Date post: | 10-Jan-2016 |
Category: |
Documents |
Upload: | mihai-bogdan |
View: | 222 times |
Download: | 0 times |
of 40
1
Basic Concepts, Laws, and Principles
TOPICS DISCUSSED
Theneedtostudyelectricalandelectronicsengineering
Behaviourofmaterialsasconductors,semiconductors,andinsulators
Conceptofcurrent,resistance,potential,andpotentialdifference
Differencesbetweenelectricfieldandmagneticfield
Ohmslaw
Effectoftemperatureonresistance
Electromagnetismandelectromagneticinduction
Lawsofelectromagneticinduction
DynamicallyandstaticallyinducedEMF
Selfandmutualinductance
Electricalcircuitelements
1.1 INTRODUCTION
Weseeapplicationsofelectricityallaroundus.Weobservethepresenceofelectricityinnature.Itisindeedamazingaswellasinterestingtoknowhowmankindhasbeenabletoputelectricityforitsuse.Allelectronicandelectricalproductsoperateonelectricity.Beityourcomputersystem,cellphones,homeentertainmentsystem,lighting,heating,andairconditioningsystemsallareexamplesofapplicationsofelectricity.Applicationofelectricityislimitlessandoftenextendsbeyondourimagination.
Electricalenergyhasbeenacceptedastheformofenergywhichiscleanandeasytotransmitfromoneplacetotheother.Allotherformsofenergyavailableinnatureare,therefore,transformedintoelectricalenergyandthentransmittedtoplaceswhereelectricityistobeusedfordoingsomework.Electricalengineering,therefore,hasbecomeadiscipline,abranchofstudywhichdealswithgeneration,transmission,distribution,andutilizationofelectricity.
Electronicsengineeringisanoffshootofelectricalengineering,whichdealswiththetheoryanduseofelectronicdevicesinwhichelectronsaretransportedthroughvacuum,gas,orsemiconductors.Themotionofelectronsinelectronicdeviceslikediodes,transistors,thyristors,etc.arecontrolledbyelectricfields.Moderncomputersanddigitalcommunicationsystemsareadvancesofelectronics.Introductionofverylargescaleintegrated(VLSI)circuitshasledtotheminiaturizationofallelectronicsystems.
Electricalandelectronicengineeringare,therefore,veryexcitingfieldsofstudy.Apersonwhoisunawareofthecontributionofthesefieldsofengineeringandthebasicconceptsunderlyingtheadvancement,willonlyhavetoblamehimselforherselffornottakinganyinitiativeinknowingtheunknown.
PREVPreface
NEXT2. DC Networks an
Basic Electrical and Electronics Engineering Recent
Topics
Tutorials
Highlights
Settings
Feedback(http://community.safaribooksonline.com)
Sign Out
Settings
10 days left in your trial. Subscribe.
Feedback(http://community.safaribooksonline.com/)
Sign Out
Enjoy Safari? Subscribe Today
Inthischapter,wewillintroducesomebasicconcepts,laws,andprincipleswhichthestudentsmighthavestudiedinphysics.However,sincetheseformthebasisofunderstandingoftheotherchaptersinthisbook,itwillbegoodtostudythemagain.
1.2 ATOMIC STRUCTURE AND ELECTRIC CHARGE
Severaltheorieshavebeendevelopedtoexplainthenatureofelectricity.Themodernelectrontheoryofmatter,propoundedbyscientistsSirEarnestRutherfordandNielBohrconsiderseverymatteraselectricalinnature.Accordingtothisatomictheory,everyelementismadeupofatomswhichareneutralinnature.Theatomcontainsparticlesofelectricitycalledelectronsandprotons.Thenumberofelectronsinanatomisequaltothenumberofprotons.
Thenucleusofanatomcontainsprotonsandneutrons.Theneutronscarrynocharge.Theprotonscarrypositivecharge.Theelectronsrevolveroundthenucleusinellipticalorbitsliketheplanetsaroundthesun.Theelectronscarrynegativecharge.Sincethereareequalnumberofprotonsandelectronsinanatom,anatomisbasicallyneutralinnature.
Iffromabodyconsistingofneutralatoms,someelectronsareremoved,therewillbeadeficitofelectronsinthebody,andthebodywillattainpositivecharge.Ifneutralatomsofabodyaresuppliedsomeextraelectrons,thebodywillattainnegativecharge.Thus,wecansaythatthedeficitorexcessofelectronsinabodyiscalledcharge.
Chargeofanelectronisverysmall.Coulombistheunitofcharge.Thechargeofanelectronisonly1.60210 Coulomb(C).Thus,wecansaythatthenumberofelectronsperCoulombisthereciprocalof1.60210whichequalsapprox.6.2810 electrons.Therefore,chargeof6.28
10 electronsisequalto1C.Whenwesaythatabodyhasapositivechargeof1C,itisunderstoodthatthebodyhasadeficitof6.2810electrons.
Anychargeisanexampleofstaticelectricitybecausetheelectronsorprotonsarenotinmotion.Youmusthaveseentheeffectofchargedparticleswhenyoucombyourhairwithaplasticcomb,thecombattractssomeofyourhair.Theworkofcombingcausesfriction,producingchargeofextraelectronsandexcessprotonscausingattraction.
Chargeinmotioniscalledelectriccurrent.Anychargehasthepotentialofdoingwork,i.e.,ofmovinganotherchargeeitherbyattractionorbyrepulsion.Achargeistheresultofseparatingelectronsandprotons.Thechargeofelectronsorprotonshaspotentialbecauseitlikestoreturnbacktheworkthatwasdonetoproduceit.
1.3 CONDUCTORS, INSULATORS, AND SEMICONDUCTORS
Theelectronsinanatomrevolveindifferentorbitsorshells.TheshellsarenamedasK,L,M,N,etc.Thenumberofelectronsthatshouldbeinafilledinnershellisgivenby2n wherenisshellnumber1,2,3,4,etc.startingfromthenearestone,i.e.,firstshelltothenucleus.Ifn=1,thefirstshellwillcontaintwoelectrons.Ifn=2,thesecondshellwillcontaineightelectrons.Thisway,thenumberofelectronsintheshellsare2,8,18,32,etc.Thefilledoutermostshellshouldalwayscontainamaximumnumberofeightelectrons.Theoutermostshellofanatommayhavelessthaneightelectrons.Asforexample,copperhasanatomicnumberof29.Thismeans,copperatomhas29protonsand29electrons.TheprotonsareconcentratedinthenucleuswhiletheelectronsaredistributedintheK,L,M,andNshellsas2,8,18,and1electrons,respectively.Theoutermostshellofacopperatomhasoneelectrononlywhereasthisshellcouldhave8electrons.
Thepositionoccupiedbyanelectroninanorbitsignifiesitsenergy.Thereexistsaforceofattractionbetweentheorbitingelectronandthenucleusduetotheoppositechargetheofelectronandtheproton.Theelectronsintheinnerorbitsarecloselyboundtothenucleusthantheelectronsoftheouteroroutermostorbit.Iftheelectronisfarawayfromthenucleus,theforceofattractionisweak,andhencetheelectronsofoutermostorbitareoftencalledfreeelectrons.Forexample,acopperatomhasonlyoneatominthelastorbitwhichotherwisecouldhaveeightelectrons.
Inacopperwireconsistingoflargenumberofcopperatoms,theatomsareheldclosetogether.Theoutermostelectronsofatomsinthecopperwirearenotsureaboutwhichatomtheybelongto.Theycanmoveeasilyfromoneatomtotheotherinarandomfashion.Suchelectronswhichcanmoveeasilyfromoneatomtotheotherinarandomfashionarecalledfreeelectrons.Itisthemovementoffreeelectronsinamateriallikecopperthatconstitutesflowofcurrent.Here,ofcourse,thenetcurrentflowwillbezeroasthemovementofthefreeelectronsisinrandomdirections.Whenweapplyapotential,whichisnothingbutaforce,itwilldirecttheflowofelectronsinaparticulardirection,i.e.,fromapointofhigherpotentialtowardsapointoflowerpotential.Thus,currentflowisestablishedbetweentwopointswhenthereexistsapotentialdifferencebetweenthepoints.
Wheninamaterialtheelectronscanmovefreelyfromoneatomtoanotheratom,thematerialiscalledaconductor.Silver,copper,gold,andaluminiumaregoodconductorsofelectricity.Ingeneral,allmetalsare
19
19 18
18
18
2
goodconductorsofelectricity.Althoughsilveristhebestconductorofelectricity,thesecondbestconductor,i.e.,copper,ismostlyusedasconductorbecauseofthecostfactor.Inelectricalandelectronicengineeringfields,thepurposeofusingaconductorascarrierofelectricityistoallowelectriccurrenttoflowwiththeminimumofresistances,i.e.,theminimumofopposition.
Inamaterialwheretheoutermostorbitoftheatomsiscompletelyfilled,thematerialiscalledaninsulator.Insulatorslikeglass,rubber,mica,plastic,paper,air,etc.donotconductelectricityveryeasily.Intheatomsofthesematerials,theelectronstendtostayintheirownorbits.However,insulatorscanstoreelectricityandcanpreventflowofcurrentthroughthem.Insulatingmaterialsareusedasdielectricincapacitorstostoreelectriccharge,i.e.,electricity.
Carbon,silicon,andgermaniumhavingatomicnumbersof6,14,and32,respectively,arecalledsemiconductingmaterial.Thenumberofelectronsintheoutermostorbitoftheiratomsisfourinsteadofthemaximumofeight.Thus,intheoutermostorbitofasemiconductormaterial,therearefourvacantpositionsforelectrons.Thesevacantpositionsarecalledholes.Inamaterial,theatomsaresoclosetogetherthattheelectronsintheoutermostorbitorshellbehaveasiftheywereorbitingintheoutermostshellsoftwoadjacentatomsproducingabindingforcebetweentheatoms.Inasemiconductormaterialtheatomsformingabonding,calledcovalentbonding,sharetheirelectronsintheoutermostorbit,andtherebyattainastablestate.Theconditionislikeaninsulatorhavingalltheeightpositionsintheoutermostorbitfilledbyeightelectrons.However,insemiconductingmaterials,withincreaseintemperatureitispossibleforsomeoftheelectronstogainsufficientenergytobreakthecovalentbondsandbecomefreeelectrons,andcausetheflowofcurrent.
1.4 ELECTRIC FIELD AND MAGNETIC FIELD
Whenchargesareseparated,aspaceiscreatedwhereforcesareexertedonthecharges.Anelectricfieldissuchaspace.Dependinguponthepolarityofthecharges,theforceiseitherattractiveorrepulsive.Therefore,wecansaythatstaticchargesgenerateanelectricfield.Anelectricfieldinfluencesthespacesurroundingit.Electricfieldstrengthisdeterminedintermsoftheforceexertedoncharges.Acapacitorisareservoirofcharge.Thetwoparallelplatesofacapacitor,whenconnectedtoavoltagesource,establishesanelectricfieldbetweentheplates.Thepositiveterminal,orpoleofthevoltagesourcewilldrawelectronsfromplate1whereasthenegativepolewillpushextraelectronsontoplate2.Voltageacrossthecapacitorwillrise.Thecapacitorgetschargedequaltothevoltageofthesource.Thecapacitanceofacapacitorisameasureofitsabilitytostorecharge.Thecapacitanceofacapacitorisincreasedbythepresenceofadielectricmaterialbetweenthetwoplatesofthecapacitor.
Acurrentcarryingconductororacoilproducesmagneticfieldaroundit.Thestrengthofthemagneticfieldproduceddependsonthemagnitudeofthecurrentflowingthroughtheconductororthecoil.Thereispresenceofmagneticfieldaroundpermanentmagnetsaswell.
Amagnetisabodywhichattractsiron,nickel,andcobalt.Permanentmagnetsretaintheirmagneticproperties.Electromagnetsaremadefromcoilsthroughwhichcurrentisallowedtoflow.Theirmagneticpropertieswillbepresentaslongascurrentflowsthroughthecoil.
Thespacewithinwhichforcesareexertedbyamagnetiscalledamagneticfield.Itistheareaofinfluenceofthemagnet.
1.5 ELECTRIC CURRENT, RESISTANCE, POTENTIAL, AND POTENTIAL DIFFERENCE
1.5.1 Electric Current
Inanyconductingmaterial,theflowofelectronsformswhatiscalledcurrent.Electronshavenegativecharge.Chargeonanelectronisverysmall.ForthisreasonchargeisexpressedintermsofCoulomb.ChargeofoneCoulombisequaltoachargeof6.2810 electrons.Theexcessordeficitofelectronsinabodyiscalledcharge.Thus,electricalcurrentisexpressedasaflowofnegativecharge,i.e.,electrons.Anysubstancelikecopper,aluminum,silver,etc.whichhasalargenumberoffreeelectrons(i.e.,looslyboundelectronsintheoutermostorbitofitsatom)willpermittheflowofelectronswhenelectricalpressureintheformofEMF(electromotiveforce,i.e.,voltage)isapplied.
Sincethesematerialsconductelectricity,theyarecalledconductors.Theyeasilyallowelectriccurrenttoflowthroughthem.Thestrengthofcurrentwilldependupontheflowofchargeperunittime.Thisisexpressedas
wherechargeQismeasuredinCoulombandtime,tinseconds.Theunitofcurrent,therefore,isCoulombpersecond,when1Cofchargeflowsin1
18
sthemagnitudeofcurrentiscalledampere,namedafterAndrMarieAmpere.
Thus,1ampereofcurrentisequivalenttotheflowofchargeof1Coulombpersecond.
Inearlieryears,currentwasassumedtoflowfrompositivetonegativeterminals.Thisconventionisusedevennowalthoughitisknownthatcurrentisduetothemovementofelectronsfromthenegativetothepositiveterminal.
1.5.2 Resistance
Electricalresistanceisthehindranceoroppositiontotheflowofelectronsinagivenmaterial.Itismeasuredinunitcalledohm.Sincecurrentistheflowofelectrons,resistanceistheoppositionofferedbyamaterial,totheflowoffreeelectrons.Resistance,R,isdirectlyproportionaltothelengthofthematerial,andinverselyproportionaltotheareaofthecrosssectionofthematerial,throughwhichcurrentflows.Theresistanceofferedbyconductingmaterialslikecopperandaluminumislowwhereasresistanceofferedbysomeotherconductingmaterialslikenicrome,tungsten,etc.isveryhigh.Allthesematerialsarecalledconductingmaterials.However,thevaluesofresistivityofthesematerialsaredifferent.Theresistance,Rofamaterialisexpressedas
whereistheresistivity,isthelengthandAisthecrosssectionalareaoftheconductingmaterial.
Theresistivity,isalsocalledthespecificresistanceofthematerial.Themostconductingmaterial,silverhasthelowestvalueofresistivity,i.e.,0.01610 ohmm.Aftersilver,copperismostconducting.Theresistivityorspecificresistanceofcopperissomewhatmorethanthatofsilver,i.e.,0.01810 ohmm.Thatistosay,copperislessconductingthansilver.Wewillseealittlelaterwhyandhowthevalueofresistancechangeswithtemperature.
1.5.3 Potential and Potential Difference
EMFproducesaforceorpressurethatcausesthefreeelectronsinabodytomoveinaparticulardirection.TheunitofEMFisvolt.EMFisalsocalledelectricpotential.Whenabodyischarged(i.e.,eitherdefficiencyofelectronsorexcessofelectronsiscreated),anamountofworkisdone.Thisworkdoneisstoredinthebodyintheformofpotentialenergy.Suchachargedbodyiscapableofdoingworkbyattractingorrepellingothercharges.Theabilityofachargedbodytodoworkinattractingorrepellingchargesiscalleditspotentialorelectricalpotential.Workdonetochargeabodyto1Cisthemeasureofitspotentialexpressedinvolts:
Whenworkdoneis1jouleandchargemovedis1C,thepotentialiscalled1volt.Ifwesaythatapointhasapotentialof6volts,itmeansthat6Joulesofworkhasbeendoneinmoving1Cofchargetothatpoint.Inotherwords,wecansaythateveryCoulombofchargeatthatpointhasanenergyof6Joules.
Thepotentialdifferenceoftwopointsindicatesthedifferenceofchargedconditionofthesepoints.SupposepointAhasapotentialof6volts,andpointBhasapotentialof3volts.WhenthepointsAandBarejoinedtogetherbyaconductingwire,electronswillflowfrompointBtopointA.WesaythatcurrentflowsfrompointAtowardspointB.Thedirectionofcurrentflowistakenfromhigherpotentialtolowerpotentialwhiletheflowofelectronsareactuallyintheoppositedirection.Theflowofcurrentfromhigherpotentialtolowerpotentialissimilartotheflowofwaterfromahigherleveltoalowerlevel.
1.6 OHMS LAW
GeorgeSimonOhmfoundthatthevoltage,Vbetweentwoterminalsofacurrentcarryingconductorisdirectlyproportionaltothecurrent,Iflowingthroughit.Theproportionalityconstant,Ristheresistanceoftheconductor.Thus,accordingtoOhmslaw
6
6
Thisrelationwillholdgoodprovidedthetemperatureandotherphysicalconditionsdonotchange.
Figure1.1(a)ShowslinearrelationshipbetweenVandI(b)VIcharacteristicsfordifferentvaluesofR
OhmslawisnotapplicabletononlineardeviceslikeZenerdiode,voltageregulators,etc.OhmslawisexpressedgraphicallyonVandIaxiesasastraightlinepassingthroughtheoriginasshowninFig.1.1(a).
TherelationshipbetweenVandIhavebeenshownfordifferentvaluesofRinFig.1.1(b).HereinV=RI,Rindicatestheslopeoftheline.ThemorethevalueofRis,themorewillbetheslopeofthelineasshowninFig.1.1(b).
1.7 THE EFFECT OF TEMPERATURE ON RESISTANCE
Resistanceofpuremetalslikecopper,aluminum,etc.increaseswithincreaseintemperature.ThevariationofresistancewithchangeintemperaturehasbeenshownasalinearrelationshipinFig.1.2.
Thechangeinresistanceduetochangeintemperatureisfoundtobedirectlyproportionaltotheinitialresistance,i.e.,R R R .Resistance(R R )alsovariesdirectlyasthetemperatureriseandthischangealsodependsuponthenatureofthematerial.Thuswecanexpressthechangeinresistanceas,
R R R t
or,R R = R t,where iscalledthetemperaturecoefficientofresistanceat0C.
Figure1.2(a)Showsthevariationofresistancewithtemperature(b)resistancesattwodifferenttemperatures
Thisexpressioncanbeappliedforbothincreaseanddecreaseintemperature.FromthegraphofFig.1.2(a)itisseenthatresistanceofthematerialcontinuestodecreasewithdecreaseintemperaturebelow0C.Ifwegoondecreasingthetemperaturetoaverylowvalue,thematerialattainsastateofzeroresistance.Thematerialatthatstatebecomessuperconducting,i.e.,conductingwithnoresistanceatall.
t 0 0
t 0
t 0 0
t 0 0 0 0
Nowsupposeaconductorisheatedfromtemperaturet tot .Theresistanceoftheconductoratt isR andatt isR ashasbeenshowninFig.1.2(b).
Usingeq.(1.5),
Usingeq.(1.5),wecanwrite
Fromfig1.2(b)usingtherelationin(1.6),wecanwrite
Thus,ifresistanceatanytemperaturet isknown,theresistanceatttemperaturecanbecalculated.
Calculation of at different temperatures
Wehaveseen,
If and arethetemperaturecoefficientsofresistanceatt andtdegrees,respectively,then
Thus,wecanwrite,
Therefore,
1 2
1 1 2 2
1 2
1 2 1 2
Temperaturecoefficientofresistance,at20CandspecificresistanceofcertainmaterialhavebeenshowninTable1.1.
Table1.1TemperatureCoefficientandSpecificResistanceofDifferentMaterials
Material Temp.coeff.ofresistance Specificresistanceinmicro
ohm
Silver 0.004 0.016
Copper 0.0039 0.018
Aluminium 0.0036 0.028
Iron 0.005 0.100
Brass 0.0015 0.070
Lead 0.0042 0.208
Tin 0.0046 0.110
Carbon 0.00045 66.67
Itistobenotedthatcarbonhasanegativetemperaturecoefficientofresistance.Thismeans,theresistanceofcarbondecreaseswithincreaseintemperature.
Bythistimeyoumustbewonderingastowhyresistanceinmostmaterialsincreaseswithincreaseintemperaturewhileresistanceinsomedecreaseswithincreaseintemperature.
Thechargedparticlesinsideamaterialisinthestateofvibration.Temperatureriseinmostmaterialsincreasesthisvibrationinsidethematerialobstructingtheflowofelectrons.Obstructiontotheflowofelectronsiscalledresistance.Atlowertemperaturesthevibrationgetsreduced,andhencetheresistance.
1.8 WORK, POWER, AND ENERGY
1.8.1 Work
Whenaforceisappliedtoabodycausingittomove,andifadisplacement,discausedinthedirectionoftheforce,then
IfforceisinNewtonsanddisinmeters,thenworkdoneisexpressedinNewtonmeterwhichiscalledJoules.
1.8.2 Power
Poweristherateatwhichworkisdone,i.e.,rateofdoingwork.Thus,
TheunitofpowerisJoules/secondwhichisalsocalledWatt.Whentheamountofpowerismore,itisexpressedinKilowatt,i.e.,kW.
1kW=110 W
Wehaveearlierseenineq.(1.3),thatelectricalpotential,Visexpressedas
20
3
ThusinacircuitifIisthecurrentflowing,andVistheappliedvoltageacrosstheterminals,power,Pisexpressedas
Thuselectricalpowercanbeexpressedas
1.8.3 Energy
Energyisdefinedasthecapacityfordoingwork.Thetotalworkdoneinanelectricalcircuitiscalledelectricalenergy.Whenavoltage,Visapplied,thecharge,Qwillflowsothat
IfpowerisinkWandtimeisinhour,theunitofenergywillbeinKilowatthourorkWh.
1.8.4 Units of Work, Power, and Energy
InSIunit,workdoneisthesameasthatofenergy.
Mechanical work or energy
WhenaForce,FNewtonactingonabodymovesitinthedirectionoftheforcebyadistancedmeters:
Workdone=FDNmorJoules
WhenaforceFNewtonisappliedtangentiallyonarotatingbodymakingaradiusrmeters,then
IfNisexpressedinrevolutionsperminute(rpm)
Whenabodyofmassmkgisliftedtoaheighthmetersagainstthegravitationalforcegm/sec ,workdoneisconvertedintopotentialenergyofthebody.
Electrical energy
Asmentionedearlier,workdoneinanelectricalcircuitisitsenergy.
IfelectricalpowerisexpressediskWandtimeinhour,then
WewillnowconvertkWhintoCalories
Since1Calorie=4.2Joules
Thermal energy
InSIunit*thermalenergyisexpressedincalories.Onecalorieindicatestheamountofheatrequiredtoraisethetemperatureof1gmofwaterby1C.Thisheatisalsocalledthespecificheat.Ifmisthemassoftheliquid,Sisthespecificheat,andtisthetemperatureriserequired,thentheamountofheatrequired,Hisexpressedas
Example1.1Acopperwirehasresistanceof0.85ohmsat20C.Whatwillbeitsresistanceat40C?Temperaturecoefficientofresistanceofcopperat0Cis0.004C.
Solution:
2
Example1.2Theheatingelementofanelectricheatermadeofnicromewirehasvalueofresistivityof110 Ohmm.Thediameterofthewireis0.2mm.Whatlengthofthisnicromewirewillmakearesistanceof100Ohms?
Solution:
Substitutingthevalues,lengthofwire,is
Example1.3Itisrequiredtoraisethetemperatureof12kgofwaterinacontainerfrom15Cto40Cin30minthroughanimmersionrodconnectedtoa230Vsupplymains.Assuminganefficiencyofoperationas80percent,calculatethecurrentdrawnbytheheatingelement(immersionrod)fromthesupply.Alsodeterminetheratingoftheimmersionrod.Specificheatofwateris4.2kiloJoules/kg/C.
Solution:
OutputorEnergyspentinheatingthewater,His
H=ms(t t )
Wheremisthemassofwaterandsisthespecificheatofwater.
Here,H=124.210 (4015)Joules
=12610 Joules
Weknow,efficiency
So,Energyinputtoimmersionrod
Thetimeofoperationoftheheaterrod
2 1
6
3
4
Currentdrawnfrom230Vsupply
P=VI=870Watts.
Example1.4Amotordrivenwaterpumplifts64m ofwaterperminutetoanoverheadtankplacedataheightof20metres.Calculatethepowerofthepumpmotor.Assumeoverallefficiencyofthepumpas80percent.
Solution:
Workdone/min=mghJoules
m=6410 kg(1m ofwaterweights1000kg)
g=9.81m/sec
h=20m
Substitutingvalues
Inputpowerofthepumpmotor
=261.3KW
Example1.5Aresidentialflathasthefollowingaverageelectricalconsumptionsperday:
1. 4tubelightsof40wattsworkingfor5hoursperday2. 2filamentlampsof60wattsworkingfor8hoursperday3. 1waterheaterrated2kWworkingfor1hourperday4. 1waterpumpof0.5kWratingworkingfor3hoursperday.
Calculatethecostofenergypermonthif1kWhofenergy(i.e.,1unitof
energy)costs
Solution:
Totalkilowatthourconsumptionofeachloadfor30daysarecalculatedas:
3
3 3
2
TotalkWhconsumedpermonth
=24kWh+28.8kWh+60kWh+45kWh
=157.5kWh
OnekWhofenergycosts
Thetotalcostofenergypermonth=157.53.50
Example1.6Anelectrickettlehastoraisethetemperatureof2kgofwaterfrom30Cto100Cin7minutes.Thekettleishavinganefficiencyof80percentandissuppliedfroma230Vsupply.Whatshouldbetheresistanceofitsheatingelement?
Solution:
Outputenergyofthekettle=mst
Supplyvoltage,V=230Volts.
Power,P=1.74kW=1740Watts.
V=230V
Example1.7Calculatethecurrentflowingthrougha60Wlampona230Vsupplywhenjustswitchedonatanambienttemperatureof25C.Theoperatingtemperatureofthefilamentmaterialis2000Canditstemperaturecoefficientofresistanceis0.005perdegreeCat0C.
Solution:
Thisresistanceofthefilamentisat2000C.LetuscallitR =881.6Ohms.
Attheinstantofswitching,resistanceisatroomtemperature,i.e.,at25C.LetuscallitasR .
WeknowR wehavetocalculateR given =0.005ohm/C.
Weknow,
Weknowtherelation,
Thecurrentflowingthroughthe60Wlampattheinstantofswitchingwillbecorrespondingtoitsresistanceat25C.
Example1.8Acoilhasaresistanceof18at20Cand20at50C.Atwhattemperaturewillitsresistancebe21Ohms?
Solution:
weknow,
Wecanwrite,
substituting,
2000
25
2000 25 0
Example1.9Theresistanceofawireincreasesfrom40at20Cto50at70C.Calculatethetemp.coefficientofresistanceat0C.
Solution:
given
or,
or,
Example1.10Aresistanceelementofcrosssectionalareaof10mmandlength10mdrawsacurrentof4Aat220Vsupplyat20C.Calculatetheresistivityofthematerial.Whatcurrentwillbedrawnwhenthetemperaturerisesto60C?Assume =0.0003/C.
Solution:
a=10mm
=1010 m
V=IR
or,
Thisresistance,wecallasR
SincewehavetocalculateR ,wehaveto
Now,
20
20
60 60
2
2
6 2
Current,
1.9 ELECTROMAGNETISM AND ELECTROMAGNETIC INDUCTION
1.9.1 Introduction
Electromagnetismisthestudyofinteractionbetweenelectriccurrentandmagneticfield,andforcesproducedthereof.Thissectionwillincludedescriptionsofmagneticfieldaroundcurrentcarryingconductors,magneticfieldproducedbyacurrentcarryingcoil,forceproducedonacurrentcarryingconductororacoilwhenplacedinamagneticfield.
ADanishscientist,Oerstedintheearlynineteenthcenturydiscoveredthattherewasamagneticfieldaroundacurrentcarryingconductor.Linesofforceintheformofconcentriccirclesexistedonaperpendicularplanearoundacurrentcarryingconductor.Thismeant,magnetismcouldbecreatedbyelectriccurrent.Itwasalsoobservedthatthedirectionoflinesofforcegotchangedwhenthedirectionofcurrentflowingthroughtheconductorwaschanged.AfewyearsafterthediscoveryofOersted,Faraday,anotherscientistfromEnglanddiscoveredthatamagneticfieldcancreateanelectriccurrentinaconductor.Whenthereisachangeinfluxlinkageinaconductororacoil,EMFisinducedinit.ThisphenomenoniscreditedtoFaradaywhoestablishedfamouslawsofelectromagneticinduction.Youwillobservethatmostoftheelectricalmachinesanddeviceshavebeendevelopedutilizingtheobservationsanddiscoveriesmadeasmentionedabove.
1.9.2 Magnetic Field Around a Current-carrying Conductor
InFig.1.3isshownaconductorcarryingacurrent,I.Linesofforceareestablishedaroundtheconductoronaperpendicularplane.InFig.1.3(a)magneticfieldaroundalongconductorhasbeenshown.Thelinesofforceareestablishedonaperpendicularplane.InFig.1.3(b)and(c),thecrosssectionalviewsofacurrentcarryingconductorhavebeenshown.Thecrossatthecentreoftheconductorindicatesthatcurrentisenteringtheconductorwhichisplacedperpendiculartotheplaneofthepaper.Thelinesofforceintheformofconcentriccirclesareontheplaneofthepaper.ThedirectionofcurrentthroughtheconductorisreversedinFig.1.3(c).Thedotatthecentreoftheconductorindicatesthatthecurrentiscomingtowardstheobserver.Thedirectionofthelinesforcearoundtheconductoralsogetreversed.
Thedirectionoffluxlinesaroundacurrentcarryingconductorisdeterminedbyapplyingthecorkscrewrulewhichisstatedbelow.
Figure1.3(a)Alongcurrentcarryingconductor(b)crosssectionalviewofaconductorwithfluxaroundit(c)crosssectionalviewoftheconductorwiththedirectionofcurrentreversed(d)resultantmagneticfieldproducedbytwocurrentcarryingconductors
CorkScrewRule:Considerarighthandscrewheldononeendofacurrentcarryingconductorandisrotatedintheclockwisedirection.Iftheadvancementofthescrewindicatesthedirectionofcurrent,thedirectioninwhichthescrewisrotatedwillindicatethedirectionofthelinesofforcearoundtheconductor.
InFig.1.3(d)hasbeenshownthattwocurrentcarryingconductorsplacedsidebysideproducearesultantmagneticfield.
1.9.3 Magnetic Field Around a Coil
Acoilisformedbywindingawireofcertaincrosssectionaroundaformer(ahollowcylindermadeofsomenonmagneticmateriallikebakelite,plastics,etc).Suchacoilisoftencalledasolenoid.Whencurrentisallowedtoflowthroughsuchacoil,amagneticfieldisproducedbythecoil.Thedirectionoffluxproducedbyacurrentcarryingcoilisdeterminedbyapplyingtherighthandgriprule.InFig.1.4(a)isshownacurrentcarryingcoil.Ifweholdthecoilbyourrighthandinsuchaway
thatthefourfingersbendtowardsthedirectionofthecurrentflowthroughthecoilturns,thethumbwillindicatethedirectionoftheresultantfluxproduced.
Figure1.4(a)Righthandgripruleappliedtodeterminedirectionoffluxproducedbyacurrentcarryingcoil(b)magneticfieldproducedbyacurrentcarryingcoil
Thefourfingersbendinthedirectionofcurrentthroughthecoil.Thedirectioninwhichthethumbpointsisthedirectionoffluxproduced.InFig.1.4(b),wehaveshownthecrosssectionalviewofthesamecoil.Forthedirectionofcurrentflowthroughthecoil,crosssectionshavebeenshownbyputtingcrossanddotconvention.Theuppersideofthecoilturns1,2,3,4,5willindicatethatcurrentisenteringwhiletheywillcomeoutfromtheothersideasshowninthebottomconductorcrosssections.ByapplyingthecorkscrewrulealsowecandeterminethedirectionoftheresultantmagneticfieldandshowthepositionsofNorthandSouthpolesformed.Ifthedirectionofcurrentflowthroughthecoilisreversed,thedirectionofthemagneticlinesofforcewillbeopposite,andhencethepositionsofNorthandSouthpoleswillchange.
IfweapplysomealternatingvoltageacrossthecoilasshowninFig.1.5,thepolarityofpowersupplywillchangeineveryhalfcycleoftheappliedvoltage.Ifasinusoidalacsupplyisprovided,boththemagnitudeaswellasthedirectionofcurrentflowwillchange.Asaresult,themagnitudeofthemagneticfieldproducedwillchangestartingfromzerovaluereachingitsmaximumvalue,thengettingreducedagaintozero,andthenbecomingnegative.Thedirectionoffluxproducedwillchangeineveryhalfcycleofcurrentflow.Suchamagneticfieldwhosemagnitudeasalsoitsdirectionchangesiscalledapulsatingalternatingmagneticfield.Incaseofdcsupply,themagneticfieldproducedwillbeofconstantmagnitudeandfixedpolarity.
Figure1.5ACsupplytoacoilproducesanalternatingmagneticfieldofvaryingmagnitude
1.9.4 A Current-carrying Conductor Placed in a Magnetic Field
Whenaconductorcarryingcurrentisplacedinamagneticfielditexperiencesaforce.Theforceactsinadirectionperpendiculartoboththemagneticfieldandthecurrent.
InFig.1.6aconductorisshownplacedperpendiculartothedirectionofmagneticfield.Suchaconductorincrosssectionalviewhasbeenshownbyasmallcircle.Thedotinsidethesmallcircleindicatesthatcurrentisflowingtowardstheobserver.Theconductorwillexperienceaforceintheupwarddirectionashasbeenshown.Ifthedirectionofcurrentthroughtheconductorisreversed,theforceontheconductorwillbeinthedownwarddirection.
Theforceontheconductorwilldependupontheflux, orfluxdensity,
B whereAistheareaofthemagneticpoles.Theforcewillalsodependupontheeffectivelengthoftheconductorinthemagneticfield,i.e.,onthemagnitudeofcurrentflowing,i.e.,I.Theforcedevelopedisexpressedas
Herethecurrentcarryingconductorandthemagneticfieldsareatrightanglestoeachother.If,however,theconductorisinclinedwiththemagneticfieldbyanangle,thenthelengthoftheconductorperpendiculartothemagneticfieldistobeconsideredasshowninFig.1.7.ThelengthoftheconductorperpendiculartothemagneticfieldisSin.Thus,thegeneralexpressionforforceFis
ThedirectionoftheforceisdeterminedbyapplyingFlemingslefthandrulewhichisstatedas:
Figure1.6Forceexperiencedbyaconductorcarryingcurrentinamagneticfield
Figure1.7Forceonacurrentcarryingconductor
Flemings left-hand rule
ThethreefingersofthelefthandarestretchedasshowninFig.1.6.Iftheforefingerpointstowardsthedirectionofthelinesofforce,andthemiddlefingerpointstowardthecurrentflowingthroughtheconductor,thenthethumbwillpointtowardsthedirectionofforceexperiencedbytheconductor.
1.9.5 A Current-carrying Coil Placed in a Magnetic Field
Nowwewillconsideracoilplacedinamagneticfield.Acoilhastwocoilsideswhichlieinthemagneticfield.Thesecoilsidesarecalledconductors.Thus,acoilhastwoconductors.Ifacoilhastwoturns,thenumberofconductorswillbefour.SeeFig.1.8(aandb).InFig.1.8(c)hasbeenshownasingleturncoilplacedinamagneticfield.Thedirectionofcurrentthroughthecoilhasalsobeenshown.ThedirectionofthemagneticfieldisfromNorthpoletoSouthpole.Thedirectionofcurrentincoilsideaisupward,i.e.,towardstheobserver.IfweapplyFlemingslefthandrule,wefindthatcoilsideawillexperienceanupwardforce.
Similarly,byapplyingthesamerule,weobservethatcoilsidewillexperienceadownwardforce.ThetwoforcesactingsimultaneouslyonthecoilwilldevelopatorquewhichwilltrytorotatethecoilalonganaxisxxintheclockwisedirectionashasbeenshowninFig.1.8(c).Thecoilwillrotatebyanangleof90.TheNorthpoleofthemagneticfieldproducedbythecurrentcarryingcoilwillfacethestationarySouthpoleasshowninFig.1.9.
ThetwomagneticfieldsgetalignedasshowninFig.1.9(b).IfitispossibletochangethedirectionofcurrentinthecoilwhenitchangesitspositionfromDDaxistoXXaxis,thecoilwillcontinuetodeveloptorqueintheclockwisedirection.Wewillgetcontinuousrotationofthecoil.Thisisthebasicprincipleofdirectcurrentelectricmotorwhichwillbediscussedindetailinaseparatechapter.
Figure1.8(a)Acoilhavingoneturn(b)acoilhavingtwoturns(c)asingleturncoilcarryingcurrentisplacedinamagneticfield(d)thecoilsidesofthecurrentcarryingcoilinthemagneticfieldexperienceforce
Figure1.9(a)Acurrentcarryingcoilinamagneticfieldexperiencesatorque(b)magneticfieldproducedbythecurrentcarryingcoilandthestationarymagneticfieldgetaligned
1.10 LAWS OF ELECTROMAGNETIC INDUCTION
Faraday,onthebasisoflaboratoryexperiments,establishedthatwheneverthereaischangeinthemagneticfluxlinkagebyacoil,EMFisinducedinthecoil.ThemagnitudeoftheEMFinducedisproportionaltotherateofchangeoffluxlinkages.Faradayslawsofelectromagneticinductionarestatedas:
Firstlaw:EMFisinducedinacoilwhenevermagneticfieldlinkingthatcoilischanged.
Secondlaw:ThemagnitudeoftheinducedEMFisproportionaltotherateofchangeoffluxlinkage.
Therateofchangeoffluxlinkageisexpressedas whereNisthenumberofturnsofthecoillinkingtheflux.Thus,theinducedEMF,eisexpressedas
TheminussignisintroducedinaccordancewithLenzslawwhichisstatedbelow.
Lenzslaw:ThislawstatesthattheinducedEMFduetochangeoffluxlinkagebyacoilwillproduceacurrentinthecoilinsuchadirectionthatitwillproduceamagneticfieldwhichwillopposethecause,thatisthechangeinfluxlinkage.
Thestudentsmayconductanexperimentinthelaboratory,similartothatdonebyFaraday,whichisexplainedbelow.
IfthemagnetshowninFig.1.10(a)isquicklybroughtnearthecoil,therewillbedeflectioninthegalvanometerindicatingEMFinducedinthecoilandcurrentflowinthecircuit.Ifthemagnetisheldstationarynearthecoil,althoughthereisfluxlinkingthecoil,therewillbenoinducedEMFsincethereisnochangeinthefluxlinkage.TheinducedEMFwillbethereonlyifthereisincreaseordecreaseinfluxlinkagebythecoil.Itwillbeobservedthatwhenthemagnetistakenawayquicklytherewillbedeflectioninthegalvanometer.ItmayalsobenotedthatEMFwillalsobeinducedinthecoilwhenthecoilismovedkeepingthemagnetstationary.
Figure1.10Faraday'sexperimentonelectromagneticinduction.(a)Amagnetissuddenlybroughtnearacoil(b)determinationofthedirectionofcurrentproducedinthecoil
Thedirectionofcurrentflowingthroughthecoilcanbedeterminedbyapplyingtherighthandgriprule.Theruleisexplainedasfollows.
Right-hand-grip rule
Holdthecoilwithyourrighthandwiththethumbopposingthedirectionofmovementofthemagnet.Theotherfourfingerswillindicatethedirectionofcurrentflowthroughthecoil.Thismeansthatthecurrentinducedinthecoilwillproducefluxinthedirectionofthethumb,thusopposingthefluxproducingtheinducedEMFinthecoil.SeeFig.1.10(b).
1.11 INDUCED EMF IN A COIL ROTATING IN A MAGNETIC FIELD
NowwewillconsideracoilrotatedinastationarymagneticfieldasshowninFig.1.11.
Hereacoil,havingtwosides(conductors)isrotatedinauniformmagneticfieldasshowninFig.1.11.Becauseoftherotationofthecoilinthemagneticfield,fluxlinkagebythecoilchanges,i.e.,thenumberoflinesofforcepassingthroughthecoilchanges.Becauseofchangeoffluxlinkage,EMFisinducedinthecoil.ThedirectionoftheinducedEMFintheconductorscanbedeterminedbyapplyingFlemingsrighthandrule(FRHR).
Figure1.11(a)EMFisinducedinacoilwhenrotatedinamagneticfield(b)determinationofdirectionofinducedEMF
FRHRstatesthatwhenwestretchthethreefingersoftherighthandperpendiculartoeachother,iftheforefingerpointstowardsthefluxlinesfromNorthpoletoSouthpole,andthethumbshowsthedirectionofmovementoftheconductor,thenthemiddlefingerwillrepresentthedirectionoftheinducedEMForcurrentintheconductor.InFig.1.11(b)isshownthedirectionoftheinducedEMFincoilsideaboftherotatingcoilabcd.Thiscoilsideisshowngoingupwards.ThemagneticfielddirectionisfromNorthpoletoSouthpole.Hence,thedirectionoftheinducedEMFwillbefrombtoaasdeterminedbyapplyingFRHR.Thestrongerthemagneticfieldis,themorewillbethemagnitudeofEMFinduced.Themorethespeedofrotationofthecoilis,themorewillbethe
magnitudeoftheEMFinduced.Thisisbecause willincreaseifboth
aswellastherateofchangeoflinkageof arechanged.ThemagnitudeoftheEMFinducedwillalsobedirectlyproportionaltothenumberofturnsoftherotatingcoil,orthenumberofcoilsconnectedinseries.TheEMFinducedcanalsobeconsideredintermsoffluxcutbyaconductor(coilside)persecond.
HereinFig.1.11,thenumberofpolesistwo.Wecanalsohavefourpoles,sixpoles,etc.Whenaconductorrotatesinsuchmagneticfield,itcutsthelinesofforce.Thenumberoflinesofforcecutbyaconductorinone
revolution,whentherearetwopoles,is2 Webers,where isthefluxperpole.IftherearesayPnumberofpoles,fluxcutbyaconductorinone
revolutionwillbeP Webers.Ifthecoilmakesnrevolutionspersecond,thetimetakenbyaconductortomakeonerevolutionwillbe1/nseconds.Thus,fluxcutpersecondwillbetheEMFinduced,ewhichis
1.12 EMF INDUCED IN A CONDUCTOR
Intermsoflengthofconductor,andvelocityoftheconductor,vinamagneticfieldoffluxdensity,B,theEMFinducedinaconductor,eiscalculatedas
Toestablishtheaboverelation,letusconsiderasingleconductorrepresentedbyasmallcircle(crosssectionalview)ismovedinamagneticfieldofstrengthBWb/m asshowninFig.1.12.
2
Figure1.12EMFinducedinaconductormovinginamagneticfield
Lettheconductorcutthefluxatrightanglesbymovingadistancedxmeter.Theareasweptbythemovingconductorisdxm .ThefluxdensityisBWb/m .
Thetimetakentomoveadistancedxmisdtseconds.
Inducedemf,e=Fluxcutpersecond
Since thelinearvelocityvoftheconductor,
IftheconductormovesinadirectionmakingananglewiththedirectionofmagneticfieldasshowninFig.1.12,theinducedEMFwillbeasstatedearlierineq.1.26.
1.13 DYNAMICALLY INDUCED EMF AND STATICALLY INDUCED EMF
Whenemfisinducedinacoilorconductorbyvirtueofmovementofeithertheconductororthemagneticfield,theemfiscalleddynamicallyinducedEMFashasbeenexplainedinsection1.11.
WhenEMFisinducedinastationarycoilbychangingitsfluxlinkageduetochangeincurrentflowthroughthecoil,suchemfiscalledstaticallyinducedEMF.
Ifacoilcarriesacurrent,fluxisestablishedaroundthecoil.Ifthecurrentischangedquickly,thefluxlinkagebythecoilwillchangeasshowninFig.1.13(a).
2
2
Figure1.13(a)ChangeinfluxlinkageinacoilduetoswitchingONandswitchingOFFofdccurrent(b)changeinfluxlinkageduetoalternatingcurrentsupply(c)inducedemfincoils1and2duetochangingfluxproducedbyalternatingcurrentflowingincoil1
InFig.1.13(a),acoilofcertainnumberofturnsiswoundonaformer,i.e.,itscore.CurrentissuppliedfromabatterybyclosingaswitchS.Iftheswitchiscontinuouslyturnedonandoff,fluxlinkagebythecoilwillchange.TherateofchangeofthefluxlinkagewillinduceEMFinthecoil.
AsimilareffectwillbethereifanacsupplyisappliedacrossthecoilasshowninFig.1.13(b).Thedirectionofcurrentinthecoilisshownforthepositivehalfcycleofthealternatingcurrent.Thedirectionofcurrentwillchangeineveryhalfcycle,andhencethedirectionoffluxproducedwillchangeineveryhalfcycle.Themagnitudeofcurrentchangescontinuouslysinceasinusoidalcurrentisflowing.Thischangingcurrentwillcreateachangingfluxlinkage,therebyinducingEMFinthecoilinboththecasesasshowninFig.1.13(a)and(b).NotethatinFig.1.13(a),iftheswitchSiskeptclosed,asteadydirectcurrent,i.e.,aconstantcurrentwillflowthroughthecoil.Thisconstantcurrentwillproduceaconstantflux.Therewillbenochangeinfluxlinkagebythecoilwithrespecttotime,andhencenoEMFwillbeinducedinthecoil.Thus,thenecessaryconditionfortheproductionofinducedEMFisthatthereshouldbeachangeinfluxlinkageandnotmerelyfluxlinkagebyacoil.
1.14 SELF-INDUCED EMF AND MUTUALLY INDUCED EMF
TheEMFinducedinacoilduetochangeinfluxlinkagewhenachangingcurrentflowsthroughthecoiliscalledselfinducedEMF.
AsshowninFig.1.13(c),whenasecondcoilisbroughtnearacoilproducingchangingflux,EMFwillbeinducedinthesecondcoilduetochangeincurrentinthefirstcoil.ThisiscalledmutuallyinducedEMF.Infact,EMFwillbeinducedinboththecoilsasboththecoilsarelinkingachangingflux.However,inthesecondcoilEMFisinducedduetochangingfluxcreatedbycoil1.ThemagnitudeoftheinducedEMFwilldependupontherateofchangeoffluxlinkageandthenumberofturnsoftheindividualcoils.TheinducedEMFinthetwocoils,e ande willbe
whereN andN arethenumberofturnsofcoil1andcoil2,respectively.
YouwillstudyinaseparatechapterhowtransformersarebuiltutilizingthebasicprincipleofmutuallyinducedEMF.
1.15 SELF-INDUCTANCE OF A COIL
ConsideracoilofNturnswoundonacoreofmagneticmaterial.LetanalternatingcurrentipassthroughthecoilasshowninFig.1.14.
1 2
1 2
Figure1.14Inductanceofacoil
Theemfinduced,ewillbe
whereisthepermeabilityofthecorematerialisthelengthoffluxpathAistheareaofthecoil.substituting,
Liscalledthecoefficientofselfinductanceorsimplyselfinductanceofthecoil.
Inductanceofacoilis,therefore,dependentuponthepermeabilityofthecorematerial.Ifweputironasthecorematerialinsteadofanynonmagneticmaterial,orairasthecore,theinductancewillincreasemanytimes.The(permcability),isexpressedas
=
where istherelativepermeabilityand isthepermeabilityoffreespace.Therelativepermeabilityofironmaybeashighas2000timesthanthatofair.Hence,anironcorecoilmayhaveaninductancevalue2000timesmorethanthatofanaircoreone,otherdimensionsremainingthesame.Again,inductance,Lisinverselyproportionaltothelengthofthefluxpathanddirectlyproportionaltotheareaofthecorematerialorthecoil.Inductanceisproportionaltothesquareofthenumberofturns.Tohaveaninductanceofalargevalue,thenumberofturnsshouldbehigh.
Theinductance,Lcanbeexpressedintermsoftherateofchangeofthefluxwithrespecttocurrentflowinginthecoilas
Forasmallincrementofdi,lettheincreaseoffluxbed .Therefore,
o r
r o
If andihavealinearrelationship,
Rememberthatreluctanceistheinverseofthepermeability.Lowreluctancewillgiverisetoahighvalueofinductance.Thatiswhyinordertoproducehighvalueinductance,thenumberofturnsshouldbehighandthereluctancetothefluxpathshouldbelow.Thecoreshouldbemadeofhighpermeabilitymateriallikeiron.
Consideringasmallincreaseofiproducingasmallincreasein asd
Inductanceisthepropertyofacoilcapableofinducingemfinitselfduetochangingcurrentthroughit.
The formulae so far derived are
1. Forceonacurrentcarryingconductorinamagneticfield
F=BINewtons.
Iftheconductorisinclinedatananglewiththemagneticfield,
F=BISinNewtons.
2. InducedEMFinacoilwherethereischangeoffluxlinkageorchangeincurrent,
3. InducedEMFinaconductorrotatinginamagneticfield,
e=P nV
wherePisthenumberofpoles, isthefluxperpoleandnistherevolutionspersecond.
4. InducedEMFinaconductormovinginamagneticfieldinaperpendiculardirection,
e=BvV
whereBisthefluxdensityinWb/m ,isthelengthoftheconductorinmandvisthevelocityinm/sec.
Iftheconductorismovingatanangleofwiththemagneticfield,theinducedEMFis
e=BvSinV.
5. InducedEMFinacoil,
Thus,wecansaythatacoilhasaninductanceof1Henryifacurrentof1Ampereflowingthroughthecoilproducesafluxlinkageof1Wbturn.
1.16 MUTUAL INDUCTANCE
ConsidertwocoilshavingN andN numberofturnsplacedneareachotherasshowninFig.1.15.Letachangingcurrent,i ,flowthroughcoil1.
Thefluxproducedbyi inN is .Sincecoil2isplacednearcoil1,a
partofthefluxproducedbycoil1willbelinkedbycoil2.Letflux
linkedbycoil2is =K whereK 1.
Ifmagneticcouplingbetweenthetwocoilsisverytight,i.e.,verygood,thewholefluxproducedbycoil1willlinkthecoil2,inwhichcasethecoefficientofthecouplingK willbe1.TheinducedEMFincoil2ise .
Figure1.15Mutualinductanceoftwocoils
From(i)and(ii),
1 2
1
1 1 1
2
2 1 1 1
1 2
2
Similarly,ifwecalculatetheinducedEMFincoil1,duetochangeincurrentincoil2,wecanfindtheinducedEMFe incoil1as
Now,multiplyingtheexpressionforMasin(iii)and(iv)above,
Againfrom(iii),
From(iii)and(vi),
1
Fromeq.(1.38)wecandefinethemutualinductanceMbetweentwocoilsasthefluxlinkageinonecircuitduetochangeperunitofcurrentintheothercircuit.
Similarly,consideringcurrentchangeinthesecondcoil
1.17 INDUCTANCE OF COILS CONNECTED IN SERIES HAVING A COMMON CORE
WehavetwocoilshavingselfinductanceL andL connectedinseries.InFig.1.16(a),theyproducefluxinthesamedirection,andinFig.1.16(b),theconnectionissuchthattheyproducefluxintheoppositedirections.
Sincethetwocoilsareconnectedinseries,thesamecurrentflowsthroughthem.
Ifthereisachangeincurrentdiamperesintimedtseconds,theEMFinducedincoil1duetoitsselfinductanceL is
Similarly,theEMFinducedincoil2duetoitsselfinductance,L is
Duetomutualinductance,theEMFinducedincoil1duetochangeincurrentincoil2andviceversaareexpressedasEMFinducedincoil1duetochangeincurrentincoil2is
Figure1.16Coilsconnectedinseriesin(a)commulativelyin(b)differentially
EMFinducedincoil2duetochangeincurrentincoil1is
1 2
1
2
Nowletthetotalequivalentinductanceofthesinglecircuitcomprisingcoil1andcoil2astheyareconnectedasinFig.1.16(a)beL
TheEMFinducedinthewholecircuitwill,therefore,be
Thus,equatingtheexpressionforein(iv)withthetotalEMFsasin(i),(ii),(iii),and(iv):
WhenthecoilsaredifferentiallyconnectedasinFig.1.16(b),theEMF
inducedincoil1duetodiintimedtincoil2,i.e., inoppositiontotheEMFinducedincoil1duetoitsselfinductance.SimilaristhecaseoftheEMFinducedincoil2duetomutualinductance.Thus,forthedifferentiallyconnectedcoil
Thus,thetotalinductanceofaninductivelycoupledseriesconnectedcoilcircuitcanbeexpressedas
Dotconventionisusedtodeterminethesignofinducedvoltage.
Ifweusedotconvention,itwillnotberequiredtoknowthewaythecoilshavebeenactuallywound.
Example1.11Thetotalinductanceoftwocoilsconnectedinseriescumulatativelyis1.6Handconnecteddifferentiallyis0.0.4H.Theselfinductanceofonecoilis0.6H.Calculate(a)themutualinductanceand(b)thecouplingcoefficient.
Solution:
Substitutingthegivenvalues
From(i)and(ii)
e
1.18 ENERGY STORED IN A MAGNETIC FIELD
Letusconsideracoilsuppliedwithanalternatingvoltagevduetowhichanalternatingcurrentflowsthroughthecoil.Whencurrentincreasesfromitszerovalue,themagneticfieldstartsincreasingandreachesitsmaximumvaluewhencurrentreachesitsmaximumvalue.Whencurrentstartsdecreasing,thefieldgoesondecreasingandgraduallybecomeszero.Then,inthenegativecycleifthecurrentflows,thefieldgetsestablishedintheoppositedirection,whichcollapseswhencurrentagainreacheszero.Thiswaythefieldisestablishedandthencollapsesineveryconsecutivehalfcycleofcurrentflow.Whenthefieldisestablished,energyintheformofamagneticfieldisstoredandwhenthefieldcollapses,thesameenergyisreturnedtothesupplysource.Assuch,noenergyisconsumedbythepurelyinductivecoil.Therefore,energystoredisequaltotheenergysupplied.
ThisinducedEMFopposestheappliedvoltagefromwhichitisproduced.ThisisduetoLenzslaw,sothat
e=vor,v=e
Thus,
foracurrentchangefrom0toI
EnergystoredinacoilofinductanceLis
Figure1.17Magneticfieldenergy
Example1.12Aconductoroflength0.5misplacedinamagneticfieldofstrength0.5Wb/m .Calculatetheforceexperiencedbytheconductorwhenacurrentof50Aflowsthroughit.Iftheforcemovestheconductoratavelocityof20m/sec,calculatetheEMFinducedinit.
Solution:
Force,Fonacurrentcarryingconductorplacedinamagneticfieldisgivenas
F=BINewton
Substitutingthevalues,
F=0.5Wb/m 500.5m=12.5N
InducedEMF,einaconductormovinginamagneticfieldisgivenas
e=BlvV
Substitutingthegivenvalues,
e=0.5Wb/m 0.5m20m/sec=5Wb/sec=5V
Example1.13Anironcoredtoroidalcoilhas100turns.Themeanlengthofthefluxpathis0.5mandthecrosssectionalareaofthecoreis10cm .Calculatetheinductanceofthecoil.Assumerelativepermeabilityofironas2000.Alsocalculatetheinducedemfinthecoilwhencurrentof5Aisreversedin10ms.
Solution:
Theexpressionforinductanceintermsofitsparametersis
Currentinthecoilischangedfrom+5Ato5Ain1010 secs.Totalchangeofcurrentis10A.
Puttingthegivenvaluesweget,
2
2
2
2
3
Example1.14Thereismutualmagneticcouplingbetweentwocoilsofnumberofturns500and2000,respectively.Only50%ofthefluxproducedbythecoilof500turnsislinkedwiththecoilof1000turns.Calculatethemutualinductanceofthetwocoils.AlsocalculatetheEMFinducedinthecoilof1000turnswhencurrentchangesattherateof10A/secondintheothercoil.Theselfinductanceofthecoilof500turnsin200mH.
Solution:
Figure1.18
Mutualinductance,
InducedEMFinthesecondcoil,e is
Example1.15Acurrentof5Aflowingthroughacoilof500turnsproducesafluxof1mWb.Anothercoilisplacednearthiscoilandcurrentinthiscoilissuddenlyreversedin10ms.Asaresult,theEMFinducedinthesecondcoilismeasuredas50V.Calculateselfandmutualinductanceofthecoilsassumingacoefficientofcouplingas60percent.
Solution:
Selfinductanceofcoil1is
Usingtheformula,M=K
Example1.16TwocoilsofnumberofturnsN =1000andN =400,respectively,areplacedneareachother.Theyaremagneticallycoupledinsuchawaythat75percentofthefluxproducedbytheoneof1000turnslinkstheother.Acurrentof6Aproducesafluxof0.8mWbinN andthesameamountofcurrentproducesafluxof0.5mWbinthecoilofNturns.DetermineL ,L ,M,andKforthecoils.
Solution:
2
1 2
1
2
1 2
Usingtherelation,M=K
substitutingvalues,
So,
Selfinductanceofcoil1=0.133H
Selfinductanceofcoil2=0.033H
Mutualinductanceofthecoils=0.04H
Coefficientofcoupling=0.606
1.19 ELECTRICAL CIRCUIT ELEMENTS
Resistors,inductors,andcapacitorsarethethreebasiccircuitparametersorcircuitcomponentsofanyelectricalnetwork.Resistorscanbewirewoundtypeorcarbonmouldedtype.Whencurrentflowsinaresistance,heatisproduced,whichisdissipated.Theheatisproducedbecausefrictionbetweenmovingfreeelectronsandatomsobstructthefreeflowofelectronsproducingelectriccurrent.Aresistorisanelementthatdissipatesenergyasheatwhencurrentflowsthroughit.
Inductorsaremadeofacoilhavinganumberofturns.Thecoreofthecoilmaybeairoramagneticmaterial,whichisplacedinsidethecoil.Whenthecoiliswoundonanironcore,theinductorformediscalledanironcoreinductorcoil.Inductanceofaninductorisdirectlyproportionaltothesquareofthenumberofturnsofthecoilused.Inductorstoresenergybecauseofcurrentflowingthroughit.
Acapacitorconsistsoftwoconductorsorconductingplatesbetweenwhichadielectricisplaced.Thecapacitanceofacapacitorisitsabilitytostoreelectriccharge.Differenttypesofcapacitorsareavailable.Theyarenamedaccordingtothedielectricplacedbetweentheconductors.Commontypesofcapacitorsareair,mica,paper,ceramic,etc.
1.19.1 Resistors
Wirewoundresistorsaremadeofwiresofconstantan,manganinornichromewoundonaceramictube.Theseresistancesareavailableinrangesvaryingfromafractionofanohmtothousandsofohms.
ThepowerratingalsovariesfromafractionofaWatttofewkiloWatts.Whilespecifyingaresistance,bothresistancevalueandpowerdissipatingvaluemustbementioned.Electroniccircuitsrequireresistorsofaccuratevalues.Thevalueofresistorsusedinelectroniccircuitsisquitehigh,oftheorderofkiloohms.Sincecarbonhashighresistivity,carbonresistorsaremadewithcopperleads.TheirpowerratingvariesfromafractionofaWatttoseveralWatts.Colorcodeisusedtoindicatethevalueofsuchresistors.
1.19.2 Inductors
TheabilityofacoiltoinduceEMFinitselfwhenthecurrentthroughitchangesiscalleditsinductance.TheunitofinductanceisHenry.1Henryofinductancecauses1Volttobeinducedwhencurrentchangesattherateof1Amperepersecond:
whereLisinHenry,eisinVolt,and isinAmperepersecond.
Whensteadydirectcurrentflowsthroughaninductor,itwillnotaffectthecircuitasthereisnochangeincurrent.Inductorsareoftwotypesvizaircoretypeandironcoretype.Inductorsarealsocalledchokes.Inductorsareavailableinallcurrentranges.Aircoreinductorsarewoundonbakeliteorcardboardrodsandareextensivelyusedinelectroniccircuitsinmillihenryandmicrohenryranges.Highvalueinductorsaremadeofironcore.Theyaremainlyusedinacpowersupplyoffrequencyof50Hz.
Thedetailsofselfandmutualinductancehavebeendiscussedearlier.
1.19.3 Capacitors
Acapacitor,initssimplestform,consistsoftwothinparallelplatesofconductingmaterialseparatedbyadielectricmaterial.Acapacitoriscapableofstoringchargewhenavoltageisappliedacrossthecapacitorplates.Ifavoltagesource,sayabattery,isconnectedacrossthetwoplatesofaparallelplatecapacitorasshowninFig.1.19,electronsfromthenegativeterminalofthevoltagesourceaccumulateonplateAofthecapacitor.TheotherplateBloseselectronsasitisconnectedtothepositiveterminalofthesourceofvoltage.Thisway,theexcesselectronsproducenegativechargeononesideofthecapacitorwhiletheoppositesidewillhavepositivecharge.Thedielectricmaterialplacedinbetweentheplatesholdthechargebecausethefreeelectronscannotflowthroughaninsulator(i.e.,thedielectricmateriallikeair,paper,ormica).Storageofchargebyacapacitormeansthatthechargeremainsinplaceevenafterthevoltagesourceisdisconnected.Capacitanceofacapacitoristheabilitytostorecharge.Charginganddischargingarethetwomaineffectsofcapacitors.Whenavoltageisapplied,thereisaccumulationofchargeinthecapacitorandasaresultvoltageisbuiltupacrosstheterminalsofthecapacitor.Thisiscalledchargingofthecapacitor.Thecapacitorvoltagebecomesequaltotheappliedvoltagewhenthecapacitorisfullycharged.Thevoltageacrossthecapacitorremainsevenafterthevoltagesourceisdisconnected.Thecapacitordischargeswhenaconductingpathisprovidedacrosstheplateswithoutanyappliedvoltageconnected.
Figure1.19Acapacitorstoreschargeinthedielectricmaterialplacedbetweentheconductingplates
Themorethechargingvoltageis,themoreistheaccumulationofchargeinthecapacitor.Theamountofcharge,Qstoredinacapacitoris,therefore,proportionaltothechargingvoltage,V.Acapacitorwithalargeareaoftheparallelplatescanstoremorecharge.Capacitanceofacapacitoralsodependsonthedistancebetweentheplatesandthetypeofdielectricusedbetweentheplates.Alargecapacitor,obviously,willstoremorecharge.Thus,wecanwrite
Q=CVCoulombs
whereQisthechargestoredinCoulombs,Visthevoltageappliedacrosstheplates,andCisthecapacitanceofthecapacitorinFarads.Thecapacitanceofaparallelplatecapacitorisexpressedas
whereistheabsolutepermittivityconstant,Cisthecapacitance,Aistheareaoftheplateanddisthedistancebetweentheplates.
Thetermabsolutepermittivityisexpressedas
=
where isthepermittivityconstantofvacuumand istherelativepermittivityofthedielectricmaterialplacedbetweenthetwoplates.
Thevalueof hasbeencalculatedexperimentallyas8.8510 Faradpermeter.
Therefore,thecapacitanceofaparallelplatecapacitorcanbeexpressedas
1.20 ENERGY STORED IN A CAPACITOR
Wehaveknownthatwhenacapacitorisswitchedontoadcsupply,thechargeqcanbeexpressedasq=Cv,whereatanyinstantqisthechange,visthepotentialdifferenceacrossthecapacitorplates,andCisthecapacitanceofthecapacitor.
PotentialdifferenceofvvoltsacrossthecapacitormeansvJoulesofworkhastobedoneintransferring1Coulombofchangefromoneplatetotheother.Ifasmallchargedqistransferredthentheworkdonedwcanbeexpressedas
dw=vdq=Cvdv
ThetotalworkdoneinraisingthepotentialofthecapacitortothesupplyvoltageofVvoltcanbeexpressedas
Thisworkdoneisstoredintheelectrostaticfieldsetupbetweentheplatesofthecapacitorintheformofenergy.Thus,theenergystored,Eisexpressedas
Example1.17Thecurrentthrougha100mHinductorchargesfrom0to200mAin4s.WhatisthevalueoftheinducedEMFintheinductororthechoke?
Solution:
Itisobservedthatahighvoltageisinducedinthechokebecauseofveryfastchangeofcurrentflowthroughit.Inatubelightcircuit,ahighvoltageisinducedinthechokebythesamemethodandisusedtoionizethegasinsidethetubelight,andthusstartthetubelight.
Example1.18SelfinductancesoftwocoilsareL =2HandL =8H.ThecoilL producesamagneticfluxof80mWbofwhichonly60WbarelinkedwithcoilL .Calculatethemutualinductanceofthetwocoils.
Solution:
Thecoefficientofcoupling,Kisgivenas
o r
o r
o
1 2
1
2
12
MutualinductanceMiscalculatedas
Example1.19Calculatethecapacitanceofacapacitormadeoftwoparallelplatesof3m havingadistancebetweentheplatesof1cm.Thedielectricisairbetweentheplates.
Solution:
=265510 F
Notethatalthoughtheareaoftheplatesislarge,thevalueofcapacitanceisverysmall.Insteadofairasthedielectric,ifweplacemicaorpaperbetweentheplates,capacitancewillincrease.Ifwealsoreducethedistancebetweentheplates,thecapacitancewillincrease.
Example1.20A25microfaradcapacitorisswitchedontoatimevaryingvoltagesource.Thevoltagewaveissuchthatvoltageincreasesattherateof10Vpersecond.Calculatethechargeaccumulatedinthecapacitoratanelapseof1secondandtheamountofenergystoredinthecapacitor.
Solution:
1.21 CAPACITOR IN PARALLEL AND IN SERIES
Whenweconnecttwocapacitorsinparallel,theplateareasareadded.Thetotalcapacitance,therefore,getsaddedup.WhencapacitancesC ,C ,C ,etc.areconnectedinparallel,thetotalcapacitanceC becomesequalto
C =C +C +C +
Figure1.20(a)Equivalentofcapacitorsconnectedinparallel(b)equivalentofcapacitorsconnectedinseries
ThisisshowninFig.1.20(a).
Seriesconnectionofcapacitors,asshowninFig.1.20(b),isequivalenttoincreasingtheeffectivedistancebetweentheplatesorthethicknessofthedielectricused.Thecombinedcapacitanceislessthantheindividualvalue.
1 2
3 r
r 1 2 3
2
12
Thevalueofacapacitorisalwaysspecifiedineithermicrofaradorpicofarad.Thereareavarietyofwaysinwhichmanufacturersindicatethevalueofacapacitor.
1.22 Review Questions
1. Giveanoverviewofthescopeofelectricalandelectronicsengineering.
2. Chargeinmotioniscalledcurrent.Explainwiththehelpofatomictheory.
3. Distinguishbetweenconductors,semiconductors,andinsulators.4. DistinguishbetweenWork,Power,andEnergy.5. Differentiatebetweentemperaturecoefficientofresistanceand
specificresistance.6. Distinguishbetweenanelectricfieldandamagneticfield.7. Definethefollowingterms:Volt,Ampere,Ohm.8. Explainwhytwoparallelcurrentcarryingconductorsattracteach
otherwhencurrentinthemflowinthesamedirection.9. StateFlemingsRightHandRule.
10. ExplainthattheEMFinducedinacoildependsuponthefluxandthespeedofrotationofthecoil.
11. DistinguishbetweenstaticallyinducedEMFanddynamicallyinducedEMF.
12. Explainwhyanironcorecoilwillhavemoreinductancethananaircorecoilofthesamenumberofturns.
13. Whatisthemeaningofcoefficientofcouplingbetweentwocoils?Whenisthisvalueequaltounityandequaltozero?
14. WhatareFaradayslawsofelectromagneticinduction?15. WhatistheLenzslaw?Giveanexample.16. Whatisthemagnitudeofforceexperiencedbyacurrentcarrying
conductorplacedinamagneticfield?17. Howdoyoudeterminethedirectionofforcedevelopedina
currentcarryingconductorplacedinamagneticfield?18. Whatarethefactorsonwhichinductanceofacoildepends?19. Whydoestheinductanceofacoilincreaseifthecorehasa
magneticmaterialinsteadofair?20. Derivethefollowingexpressionforselfinductanceofacoil
21. Youhavetomakeaninductanceofhighvalue.Howwillyouproceed?
22. WhatisFlemingsRightHandRule?Whereisitused?23. Whatruledoyouapplytodeterminethedirectionofforceona
currentcarryingconductorplacedinmagneticfield?24. Whatisthemagnitudeofforceonacurrentcarryingconductor
placedinamagneticfield?25. Showthattheenergystoredinamagneticfieldproducedbyan
inductoris .26. Distinguishbetweenselfinductanceandmutualinductance.27. Explainwhyinductanceofacoilincreasesifanironpieceforms
itscoreinsteadofairoranynonmagneticmaterial.
28. Establishtherelation, fortwoadjacentcoilslinkingflux.
29. Onwhatfactorsdoesthereluctanceofamagneticmaterialdepend?
30. Whatisthecorkscrewrule?Wheredoyouuseit?31. Twoadjacentconductorsarecarryingcurrentintheopposite
directions.Showthattherewillbeforceofrepulsionbetweentheconductors.
32. Whencapacitorsareconnectedinparallel,theirequivalentcapacitanceisincreased.Explainwhy?
33. Explainwhycapacitorsarecalledenergystoragedevices.34. Whatisthemeaningofrelativepermittivityordielectric
constant?Whatisitsunit?35. Writethreeformulaeofelectricalpower.36. Provethat1kWhisequalto3.610 Joules.37. Themostimportantpropertyofacapacitorisitsabilitytoblock
steadydcvoltagewhilepassingacsignals,explain.38. DefinetheFaradunitofcapacitance.39. Howisenergystoredinacapacitor?Onwhatfactorsdoesit
depend?40. Whatarethephysicalfactorsthataffectthecapacitanceofa
capacitor?41. TwocoilsofN =50andN =500turns,respectively,arewound
sidebysideonanironringofcrosssectionalareaof50cm andmeanlengthof120cm.Calculatethemutualinductancebetweenthecoils,selfinductanceofthecoils,andthecoefficientofcouplingassumingpermeabilityofironas1000.
1 2
6
2
[Ans0.13H,0.013H,1.3H,1.0]
42. TwocoilsofN =1500andN =200turnsarewoundonacommonmagneticcircuitofreluctance2510 AT/Wb.Calculatethemutualinductancebetweenthecoils.
[Ans1.2H]
43. Twocoilshaveamutualinductanceof400mH.CalculatetheEMFinducedinonecoilwhencurrentinthesecondcoilvariesatarateof6000Amperespersecond.
[Ans2.4V]
44. Twosimilarcoilshaveacouplingcoefficientof0.4.Whenthecoilsareconnectedinseriescumulatively,thetotalinductancebecomesequalto140mH.Calculatetheselfinductanceofeachcoil.
[Ans50mH]
45. Twocoilswhenconnectedinseriescumulatativelyshowtohaveatotalinductanceof2.4Handwhenconnectedinseriesbutdifferentiallyshowatotalinductanceof0.4H.Theinductanceofonecoilwhenisolatediscalculatedasequalto0.8H.Calculate(a)themutualinductanceand(b)thecoefficientofcouplingbetweenthecoils.
[AnsM=0.5H,0.75]
46. Calculatetheinductanceofacoilhaving100turnswoundonamagneticcoreofpermeabilityequalto1000,meanlengthof0.25m,andcrosssectionalareaof10cm .
[AnsL=50.24mH]
47. Aconductoroflength25cmisplacedinauniformmagneticfieldofstrength0.5Wb/m .CalculatetheEMFinducedintheconductorwhenitismovedattherateof10m/sec(a)paralleltothemagneticfield,(b)perpendiculartothemagneticfield.
[Ans(a)0V(b)1.25V]
MultipleChoiceQuestions
1. ThenumberofelectronsperCoulombisequalto1. 1.602102. 6.28103. 1.602104. 6.2810 .
2. Ininsulatorstheoutermostorbitoftheiratomsisfilledwith1. 4electrons2. 8electrons3. 1electron4. 18electrons.
3. Intheatomsofsemiconductingmaterialslikesiliconandgermaniumtheoutermostorbithas
1. 1electron2. 2electrons3. 8electrons4. 4electrons.
4. Whichofthefollowingexpressionsisincorrect?
1. Current,2. Charge=currenttime
3.4. Volt=joulesperCoulomb.
5. Whichisthefollowingexpressionsdoesnotrepresentpower?1. I R
2.3. VI
4. .6. Whichofthefollowingisnottheunitofpower?
1 24
2
2
19
18
18
19
2
1. Joules/second2. Watthour3. KW4. VoltAmpere
7. AconductoroflengthanddiameterdhasresistanceofRohms.Ifthediameterisreducedtoonethirdandlengthincreasedbythreetimes,theresistanceoftheconductorwillbe
1. 3R2. 6R3. 9R4. 27R.
8. Whichofthefollowingexpressionsisincorrect?
1.
2.
3.
4. .9. Whichofthefollowingexpressionsisincorrect?
1.
2.
3.
4. .10. Inductanceofanaircorecoilwillincreaseifthecoreismadeof
1. copper2. aluminium3. iron4. porcelain.
11. Whichofthefollowingstatementsisnottrue?1. Inductanceofacoilwillincreasebyfourtimesifthe
numberoftermseqisdoubled2. inductanceofacoilwillincreaseiftheareaofcross
sectionofthecoil,i.e.,thefluxpathisincreased3. inductanceofacoilwillincreasedifthelengthofflux
pathisincreased4. inductanceofacoilwillincreaseifthecoreismadeupof
materialhavinghigherpermeability.12. ThedirectionoftheinducedEMFinthecoilsidesofacoil
rotatinginamagneticfieldcanbedeterminedbyapplying1. Flemingslefthandrule2. Righthandgriprule3. Flemingslefthandrule4. Corkscrewrule.
13. Whichofthefollowingisnottheunitofenergy?1. kWh2. Joules/second3. Watthour4. Joules.
14. Selfinductanceoftwomagneticallycoupledcoilsare8Hand2H,respectively.Whatcoefficientofcouplingwillmaketheirmutualinductanceequalto4H?
1. K=0.52. K=0.253. 0.14. 1.0.
15. Whichofthefollowingeq.isincorrectwithrespectofincreaseinresistancewithincreaseintemperatureofaconductingmaterial?
1.
2.
3.
4. .
AnswerstoMultipleChoiceQuestions
1. (b)2. (b)3. (d)4. (c)5. (d)6. (b)7. (d)8. (b)9. (a)
10. (c)11. (c)
Recommended / Queue / Recent / Topics / Tutorials / Settings / Blog(http://blog.safaribooksonline.com) /Feedback(http://community.safaribooksonline.com/) / Sign Out 2015 Safari(http://www.safaribooksonline.com/). Terms of Service / Privacy Policy
12. (c)13. (b)14. (d)15. (d)
People who finished this also enjoyed:
Time for action coloring andtexturing the sloop hullfrom: Blender 3D Basics by Gordon FisherReleased: June 20128 MINS
Digital Media
BOOK SECTION
Chapter 15: Artificial IntelligenceTechniques for Solar Energy andPhotovoltaic Applicationsfrom: Handbook of Research on Solar EnergySystems and Technologies by Sohail Anwar...
Released: August 2012156 MINSEngineering
BOOK SECTION
Synchronous Motorsfrom: Electrical Machines, 2nd Edition by SmarajitGhoshReleased: March 201275 MINS
Engineering
BOOK SECTION
Chapter 8. Number Systemsfrom: Circuit Design: Know It All by Darren Ashby...Released: August 200829 MINS
DIY & Hardware / Engineering
BOOK SECTION
Chapter 12: Measuring and AnalyzingCircuitsfrom: Electronics For Dummies, 2nd Edition byCathleen Shamieh...Released: September 2009
33 MINSDIY & Hardware
BOOK SECTION
PREVPreface
NEXT2. DC Networks an