Download pdf - INDUCTANCE The study of inductance presents a very challenging but rewarding segment of electricity. It is Challenging in the sense that, at first, it will seem that new concepts are being introduced. You

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


15/33

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.


16/33

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.


17/33

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.


25/33


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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.


Decreasesinductance.



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.