanLRcircuitis10amperes.Asyouknow,whenthecircuitis
energized,ittakestimeforthecurrenttogofromzeroto10amperes.Whenthefirsttime
constantiscompleted,thecurrentinthecircuitisequalto63.2%of10amperes.Thusthe
amplitudeofcurrentattheendof1timeconstantis6.32amperes.
Figure211. L/Rtimeconstant.
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Duringthesecondtimeconstant,currentagainincreasesby63.2%(.632)ofthedifferencein
valuebetweenthecurrentflowingintheinductorandthemaximumcurrent.Thisdifferenceis
10amperesminus6.32amperesandequals3.68amperes;63.2%of3.68amperesis2.32
amperes.Thisincreaseincurrentduringthesecondtimeconstantisaddedtothatofthefirst
time
constant.
Thus,
upon
completion
of
the
second
time
constant,
the
amount
of
current
in
theLRcircuitis6.32amperes+2.32amperes=8.64amperes.
Duringthethirdconstant,currentagainincreases:
Duringthefourthtimeconstant,currentagainincreases:
Duringthefifthtimeconstant,currentincreasesasbefore:
Thus,thecurrentattheendofthefifthtimeconstantisalmostequalto10.0amperes,the
maximumcurrent.Forallpracticalpurposestheslightdifferenceinvaluecanbeignored.
WhenanLRcircuitisdeenergized,thecircuitcurrentdecreases(decays)tozeroinfivetime
constantsatthesameratethatitpreviouslyincreased.Ifthegrowthanddecayofcurrentinan
LRcircuitareplottedonagraph,thecurveappearsasshowninfigure211.Noticethatcurrent
increasesanddecaysatthesamerateinfivetimeconstants.
Thevalueofthetimeconstantinsecondsisequaltotheinductanceinhenrysdividedbythe
circuitresistanceinohms.
TheformulausedtocalculateoneL/Rtimeconstantis:
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Q.9WhatistheformulaforoneL/Rtimeconstant?
Q.10Themaximumcurrentappliedtoaninductoris1.8amperes.Howmuchcurrentflowedin
theinductor3timeconstantsafterthecircuitwasfirstenergized?
WhatistheminimumnumberoftimeconstantsrequiredforthecurrentinanLRcircuitto
increasetoitsmaximumvalue?
Acircuitcontainingonlyaninductorandaresistorhasamaximumof12amperesofapplied
currentflowinginit.After5L/Rtimeconstantsthecircuitisopened.Howmanytimeconstants
isrequiredforthecurrenttodecayto1.625amperes?
POWERLOSSINANINDUCTOR
Sinceaninductor(coil)consistsofanumberofturnsofwire,andsinceallwirehassome
resistance,everyinductorhasacertainamountofresistance.Normallythisresistanceissmall.
Itisusuallyneglectedinsolvingvarioustypesofaccircuitproblemsbecausethereactanceof
theinductor(theoppositiontoalternatingcurrent,whichwillbediscussedlater)issomuch
greaterthantheresistancethattheresistancehasanegligibleeffectonthecurrent.
However,sincesomeinductorsaredesignedtocarryrelativelylargeamountsofcurrent,
considerablepowercanbedissipatedintheinductoreventhoughtheamountofresistancein
theinductorissmall.ThispoweriswastedpowerandiscalledCOPPERLOSS.Thecopperlossof
aninductorcanbecalculatedbymultiplyingthesquareofthecurrentintheinductorbythe
resistanceofthewinding(I2R).
Inadditiontocopperloss,anironcorecoil(inductor)hastwoironlosses.Thesearecalled
HYSTERESISLOSSandEDDYCURRENTLOSS.Hysteresislossisduetopowerthatisconsumedin
reversingthemagneticfieldoftheinductorcoreeachtimethedirectionofcurrentinthe
inductorchanges.
Eddycurrentlossisduetoheatingofthecorebycirculatingcurrentsthatareinducedinthe
ironcorebythemagneticfieldaroundtheturnsofthecoil.Thesecurrentsarecallededdy
currentsandcirculatewithintheironcoreonly.
Alltheselossesdissipatepowerintheformofheat.Sincethispowercannotbereturnedtothe
electricalcircuit,itislostpower.
Q.11Statethreetypesofpowerlossinaninductor.
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MUTUALINDUCTANCE
Whenevertwocoilsarelocatedsothatthefluxfromonecoillinkswiththeturnsoftheother
coil,achangeoffluxinonecoilcausesanemftobeinducedintheothercoil.Thisallowsthe
energyfromonecoiltobetransferredorcoupledtotheothercoil.Thetwocoilsaresaidtobe
coupledor
linked
by
the
property
of
MUTUAL
INDUCTANCE
(M).
The
amount
of
mutual
inductancedependsontherelativepositionsofthetwocoils.Thisisshowninfigure212.Ifthe
coilsareseparatedaconsiderabledistance,theamountoffluxcommontobothcoilsissmall
andthemutualinductanceislow.Conversely,ifthecoilsareclosetogethersothatnearlyall
thefluxofonecoillinkstheturnsoftheother,themutualinductanceishigh.Themutual
inductancecanbeincreasedgreatlybymountingthecoilsonacommonironcore.
Figure212. Theeffectofpositionofcoilsonmutualinductance(M).
Two
coils
are
placed
close
together
as
shown
in
figure
2
13.
Coil
1
is
connected
to
a
battery
throughswitchS,andcoil2isconnectedtoanammeter(A).WhenswitchSisclosedasin
figure213(A),thecurrentthatflowsincoil1setsupamagneticfieldthatlinkswithcoil2,
causinganinducedvoltageincoil2andamomentarydeflectionoftheammeter.Whenthe
currentincoil1reachesasteadyvalue,theammeterreturnstozero.IfswitchSisnowopened
asinfigure213(B),theammeter(A)deflectsmomentarilyintheoppositedirection,indicating
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amomentaryflowofcurrentintheoppositedirectionincoil2.Thiscurrentincoil2isproduced
bythecollapsingmagneticfieldofcoil1.
Figure213. Mutualinductance.
Q.12Definemutualinductance.
FACTORSAFFECTINGMUTUALINDUCTANCE
Themutualinductanceoftwoadjacentcoilsisdependentuponthephysicaldimensionsofthe
twocoils,thenumberofturnsineachcoil,thedistancebetweenthetwocoils,therelative
positionsoftheaxesofthetwocoils,andthepermeabilityofthecores.
TheCOEFFICIENTOFCOUPLINGbetweentwocoilsisequaltotheratioofthefluxcuttingone
coiltothefluxoriginatedintheothercoil.Ifthetwocoilsaresopositionedwithrespectto
eachothersothatallofthefluxofonecoilcutsalloftheturnsoftheother,thecoilsaresaidto
haveaunitycoefficientofcoupling.Itisneverexactlyequaltounity(1),butitapproachesthis
valueincertaintypesofcouplingdevices.Ifallofthefluxproducedbyonecoilcutsonlyhalf
theturnsoftheothercoil,thecoefficientofcouplingis0.5.Thecoefficientofcouplingis
designatedbytheletterK.
Themutualinductancebetweentwocoils,L1andL2,isexpressedintermsoftheinductanceof
eachcoil
and
the
coefficient
of
coupling
K.
As
aformula:
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Exampleproblem:
One10Hcoilandone20Hcoilareconnectedinseriesandarephysicallycloseenoughtoeach
othersothattheircoefficientofcouplingis0.5.Whatisthemutualinductancebetweenthe
coils?
Q.13Whenaretwocircuitssaidtobecoupled?
Q.14Whatismeantbythecoefficientofcoupling?
SERIESINDUCTORSWITHOUTMAGNETICCOUPLING
When
inductors
are
well
shielded
or
are
located
far
enough
apart
from
one
another,
the
effect
ofmutualinductanceisnegligible.Ifthereisnomutualinductance(magneticcoupling)andthe
inductorsareconnectedinseries,thetotalinductanceisequaltothesumoftheindividual
inductances.Asaformula:
whereLTisthetotalinductance;L1,L2,L3aretheinductancesofL1,L2,L3;andLnmeansthatany
number(n)ofinductorsmaybeused.Theinductancesofinductorsinseriesareaddedtogether
liketheresistancesofresistorsinseries.
SERIESINDUCTORSWITHMAGNETICCOUPLING
Whentwoinductorsinseriesaresoarrangedthatthefieldofonelinkstheother,the
combinedinductanceisdeterminedasfollows:
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TheplussignisusedwithMwhenthemagneticfieldsofthetwoinductorsareaidingeach
other,asshowninfigure214.TheminussignisusedwithMwhenthemagneticfieldofthe
twoinductorsopposeeachother,asshowninfigure215.Thefactor2Maccountsforthe
influenceofL1onL2andL2onL1.
Figure214. Seriesinductorswithaidingfields.
Figure215. Seriesinductorswithopposingfields.
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Exampleproblem:
A10Hcoilisconnectedinserieswitha5Hcoilsothefieldsaideachother.Theirmutual
inductanceis7H.Whatisthecombinedinductanceofthecoils?
Q.15Twoseriesconnected7Hinductorsareadjacenttoeachother;theircoefficientof
couplingis0.64.WhatisthevalueofM?
Q.16A
circuit
contains
two
series
inductors
aligned
in
such
away
that
their
magnetic
fields
opposeeachother.Whatformulashouldyouusetocomputetotalinductanceinthiscircuit?
Q.17Themagneticfieldsoftwocoilsareaidingeachother.Theinductanceofthecoilsare3H
and5H,respectively.Thecoils'mutualinductanceis5H.Whatistheircombinedinductance?
PARALLELINDUCTORSWITHOUTCOUPLING
Thetotalinductance(LT)ofinductorsinparalleliscalculatedinthesamemannerthatthetotal
resistanceofresistorsinparalleliscalculated,providedthecoefficientofcouplingbetweenthe
coilsiszero.Expressedmathematically:
SUMMARY
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Theimportantpointsofthischapteraresummarizedbelow.Studythisinformationbefore
continuing,asthisinformationwilllaythefoundationforlaterchapters.
INDUCTANCE Thecharacteristicofanelectricalcircuitthatopposesachangeincurrent.The
reaction(opposition)iscausedbythecreationordestructionofamagneticfield.Whencurrent
startsto
flow,
magnetic
lines
of
force
are
created.
These
lines
of
force
cut
the
conductor
inducingacounteremfinadirectionthatopposescurrent.
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SELFINDUCTANCE Theprocessbywhichacircuitinducesanemfintoitselfbyitsownmoving
magneticfield.Allelectricalcircuitspossessselfinductance.Thisopposition(inductance),
however,onlytakesplacewhenthereisachangeincurrent.InductancedoesNOToppose
current,onlyaCHANGEincurrent.Thepropertyofinductancecanbeincreasedbyformingthe
conductor
into
a
loop.
In
a
loop,
the
magnetic
lines
of
force
affect
more
of
the
conductor
at
one
time.Thisincreasestheselfinducedemf.
INDUCTANCEOFACOIL Thepropertyofinductancecanbefurtherincreasediftheconductor
isformedintoacoil.Becauseacoilcontainsmoreloops,moreoftheconductorcanbeaffected
bythemagneticfield.Inductors(coils)areclassifiedaccordingtocoretype.Thecorematerialis
normallyeitherairorsoftiron.
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FACTORSAFFECTINGCOILINDUCTANCE Theinductanceofacoilisentirelydependentuponits
physicalconstruction.Someofthefactorsaffectingtheinductanceare:
Thenumber
of
turns
in
the
coil.
Increasing
the
number
of
turns
will
increase
the
inductance.
Thecoildiameter.Theinductanceincreasesdirectlyasthecrosssectionalareaofthecoil
increases.
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Thelengthofthecoil.Whenthelengthofthecoilisincreasedwhilekeepingthenumberof
turnsthesame,theturnspacingisincreased.Thisdecreasestheinductanceofthecoil.
Thetypeofcorematerial.Increasingthepermeabilityofthecoreresultsinincreasingthe
inductanceofthecoil.
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Windingthecoilinlayers.Themorelayersusedtoformacoil,thegreatereffectthemagnetic
fieldhasontheconductor.Bylayeringacoil,youcanincreasetheinductance.
UNITOFINDUCTANCE Inductance(L)ismeasuredinhenrys(H).Aninductorhasaninductance
ofonehenry(H)ifanemfofonevoltisinducedintheinductorwhenthecurrentthroughthe
inductorischangingattherateof1amperepersecond.Commonunitsofinductancearehenry
(H),millihenry(mH)andthemicrohenry(H).
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GROWTHANDDECAYOFCURRENTINANLRCIRCUIT Therequiredforthecurrentinan
inductortoincreaseto63.2percentofthemaximumcurrentortodecreaseto36.8percentof
themaximumcurrentisknownasthetimeconstant.ThelettersymbolforanLRtimeconstant
isL/R.Asaformula:
ThetimeconstantofanLRcircuitmayalsobedefinedasthetimerequiredforthecurrentin
theinductortogrowordecaytoitsfinalvalueifitcontinuedtogrowordecayatitsinitialrate.
Forallpracticalpurposes,thecurrentintheinductorreachesamaximumvaluein5"Time
Constants"anddecreasestozeroin5"TimeConstants".
POWERLOSSESINANINDUCTOR Sinceaninductor(coil)containsanumberturnsofwire,and
allwire
has
some
resistance,
the
inductor
has
acertain
amount
of
resistance.
This
resistance
is
usuallyverysmallandhasanegligibleeffectoncurrent.However,therearepowerlossesinan
inductor.Themainpowerlossesinaninductorarecopperloss,hysteresisloss,andeddy
currentloss.Copperlosscanbecalculatedbymultiplyingthesquareofthecurrentbythe
resistanceofthewireinthecoil(I2R).Hysteresislossisduetopowerthatisconsumedin
reversingthemagneticfieldofthecoreeachtimethecurrentdirectionchanges.Eddycurrent
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lossisduetocoreheatingcausedbycirculatingcurrentsinducedinanironcorebythe
magneticfieldofthecoil.
MUTUALINDUCTANCE Whentwocoilsarelocatedsothatthefluxfromonecoilcutstheturns
oftheothercoil,thecoilshavemutualinductance.Theamountofmutualinductancedepends
uponseveral
factors:
the
relative
position
of
the
axes
of
the
two
coils;
the
permeability
of
the
cores;thephysicaldimensionsofthetwocoils;thenumberofturnsineachcoil,andthe
distancebetweenthecoils.ThecoefficientofcouplingKspecifiestheamountofcoupling
betweenthecoils.Ifallofthefluxfromonecoilcutsalloftheturnsoftheothercoil,the
coefficientofcouplingKis1orunity.Ifnoneofthefluxfromonecoilcutstheturnsofthe
othercoil,thecoefficientofcouplingiszero.Themutualinductancebetweentwocoils(L1and
L2)maybeexpressedmathematicallyas:
COMPUTINGTHEINDUCTANCEOFACIRCUIT Whenthetotalinductanceofacircuitis
computed,theindividualinductivevaluesaretreatedthesameasresistancevalues.The
inductancesofinductorsinseriesareaddedliketheresistancesofresistorsinseries.Thatis,
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Theinductancesofinductorsinparallelarecombinedmathematicallyliketheresistancesof
resistorsinparallel.Thatis,
Bothoftheaboveformulasareaccurate,providingthereisnomutualinductancebetweenthe
inductors.
ANSWERSTOQUESTIONSQ1.THROUGHQ17.
A1.Thehenry,H.
A2.Magneticfield.
A3.Inductanceisthepropertyofacoil(orcircuit)whichopposesanyCHANGEincurrent.
A4.Induced
emf
is
the
emf
which
appears
across
aconductor
when
there
is
relative
motion
betweentheconductorandamagneticfield;counteremfistheemfinducedinaconductor
thatopposestheappliedvoltage.
A5.Theinducedemfinanycircuitisinadirectiontoopposetheeffectthatproducedit.
A6.Noeffect.
Inductanceopposesanychangeintheamplitudeofcurrent.
A7.Thenumbersofturnsinacoil.
Thetypeofmaterialusedinthecore.
Thediameterofthecoil.
Thecoil
length.
Thenumberoflayersofwindingsinthecoil.
Increasesinductance.
Increasesinductance.
Decreasesinductance.
Increasesinductance.
Increasesinductance.
A8.Inductancecausesaverylargeoppositiontotheflowofcurrentwhenvoltageisfirst
appliedto
an
LR
circuit;
resistance
causes
comparatively
little
opposition
to
current
at
that
time.Zero.Duringcurrentbuildup,thevoltageacrosstheresistorgraduallyincreasestothe
samevoltageasthesourcevoltage;andduringcurrentdecaythevoltageacrosstheresistor
graduallydropstozero.
A9.
A10.1.71amperes.
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5timeconstants.
2timeconstants.
A11.Copperloss;hysteresisloss;eddycurrentloss.
A12.Mutualinductanceisthepropertyexistingbetweentwocoilssopositionedthatfluxfrom
one
coil
cuts
the
windings
of
the
other
coil.
A13.Whentheyarearrangedsothatenergyfromonecircuitistransferredtotheothercircuit.
A14.Theratioofthefinesofforceproducedbyonecoiltothelinesofforcethatlinkanother
coil.Itisnevergreaterthanone.
A15.
A16.
A17.